Rational points on algebraic varieties

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DIAGONAL CUBIC EQUATIONS IN FOUR VARIABLES WITH PRIME COEFFICIENTS Carmen Laura Basile Department of Mathematics, Imperial College of Science, Technology and Medicine, London ([email protected])

Thomas Anthony Fisher Department of Pure Mathematics and Mathematical Statistics, Sidney Sussex College, Cambridge ([email protected])

November 2000

Abstract The aim of this paper is to give an alternative proof of a theorem of R. HeathBrown [3] regarding the existence of non-zero integral solutions of the equation

pX 1

3 1

+

pX 2

3 2

+

pX 3

3 3

+

pX 4

3 4

= 0 , where the

pj are prime integers  2 (mod 3).

We start by presenting the main result of this paper. This result has been proved by Roger Heath-Brown [3] under the conjecture that the dierence s(A) r(A) between the Selmer rank and the arithmetic rank of the elliptic curve X 3 + Y 3 = AZ 3 is even. In this note we will show that we do not need this assumption and give a detailed proof of this result. Theorem 1 Let p1 ; p2 ; p3 ; p4 be prime integers such that pi  2 (mod 3) (1 Then the equation p1 X13 + p2 X23 + p3 X33 + p4 X43 = 0

 i  4).

has non-zero integral solutions, assuming the conjecture that the Tate-Shafarevich group of the elliptic curve X 3 + Y 3 = AZ 3 over Q is nite.

A much stronger result has been recently proved by Sir Peter Swinnerton-Dyer [7]. Note that in order to prove the above theorem, it is suÆcient to prove that the equations

p1 X13 + p2 X23 = p p3 X33 + p4 X43 = p  Supported

by INdAM (Istituto Nazionale di Alta Matematica \F.Severi"), Italy.

2 have non-zero rational solutions for some prime integer p. So it suÆces to prove that the equation p1 X 3 + p2 Y 3 = p has non-zero solutions in Q or, equivalently, in a quadratic extension of Q . Notation Let ! be a primitive cube root of unity and let k = Q (! ). Let A 2 Z n f 1; 0; 1g be a cube-free integer. We denote by EA the elliptic curve

EA : X 3 + Y 3 = AZ 3 : For any  2 k , let CA; be the smooth projective curve given by the equation

CA; : X 3 +  1 Y 3 = AZ 3 : For  2 k , the curves CA; are principal homogeneous spaces over EA . Moreover, it is clear that if  and belong to the same class modulo (k )3 then the curves CA; and CA; are isomorphic. Let us consider the multiplication-by-

p

3:

EA

(X; Y; Z )

! EA 7 ! (! X 2

3

p

+ !Y 3

and the following diagram with exact row 0

! EA(k)=

p

3EA (k)

! H (k; EA 1

3 endomorphism on EA (see [1]), given by

AZ 3 ; !X 3 + ! 2Y 3

p





3)

! &

AZ 3 ;

WC (EA =k) Q

3XY Z );

p



#

v WC (EA =kv )

3

p





!0 

3

where WC (EA =k) is the Weil-Ch^atelet group ofpEA =k  and v runs over all the places of  k. It is readily veri ed that the group EA (k) 3 is isomorphic to the group 3 (k)  )-module. It follows from Kummer theory that of cube roots p of  unity, as a Gal(k=k 1 H (k; EA 3 ) is isomorphic to k =(k )3 and so we get the exact sequence 0

p

! EA(k)=

3EA (k)

! k =(k)

3

f

! g&

WC (EA =k) Q

#

p



v WC (EA =kv )

3

p





!0 

3

where thep map f sends an element (k )3 to the curve CA; . We denote by S (A) the Selmer group S ( 3) (EA =k) (which is de ned to be the kernel of the map g in the diagram above) and by C (A) the kernel of the map f . Obviously C (A) is a subgroup of S (A); we can write them explicitly S (A) = f(k )3 : CA; has k points for any prime  2 k;  2 k g; C (A) = f(k )3 : CA; has k points;  2 k g:

p

Observe also that C (A) is isomorphic to EA (k)= the diagram above).

3EA (k) (this follows immediately from

In the following two lemmas we will determine the structure of S (A) when A satis es certain conditions.

3 Lemma 2 Let  2 k be a prime above 3, and let a; b; c 2 k such that a; b; c are congruent to 1 modulo 2 . If the projective curve

E : aX 3 + b!Y 3 + c! 2Z 3 = 0 has a point over k , then abc  1 (mod 3 ). Proof.

Let (x; y; z ) be a k -point of the curve E and suppose that minfval (x) ; val (y ) ; val (z )g = 0:

Say

with (x0 ; y0 ; z0 ) 6= (0; 0; 0).

x = x0 + x1  + x2 2 + : : : ; y = y0 + y1  + y2 2 + : : : ; z = z0 + z1  + z2 2 + : : :

>From the condition ax3 + b!y 3 + c! 2z 3 = 0 it follows that x30 Therefore we may assume (x0 ; y0; z0 ) = (1; 1; 1) and hence we get

y z 3 0

3 0

(mod 2 ).

(x3 ; y 3 ; z 3 )  (1; 1; 1) (mod 3 ): Let a  1 + a2 2 (mod 3 ), b  1 + b2 2 (mod 3 ) and c from a + b! + c! 2  0 (mod 3 ) it follows a2 + b2 + c2 abc  1 + (a2 + b2 + c2 )2  1 (mod 3 ). Lemma 3 Let A = p1 p2 N( ) where p1 ; p2

(mod 3) is a prime in Z[! ] such that A

6

 1 + c  (mod  ). Then  0 (mod ) and therefore 2 2

2

3



2 (mod 3)  are  integer   1  primes,    = 6= 1. Then 1 (mod 9) and

S (A) is isomorphic to Z=3  Z=3 as an abelian group.

p1

3

p2

3

We will prove that S (A) is generated by the elements A(k )3 and p1 p22 (k )3 . The curve CA;A has a k-point, namely (1; 0; 1); hence A(k )3 2 S (A). Suppose now  6= A. Note that, in order to determine for which elements  the coset (k )3 belongs to S (A), it is suÆcient to test the elements  where  is a cube-free integer in Z[! ] such that Proof.

(i)  is composed of primes dividing A; (ii) we can x a prime in fp1 ; p2 ; ;  g, say  , such that  does not divide . Indeed:

4 (i) suppose  contains a prime factor  which does not divide A; it is then easy to verify that CA; contains no k -points. To get a contradiction, we suppose CA; contains a k -point (X; Y; Z ); so we have

2 X 3 + Y 3 = AZ 3 : Considering the -adic valuation of the right hand side and of the left hand side, and noting val (2 X 3 ) 6= val (Y 3 ), we get











val AZ 3 = val 2 X 3 + Y 3 = minfval 2 X 3 ; val Y 3 g:

But this is impossible since, on the one hand, val (AZ 3 ) = i + 3j , where i 2 f1; 2g is de ned to be the -adic valuation of  and j is an integer, and on the other hand


minfval 2 X 3 ; val Y 3




g = minf2i + 3h; 3lg

with h; l integers; (ii) suppose  j divides  (with j = 1; 2); then, since S (A) is a group and A(k )3 belongs to S (A), instead of  we may consider the cube-free integer representative of the coset (=Aj ) (k )3 . As a result, we may assume  = ! m pn1 1 pn2 2  n where m; n1 ; n2 ; n 2 f0; 1; 2g. In fact it is not diÆcult to prove that if m 6= 0 then  does not belong to S (A). Indeed, to get a contradiction suppose that m 6= 0 and that CA; contains a k -point where  2 k is a prime above 3. Then from Lemma 2 it follows that A  1 (mod 3 ) and hence A  1 (mod 9) which contradicts our hypotheses. Therefore we may assume m = 0. Let  2 Z[! ] be a prime. If  does not divide 3A then CA; contains points over k for any . Indeed, a smooth curve of genus 1 always contains points over any nite eld (see [2]); moreover if  does not divide 3A then CA; is non-singular over Z[! ]= and therefore it contains a non-singular point over Z[! ]= which, by Hensel's lemma, can be lifted to a point over k .

Suppose now that  divides 3A; hence we have to consider the two cases  j A and  j 3.

(1) Let  divide A, so  belongs to fp1 ; p2 ; ;  g. We can consider three further subcases. (1a) Let  divide  exactly; then the projective curve

X 3 +  1 Y 3 = AZ 3 can also be written as

2 A (X )3 + Y 3 = 2 (Z )3 : 2   Hence, considering the transformation X ! X , Z ! Z and reducing modulo , we get the curve A 3 X3 Z =0 

5 which contains a smooth point over F  if and only if 

A if 

1



A 2 (F  )3, i.e. if and only 

= 1. By Hensel's lemma, we can lift this smooth point over F  to 3

a point over k . (1b) Suppose now 2 divides exactly . Similarly, considering a suitable transformation and reducing modulo , we get the curve

A 3 Z =0 3

Y3

A 3

which contains a smooth point over F  if and only if 



A 3 only if = 1.  3 (1c) Similarly, if  does not divide , we get the curve

2 (F  ) , i.e. 3

2 X 3 + Y 3 = 0 which contains a smooth point over F  if and only if  2 (F  )3 , i.e. 2

Suppose  = pj with j = 1; 2.



A If pj divides exactly , then by case (1a) we must have pj ! 1 pj = 1; indeed y pj 3 

A pj

1





p = h pj 3

 3

p pj

 3



pj 1  1 pj = pj 3 pj

where h 2 f1; 2g; h 6= j and p = N( ).

!



p = h pj 3

 3

p pj

 3

pj 2 pj

where again h 2 f1; 2g and h 6= j .

y If

!

 

 = 1.  3



= 1 and hence 3

!

Apj 3 If p2j divides exactly , then by case (1b) we must have pj ! pj 2 = 1, since pj 3 Apj 3 pj

1

1

3

!

= 1 and therefore 3

!

pj 2 = ; p j 3 3 



 If pj does not divide , then by case (1c) we must have = 1. pj 3   6= 3 is a prime integer with #Fq 6 0 (mod 3) then = 1 for any 2 Z such that (

q

n q

n

3

if and

n; q

) = 1.

6 Finally, we obtain the two following conditions which must be satis ed in order to have (k )3 2 S (A):  n2 n  p  2

p1

Since 

and

= 1 and

 p1

 3

p1

1

p2

3

 n2 n  p  2

 n1 n  p 



p = 2 p1 3

n2  3

 p1

n 3

= 1:

3



 = p1

n 3

6= 1 by hypothesis, the rst condition is satis ed if and only if n = 0.

It is easy to verify that for n = 0 the second condition is satis ed as well. Hence  n 1 n2  = p1 p2 . So, by case (1c), it follows that we must have = 1; therefore, since p 

p 



3

by the Cubic Reciprocity Law = 6= 1, we get the condition n1 + n2  0  3  3 2 (mod 3). We may suppose n1 = 1; then  = p1 p2 or, equivalently,  = p1 p2 1 . 1

2

(2) Suppose now that  divides 3; in other words, since 3 = ! 2 (1 ! )2 , suppose  = 1 ! . Recall that now we have A = p1 p2 p and  = p1 p2 1 ; therefore the curve

X 3 +  1 Y 3 = AZ 3 is isomorphic to the curve

E : p1 X 3 + p2 Y 3 = pZ 3 : We have to prove that E contains a point over k . Since Q 3 is contained in k , it suÆces to nd a point over Q 3 . By hypothesis p  1 (mod 3) and p1 ; p2  2 (mod 3), so let

p  1 + 3b (mod 9)

and

pj  1 + 3aj (mod 9)

for j = 1; 2; since A 6 1 (mod 9) by hypothesis, we have a1 + a2 b 6 0 (mod 3). Thus the integers a1 ; a2 ; b cannot be all dierent modulo 3; wlog we may suppose a1 = a2 . Therefore we have p1 p2 1  1 (mod 9); in other words p1 p2 1 belongs to 1 + 9Z3 which is contained in (Z3)3 . It follows that E contains the point (1; y; 0) where y 2 Z3 is a cube root of p1 p2 1 . More precisely, if p1 p2 1  1 + 9l (mod 27), then we use Hensel's lemma to construct y 2 Z3 such that y  1 + 3l (mod 9) and y 3 = p1 p2 1 . Hence E contains a k -point. In conclusion, we have proved that S (A) is generated by the two elements of order 3 A(k )3 and p1 p22 (k )3 and thus it is isomorphic to Z=3  Z=3. 2 The following two lemmas allow us to conclude that, under the hypotheses of Lemma 3, the two groups C (A) and S (A) coincide. This provides a local-to-global principle for the

7 curves CA; when A is as in Lemma 3. As in the statement of Theorem 1, our work here is conditional on the niteness of the Tate-Shafarevich group (EA =Q ).

X

To get a contradiction, let us suppose that C (A) is strictly included in S (A). Note that C (A) cannot be the trivial group as A(k )3 belongs to C (A); then C (A) has order 3. From the exactness of the sequence 0

p 3EA (k) ! k =(k ) p

! EA(k)=

3

f

! WC (EA=k)

p





! 0;

3

it follows that EA (k)= 3EA (k) and Z=3 are isomorphic as abelian groups (recall that C (A) is the kernel of the map f ). Hence from Lemma 3 and the exact sequence p p p  0 ! EA (k)= 3EA (k) ! S ( 3) (EA =k) ! (EA =k) 3 !0

p

X

X





3 is isomorphic to Z=3 and this is impossible, as we will we deduce that (EA =k) show in Lemma 5. But rst we need one more result. Lemma 4 Let E=L be an elliptic curve over a number eld L. Let K be a Galois extension of L of degree n. Let m be a positive integer such that (m; n) = 1. Then:

X(E=L)[m] = X(E=K )[m] : In particular, assuming the niteness of X(E=L), the order of X(E=K )[m] K=L)

Gal(

K=L)

Gal(

be a square.

must

Let us consider the following commutative diagram with exact rows and columns where the rows are obtained by the multiplication-by-m endomorphism and the columns are restriction-in ation sequences: 0 0 Proof.

?

H (Gal(K=L); E (K )[m]) 1

?

H (Gal(K=L); E (K ))[m] 1

0

- E (L)=mE (L)

? - H (L; E [m])

? - H (L; E )[m]

-0

0

? - E (K )=mE (K )

? - H (K; E [m])

? - H (K; E )[m]

- 0.

1

1

1

1

Since Gal(K=L) has order n, every element of H 1(Gal(K=L); E (K )) has order dividing n (this follows from properties of the restriction and corestriction maps; see [5]). Hence, as m and n are coprime, H 1(Gal(K=L); E (K ))[m] = 0. Thus from the diagram above it follows that H 1(L; E )[m] injects into H 1(K; E )[m]. >From the exactness of the second row of the diagram we get the exact sequence 0

! E (K )=mE (K )

! H (K; E )[m] 1

K=L)

Gal(

K=L)

Gal(

! H (K; E [m]) 1

K=L)

Gal(

!

! H (Gal(K=L); E (K )=mE (K )) = 0 1

8 where H 1(Gal(K=L); E (K )=mE (K )) is the zero group because it is killed by m and by n which are coprime. On the other hand, from the exact sequence of low degree terms of the Hochschild-Serre spectral sequence, we get the exact sequence

H 1(Gal(K=L); E (K )[m])

! H (L; E [m]) '! H (K; E [m])

!H

1

2

1

K=L)

Gal(

(Gal(K=L); E (K )[m])

!

where the rst and the last term are trivial because, again, they are killed by coprime integers. Hence the map ' is an isomorphism. The following diagram of exact rows and columns summarizes the information we have obtained so far 0 0

#

H (L; E [m]) 1

#' #

#

H (L; E )[m]

#'

!

0

!

0

0

H (K; E [m]) 1

!

1

K=L)

Gal(

'

! H (K; E )[m] 00

1

K=L)

Gal(

0:

It is immediate to verify that the injective map '0 is also surjective because of the surjectivity of the maps '00 and '. Therefore we obtain

H 1(L; E )[m] = H 1(K; E )[m]Gal(K=L) : Let us consider now a place v of L; since K is a Galois extension of L, for any place w of K over v the degrees of the local extensions Kw =Lv divide n and therefore they are coprime to m. Hence the reasoning above can be applied also to the extensions Kw =Lv and thus we obtain H 1(Lv ; E )[m] = H 1(Kw ; E )[m]Gal(Kw =Lv ) : Considering the corresponding Tate-Shafarevich groups, we get

X(E=L)[m] = X(E=K )[m] : Furthermore, assuming the niteness of X(E=L), it follows from the existence of the Cassels alternating bilinear pairing on X(E=L) that the order of X(E=L)[m] is a perfect square and hence the order of X(E=K )[m] is a square too. 2 p 3 cannot have order 3. Lemma 5 If X(E =Q ) is nite, then X(E =k) p To get a contradiction, assume that X(E =k) 3 and Z=3 are isomorphic as K=L)

Gal(

K=L)

Gal(

A

Proof.

A



A







abelian groups. Let E~A be the quadratic twist of EA corresponding to the class of 3 in H 1(Q ; Z=2) = 2 Q  =Q  ; E~A has equation E~A : 3Y 2 Z = X 3 432A2 Z 3

9 and is isomorphic to EA over k through the map :

EA (X; Y; Z )

! E~A 7 ! (12AZ; p36A3 (X Y ); X + Y ):

Let us consider the dualp isogenies 1 : EpA ! E~A and 2 : E~A ! EA given by the compositions 1 = Æ 3 and 2 = 3 Æ 1 ; they are de ned over Q and their composition 2 Æ 1 gives the multiplication-by-3 map on EA .

X

To obtain a contradiction we have assumed that (EA =k) as an abelian group; this is equivalent to the assumption that to Z=3.

p





3 is isomorphic to Z=3 (EA =k)[1 ] is isomorphic

X

X

Let Gal(k=Q ) = h i. We have two possibilities: either  acts trivially on (EA =k)[1 ] and therefore (EA =k)[] and Z=3 are isomorphic as Gal(k=Q )-modules; or  exchanges the two non-trivial elements of (EA =k)[1 ] and so (EA =k)[1 ] is isomorphic to 3 . If (EA =k)[1 ] is isomorphic to Z=3 as a Gal(k=Q )-module then (E~A =k)[2 ] is isomorphic to 3 as a Gal(k=Q )-module and vice versa. Indeed, if (EA =k)[1 ] is composed of the cohomology classes 0 ; 1 ; 2, then (E~A =k)[2 ] is composed of 0 ; 1; 2 ; moreover  = . So, if  acts trivially on (EA =k)[1 ] then it does not on (E~A =k)[2 ] and vice versa.

X

X

X

X

X

X X

X

X

X

Suppose that (EA =k)[1 ]  = Z=3 as a Gal(k=Q )-module and consider the exact sequence of Gal(k=Q )-modules 0

! X(EA=k)[ ] ! X(EA=k)[3] ! X(E~A =k)[ ] 1

2

where the rst map is the natural inclusion and the second one is induced by 1 . >From this sequence we get the exact sequence 0

! X(EA =k)[ ] 1

X

k=Q)

Gal(

! X(EA=k)[3]

! X(E~A=k)[ ] k=Q  = 0.

k=Q)

Gal(

X ! X(E =k)[3] ! X(E =k)[ ]

2

k=Q)

Gal(

where (EA =k)[1 ]Gal(k=Q)  = Z=3 and (E~A =k)[2 ]Gal( ) If (EA =k)[1 ]  = 3 , it is suÆcient to consider the sequence

X

0

! X(E~A=k)[ ]

instead of that above.

2

A

X

A

1

In both cases, we can conclude that (EA =k)[3]Gal(k=Q) is isomorphic to Z=3. This contradicts Lemma 4 which that the order of (EA =k)[3]Gal(k=Q) must be a square.  p claims 3 cannot have order 3. 2 As a result, (EA =k)

X

X

In conclusion, we have proved that C (A) = S (A) for A = p1 p2 p. Moreover, in the proof of Lemma 3 we have shown p1 p22 (k )3 belongs to S (A); it follows that the curve p1 X 3 + p2 Y 3 = p has a k point for any prime  of k and hence, by the local-to-global

10

REFERENCES

principle, it has a point over the quadratic extension k of Q . Therefore it has a point over Q. In order to prove Theorem 1, it only remains to be shown that given the prime integers

p1 ; p2 ; p3 ; p4  2 (mod 3) there exists a prime  such that the hypotheses of Lemma 3 are satis ed for each of the triples p1 ; p2 ;  and p3 ; p4 ;  . Lemma 6 Let p1 ; p2 ; p3 ; p4 be prime integers congruent to 2 modulo 3. Then there exists a prime  2 Z[! ] such that (i)   1 (mod 3); (ii) p1 p2 N( ) and p3 p4 N( ) are not congruent to 1 modulo 9; 

(iii) Proof.















    = = = 6 1. = p1 3 p2 3 p3 3 p4 3 Let B 2 f1; 4; 7g such that 

p1 p2 B 6 1 (mod 9) p3 p4 B 6 1 (mod 9):

Take a prime  2 Z[! ] such that N( )  B (mod 9). This condition can be satis ed by taking   (mod 9), where is an element of Z[! ] congruent to 1; 2 or 1+3! modulo 9 if B = 1; 4 or 7, respectively. Hence we have that pi pj B 6 1 (mod 9) if and only if pi pj N( ) 6 1 (mod 9). Therefore conditions (i) and (ii) are satis ed. As far as condition (iii) is concerned, in order for  to satisfy 















    = = = 6 1 = p1 3 p2 3 p3 3 p4 3

it is suÆcient to take  belonging to a suitable congruence class modulo p1 p2 p3 p4 . The Chinese Remainder Theorem allows us to determine a suitable residue class modulo 9p1 p2 p3 p3 such that, if   (mod 9p1 p2 p3 p3 ), then  satis es the required conditions. The existence of such a prime  is assured by Dirichlet's Theorem. 2 Hence Theorem 1 is proved. We thank Alexei Skorobogatov for his help in the preparation of this note.

References [1] J.W.S. Cassels. Arithmetic on curves reine angew. Math. 202 (1959) 52{99.

of genus

1.

. J.

I. On a conjecture of Selmer

11

REFERENCES

[2] J.W.S. Cassels. Lectures on Elliptic Curves. Cambridge University Press, 1991. [3] R. Heath-Brown. The solubility Soc. (3) 79 (1999), 241{259.

. Proc. London Math.

of diagonal cubic equations

[4] J.-P. Serre. A course in Arithmetic. New York: Springer{Verlag, 1973. [5] J.-P. Serre. Galois cohomology. Springer{Verlag Berlin Heidelberg, 1997. [6] J.H. Silverman. The Arithmetic of Elliptic Curves. New York: Springer{Verlag, 1986. [7] H.P.F. Swinnerton{Dyer. The solubility of diagonal cubic surfaces. Preprint (August 2000).

RATIONAL POINTS ON CUBIC SURFACES NIKLAS BROBERG . Let k be an algebraic number eld and F (x0 ; x1 ; x2 ; x3 ) a non{singular cubic form with coeÆcients in k. Suppose that the projective cubic k{surface X  P3k given by F = 0 contains three coplanar lines de ned over k, and let U (k) be the set of k{points on X which does not lie on any line on X . We show that the number of points in U (k), with height at most B , is OF;" (B 4=3+" ) for any " > 0. Abstract

1. Introduction It has been known since the 19th century that there are exactly 27 lines on any non{singular projective cubic surface X . If the form F de ning X has coeÆcients in a eld k which is not algebraically closed, then some of the lines may not be de ned over k. One of the more interesting problems of diophantine geometry is to count the number nX (B ) of rational points of height at most B on such a surface, and to study the asymptotic growth of this counting function as B tends to in nity. It appears that the lines on X de ned over k play the dominant role in determining the behaviour of nX (B ). It is known by a theorem of Schanuel [11] that if Z is such a rational line on X and nZ (B ) is the number of rational points on Z of height not exceeding B , then nZ (B ) = cB 2 + O(B log B ) for some positive constant c which depends on k and the choice of coordinates. In this formula, the eld k is a nite extension eld of Q , and the height function used in the de nition of nZ (B ) is the [k : Q ] power of the absolute height on the set of algebraic points P3 (Q ) in projective 3{space. When it comes to the points outside the lines, very little is actually known. It has been conjectured by Manin (see [3], for example) that if U is the open subset of X given by the complement of the 27 lines, then nU (B ) = O(B 1+") for any " > 0. In contrast to this hypothesis, the best general result of this kind seems to be nU (B ) = O(B 7=3+" ), due to Pila (see [6]). In some special cases, sharper estimates have been established. In the case where all 27 lines are de ned over the ground eld, it was shown by Manin and Tschinkel [9] that nU (B ) = O(B 5=3+" ). Another, much older, result is due to Hooley [7]. He proved, by means of sieve methods, that nU (B ) = O(B 5=3+" ) for the surface x30 + x31 + x32 + x33 = 0 over Q . This result is not covered by the theorem of Manin and Tschinkel, since 24 of the lines are not de ned over Q . A more elementary proof of Hooley's theorem was given by Wooley [14], and the result was improved and generalized by Heath{Brown [5], who 1991 Mathematics Subject Classi cation. 11G35, 11E20. Key words and phrases. Cubic surfaces, Rational Points, Quadratic Forms.

1

2 showed that nU (B ) = O(B 4=3+" ) for cubic surfaces over Q containing three coplanar rational lines. The aim of this paper is to extend Heath{Brown's estimate to cubic surfaces over arbitrary number elds. In order to achieve this we shall combine Heath{Brown's arguments with arguments from algebraic geometry and the arithmetic of ternary quadratic forms over arbitrary number elds. The main theorem of this paper is the following.

Theorem 1. Let k be an algebraic number eld and F (x0 ; x1 ; x2 ; x3 ) a non{ singular cubic form with coeÆcients in k. Suppose that the projective cubic k{surface X  P3k given by F = 0 contains three coplanar lines de ned over k. Then the number of rational points on X , not lying on any line, and with height not exceeding B is O(B 4=3+" ). The content of the paper is as follows. In section 2 we x the notation. We choose normalized absolute values, de ne height functions, and look brie y at the notion of lattices. We also state some preliminary results, in most cases without proof. One of these results is the adelic version of Minkowski's second theorem about successive minima, due to Bombieri and Vaaler [1]. Section 3, which is the larger part of the paper, is devoted to the arithmetic of ternary quadratic forms. We generalize two theorems of Heath{ Brown. Both results give uniform estimates for the number of points of bounded height on a conic. By uniform we mean that the estimates only depend on a few parameters of the form de ning the conic. In section 4 we prove theorem 1. The assumption that X contains three rational coplanar lines makes it possible to de ne three conic bundle morphisms fi : X ! P1k , one for each line. The general theory of height functions shows that if the height of x 2 X (k) is  B , then there is an index i such that x 2 fi 1(a; b) for some (a; b) 2 P1 (k) of height O(B 2=3 ). The results from section 3 give estimates for the number of rational points on the bres fi 1(a; b) of height  B . By uniformity, the estimates only depend on B and the parameter (a; b). By choosing the most favourable estimate for each (a; b) and summing over all such estimates, we get the required bound O(B 4=3+" ) for nU (B ).

Acknowledgements. I wish to thank my supervisor Per Salberger for his guidance and support during the course of this work. 2. Notations and preliminaries In this section we x the notation and state some preliminary results. For any set A, we denote by jAj or #A the cardinality of A. If A and B are commensurable abelian subgroups of some group, then [A : B ] is the quotient [A : A \ B ]=[B : A \ B ], where [A : A \ B ] and [B : A \ B ] are the index of A \ B in A and B , respectively. If f and g are two non{negative functions such that f  cg for some positive constant c on some common domain of f and g, then we write f  g or f = O(g). If the constant c is not absolute, then we may indicate in subscript the parameters on which it depends.

3 For any integral domain R and elements x1 ; : : : ; xn in the quotient eld of R, we let hx1 ; : : : ; xn i be the fractional ideal of R generated by x1 ; : : : ; xn .

Algebraic number elds. In general we denote a number eld by k, and any of its places by v. If k = Q , then by abuse of notation we write p for the non{archimedean ( nite) place corresponding to the prime number p, and 1 for its archimedean (in nite) place. If w is a place of Q obtained by restriction of a place v on k, we write v j w and dv = [kv : Q w ] for the local degree of the extension Q w  kv of local elds. Since the number of in nite places of k will occur frequently we denote this number by sk . The normalized absolute value j jv on k is the one which induces the ordinary absolute value on R if v j 1, and the p{adic absolute value jpjv = 1Q=p if v j p. We set kxkv = jxjdvv , so that we have the product formula  v kxkv = 1, for all x 2 k . We denote by ok = o and ov the ring of integers of k and kv , respectively, and write Nk (a) for the index [o : a], where a is any fractional ideal of o. Note that if v is a nite place of k corresponding to the prime ideal p of o and  is a generator of the maximal ideal of ov , then kkv 1 = Nk (p). We also have that Nk (hxi) is the absolute value of the norm Nk= (x) for x 2 k. In this case we write Nk (x) for short. We will not use the notion of measure in the last two sections, but we need it in the formulation of Bombieri and Vaaler's theorem in this section. If v j 1 and kv = R, then dv is the ordinary Lebesgue measure on R. If v j 1 and kv = C , then dv is the ordinary Lebesgue measure on the complex plane multiplied by 2. If v - 1, then dv is the measure normalized so that v (ov ) = kDv k1v=2 , where Dv is the local dierent of k at v. The following two results are well{known, in one form or another, and may be found in almost any book on algebraic number theory (see [8, V, x1], for example). Theorem 2 (Unit theorem). Let Uk be the group of units of o and Wk the subgroup of Uk consisting of the roots of unity in k. The image of the Q

regulator map

l : Uk ! Rsk ; x 7! (log kxkv )vj1

1){dimensional lattice in Rsk and the kernel is Wk . Let r 2 (R>0 )sk be a vector with components rv for v j 1, and let L(r) be the set of all x 2 o such that Q kxkv  rv . The size kr k of the vector r is de ned to be the product vj1 rv . Proposition 1. For any r 2 (R>0 )sk we have krk k jL(r)j k supf1; krkg: Height functions. Let k be a number eld and Pn (k) the set of k{points in projective n{space Pnk . Since k is equipped with a product formula there is a well{de ned height function Hk : Pn (k) ! R1 , given by Y Y (x0 ; : : : ; xn ) 7! sup kxi kv = Nk (hx0 ; : : : ; xn i) 1 sup kxi kv : 0  i  n 0  v vj1 in is a (sk

4 Note that Hk is not a restriction of the absolute height on Pn (Q ), where Q is an algebraic closure of Q . If K is an extension eld of k, then HK (x) = Hk (x)[K :k] for all x 2 Pn (k). We mention this because Northcott's niteness theorem below is formulated in terms of the absolute height on Pn (Q ). For a detailed discussion on heights and for proofs of the following two results see [12]. Theorem 3 (Northcott's niteness theorem). Let n, d, and B be positive integers. Then there are only nitely many points in Pn (Q ) of absolute height not exceeding B and of degree less or equal to d. Proposition 2. Let 0 ; : : : ; r be homogeneous polynomials of degree m and with coeÆcients in k. If the polynomials are not simultaneously zero on k (an algebraic closure of k), then log Hk (0 (x); : : : ; r (x)) = m log Hk (x) + O(1); n for all x 2 P (k). Note. A special, but important, case of this proposition is when  : kn+1 ! kn+1 is an automorphism. Then Hk (x)  Hk ((x))  Hk (x) for all x 2 n P (k ). The next result may also be found in [12, 13.2]. We include a proof since we will refer to it later. Proposition 3. Every point x 2 Pn (k) is representable by homogeneous coordinates (x0 ; : : : ; xn ) 2 on+1 such that sup0in kxi kv k;n Hk (x)1=sk for all v j 1. Proof. Suppose that a1 ; : : : ; ah are ideals representing the ideal classes of o. Then any point x 2 Pn (k) has coordinates (y0; : : : ; yn) 2 on+1 for which hy0; : : : ; yni = aj for some j . In particular, if (x0 ; : : : ; xn ) = (u 1 y0 ; : : : ; u 1 yn ) for some unit u of o, then 1 Y sup kx k : Hk (x) = Nk (aj ) vj1 0in i v It is therefore suÆcient to nd a unit u such that Y sup kxi kv  sup kxi k1v=sk 0in vj1 0in for all v j 1. Let Y yv = sup kyi kv = sup kyik1v=sk ; 0in vj1 0in so that the inequalities above may be written as yv  kukv . If Y  Q (R>0 )sk is the locally compact subgroup of elements y = (yv ) which satisfy v yv = 1, then the image of Uk in Y under u 7! (kukv ) is a discrete cocompact multiplicative lattice, by theorem 2. Hence there exists u of the desired type.

5 We have not found the next result in the literature so we include a proof. The height hk (x) of an element x 2 k is de ned to be Hk (1; x). Proposition 4. If d is the degree of k over Q , then the number of units u 2 Uk such that hk (u)  B is d supf1; (log B )sk g. Note. The exponent sk can be reduced to sk 1, but for our application the actual order is not very signi cant. Proof. First note that hk (u) = hk (u 1 ) for all u 2 Uk , so if hk (u)  B for some u 2 Uk , then supvj1 jlog kukv j  log B . By theorem 2, it is therefore suÆcient to show that there are d supf1; (log B )sk g lattice points of l(Uk )  Rsk in the box centred at the origin with side length 2 log B . From theorem 3 we have that there is a constant c > 1, only depending on d, such that u 2 Wk whenever u 2 Uk and hk (u)  c. This implies that any box with side length s2k log c contains at most one of the points of l(Uk ). By comparing volumes we thus have that the number of points of l(Uk ) in the box jlog kukv j  log B is  (log B )sk =(log c)sk d (log B )sk , providing that B  c. Lattices over number elds. Let k be a number eld of degree d over Q , and let n be a positive integer. One usually says that an o{module in kn is an o{lattice in kn if it is nitely generated and contains a basis of kn over k. It is not hard to see that if  is an o{lattice in kn and v is a nite place of k, then v = ov o  is a free ov {module in (kv )n such that v contains a basis for (kv )n over kv . We say that such an ov {module is an ov {lattice in (kv )n . The next result states that an o{lattice is known if it is known locally everywhere. For a proof see, for example, [10, 5.3]. Theorem 4. For each nite place v of k, let L in (kv )n Tv be an ov {lattice n n such that Lv = (ov ) for almost all v. If  = v 1 Lv \ k , then  is the unique o{lattice in kn such that v = Lv for all v. -

Proposition 5. If L   are o{lattices in kn , then there is an element a 2 k such that   aL and [L : ] k Nk (a). Proof. By the invariant factor theorem (see [10, 4.14], for example), there are elements u1 ; : : : ; un 2 , fractional o{ideals a1 ; : : : ; an , and o{ideals b1; : : : ; bn such that =

n M i=1

ai ui

L=

n M

ba

i i ui : i=1 Qn n with Nk (b) k i=1 Nk ( i ) and [L : ] k Nk (a).

and

Then, if we choose b 2 b1


b b = [ : L], and put a = b 1 , we have   aL Next we formulate Bombieri and Vaaler's adelic version of Minkowski's second theorem [1]. Later, when it is needed, we will use a slightly reformulated version of this result. We give this version as a corollary of the theorem. However, before we can formulate the result we have to make some de nitions. For each nite place v of k, let Lv be an ov {lattice in (kv )n such that Lv = (ov )n for almost all v. For each in nite place v of k, let Sv be a

6 nonempty, open, convex, symmetric, bounded subset of (kv )n . By symmetric we mean that Sv = Sv . Then Y Y = Sv  Lv v1 vj1 is a subset of (k )n , the n{fold product of adeles over k. In fact, is an open neighbourhood of 0 and the closure of is compact. Thus the volume Y Y n Vol( ) = nv (Sv ) v (Lv ) vj1 v1 exists as a nite number. Let  be the unique o{lattice such Q that v = Lv for all v - 1. The i:th successive minima i for S = vj1 Sv , with respect to  is de ned to be the in mum of all positive reals  such that  \ S contains i linearly independent vectors. It is obvious that 1 


 n . It is also evident from the de nition that there exist n linearly independent vectorsQui 2  such that ui 2= i Sv for some v, but fu1 ; : : : ; ui g  i S , where S = vj1 S v is the closure of S . We will use this observation later. For a proof of the following theorem see [1]. Theorem 5. The successive minima 1 


 n satisfy the inequality (1


n )d Vol( )  2dn : Corollary. The successive minima satisfy the relation Y (1


n )d Vol(Sv ) k;n [on : ]: vj1 Proof. All weQhave to do is to calculate the volume Vol( ). By de nition the product vj1 nv (Sv ) is the product of the volumes Vol(Sv ) multiplied by 2d sk (d sk being the number of complex places of k). By additivity and translation invariance of nv we obtain nv (v ) = [v : v \ (ov )n ] nv (v \ (ov )n ) = [v : (ov )n ] nv ((ov )n ): But then we are done since Y [(ov )n : v ] = [on : ]; v1 and Y Y v (ov ) = kDv k1v=2 = j
k j 1=2 ; v1 v1 where
k is the discriminant of k. -

A

-

-

-

-

3. Ternary quadratic forms In this section we estimate the number of rational points of bounded height on conics over number elds. The main estimates are uniform in the sense that the implied constants only depend on a few invariants of the coeÆcients of the quadratic form de ning the conic. The ground eld k is assumed to be xed from now on, so any implicitly given constants may depend on k, even if this is not stated explicitly. Before

7 we begin our discussion, we also want to stress that most of the arguments in this and the next section are generalizations of Heath{Brown's arguments in the case k = Q (see [5]). Let M be any invertible n  n{matrix with entries in k. We de ne
(M ) and
0 (M ) to be the fractional ideal of o generated by the determinant of M and the fractional ideal generated by the (n 1)  (n 1){minors of M , respectively. Lemma 1.
(MN ) =
(M )
(N ) and
0 (MN ) 
0 (M )
0 (N ). Proof. The equality
(MN ) =
(M )
(N ) is well{known, and the inclug = Ne M f besion
0 (MN ) 
0 (M )
0 (N ) follows from the relation MN tween the cofactor matrices of M , N , and MN . The main result of this section is the following theorem. Theorem 6. Let q be a ternary quadratic form with matrix M 2 M3 (o), and let r1 , r2 , r3 2 (R>0 )sk be given. If q is non{singular, then there are s ! k r1 k kr2 k kr3 k Nk (
0 (M ))2 " 1 + Nk (
(M ))" Nk (
(M )) points (x1 ; x2 ; x3 ) 2 P2 (k) on the conic q = 0 such that xi 2 L(ri ). Note. By proposition 3 we know that for any point x 2 P2 (k) with Hk (x)  B , we can nd homogeneous coordinates (x1 ; x2 ; x3 ) 2 o3 such that kxi kv  B 1=sk for all v j 1. The theorem then says that there are Oq (B 3=2 ) points on the conic q = 0 of height at most B . Intuitively this estimate is not what one expects (q is of degree 2). One can of course do better if one does not require uniformity in q, as the following proposition shows. Proposition 6. If q is a non{singular quadratic form, then there are Oq (B ) points x 2 P2 (k) on the conic q(x) = 0 such that Hk (x)  B . Proof. If q is isotropic, then q is \k{equivalent" to (x20 x1 x2 ) for some  2 k . But (y0 ; y1 ) 7! (y0 y1 ; y02 ; y12 ) is a parametrization of the solutions of x20 x1 x2 = 0, and Hk (y0 y1 ; y02 ; y12 ) = Hk (y0 ; y1 )2 for all (y0 ; y1 ) 2 P1 (k), so by Schanuel's theorem,  # x 2 P2 (k) : q(x) = 0; Hk (x)  B q n

# y 2 P1 (k) : Hk (y)  B 1=2

o

 B:

The proof of the theorem will be given in several steps. In lemma 4 we look at the equation q = 0 at each nite place of k and nd that each solution (x1 ; x2 ; x3 ) 2 o3 must belong to one of not too many o{lattices in k3 . We also nd lower bounds for the indices of the lattices in o3 . By using the theory of successive minima we then use this information to reduce the proof to a problem of counting the number of points on a conic in a \bounded domain" of P2 (k). We obtain a solution to this problem in lemma 5. First, however, we formulate some minor results which are included for completeness and which we need in the proof of lemma 4.

8 For an ideal a of o, let (a) be the number of prime ideals containing a and  (a) the number of ideals containing a. Recall that an arithmetical function f : Z>0 ! C is said to be multiplicative if f (mn) = f (m)f (n) whenever gcd(m; n) = 1. For a proof of the following result see, for example, [13, I. x5.1]. Theorem 7. Let f be a multiplicative function. If limp !1 f (p ) = 0, where p are prime powers, then limn!1 f (n) = 0. Corollary. If c is a positive number, then c(a)  (a) c;" Nk (a)" for all ideals

a of o.

Proof. Let (n) and  (n) be the number of primes dividing n 2 Z>0 and the

number of divisors of n, respectively. Then we have (a)  d (Nk (a)) and  (a)   (Nk (a))d for all ideals a of o, where d is the degree of k over Q . Thus c(a)  (a)  cd(Nk (a))  (Nk (a))d c;" Nk (a)" , since f (x) = (c(x)  (x))d =x" is a multiplicative function and f (p ) = (c ( + 1))d p " ! 0 as p ! 1. If M , N , and T are square matrices of the same size, with entries in some ring R, and T is unimodular with T t MT = N , then we say that M and N are R{equivalent. This means that M and N only diers by a nite number of simultaneous elementary row{ and column{operations. The next lemma says that over a discrete valuation ring any symmetric matrix is almost equivalent to a diagonal matrix. Lemma 2. Let v be a discrete valuation on some eld, and let R be the corresponding valuation ring. Then for any symmetric matrix M 2 Mn (R) there exists an element x 2 R and matrices P , D 2 Mn (R) such that D is diagonal, P t DP = xM , and v(det P ) = v(x)  (n 1)v(2). Proof. Let n be a positive integer and M any symmetric n  n{matrix with entries in R. It is suÆcient to show that we can nd an element x 2 R and a matrix P 2 Mn (R) such that v(det P ) = v(x)  v(2) and P t DP = xM , where D is some matrix with zeros in its rst row and column, except perhaps on the diagonal. The lemma then follows by induction on the size of the matrices. Since the existence of the objects x and P is trivially veri able when n = 1 or M = 0, we may assume that n  2 and that at least one of the entries of M is a unit in R. Then M is equivalent to a matrix with its rst row equal to (a; u; 0; : : : ; 0) or (u; 0; 0; : : : ; 0), where u is a unit and a is some element of R such that 0  v(a)  v(2). If v(2) = 0, then a matrix of the rst shape is obviously equivalent to a matrix of the second shape. If, on the other hand, v(2) > 0, we can use the identity  a u t   0 0   a u    1 0 0 =a a x ; 2 0 01 0 0 1 0 u 0 I 0 aM 0 0 I xt M 0 0 0 where M 0 is a symmetric (n 1)  (n 1){matrix and x = (u; 0; : : : ; 0). This establishes the existence of the objects x and P . Before the next lemma we just recall that if v is a nite place of k lying over a prime number p, then kpkv = kkrv , where  is a generator of the maximal ideal of ov and r is the rami cation index of Q p  kv .

9 Lemma 3. For a nite place v of k and any integer n  1, let Rn be the ring ov =hn i. If " 2 Rn , then the number of solutions of the equation x2 = " is less or equal to 2 k2kv 1 . Proof. Since (x=y)2 = 1 if x2 = y2 = " for " 2 Rn , we only need to study the case " = 1. Let r be the least integer such that 2 2= hr+1 i. Then either x +1 2= hr+1 i or x 1 2= hr+1 i for x 2 ov . Hence, if x2 1 = (x +1)(x 1) 2 hni, we must have x + 1 2 hn r i or x 1 2 hn r i. There are thus at most 2 jhn r i=hn ij = 2 j(R1 )r j solutions of x2 = " in Rn . Moreover, r = 0 if k2kv = 1, and r is the rami cation index of Q 2  kv if k2kv < 1. We now have enough information to prove the rst main lemma in the proof of theorem 6.

Lemma 4. Let q be non{singular ternary quadratic form with matrix M 2 M3 (o). For each nite place v of k, let v be a generator of the maximal ideal of ov , and let av and bv be the non{negative integers de ned by k
(M )kv = kv kavv and k
0(M )kv = kv kbvv , respectively. (a) If av > 0 and q(x) = 0 for some x 2 (ov )3 , then x belongs to at least one of at most O(av ) ov {lattices  in (kv )3 , each satisfying [(ov )3 : ]  k2k8 k
(M )k 1 k
0 (M )k2 : v

v

v

Moreover, the implied constant in O(av ), let us call it c, depends only on k and may be chosen to be the same for all v. (b) If q(x) = 0 for some x 2 o3 , then x belongs to at least one of at most O(Nk (
(M ))" ) o{lattices  in k3 , each satisfying

[o3 : ]  Nk (
(M )) Nk (
0 (M )) 2 :

Proof. It is (b) that really interests us, so assume for the moment that

we have a proof of (a). If q(x) = 0 for some x 2 o3 , then x belongs to an o{lattice  in k3 with v = (ov )3 if av = 0, and v equal to one of the ov {lattices from (a) if av > 0. This is a consequence of theorem 3 4. It follows immediately from the local nature of index Q that [o : ]  2 Nk (
(M )) Nk (
0 (M )) . Moreover, there are at most av >0 cav such lattices , and Y cav  c(
(M ))  (
(M ))  Nk (
(M ))" av >0 by the corollary of theorem 7. It is thus suÆcient to nd a proof of (a). Let v be a nite place of k such that av > 0. To keep the notations as simple as possible we skip the indices and write , a, and b for v , av , and bv , respectively. By lemma 2 there is an element x 2 ov such that x q(x) may be diagonalized over the ring ov , using a matrix P with kdet P kv = kxkv  k2k2v . Let D be the matrix of this diagonalized form Q(y1 ; y2 ; y3 ) = "1 1 y12 + "2 2 y22 + "3 3 y32 ; where 1  2  3  0 are integers and "1 , "2 , "3 are units. From the identity
(xM ) =
(P )2
(D) and the inclusion
0 (xM ) 
0 (P )2
0 (D), we

10 have kxkv kkav = kkv 1 +2 +3 and kxk2v kkbv  kkv 2 +3 , respectively. If we combine these two relations and use the fact kxkv  k2k2v , we get kkv 1 +2 +3  k2k8v k
(P )kv 1 kk2vb a : This inequality makes the hypothesis of the lemma more understandable. All we have to prove is that any solution y 2 (ov )3 of Q = 0 belongs to at least one of at most O(a) lattices, each with index  kkv 1 +2 +3 in (ov )3 . Now suppose that Q(y1 ; y2 ; y3 ) = 0 for some (y1 ; y2 ; y3 ) 2 (ov )3 . Then we nd that (3.1) "2 2 3 y22 + "3 y32 2 h1 3 i: When 2 and 3 have opposite parities this implies that y22 2 h1 2 i and y32 2 h1 3 i. It follows that (y1 ; y2 ; y3 ) belongs to the lattice 1 2 +1 1 3 +1  = ov  h[ 2 ] i  h[ 2 ] i: Since [(ov )3 : ]  kkv 1 +2 +3 , this completes the proof in this case. The case in which 2 3 = 2h, for some h, needs slightly more work. We can assume that y22 2= h1 2 i and y32 2= h1 3 i. Otherwise (y1 ; y2 ; y3 ) would belong to the lattice  and we would not get anything new. By studying (3.1) we see that y2 = u2 n and y3 = u3 n+h for some non{ negative integer n < [ 1 2 2 +1 ] and some units u2 and u3 , which satisfy "2 u22 + "3 u23 2 h1 2 2n i. By lemma 3 there are at most 2 k2kv 1 solutions of the equation r2 +"2 ="3 = 0 in the ring ov =h1 2 2n i. If r 2 ov represents the same solution as u2 =u3 , then r hy2 y3 2 h1 2 +h ni, so 0 1 0 10 1 1 0 0 y1 z1 @y2 A = @1 n A @z2 A 0 y3 z3 1 rh+n 1 2 +h n for some (z1 ; z2 ; z3 ) 2 (ov )3 . Thus (y1 ; y2 ; y3 ) belongs to at least one of at most 2 k2kv 1 [ 1 22 +1 ] lattices, each with index kkv 2 1 h  kkv 1 +2+3 in (ov )3 . To complete the proof we have to show that (2 + 1 2 ) k2kv 1 + 1  ca for some constant c which only depends on k, but this is obviously true. The next result is the second tool in the proof of the theorem. The significance of this result is that the bounds are completely independent of the form involved. Lemma 5. Let f be a ternary form of degree d with no linear factor, and let r1 , r2 , r3 2 (R>0 )sk be given. Then there are p d 1 + kr1k kr2 k kr3k points (x1 ; x2 ; x3 ) 2 P2 (k) on the curve f = 0 such that xi 2 L(ri ). Proof. Let riv be the v:th component of the vector ri , and set Ri = kri k. We begin by showing that any point (x1 ; x2 ; x3 ) 2 P2 (k) with xi 2 L(ri ) lies on a line a1 x1 + a2 x2 + a3 x3 = 0; where the coeÆcients ai 2 o are not all p 1 zero and kai k  r r1v r2v r3v . This is a simple application of the box v

iv

11

3sk pR1 R2 R3 sets of coeÆcients principle. By proposition 1 there are  Y p with kaki  Y riv1 r1v r2v r3v . The corresponding values a1 x1 + a2 x2 + a3 x3 satis es ka1x1 + a2x2 + a3x3 kv  Y pr1v r2v r3v ; p so by the same proposition there are  supf1; Y sk R1 R2 R3 g such values. Hence, if Y is suÆciently large, then two such values must agree, and this for a Y independent of R1 , R2 , R3 (for the case R1 R2 R3 < 1 see below). Since f does not have any linear factors, each of the above lines has at most d points in common with f = 0. Moreover, the number of lines is 3 Y p p  supf1; Ri 1 R1R2 R3g  R1R2 R3; i=1 p if Ri 1 R1 R2 R3  1 for all i. If this last condition is not satis ed, then p

Ri Rj = inf fR1 R2 ; R1 R3 ; R2 R3 g  R1 R2 R3 : Since each pair of elements xi 2 L(ri ), xj 2 L(rj ) produces at most d solutions of f = 0, and the number of such pairs is  supf1; Ri g supf1; Rj g; we can use the estimate O(Ri Rj ) in this case. Of course we have to assume that Ri  1 and Rj  1, but L(ri ) = f0g or L(rj ) = f0g in any other case, and then there are at most d solutions.

To prove theorem 6 we now use the theory of successive minima to combine these two last lemmas. Recall that q is a non{singular ternary quadratic form with matrix M 2 M3 (o), and r1 , r2 , r3 2 (R>0 )sk are vectors with components riv for v j 1. In order to have the same notation as in section 2, we de ne the nonempty, open, convex, symmetric, bounded subsets Sv of (kv )3 by n o Sv = (x1 ; x2 ; x3 ) 2 (kv )3 : jxi j < riv1=dv ; Q and put S = vj1 Sv . Obviously o3 \ S = L(r1 )  L(r2 )  L(r3 ), where Q S = vj1 S v is the closure of S . If d is the degree of k over Q and 1  2  3 are the successive minima of S with respect to one of the lattices  from lemma 4, then Y (1 2 3 )d Vol(Sv )  [o3 : ]; vj1 by the corollary of theorem 5. Moreover, Y

vj1

Vol(Sv )  kr1 k kr2 k kr3 k

by the de nition of the sets Sv . We mentioned in the preparations for theorem 5 that one can nd a basis u1 , u2 , u3 of k3 over k such that ui 2 i S , but ui 2= i Sv for some v. If uij is the j :th component of ui , then these conditions imply that kuij kv  di v rjv . Thus, if (x1 ; x2 ; x3 ) =

12 y1 u1 + y2 u2 + y3 u3 2 S for some (y1 ; y2 ; y3 ) 2 k3 , and U is the matrix with columns ui , then

0 1

x u u 1 21 31

r3v (2 3 )dv ky1kv = kdet1U k

det @x2 u22 u32 A

 r1v r2kvdet U kv v x3 u23 u33 v by Cramer's rule. There are of course analogous estimates for ky2 kv and ky3kv . Note that u1, u2, u3 does not constitute a basis for  over o. Presumably the lattice  is not even free. However, if L is the free o{lattice with u1 , u2 , u3 as generators, then by proposition 5 we have L    aL for some a 2 k such that Nk (a)  [L : ]. Thus, any element (x1 ; x2 ; x3 ) 2  \ S may be written as y1 (au1 )+ y2 (au2 )+ y3 (au3 ) for some (y1 ; y2 ; y3 ) 2 o3 . This shows that if q0 is the non{singular quadratic form with matrix U t MU , then any solution (x1 ; x2 ; x3 ) 2  \ S of q = 0 gives a solution (y1 ; y2 ; y3 ) 2 o3 of q0 = 0 which satis es   r1v r2v r3v 1 2 3 dv kyikv  kak kdet U k = tiv : i v v Moreover, there are gets Y i;v

 1+

qQ

i;v tiv

such solutions by lemma 5, and one

 k r1 k kr2 k kr3 k 3 (1 2 3 )2d  tiv = N (a)N (det U ) 

k



k

kr1k kr2 k kr3k 3 

2 [o3 : ] kr1k kr2 k kr3k [L : ][o3 : L] kr1k kr2 k kr3k = [o3 : ] from the de nitions and estimates above. Referring to lemma 4, this completes the proof of the theorem. Next we prove a result that we can use when the determinant Nk (
(M )) of the form q is small. In that case the estimate from theorem 6 is obviously not very good. Proposition 7. Let q be a non{singular ternary quadratic form such that the binary form q(0; x2 ; x3 ) is also non{singular. Let r 2 (R>0 )sk and R  1 be given, and denote by kqk the supremum of kkv , where  ranges over the coeÆcients of q and v over the in nite places of k. Then there are  supf1; krk (krk kqk R)"g points (x1 ; x2 ; x3 ) 2 P2 (k) on the conic q = 0, satisfying (x1 ; x2 ; x3 ) 2 L(r)  o  o and supvj1 kxikv  R for i = 2; 3. Here r 2 (R>0 )sk is the vector with the v:th component equal to rv if rv  1 and 1 otherwise. As before, rv denotes the v:th component of r. Proof. Since q(0; x2 ; x3 ) is non{singular there is an invertible matrix P 2 M3 (k) with rst row (1; 0; 0) such that q(x) = d(P x) for some diagonal form d. Moreover, we may choose P so that the entries pij satis es supfkpij kv ; kpij kv 1 g  kqkA for all v j 1 and some xed exponent A. The equation q(x1 ; x2 ; x3 ) = 0 then becomes (3.2) L1 (x1 ; x2 ; x3 )2 + L2 (x1 ; x2 ; x3 )2 = x21

13 with non{zero coeÆcients , , , and linear forms Li such that L1 (0; x2 ; x3 ) and L2 (0; x2 ; x3 ) are linearly independent. By multiplying the equation with a suitable factor, we can assume that all coeÆcients are integers in k and satisfy supvj1 kkv  kqkA , possibly with a new constant value of A. By the same argument, we may also assume that  is p a square. The left hand side of (3.2) then factorizes over the eld K = k( ), and we have (3.3) L3 (x1 ; x2 ; x3 ) L4 (x1 ; x2 ; x3 ) = x21 for some linear forms Li with coeÆcients in oK . Now suppose that (x1 ; x2 ; x3 ) 2 L(r)  o  o satis es (3.3), and that supvj1 kxi k  R for i = 2; 3. If kr k  1, then there are at most two such solutions, considered as elements of P2 (k), since in that case L(r) only consists of the zero element. If kr k  1, on the other hand, the cardinality of L(r) is  krk, and in this case we show that there are  (krk kqk R)" possible pairs x2 , x3 for each x1 . That is, we show that there are  (kr k kqk R)" possible factorizations of the element y12 in the ring oK with certain conditions on the factors induced by the conditions on x1 , x2 , x3 stated above. Let y1 = x1 , y2 = L3 (x1 ; x2 ; x3 ), and y3 = L4 (x1 ; x2 ; x3 ) so that we may write (3.3) as y2 y3 = y12 . If w is an in nite place of K , lying over the place v of k, then kyikw  (kqkA supfrv ; Rg)dw =dv  kqk2A supfrv2; R2 g = tw for i = 2; 3. This follows from the conditions on x1 , x2 , x3 and the conditions on the coeÆcients of the forms L3 and L4 . Remember that dv = [kv : Q v ] is the local degree of k at v. We have seen earlier that there are  Nk ( y12 )" ideals of o which contain the element y12 (c = 1 in the corollary of theorem 7). Since a prime of o can split in at most two primes of oK , there are thus  Nk ( y12)"  (kqk krk)" ideals of oK containing y12. To complete the proof, then, it is suÆcient to show that there are  (kr k kqk R)" possible generators of any principal ideal hyi  oK such that kykw  tw . But if uy is another such generator for some unit u of oK , then Y hK (u)  tw = kykw  (kr k kqk R)2AsK ; wj1 and by proposition 4 there are  (kr k kqk R)" such units. Note. In contrary to theorem 6, this proposition gives the expected estimate O(B 1+" ) for the number of points of height  B on the conic q = 0. 4. Proof of the main theorem We have now come to the proof of theorem 1. We will begin by rede ning the objects of interest and formulate the hypothesis once again. As in the previous section we assume that the ground eld k is xed. Let F be a non{singular quaternary cubic form such that the k{surface X  P3k given by F = 0 contains three coplanar lines de ned over the ground eld, and let U  X be the complement of all the lines on X . We de ne n(B ) to be the number of rational points in U with height not exceeding B (note that n(B ) is nite by theorem 3). Our claim is that n(B ) = O(B 4=3+" ).

14 To show this we will proceed in several steps. Clearly n(B ) depends on the choice of coordinates. But it follows from proposition 2 that the validity of the statement n(B )  B 4=3+" is independent of the choice of coordinates. Proposition 2 says that Hk ((x))  Hk (x) if  is an automorphism of k4 . The rst thing we do, therefore, is to nd appropriate coordinates in which to study F . By an initial linear change of variables we may assume that the plane containing the three lines is x0 = 0. Then F = L1 L2 L3 x0 Q for some non{zero quadratic form Q and some non{zero linear forms Li (x1 ; x2 ; x3 ). After re{scaling both the variables and the form F appropriately we may moreover assume that Q and the Li have coeÆcients in o. The reduction which is about to follow is the most important part of our proof, and it is the reason why we assume that X contains three coplanar rational lines. The reduction is a generalization of an argument of Wooley and Heath{Brown. From the shape of F we see that the projections (

(x0 ; L1 ) if (x0 ; L1 ) 6= (0; 0) f1 (x0 ; x1 ; x2 ; x3 ) = (L2 L3 ; Q) otherwise; (

(x0 ; L2 ) if (x0 ; L2 ) 6= (0; 0) f2 (x0 ; x1 ; x2 ; x3 ) = (L1 L3 ; Q) otherwise; (

(x0 ; L3 ) if (x0 ; L3 ) 6= (0; 0) f3 (x0 ; x1 ; x2 ; x3 ) = (L1 L2 ; Q) otherwise de ne three conic bundle morphisms fi : X ! P1k . Lemma 6. For all x 2 X (k), H1 (x)H2 (x)H3 (x) F H (x)2 ; where Hi(x) = Hk (fi (x)), and H (x) = Hk (x). Proof. Let f : X (k) ! P1 (k)  P1 (k)  P1 (k) be the morphism given by (f1 ; f2 ; f3 ), and let : P1 (k)  P1 (k)  P1 (k) ! P7 (k) be the trilinear Segre embedding. Then from the equation L1 L2 L3 = x0 Q one sees that f (x0 ; x1 ; x2 ; x3 ) is given by (y0 ; y1 ; y2 ; y3 ; y4 ; y5 ; y6 ; y7 ) = (x20 ; x0 L3 ; x0 L2 ; L2 L3 ; x0 L1 ; L1 L3 ; L1 L2 ; Q): Hence, Y Y H1 (x)H2 (x)H3 (x) = sup kyikv  sup kxi xj kv = H (x)2 v 0i7 v 0i;j 3 for all x = (x0 ; x1 ; x2 ; x3 ) 2 P3 (k). Note. The converse H (x)2  H1 (x)H2 (x)H3 (x) is also true. The assertion that 2 log H = log H1 + log H2 + log H3 + O(1) on X (k) is in fact a special case of a standard result in the theory of heights (cf. e.g. [12, 2.8]). According to this lemma, we can choose a positive constant c1 such that H1 (x)H2 (x)H3 (x)  c31 H (x)2 on X (k). If ni (B ) is the number of points x 2 U (k) such that H (x)  B and Hi(x)  c1 B 2=3 , then n(B )  n1 (B ) + n2 (B ) + n3 (B ):

15 It is thus suÆcient to show that ni(B )  B 4=3+" for i = 1, 2, 3. We will concentrate on the proof of the statement n1 (B )  B 4=3+" . In this particular case, it is convenient to have L1 = x1 and L2 = x2 . Of course, this change of variables will aect the counting function n1 (B ), but as pointed out above, the validity of the statement n1 (B )  B 4=3+" is independent of the choice of coordinates. The next step in the proof is to look at the bres of f1 : X ! P1k in U . If we de ne n1 (a; b; B ) to be the number of rational points in f1 1(a; b) \ U with height not exceeding B , then we have X (4.1) n1 (B )  n1 (a; b; B ): (a;b)2 1(k) H (a;b)c1 B 2=3 From the de nition of F we see that (ay1 ; by1 ; y2 ; y3 ) 2 f1 1(a; b) \ U only if y1 6= 0 and q(y1 ; y2 ; y3 ; a; b) = 0, where q(y1 ; y2 ; y3 ; a; b) = 2 b y2 L3 (by1 ; y2 ; y3 ) 2 a Q(ay1 ; by1 ; y2 ; y3 ): In order to estimate the counting functions n1 (a; b; B ) in the sum (4.1), we shall apply the results from the previous section to the quadratic forms q(y1 ; y2 ; y3 ; a; b). However, the results in the previous section were formulated for quadratic forms, de ned over the ring of integers of k, and points in projective space, with conditions on their integral coordinates. Before we can continue, we therefore have to look more closely at the de nition of n1 (a; b; B ) and make some adjustments to be able to apply these results. Let a1 ; : : : ; ah be ideals representing the ideal classes of o. By the proof of proposition 3, we can choose coordinates (a; b) 2 o2 for each point in 1 1=sk P (k ) such that ha; bi = aj for some j and supfkak ; kbk g  H (a; b) v v for all v j 1. We let A be the set of all such coordinates and A(B ) be the subset consisting of the elements (a; b) with H (a; b)  c1 B 2=3 . By the same proposition, each point on the bres of f1 has coordinates (ay1 ; by1 ; y2 ; y3 ) 2 o4 , where (a; b) 2 A and (4.2) supfkay1 kv ; kby1 kv ; ky2 kv ; ky3 kv g  H (ay1 ; by1 ; y2 ; y3 )1=sk : A choice of (y1 ; y2 ; y3 ) under these restrictions does not guarantee that y1 2 o. However, by multiplying Qh (2y1 ; y2 ; y3 ) with a suitable factor, say, a generator of the principal ideal i=1 ai , and by changing the implied constant of (4.2) accordingly, we may assume that (y1 ; y2 ; y3 ) 2 o3 . If we introduce the symbol c2 for this modi ed implied constant, then we see that n1 (a; b; B ) is bounded from above by the number of points (y1 ; y2 ; y3 ) 2 P2 (k) satisfying yi 2 o, (ay1 ; by1 ; y2 ; y3 ) 2 U (k), and supfkay1 kv ; kby1 kv ; ky2 kv ; ky3 kv g  c2 B 1=sk : In fact, we rede ne n1 (a; b; B ) to be this cardinality. Let r 2 (R>0 )sk be a vector with components rv . Then by the de nition of the set L(r)  o we have that an integer x of k belongs to L(r) if and only if kxkv  rv for all v j 1. In agreement with the observations above, we now see that we can de ne n1 (a; b; B ) to be the number of points (y1 ; y2 ; y3 ) 2 P2 (k) on the conic q(y1 ; y2 ; y3 ; a; b) = 0, not lying on any line on X , and such that yi 2 L(ri ), where r1 is the vector with components P

16 c2 B 1=sk = supfkakv ; kbkv g and r2 = r3 the vectors with components c2 B 1=sk . With this de nition of n1 (a; b; B ), we are now in a position to use the results from the previous section to handle the sum (4.1). Note that with our new de nitions we may write this sum as X n1 (a; b; B ): (a;b)2A(B) Also note that we only have to sum over those (a; b) 2 A(B ) for which q(y1 ; y2 ; y3 ; a; b) is non{singular. By de nition, n1 (a; b; B ) = 0 whenever the components of f1 1 (a; b) are lines on X . On the hypothesis that q(y1 ; y2 ; y3 ; a; b) is non{singular, theorem 6 gives ! B 3=2 Nk (
0 (a; b)) (4.3) n1 (a; b; B )  1 + Nk (
(a; b))" ; H (a; b)1=2 Nk (
(a; b))1=2 where
(a; b) is the determinant of q(y1 ; y2 ; y3 ; a; b) and
0 (a; b) the ideal generated by all the 2  2{minors of the matrix of q(y1 ; y2 ; y3 ; a; b). If, in addition, q(0; y2 ; y3 ; a; b) is non{singular, proposition 7 gives (4.4) n1 (a; b; B )  B 1+"=H (a; b): In order for these estimates to be useful we need the following two lemmas. Lemma 7. Nk (
0 (a; b)) F 1 for all (a; b) 2 A. Proof. Let M (a; b) be the matrix of q(y1 ; y2 ; y3 ; a; b). The ij :th minor of M (a; b) will be a certain integral form Mij (a; b). If these forms were to have a common factor ( a b) over some algebraic closure of k, then the rank of q(y1 ; y2 ; y3 ; ; ) would be at most one. But f1 1 (; ) is not allowed to be a double line since X is non{singular. Thus the forms Mij (a; b) do not have a common factor. According to Hilbert's Nullstellensatz, P there is a n positive P integer n and polynomials fij , gij such that a = fij Mij and bn = gij Mij . Then by clearing denominators we see that an , bn 2
0 (a; b) for some 2 o. Hence, Nk (
0 (a; b))  Nk (han ; bn i) = Nk (aj )n for some j = 1; 2; : : : ; h. The lemma tells us that we can forget about the factor
0 (a; b) in (4.3). Lemma 8. q(0; y2 ; y3 ; a; b) is non{singular for almost all (a; b) 2 A. Proof. By de nition q(0; y2 ; y3 ; a; b) = 2 b y2 L3 (0; y2 ; y3 ) 2 a Q(0; 0; y2 ; y3 ): If L3 (0; y2 ; y3 ) = y2 + y3 and Q(0; 0; y2 ; y3 ) = y22 + y2 y3 + y32 for some , , , ,  2 o, then q(0; y2 ; y3 ; a; b) = 2 (b a) y22 + 2 (a b ) y2 y3 2 a y32 : Hence, q(0; y2 ; y3 ; a; b) is singular precisely when (4 2 ) a2 + 2 (  2 ) a b  2 b2 = 0: It is straightforward to check that X would be singular if this form were to vanish identically. Thus q(0; y2 ; y3 ; a; b) is singular for at most two pairs (a; b) 2 A.

17 From this last lemma, we see that if q(0; y2 ; y3 ; a; b) is singular, then we may use the estimate n1 (a; b; B )  B given by proposition 6. It thus remains to sum n1 (a; b; B ) over those (a; b) 2 A(B ) for which both q(y1 ; y2 ; y3 ; a; b) and q(0; y2 ; y3 ; a; b) are non{singular. In this sum, the constant term of (4.3) gives the contribution O(B 4=3+" ). Simply because the cardinality of A(B ) is O(B 4=3+" ). We therefore only have to account for the second term (4.5) B 3=2+" H (a; b) 1=2 Nk (
(a; b)) 1=2 in the contribution to n1 (a; b; B ). We divide the ranges of H (a; b) and Nk (
(a; b)) into intervals (R; 2R] and (S; 2S ], respectively. Since
(a; b) is a form of degree 5, the next lemma states that there are OF (S 1=5 R1+" ) elements (a; b) 2 A which satisfy H (a; b)  2R and Nk (
(a; b))  2S . The fact that
(a; b) is of degree 5 also implies that Nk (
(a; b))  H (a; b)5  B 10=3 . If S  B 5=3 , then X n1 (a; b; B )  S 1=5 R1+"B 1+" R 1  B 4=3+" H (a;b)2(R;2R] Nk (
(a;b))2(S;2S ] by (4.4), and if S  B 5=3 , then X n1 (a; b; B )  S 1=5 R1+"B 3=2 R 1=2 S 1=2+"  B 4=3+" H (a;b)2(R;2R] Nk (
(a;b))2(S;2S ] by (4.5). This completes the proof of the theorem, on summing over appropriate values of R and S . Lemma 9. Let G(x; y) be a form of degree n with coeÆcients in o, and let R, S  1 be given. Then there are OG (S 1=n R1+" ) elements (x; y) 2 o2 such that 0 < Nk (G(x; y))  S and supfkxkv ; kykv g  R1=sk for all v j 1. Proof. First we note that we may assume that G(x; y) is irreducible over k. If not, G(x; y) = G1 (x; y) G2 (x; y) for some forms of positive degrees n1 and n2 , respectively. If Nk (G(x; y))  S for some (x; y) 2 o2 , then either Nk (G1 (x; y))  S n1 =n or Nk (G2 (x; y))  S n2 =n . On the assumption that the lemma is valid for all forms of degree less than n, we thus have that it is valid for all reducible forms of degree n. Now assume that G(x; y) is irreducible and let v : k ! C be embeddings such that jxjv = jv (x)j for all x 2 k. Then there are non{zero complex numbers vi and vi such that

v (G(x; y)) =

n Y i=1

(vi v (x) vi v (y)):

If (x; y) 2 o2 satis es Nk (G(x; y))  S , then Y (4.6) jvi v (x) vi v (y)jdv  S 1=n vj1 for some i. It is thus suÆcient to show that there are O(S 1=n R1+" ) elements (x; y) 2 o2 satisfying the assumptions of the lemma and (4.6) for a xed i. Note that there are O(R) elements y 2 o such that sup kykv  R1=sk .

18 We therefore assume that y is xed and show that there are O(S 1=n R") elements x 2 o such that (x; y) have the required properties. In order to accomplish this, we choose one place w of k and divide the ranges of jvi v (x) vi v (y)jdv for the places v 6= w into intervals (Tv ; 2Tv ]. Note that R1=sk n  jvi v (x) vi v (y)jdv  R1=sk for all (x; y) 2 o2 satisfying the assumptions of the lemma. We get the lower bounds from the above bounds and the fact 1  Nk (G(x; y)). Now, if jvi v (x) vi v (y)jdv 2 (Tv ; 2Tv ] and (4.6) holds, then Y jwi w (x) wi w (y)jdw  S 1=n= Tv : v6=w Since kx x0 kv  Y= jvi jdv whenever jvi v (x) vi v (y)jdv  Y and jvi v (x0) vi v (y)jdv  Y , proposition 2 gives the estimate OG(S 1=n ) for the number of elements x 2 o which satis es the above conditions. By summing over all the intervals (Tv ; 2Tv ], we get the required estimate OG (S 1=n R" ), and this completes the proof. References

[1] E. Bombieri and J. Vaaler. On Siegel's lemma. Invent. Math., 73:11{32, 1983. [2] J. W. S. Cassels. An Introduction to the Geometry of Numbers. Springer{Verlag, 1959. [3] J. Franke, Yu. I. Manin, and Yu. Tschinkel. Rational points of bounded height on Fano varieties. Invent. Math., 95:421{435, 1989. [4] R. Hartshorne. Algebraic Geometry. Springer{Verlag, 1977. [5] D. R. Heath-Brown. The density of rational points on cubic surfaces. Acta Arith., 79(1):17{30, 1997. [6] D. R. Heath-Brown. Counting rational points on cubic surfaces. In E. Peyre, editor, Nombre et Repartition de points de hauteur bornee, number 251 in Asterisque, pages 13{29. Societe mathematique de france, 1998. [7] C. Hooley. On the numbers that are representable as the sum of two cubes. Angew. Math., 314:146{173, 1980. [8] S. Lang. Algebraic Number Theory. Springer{Verlag, second edition, 1994. [9] Yu. I. Manin and Yu. Tschinkel. Points of bounded height on del Pezzo surfaces. Compositio Math., 85:315{332, 1993. [10] I. Reiner. Maximal Orders. Academic Press Inc., 1975. [11] S. H. Schanuel. Heights in number elds. Bull. Soc. Math. Fr., 107:433{449, 1979. [12] J.-P. Serre. Lectures on the Mordell{Weil Theorem. Wieweg, 1989. [13] G. Tenenbaum. Introduction to analytic and probabalistic number theory. Cambridge university press, 1995. [14] T. D. Wooley. Sums of two cubes. Internat. Math. Res. Notices, (4):181{185, 1995.  teborg UniDept. of Mathematics, Chalmers University of Technology, Go  teborg, Sweden versity, SE{412 96 Go

E-mail address: [email protected]

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS par

Antoine Chambert-Loir & Yuri Tschinkel

Table des matières Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations et conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ÿ 1. Torseurs arithmétiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Dénitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Propriétés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Métriques adéliques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Fonctions L d'Arakelov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ÿ 2. Espaces brés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Groupe de Picard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Métriques hermitiennes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Torsion des métriques adéliques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Nombres de Tamagawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Torseurs trivialisants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Exemples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 4 4 6 8 10 12 12 14 17 19 22 24 25 27

Introduction Cet article est le premier d'une série dont le thème principal est l'étude des hauteurs sur certaines variétés algébriques sur un corps de nombres. On voudrait notamment comprendre la distribution des points rationnels de hauteur bornée. Précisément, soient X une variété algébrique projective lisse sur un corps de nombres F , L un bré en droites sur X et HL : X (F ) ! R+ une fonction hauteur (exponentielle) pour L . Si U est un ouvert de Zariski de X , on cherche à estimer le nombre

NU (L ; H ) = #fx 2 U (F ) ; HL (x)  H g

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

2

lorsque H tend vers +1. L'étude de nombreux exemples a montré que l'on peut s'attendre à un équivalent de la forme

NU (L ; H ) = (L )H a(L ) (log H )b(L ) 1 (1 + o(1)); H ! +1 pour un ouvert U convenable et lorsque par exemple L et !X1 (bré anticanonique) sont amples. On a en eet un résultat de ce genre lorsque X est une variété de drapeaux [12], une intersection complète lisse de bas degré (méthode du cercle), une variété torique [5], une variété horosphérique [23], etc. On dispose de plus d'une description conjecturale assez précise des constantes a(L ) et b(L ) en termes du cône des diviseurs eectifs [2] ainsi que de la constante (L ) ([16], [6]). ()

En fait, on étudie plutôt la fonction zêta des hauteurs, dénie par la série de Dirichlet

ZU (L ; s) =

X

HL (x)

s

2 ( ) à laquelle on applique des théorèmes taubériens standard. Sur cette série, on peut se poser les questions suivantes : domaine de convergence, prolongement méromorphe, ordre du premier pôle, terme principal, sans oublier la croissance dans les bandes verticales à gauche du premier pôle. Cela permet de proposer des conjectures de précision variable. Il est naturel de vouloir tester la compatibilité de cette conjecture avec les constructions usuelles de la géométrie algébrique. Par exemple, on n'arrive pas à démontrer cette conjecture pour un éclatement X 0 d'une variété X pour laquelle cette conjecture est connue. Même pour un éclatement de 4 points dans le plan projectif, on n'a pas de résultat complet ! x U F

Dans cet article, nous considérons certaines brations localement triviales construites de la façon suivante. Soient G un groupe algébrique linéaire sur F agissant sur une variété projective lisse X , B une variété projective lisse sur F et T un G-torseur sur B localement trivial pour la topologie de Zariski. Ces données dénissent une variété algébrique projective Y munie d'un morphisme Y ! B dont les bres sont isomorphes à X . On donne au Ÿ 2.7 de nombreux exemples  concrets  de variétés algébriques provenant d'une telle construction. Le c÷ur du problème est de comprendre le comportement de la fonction hauteur lorsqu'on passe d'une bre à l'autre, comportement vraiment non trivial bien qu'elles soient toutes isomorphes. Pour dénir et étudier de façon systématique les fonctions hauteurs sur Y , on est amené à dégager de nouvelles notions dans l'esprit de la géométrie d'Arakelov. Apparaissent notamment les notions de G-torseur arithmétique au Ÿ 1.1.3, ainsi que la dénition de la fonction L d'Arakelov attachée à un tel torseur arithmétique et à une fonction sur le groupe adélique G(AF ) invariante par G(F ) et par un sous-groupe compact convenable (Ÿ 1.4). Elles généralisent les notions usuelles de bré inversible métrisé ainsi que la fonction zêta des hauteurs introduits par S. Arakelov [1]. Ceci fait, on peut voir que les fonctions hauteurs d'une bre Yb de la projection Y ! B dièrent de la fonction hauteur sur X par ce que nous appelons torsion

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS

3

adélique, dans laquelle on retrouve explicitement la classe d'isomorphisme du G-torseur

arithmétique Tb sur F (Ÿ 2.4). Dans un deuxième article, nous appliquerons ces considérations générales au cas d'une bration en variétés toriques provenant d'un torseur sous un tore pour l'ouvert U déni par le tore. Le principe de l'étude généralise [23] et est le suivant. On construit les hauteurs à l'aide d'un prolongement du torseur géométrique en un torseur arithmétique, ce qui correspond en l'occurence au choix de métriques hermitiennes sur certains brés en droites. On écrit ensuite la fonction zêta comme la somme des fonctions zêta des bres

ZU (L ; s) =

X

X

2 ( ) 2 ( )

b B F x Ub F

HL (x) s =

X

2 ( )

ZUb (L jUb ; s):

b B F

Chaque Ub est isomorphe au tore et on peut récrire la fonction zêta des hauteurs de Ub à l'aide de la formule de Poisson adélique. De cette façon, la fonction zêta de U apparaît comme une intégrale sur certains caractères du tore adélique de la fonction L d'Arakelov d'un torseur arithmétique sur B . Cette expression nous permettra d'établir un théorème de montée : supposons que B vérie une conjecture, alors Y la vérie. Bien sûr, la méthode reprend les outils utilisés dans la démonstration de ces conjectures pour les variétés toriques ([5, 3, 4]). Alors que le présent article contient des considérations générales de  théorie d'Arakelov équivariante  dont on peut espérer qu'elles seront utiles dans d'autres contextes, le deuxième verra intervenir des outils de théorie analytique des nombres (formule de Poisson, théorème des résidus, estimations, etc.). .  Nous remercions J.-B. Bost pour d'utiles discussions. Pendant la préparation de cet article, le second auteur() était invité à l'I.H.E.S. et à Jussieu ; il est reconnaissant envers ces institutions pour leur hospitalité. Remerciements

Notations et conventions Si X est un schéma, on désigne par QCoh(X ) et Fibd (X ) les catégories des faisceaux quasi-cohérents (resp. des faisceaux localement libres de rang d) sur X . On note Pic(X ) le groupe des classes d'isomorphisme de faisceaux inversibles sur X . Si F est un faisceau localement libre sur X , on note V(F ) = Spec Sym F et P(F ) = Proj Sym F les brés vectoriels et projectifs associés à F . On note d Fibd (X ) la catégorie des brés vectoriels hermitiens sur X (c'est-à-dire des faisceaux localement libres de rang d munis d'une métrique hermitienne continue c X ) le groupe des sur X (C) et invariante par la conjugaison complexe). On note Pic( classes d'isomorphisme de brés en droites hermitiens sur ()

partially supported by the N.S.A.

X.

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

4

Si X est un S -schéma, et si  2 S (C), on désigne par X le C-schéma X  C. Cette notation servira lorsque S est le spectre d'un localisé de l'anneau des entiers d'un corps de nombres F , de sorte que  n'est autre qu'un plongement de F dans C. Si G est un schéma en groupes sur S , X  (G) désigne le groupe des S -homomorphismes G ! Gm (caractères algébriques). Si X =S est lisse, le faisceau canonique de X =S , noté !X =S est la puissance extérieure maximale de 1X =S .

Ÿ1 Torseurs arithmétiques 1.1. Dénitions Rappelons la dénition d'un torseur en géométrie algébrique. Dénition 1.1.1.  Soient S un schéma,

B un S -schéma et G un S -schéma en groupes

plat et localement de présentation nie. Un G-torseur sur un B est un B -schéma  : T ! B dèlement plat et localement de présentation nie muni d'une action de G au-dessus de B , m : G S T ! T , de sorte que le morphisme

(m; p2 ) : G S T

! T B T

soit un isomorphisme. On le suppose de plus localement trivial pour la topologie de Zariski.

On note H1 (B ; G) l'ensemble des classes d'isomorphisme de G-torseurs sur

Situation 1.1.2.

B.

 Supposons que S est le spectre de l'anneau des entiers d'un corps

de nombres F et que G est un S -schéma en groupes linéaire connexe plat et de présentation nie. Fixons pour tout plongement complexe de F ,  2 S ( ), un sous-groupe compact maximal K de G( ) et notons 1 la collection (K ) . On suppose que pour deux plongements complexes conjugués, les sous-groupes compacts maximaux correspondants sont échangés par la conjugaison complexe.

C

K

C

(G; K1)-torseur arithmétique sur B la donnée d'un G-torseur T sur B ainsi que pour tout  2 S (C), d'une section du K nG (C)-bré sur B (C) quotient à T (C) par l'action de K . On suppose de plus que pour deux Dénition 1.1.3.  On appelle

plongements complexes conjugués, les sections sont échangées par la conjugaison complexe. b 1 (B ; (G; K1 )) l'ensemble des classes d'isomorphisme de (G; K1 )-torseurs On note H arithmétiques sur B . b 0 (B ; (G; K1 )) l'ensemble des sections g 2 H0 (B ; G) telles que pour On note aussi H toute place à l'inni  , g dénisse une section B (C) ! K .

K nG (C)-bré associé à T (C) sur B (C) revient à xer dans un recouvrement ouvert (Ui) pour la topologie complexe Remarque 1.1.4.  Se donner une section du

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS

5

les fonctions de transition gij 2 (Ui \ Uj ; G) à valeurs dans K . Il en existe car G (C) est homéomorphe au produit de K par un R-espace vectoriel de dimension nie, cf. par exemple [7]. D'autre part, on choisit dans cet article de supposer la section continue. Dans certaines situations, il pourrait être judicieux de la supposer indéniment diérentiable. La dépendance deQ cette notion en les sous-groupes maximaux xés est la suivante : toute famille (x ) 2  G (C) telle que K0 = x K x 1 détermine une bijection canonique b 1 (B ; (G; K1 )) ' H b 1 (B ; (G; K0 )): H 1 (Rappelons que deux sous-groupes compacts maximaux sont conjugués.) 1.1.5. Variante adélique.  Il existe une variante adélique des considérations précédentes qui supprime en apparence la référence à un modèle sur Spec oF . En eet, si est propre sur Spec oF , remarquons que pour toute place nie de F , un G-torseur arithmétique sur induit une section du morphisme G(ov )n (Fv ) ! (Fv ).

B

B

T

B

GF un FQ -schéma en groupes de type ni et xons un sous(1) groupe compact maximal K = v Kv du groupe adélique G(AF ). Soit BF un F Dénition 1.1.6.  Soit

schéma propre. On appelle (GF ; K)-torseur adélique sur BF la donnée d'un GF -torseur TF ! BF , ainsi que pour toute place v de F , d'une section continue de Kv nTF (Fv ) ! BF (Fv ). On suppose de plus qu'il existe un ouvert non vide U de Spec oF , un U -schéma en groupes plat et de présentation ni G, un U -schéma B propre, plat et de type ni, ainsi qu'un G-torseur T ! B qui prolongent respectivement GF , BF et TF et vériant : pour toute place nie v de F dominant U , G(ov ) = Kv et la section continue de Kv nTF (Fv ) ! BF (Fv ) est celle fournie par le modèle T ! B . On note H1 (BF ; (GF ; K)) l'ensemble des classes d'isomorphisme de (GF ; K)-torseurs adéliques sur BF . Bien sûr, si B est un oF -schéma propre et G un oF -schéma en groupes plat et de présentation nie, tout (G; K1)-torseurQarithmétiqueQsur B dénit un (GF ; K)-torseur adélique où K est le compact adélique v nie G(ov )  K . 1.1.7. Exemples.  a) Quand G = GL(d), le torseur

T

correspond naturellement à la donnée d'un bré vectoriel E de rang d sur B par la formule T = Isom(OBn ; E ). Si l'on choisit K = U(d), une section du U(d; C)n GL(d; C)-bré associé correspond à une métrique hermitienne (continue) sur E . Ainsi, les (GL(d); U(d))-torseurs arithmétiques sont en bijection naturelle avec les brés vectoriels hermitiens. b) En particulier, lorsque G = Gm , la famille des sous-groupes compacts maximaux K1 est canoniquement dénie (ce qui permet de les omettre dans la notation) et b 1 (B ; Gm ) = Pic( c B ), le groupe des classes d'isomorphisme de brés en droites sur H Cela signiera pour nous que les Kv sont des sous-groupes compacts ouverts aux places nies, et maximaux aux places innies. (1)

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

6

B munis d'une métrique hermitienne continue compatible à la conjugaison complexe.

Les Gm -torseurs adéliques s'identient de même aux brés inversibles munis d'une métrique adélique. Nous rappelons cette théorie au paragraphe 1.3 c) Dans ce texte, nous ne considérons que des G-torseurs localement triviaux pour la topologie de Zariski. Néanmoins, lorsque G=S est un S -schéma abélien, un exemple de G-torseur localement trivial pour la topologie étale sur B est fourni par un schéma abélien A =B obtenu par torsion de G=S , c'est-à-dire tel qu'il existe un revêtement étale B 0 ! B de sorte que A B B 0 soit isomorphe à G S B 0 (famille de schémas abéliens à module constant). De tels exemples devraient bien sûr faire partie d'une étude plus générale de la géométrie d'Arakelov des torseurs que nous reportons à une occasion ultérieure.

1.2. Propriétés

Les ensembles de classes d'isomorphisme de (G; K1)-torseurs arithmétiques vérient un certain nombre de propriétés formelles, dont les analogues algébriques sont bien connus. Leur démonstration est standard et laissée au lecteur. Proposition 1.2.1.

 L'oubli de la structure arithmétique induit une application b 1 (B ; (G; K1 )) ! H1 (B ; G): H

On a aussi une suite exacte d'ensembles pointés : b 0 (B ; (G; K1 )) ! H0 (B ; G) ! 1!H

(F1 désigne la conjugaison complexe et complexe.)

M 

!F1

(B (C); K nG (C))

!

! Hb 1 (B; (G; K1)) ! H1(B; G) ! 1:

(
)F1 la partie invariante par la conjugaison

, en identiant Gm (C)=K à R+, nous retrouvons c et Pic (cf. [14], 3.3.5 ou 3.4.2). la suite exacte bien connue pour Pic D'autre part, on devrait pouvoir interpréter cette suite exacte à l'aide de la mapping cylinder category introduite par S. Lichtenbaum dans son étude des valeurs spéciales des fonctions zêta des corps de nombres (exposé à Paris 6, 1998). En eet, cette catégorie est ( ? !) la catégorie des faisceaux en groupes abéliens sur, disons Spec Z [ f1g. Remarque 1.2.2.  Lorsque G =

G

m

 Supposons que le groupe G est commutatif. Alors, les sousb 1 (B ; (G; K1 )) hérite d'une groupes compacts maximaux sont uniques et l'ensemble H Proposition 1.2.3.

B; G).

structure de groupe abélien compatible avec la structure de groupe abélien sur H1 ( Dans ce cas, la suite exacte 1.2.1 est une suite exacte de groupes abéliens.

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS

7

 (Changement de base) Tout morphisme de S -schémas B 0 ! B induit un foncteur des (G; K1)-torseurs arithmétiques sur B vers les (G; K1)torseurs arithmétiques sur B 0 , compatible à l'oubli des structures arithmétiques et aux Proposition 1.2.4.

classes d'isomorphisme.

(Changement du corps de base) Si F 0 est une extension de F , S 0 = Spec oF 0 et si on choisit pour tout plongement complexe  0 de F 0 K0 = K0 jF , on dispose d'un foncteur des (G; K1 )-torseurs arithmétiques sur B vers les (G S S 0 ; K1 )-torseurs arithmétiques sur B S S 0 , compatible à l'oubli des structures arithmétiques et aux classes d'isomorphisme. Proposition 1.2.5.

 (Changement de groupe) Si p : G

! G0

est un morphisme de S -schémas en groupes et que les sous-groupes compacts maximaux 1 et 01 sont choisis de sorte que pour tout plongement complexe  , tel que p(K0 )  K0 , il y a un foncteur des (G; 1)-torseurs arithmétiques vers les (G0 ; 01 )-torseurs arithmétiques, compatible à l'oubli des structures arithmétiques et aux classes d'isomorphisme. (Suite exacte courte) Soit

K

K

K

K

 1 ! G00 ! G ! G0 ! 1 p

K K

K

une suite exacte de S -schémas en groupes. Soient 1 , 01 et 001 des familles de sous-groupes compacts maximaux pour G, G0 et G00 aux places archimédiennes choisis de sorte que K00 =  1 (K ) et p(K ) = K0 pour toute place  . Si p admet localement une section (comme S -schéma), alors on a une suite exacte courte canonique d'ensembles pointés : p  Æ b 0 (B ; (G00 ; K00 )) ! b 0 (B ; (G; K1 )) ! b 0 (B ; (G0 ; K0 )) ! 1!H H H 1 1 ! Hb 1(B; (G00; K001)) ! Hb 1(B; (G; K1)) !p Hb 1 (B; (G0; K01)):

Sur Spec oF , l'ensemble des classes d'isomorphisme de (G; K1)-torseurs arithmétiques a une description très simple, similaire à la description classique des classes d'isomorphisme de G-torseurs sur une courbe projective sur un corps ni. Cela générac lise la description analogue du groupe Pic(Spec oF ) (cf. [14], 3.4.3, p. 131, où le groupe 1 c correspondant est noté CH (Spec oF )). Proposition 1.2.6.

où

K

G

 On a des isomorphismes canoniques

b 1 (Spec oF ; (G; K1)) ' G(F )nG(AF )=KG ; H Q Q désigne le produit G(ov ) K . v

nie



innie

K

A

De même, pour un sous-groupe compact maximal de G( F ), on a un isomorphisme canonique H1 (Spec F; (G ; )) ' G(F )nG( )= : F

K

A K F

(G; K)-torseur arithmétique sur Spec(oF ), localement trivial pour la topologie de Zariski. Commençons par xer un section F 2 T (F ). Si v

T

Démonstration.  Soit c un

8

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

est une place nie de F , comme H1 (Spec ov ; G) = 0, il existe une section v 2 T (ov ), unique modulo l'action de G(ov ). Cette section se relie à F par un élement bien déni gv 2 G(Fv )=G(ov ) tel que gv 1
F = v . Comme F s'étend en une section de T sur un ouvert de Spec oF , on a gv 2 Kv pour presque toute place v . D'autre part, si  est une place innie, la section de K nT (C) donnée par la structure de (G; K1)-torseur arithmétique est de la forme K g 1 F , pour un unique g 2 G(C)=K . On a ainsi déni un élement g dans G(AF )=KG . Il dépend de la section F , mais si on choisit une autre section, elle sera de la forme gF F , ce qui revient à changer l'élément g par gF 1g. Nous avons donc attaché au (G; K1)-torseur arithmétique Tc un élément dans G(F )nG(AF )=KG qui visiblement ne dépend que de la classe d'isomorphisme de Tc. Pour la bijection réciproque, on choisit un représentant de g 2 G(F )nG(AF )=KG où pour toute place nie v , gv 2 G(F ), et où presque tous les gv valent 1. Soit alors U le plus grand ouvert de Spec oF tel que pour toute place nie v , gv 2 G(U ) ; si v est une place nie qui ne domine pas U , soit Uv = U [ fv g. On dénit un G-torseur T sur Spec oF comme isomorphe à G sur U et sur chaque Uv , les isomorphismes de transition étant xés par l'isomorphisme entre T jU = GjU et T jUv  U = GjU induit par la multiplication à gauche par gv 1 . On munit ce G-torseur de la K -classe à gauche K g 1 dans la trivialisation canonique sur l'ouvert U qui contient Spec F , d'où un (G; K1)-torseur arithmétique sur Spec oF . On laisse au lecteur le soin de vérier plus en détail que la classe d'isomorphisme du (G; K1)-torseur arithmétique ainsi construit est indépendante du représentant choisi, et que cela dénit eectivement la bijection réciproque voulue. La variante adélique H1 (Spec F; (GF ; K)) se traite de même (et plus facilement car on n'a pas de torseur à construire !). Remarque 1.2.7.  On aurait aussi pu construire le

G-torseur

T

associé à un point adélique (gv ) en décrétant que les sections de T sur un ouvert U de Spec oF sont les

2 G(F ) tels que pour toute place nie v dominant U , gv 2 G(ov ).

1.3. Métriques adéliques Pour la commodité du lecteur, nous rappelons la théorie des métriques adéliques sur les brés en droites. C'est un cas particulier bien connu des constructions précédentes lorsque le groupe est Gm , mais l'exposer nous permettra de xer quelques notations.

L un bré en droites sur X . Une métrique sur L est une application continue V(L _ )(F ) ! R+ de sorte que pour tout x 2 X (F ), la restriction de cette application à la bre en x (identiée naturellement à F ) soit une norme. Dénition 1.3.1.  Soient F un corps valué, X un schéma de type ni sur F et

Soient F un corps de nombres, X un schéma projectif sur F et L un bré en e sur le spectre S = Spec oF droites sur X . La donnée d'un schéma projectif et plat X de l'anneau des entiers de F dont la bre générique est X dénit pour toute place non-archimédienne v de F une métrique sur le bré en droites L Fv sur X  Fv .

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS Dénition 1.3.2.  On appelle métrique adélique sur

9

L

toute collection de métriques (k
kv )v sur L Fv pour toutes les places v de F qui est obtenue de cette façon pour presque toutes les places (non-archimédiennes) de F . On note Pic(X ) = H1 (X; Gm ) le groupe des classes d'isomorphisme de brés en droites sur X munis de métriques adéliques.

Donnons nous une métrique adélique sur L . Tout morphisme f : Y ! X de F schémas projectifs fournit par image réciproque une métrique adélique sur f  L . Si Y n'est pas projective, on obtient tout de même de la sorte une collection de métriques pour toutes les places de F . Dénition 1.3.3.  Si

L

= (L ; (k
kv )v ) est un bré en droites sur X muni d'une

métrique adélique, on appelle fonction hauteur (exponentielle) associée à L la fonction

H (L ;
) : X (F ) ! R+ ; x 7!

Y

kskv (x) 1;

v

étant une section non nulle arbitraire de L jx ' F . Si s est une section globale non nulle de L , on dénit une fonction hauteur (exponentielle) sur les points adéliques de X en posant s

H (L ; s;
) : X (AF ) n j div(s)j ! R+ ;

x = (x ) 7!

Y

v v

kskv (xv ) 1 :

v

(Dans les deux cas, le produit converge en eet car il n'y a qu'un nombre ni de termes diérents de 1.) D'autre part, elle est multiplicative en le bré en droites (resp. en la section), ce qui permettra de l'étendre aux groupes de Picard tensorisés par C. c Comme on a un isomorphisme canonique Pic(Spec F ) = Pic(Spec oF ), on remarque que d L jx ) H (L ; x) = exp(deg

d : Pic(Spec c où deg oF ) ! R est l'homomorphisme  degré arithmétique  déni dans [14], 3.4.3, p. 131. Par l'isomorphisme de loc. cit., c Pic(Spec o ) ! F  nA =K; F

d correspond à l'inverse de la norme. exp Ædeg

F

X une variété sur F . Si U  X est un ouvert de Zariski, la fonction zêta des hauteurs de U est la fonction sur Pic(X )C (le groupe abélien des brés inversibles sur X munis d'une métrique adélique tensorisé par C) à valeurs dans Dénition 1.3.4.  Soit

C qui associe à L

la somme,

ZU (L ) =

X

2 ( )

H (L ; x) 1 ;

x U F

quand elle existe. Remarque 1.3.5.  La convergence absolue de la série ne dépend que de la partie réelle

de L dans Pic(X )R (on peut comparer deux métriques adéliques). De plus, l'ensemble

10

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

des L 2 Pic(X )R pour lesquels la série converge est une partie convexe (inégalité arithmético-géométrique). Enn, si L est ample, alors ZU (sL ) converge pour 0. Il est aussi bien connu comment utiliser cette équation pour en déduire que la fonction dénie par b s) = (E;

q

b s) vol(Eb) (E;

2 (s=2)

s=

possède un prolongement méromorphe à C,qavec des pôles simples en s = 0 et s = d q de résidus respectivement 2 vol(Eb ) et 2= vol(Eb ) et vérie l'équation fonctionnelle b s) = (E b _; d (E;

s):

Sur un corps de nombres quelconque, il faudrait tenir compte de la diérente, comme dans l'article récent de van der Geer et Schoof [13]. Selon ces mêmes auteurs, l'invariant b 1) mesure l'eectivité du bré vectoriel hermitien E b . Ils interprètent en particulier (E; l'équation fonctionnelle de la fonction  comme une formule de RiemannRoch.

E

B). On peut

1.4.5. Exemples exotiques de fonctions L.  Soit maintenant b 2 d Fibd (

dénir des fonctions L d'Arakelov (pour une partie U

(Eb; s) = L(Eb; U; (
; 1) vol(
)s ) =

X

 B(F ) xée) (Ebjb ; 1) vol(Ebjb )s

2 B(F )

b U

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

12

et

Z (Eb; s) = L(Eb; U;  (
; ds) vol(
)s ) =

X

 (Ebjb ; ds) vol(Ebjb )s

2 B(F ) et l'on a les égalités, où chacun des membres converge absolument quand l'autre converge absolument, (Eb; s) = (Eb_ ; 1 s) et Z (Eb; s) = Z (Eb_ ; 1 s): b U

Par exemple, pour B = P1Z et Eb = OP (1) avec la métrique  max. des coordonnées , on a X

(Eb; s) =

N

1

2(1 + 2'(N ))(N 2 )N 1 s ;

expression qui converge pour 3 et dans laquelle  désigne la fonction thêta de Riemann.

Ÿ2 Espaces brés 2.1. Constructions Situation 2.1.1.

 Soient S un schéma, G un S -schéma en groupes linéaire et plat,

X

dont on suppose pour simplier les bres géométriquement connexes f : ! S un S schéma plat (quasi-compact et quasi-séparé), muni d'une action de G=S . Soient aussi g : ! S un S -schéma plat ainsi qu'un G-torseur ! localement trivial pour la topologie de Zariski.

B

T

B

 On dénit un S -schéma Y , muni d'un morphisme  : Y

! B localement isomorphe à X sur B, par le changement de groupe structural G ! AutS (X ). En eet, soit (Ui )i2I un recouvrement ouvert de B tel qu'il existe une trivialisation  'i : G S Ui ! T jUi . Si i; j 2 I , soit gij 2 (Ui \ Uj ; G) l'unique section telle que 'i = gij 'j sur Ui \ Uj . En particulier, les gij donnent un cocycle dont la classe dans H1 (B ; G) représente la classe d'isomorphisme du G-torseur T . Posons Yi = X S Ui ; alors, gij agit sur X S (Ui \ Uj ) et induit un isomorphisme 'ij : Yj jUi \Uj ' Yi jUi \Uj que l'on utilise pour recoller les Yi . On laisse vérier que Y est un B -schéma bien déni, c'est-à-dire qu'il ne dépend Construction 2.1.2.

pas à isomorphisme canonique près du choix des trivialisations locales que l'on a fait. Lemme 2.1.3.  On a  OY = g  f OX . Remarque 2.1.4.  Dans certains cas,

notamment quand G est commutatif.

Y

hérite d'une action d'un sous-groupe de G,

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS Construction 2.1.5.

13

 Il résulte de la construction précédente une application

# : Zd;G(X ) ! Zd (Y )

des cycles .

Y

G-invariants de codimension d sur X dans les cycles de codimension d sur

G-linéarisation d'un faisceau quasi-cohérent F sur X est une action de G sur V(F ) qui relève l'action de G sur X . Un morphisme (resp. le produit tensoriel, le dual, la somme directe, le faisceau des homomorphismes, des extensions, etc.) de faisceaux quasi-cohérents G-linéarisés est G déni naturellement. On note QCohG (X ) (resp. FibG d (X ), resp. Pic (X )) la catégorie des faisceaux quasi-cohérents (resp. de brés vectoriels de rang d, resp. des classes d'isomorphisme de brés inversibles) G-linéarisés sur X . Dénition 2.1.6.  Une

Construction 2.1.7.

 On construit un foncteur

# : QCohG (X ) ! QCoh(Y ) qui est compatible avec les opérations standard sur les faisceaux quasi-cohérents.

Soit F un faisceau quasi-cohérent G-linéarisé sur X . Reprenons les notations de la construction 2.1.2 de Y . Posons Fi le faisceau quasi-cohérent sur Yi = X S Ui image réciproque de F par la première projection. Grâce à la G-linéarisation sur F , les gij induisent des isomorphismes 'ij Fj jX (Ui \Uj ) ' Fi jX (Ui \Uj ) qui fournissent par recollement un faisceau quasi-cohérent sur Y . On laisse vérier que ce foncteur est bien déni, c'est-à-dire, est indépendant des choix que l'on a fait. Si F est un bré vectoriel G-linéarisé de rang d sur X , il est clair que le faisceau obtenu sur Y est aussi un bré vectoriel de rang d. On laisse vérier que cette application est compatible aux opérations standard, et en particulier qu'elle descend en une application sur les classes d'isomorphisme. Un cas particulier des constructions précédentes est obtenu lorsque X = S , auquel cas Y = B . On notera T l'application qui en résulte des faisceaux quasi-cohérents sur S avec action de G=S vers les faisceaux quasi-cohérents sur B . Bien sûr, T : Repd (G) ! Fibd (B ) n'est autre que l'application usuelle de changement de groupe structural (passage d'un G-torseur à un GL(d)-torseur). Proposition 2.1.8.

 Le faisceau 1X =S est muni d'une linéarisation canonique de

G. Par la construction 2.1.7, on obtient le faisceau 1X =B . Supposons en particulier que X et B sont lisses sur S ; le faisceau canonique sur X =S est alors automatiquement G-linéarisé et on a un isomorphisme !Y =S ' #(!X =S )   !B=S :

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

14

Démonstration.  Si (' ;  ) :  1 (U ) ' i

i

naturel

X

(Ui ) est un recouvrement ouvert de B avec des isomorphismes S Ui comme dans la construction 2.1.2, on a un isomorphisme

1Y =B j 1 (Ui ) = 1 1 (Ui )=Ui ' 'i 1X =S

qui se recollent précisément comme dans la construction 2.1.7. Dans le cas où X =S et B =S sont lisses, la suite exacte

0 ! 1Y =B ! 1Y =S ! 1B=S ! 0

implique que

!Y =S ' det 1Y =B   !B=S ' #(!X =B )   !B=S :

Lemme 2.1.9.

 #(F )

 Si

F 2 QCoh (X ), fF G

est muni d'une action naturelle de est canoniquement isomorphe à T (f ).

F

G et

Démonstration.  Laissée au lecteur. Proposition 2.1.10.

 Soient (; ) 2 PicG (X )  Pic(B ). Le bré en droites #()

   sur Y est eectif si et seulement si le bré vectoriel sur B T (f )  est eectif. Cela implique que  est eectif. Démonstration.  On a

 (#()   ) =  (#())  = T (f ) 

d'après le lemme 2.1.9.

Notons  le morphisme de groupes naturel X  (G) ! PicG (X ) qui associe à un caractère  le bré trivial muni de la linéarisation telle que G agit par  sur le second facteur de X S A1S . Proposition 2.1.11.

de faisceaux inversibles

 Pour tout caractère , il existe un isomorphisme canonique

#(()) '   T ():

B

Démonstration.  Soit (Ui ) un recouvrement ouvert de avec des isomorphismes 1 ('i ;  ) :  (Ui ) ' S Ui ; notons gij 2 G(Ui \ UJ ) tel que 'i = gij
'j :  1 (Ui \ Uj ) ! . Alors, le bré en droites #(()) est obtenu en recollant 1   Ui et 1  Uj par le morphisme (t; x; u) 7! ((gij )t; gij
x; u). D'autre part, T () est un bré en droite sur obtenu en recollant 1  Ui et 1  U par (t; u) 7! ((g )t; u).

A A

X X j

X

A X

B

A

ij

2.2. Groupe de Picard Dans ce paragraphe, on suppose que S est le spectre d'un corps F de caractéristique 0. On cherche à exprimer le groupe de Picard de Y en fonction de ceux de X et B . Pour cela, on se place sous les hypothèses suivantes :

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS

15

X. 

On suppose que X est propre, lisse, géométriquement intègre ; H1 (X ; OX ) = 0 ; X (F ) est non vide ; tout bré en droites sur X est G-linéarisable, et de même après toute extension algébrique de F ; Pic(XF ) est sans torsion.

2.2.1. Hypothèses sur

1. 2. 3. 4. 5.

X sont vériées lorsque X est une variété torique projective déployée sur F , ou bien un espace de drapeaux généralisé pour un groupe algébrique déployé sur F . Elles entraînent que les groupes de Picard et de Néron-Séveri de XF coïncident (voir la preuve du lemme 2.2.3 plus bas). En particulier, Pic(XF ) est sous ces hypothèses un Z-module libre de rang ni. D'autre part, il est prouvé dans [15], Cor. 1.6, p. 35, que sous l'hypothèse (i), tout bré en droites sur X admet une puissance G-linéarisable. (Rappelons que G est connexe.) Le lecteur qui désirerait s'aranchir de cette hypothèse vériera que de nombreux résultats de la suite de ce texte restent vrais, au moins après tensorisation par Q. Remarque 2.2.2.  Ces hypothèses concernant

Lemme 2.2.3.

 Si les hypothèses 2.2.1 sont satisfaites, on a les deux assertions : 0  H (X ; OX ) = F ;  pour tout F -schéma connexe U possédant un point F -rationnel, l'homomorphisme naturel

Pic(X )  Pic(U ) ! Pic(X

F U )

est un isomorphisme. Démonstration.  La première proposition découle de la factorisation de Stein. Pour

la seconde, on a d'après [8, 8.1/4] une suite exacte

0 ! Pic(U ) ! Pic(X F U ) ! PicX =F (U ) ! 0: En particulier, Pic(X ) = PicX =F (F ). La nullité de H1 (X ; OX ) implique que PicX =F est de dimension 0, donc que sa composante neutre Pic0X =F = 0 puisque F est de caractéristique nulle. Ainsi, PicX =F est discret. Alors, tout point rationnel u 2 U (F ) dénit un homomorphisme u : PicX =F (U ) ! PicX =F (F ) qui par connexité est l'inverse de l'homomorphisme naturel PicX =F (F ) ! PicX =F (U ). G Théorème 2.2.4.  Si  désigne le morphisme de groupes X  (G) ! Pic (X ) introduit au paragraphe précédent, considérons l'homomorphisme

PicG (X )  Pic(B ) ! Pic(Y ); (; ) 7! #()   : Si les hypothèses 2.2.1 sont satisfaites et si B (F ) est Zariski-dense dans suite

0 ! X  (G)

(; T )

! PicG(X )  Pic(B)

# 



! Pic(Y ) ! 0

B, alors la

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

16

est exacte.

X ), il résulte de ce que H0(X ; OX ) =

Démonstration.  Si () est trivial dans PicG (

F que  est nécessairement le caractère trivial. En particulier, le premier homomorphisme est injectif. La proposition 2.1.11 implique que la composition des deux premiers homomorphismes est nulle. Si  est un bré en droites G-linéarisé sur X et  est un bré en droites sur B , #()    est un bré en droites sur Y dont la classe d'isomorphisme ne dépend que des classes d'isomorphismes de  dans PicG (X ) et  dans Pic(B ). Supposons qu'elle soit triviale. Soit b un point F -rationnel de B . En restreignant #()    à  1 (b), la construction 2.1.5 de #() implique que  est trivial. La Glinéarisation de  est ainsi donnée par un caractère  de G et  = (). D'après la proposition 2.1.11, on a #() =   T (). Par suite,    '   T () 1 , ce qui prouve l'exactitude au milieu. Montrons alors que la dernière èche est surjective. Soit L un bré en droites sur Y . On peut recouvrir B par des ouverts connexes non vides Ui assez petits de sorte que

 1 (Ui ) ' X La restriction de

L

F Ui:

à  1 (Ui ) fournit alors pour tout i un élément de

Pic(X

F Ui ) = Pic(X )  Pic(Ui )

puisque chaque Ui a un point F -rationnel. On en déduit d'abord pour tout i un élément de Pic(X ) qui, comme on le voit en les restreignant à Ui \Uj , ne dépend pas de i. Notons le . Finalement, il existe un faisceau inversible i 2 Pic(Ui ) tel que la restriction de L à  1(Ui ) ' X F Ui est isomorphe à p1 p2i. Quitte à raner le recouvrement (Ui ), on peut de plus supposer que i ' OUi . Choisissons une G-linéarisation sur . On constate que la restriction de L #() 1 à  1 (Ui ) est triviale. Si l'on choisit des trivialisations on obtient en les comparant sur  1 (Ui \ Uj ) un élément de

( 1 (Ui \ Uj ); OY ) = (Ui \ Uj ; OB) car H0 (X ; OX ) = F . Ces éléments dénissent un 2-cocycle de ƒech sur dans le faisceau OB , d'où un bré en droites  2 Pic(B ) tel que

L #() 1 ' : Autrement dit, L appartient à l'image de l'homomorphisme #   . Le théorème est ainsi démontré.

B à valeurs

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS Corollaire 2.2.5.

17

 Supposons vériées les hypothèses 2.2.1 et supposons que B (F )

B. On dispose alors de suites exactes de Z[Gal(F =F )]-modules : 0 ! X  (G ) ! Pic (X ) ! Pic(X ) ! 0 0 ! X  (G ) ! Pic (X )  Pic(B ) ! Pic(Y ) ! 0  0 ! Pic(B ) ! Pic(Y ) ! Pic(X ) ! 0:

est Zariski-dense dans

(2.2.6)

G

F

(2.2.7)

G

F

F

F

F

F

F



(2.2.8)

F

F

F

Démonstration.  Il sut d'appliquer le théorème 2.2.4 sur F , et de constater que la suite exacte obtenue est Gal(F =F )-équivariante. Théorème 2.2.9.

B

 Supposons vériées les hypothèses 2.2.1, que

B(F ) est Zariski-

dense dans , et supposons de plus que G est un groupe algébrique F -résoluble(2) , un bré en droites sur est alors eectif si et seulement s'il s'écrit comme l'image d'un G couple (; ) 2 Pic ( )  Pic( ) où  et  sont eectifs.

Y X

B

 2 PicG (X ) et  2 Pic(B ) eectifs. On veut montrer que #()    est eectif. Il sut de prouver que #() est eectif, et pour cela, il sut de prouver qu'il existe un diviseur de Cartier G-invariant D sur X tel que l'on ait un isomorphisme de brés en droites G-linéarisés,  ' O (D). Autrement dit, il faut montrer que la représentation de G sur f  admet une F -droite stable, ce qu'implique le théorème de point xe de Borel puisque G est F -résoluble. Soit maintenant L un bré en droites eectif sur Y . Comme G est connexe et Pic(G) = 0, la démonstration de la proposition 1.5, p. 34, de [15] implique que tout bré inversible sur X est G-linéarisable. Le théorème 2.2.4 implique donc qu'il existe  2 PicG (X ) et  2 Pic(B ) tels que L = #()   . D'après la proposition 2.1.10, T (f )  est eectif. Comme G est F -résoluble, toute représentation linéaire de G est extension successive de représentations de dimension 1. Cela implique que T (f ) est extension successive de brés en droites ; notons les i . Alors, T (f )  est extension des i , et l'eectivité de L implique que l'un au moins des i  est Démonstration.  Soient

eectif. Or, i est associé à un caractère i de G ; si on remplace  par le bré en droite G-linéarisé  (i ) 1 où l'action a été divisée par i , on représente ainsi L sous la forme

L ' #( ( ) 1 ) ( ); ce qui conclut la démonstration,  ( ) 1 étant isomorphe à  comme bré en droites, i

donc eectif.

i

i

2.3. Métriques hermitiennes Dans ce paragraphe, nous étendons la construction 2.1.7 en supposant que S est le spectre d'un corps de nombres et en faisant intervenir des métriques hermitiennes. Cela signie que G est extension itérée de sur F . (2)

Gm et Ga, autrement dit, que G est résoluble et déployé

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

18

G un groupe de Lie connexe sur C ; xons un sous-groupe compact maximal K de G. Soit X une variété analytique complexe munie d'une action de G. Si E est un bré vectoriel complexe G-linéarisé sur X , on dit qu'une métrique hermitienne est K -invariante si l'action de K sur V(E )  X est isométrique. Dénition 2.3.1.  Soit

On remarquera que les constructions usuelles (tensorielles) de brés hermitiens préservent la K -invariance des métriques hermitiennes. Remarque 2.3.2.  Avec les notations de la dénition précédente, tout bré vectoriel

sur X admet une métrique hermitienne K -invariante : si k
k0 est une métrique hermitienne sur E , on peut en eet choisir une mesure de Haar sur K et poser pour toute section s, Z

ksk2 (x) =

K

kk
sk (x)2 dk:

Rappelons l'énoncé de la situation 1.1.2 : Situation.

 Supposons que S est le spectre de l'anneau des entiers d'un corps de

nombres F et que G est un S -schéma en groupes linéaire connexe. Fixons pour tout plongement complexe de F  2 S ( ) un sous-groupe compact maximal K de G( ) et notons 1 la collection (K ) .

C

K

C

Dénition 2.3.3.  Supposons que G agit sur un S -schéma plat bré vectoriel hermitien (G; 1)-linéarisé un bré vectoriel sur

K

E

X . On appelle X muni d'une G-

linéarisation et, pour tout  2 S (C), d'une métrique hermitienne sur le bré vectoriel E  C sur XG;(CK) qui est K -invariante. On note d Fibd 1 (X ) la catégorie des brés vectoriels hermitiens (G; K1)-linéarisés c G;K1 (X ) le groupe des classes d'isomorphisme de rang d sur X . Si d = 1, on notera Pic de brés vectoriels hermitiens de rang 1 (G; K1)-linéarisés sur X .

X

 Plaçons-nous dans la situation 1.1.2. Soit f : ! S un S schéma plat, muni d'une action de G=S . Soient aussi g : ! S un S -schéma plat ainsi qu'un (G; )-torseur arithmétique c sur (voir la dénition 1.1.3). Situation 2.3.4.

K

B

T

B Construction 2.3.5.  Le foncteur # : Fib (X ) ! Fib (Y ) s'étend en un foncG d

teur

d

dG;K1 (X ) ! Fib dd (Y ) # : Fib d

qui est compatible avec les opérations tensorielles standard sur les brés vectoriels hermitiens (G; 1)-linéarisés (resp. les brés vectoriels hermitiens).

K

Soit F un bré vectoriel hermitien (G; K1)-linéarisé sur X . Soit  2 S (C). De manière analogue à ce qu'on a fait dans la construction 2.1.5, choisissons un recouvrement ouvert (Ui ) de B (C) pour la topologie complexe de sorte que la restriction du torseur T à Ui est triviale et qu'il existe des trivialisations dont les fonctions de transistions associés gij 2 (Ui \ Uj ; G) soient à valeurs dans K . Le choix de telles trivialisations

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS

19

induit des isomorphismes

 1 (Ui ) ' X (C)  Ui ;

#(F )j 1 (Ui ) ' p1 F : Pour tout i, on a ainsi une métrique hermitienne naturelle sur #(F )j 1 (Ui ) par image réciproque de la métrique hermitienne sur F . Comme gij 2 K et comme la métrique hermitienne sur F est K -invariante, les métriques hermitiennes sur #(F )jUi \Uj induites par Ui et par Uj coïncident, d'où une métrique hermitienne bien dénie sur #(F ). Enn, la proposition 2.1.11 admet une généralisation avec métriques hermitiennes : Proposition 2.3.6.  Pour tout caractère  2 X  (G), l'isomorphisme canonique de la proposition 2.1.11 est une isométrie.

Démonstration.  Si l'on reproduit la démonstration de la proposition 2.1.11 pour

un recouvrement ouvert pour la topologie complexe (les gij étant donc dans le sousgroupe compact maximal), chacun des brés est déni par recollement de la même manière, et les métriques sur ces brés sont dénies de sorte que cette identication soit une isométrie. Il en résulte que l'isomorphisme de cette proposition, qui consistait en l'application évidente sur les ouverts X  Ui est une isométrie.

2.4. Torsion des métriques adéliques

Plaçons nous alors dans la situation 2.3.4, toujours avec S = Spec oF . Soit L un bré en droites hermitien (G; K1)-linéarisé sur X . La restriction de L à XF est ainsi munie d'une métrique adélique naturelle.

g

A

 Soit = (gv )v 2 G( F ). On dénit une métrique adélique sur , appelée métrique adélique tordue par en posant pour toute place v de F , tout point x 2 (Fv ) et toute section s 2 x , Proposition-Définition 2.4.1.

L

X ksk0 (x) = kg
sk v

v

v

(gv
x):

L

g

Démonstration.  Il est clair que pour toute place v , on a déni une métrique v -adique. L'ensemble des places non-archimédiennes v telles que gv 2 G(ov ) est par dénition de

complémentaire ni. Pour ces places, ksk0v (x) = kskv (x) car gv étant un automorphisme de L sur Spec ov , la section gv
s est entière en gv
x si et seulement si la section s est entière en x. Ainsi, hors d'un nombre ni de places, la nouvelle collection de métriques v -adiques est dénie par un modèle entier. Elle dénit donc une métrique adélique. Remarquons que G(AF ) n'agit en fait qu'à travers G(AF )=KG. Exemple 2.4.2.  Soit E un F -espace vectoriel de dimension nie et notons

P l'espace

projectif des droites de E . Faisons agir GL(E ) de manière naturelle sur P. Le faisceau OP(1) possède une GL(E )-linéarisation naturelle dès qu'on a remarqué qu'une section de OP ( 1) en un point x 2 P correspond à un point de la droite Dx dénie par x. De manière explicite, l'espace vectoriel des sections globales de O (1) sur P s'identie

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

20

au dual E  de E sur lequel la GL(E )-linéarisation sur O (1) induit la représentation contragrédiente ' 7! ' Æ g 1. Supposons que E est muni d'une métrique adélique. On a alors une métrique adélique sur OP (1) par la formule

k'k (x) = j'ke(ke)j ; ' 2 E  ; e 2 Dx n f0g: v

Il résulte de la formule du produit que la hauteur exponentielle d'un point est donnée par la formule

H (x) =

Soit alors (gv )v ainsi donnée par

Y v

x 2 P(F )

kekv ; e 2 Dx n f0g:

2 GL(E )(AF ). La métrique v-adique tordue par gv

sur

OP(1) est

k'k0 (x) = kgj'(
ee)kj ; ' 2 E ; e 2 Dx n f0g: v

v

Autrement dit, la hauteur exponentielle tordue de x 2 P(F ) est dénie par l'expression Y H 0 (x) = kg
ek ; e 2 D n f0g: v

x

v

v

Cette formule était donnée comme dénition de la hauteur tordue par Roy et Thunder dans [19]. Dans certains cas, on peut comparer la métrique adélique initiale sur trique adélique tordue.

L

Proposition 2.4.3.

L

et la mé-

X

 Supposons que s est une section globale de sur F dont le diviseur est G-invariant. Il existe alors un unique caractère  2 X  (G) F -rationnel (le poids de s) tel que pour tout g 2 G, g
s = (g )s. Soit 2 G( F ), et considérons 0 la métrique adélique tordue par . Si x 2 X (F ) n'appartient pas au diviseur de s, on a la formule Y H ( 0 ; x) = j(g )j 1 H ( ; s;
x):

g

A

g

L

L

v

L g

v

v

est G-invariant, il existe pour tout g 2 G un élément (g ) 6= 0 tel que g
s = (g )s. Il est alors clair que g 7! (g ) dénit un caractère F -rationnel (algébrique) de g . D'autre part, on a pour toute place v de F , ksk0 (x) = kg
sk (g x) = k(g )sk (g x) = j(g )j ksk (g x): Démonstration.  Comme le diviseur de

v

v

v

v

s

v

v

v

v

v

v

v

La proposition en découle en prenant le produit. Remarque 2.4.4.  Bien sûr, dans l'énoncé précédent, il sut de supposer que la sec-

tion s est propre pour les éléments gv . En particulier, si G0 est un sous-groupe de G tel

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS

21

que div(s) est invariant par G0 , on aura une formule du même type pour les métriques adéliques tordue par un élément de G0 (AF ). Remarque 2.4.5 (Choix des sections).  La formule précédente permet de comparer

la restriction à G(AF )X (F ) des hauteurs sur les points adéliques associées à deux sections s1 et s2 de poids respectivement 1 et 2 . En eet, si x = g
x 2 G(AF )X (F ), on a, L 0 désignant la métrique adélique tordue par g,

H (L ; s1 ; x) =

Y v

j1 (gv )jv H (L 0; x) =

Y 1  1 (gv )

2

v

v

H (L ; s2 ; x):

Appliquée à des sections de même poids , cela permet d'étendre les fonctions H (L ; s;
) au complémentaire dans G(AF )X (F ) de l'intersection des diviseurs des sections de poids .

X est une variété torique, compactication équivariante lisse d'un tore G, tout bré en droites eectif L qui est G-linéarisé possède une unique droite F -rationnelle de sections pour lesquelles G agit par le caractère trivial. On peut utiliser cette section pour dénir une hauteur sur les points adéliques du complémentaire de son diviseur, donc en particulier sur G(AF ). Remarque 2.4.6.  Lorsque

Expliquons maintenant comment la torsion des métriques adéliques intervient dans nos constructions. Nous allons préciser un peu la situation 2.3.4 en faisant désormais l'hypothèse suivante : Situation 2.4.7.

 Nous faisons les hypothèses contenues dans la situation 2.3.4.

S est le spectre de l'anneau des entiers de corps de nombres F . De plus, et B sont propres sur S . Soit b un point F -rationnel de B . Comme B est propre sur S , il en résulte une unique section "b : S ! B qui prolonge b. Toute trivialisation du GF -torseur GF ' T jb sur Spec F (il en existe car c'est un torseur pour la topologie de Zariski) induit un isomorphisme XF ' Y jb . Fixons un tel isomorphisme '. Si  2 PicG (X ), ' #() est un bré en droite sur XF canoniquement isomorphe à . En revanche, les métriques En particulier, supposons que

X

(adéliques) sont en général distinctes. Soit v une place nie de F , notons ov le complété de l'anneau local de oF en v . Soit "v : Spec ov ! B la restriction de "b à Spec ov . Alors, "v T est un G ov -torseur sur Spec ov , et est donc trivialisable. Ainsi, "v Y est isomorphe à X ov . Fixons un isomorphisme 'v induit par une trivialisation du torseur. Il existe par dénition gv 2 G(Fv ) tel que

' = 'v Æ [gv ];

X F !Yj F ; v

b

v

[gv ] désignant l'automorphisme de X Fv déni par gv . La dénition de la métrique v -adique associée à un modèle montre que 'v est une isométrie. Ainsi, en tant que bré inversible métrisé sur X Fv , ' (#()) est isomorphe (isométrique) à [gv ] .

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

22

Soit maintenant v une place à l'inni. Comme on s'était xé une trivialisation du G(C)=Kv -bré sur B (C), on dispose d'un isomorphisme 'v bien déni modulo Kv qui par dénition ne modie pas les métriques. La comparaison entre ' et 'v se fait comme précédemment par un élément gv 2 G(C). Il en résulte le théorème :

g = (g ) 2 G(A

) l'élément du groupe adélique que nous venons d'introduire. Il représente la classe de la restriction à b du (G; K1 )-torseur Théorème 2.4.8.

 Soit

v v

F

T

arithmétique c dans l'isomorphisme de la proposition 1.2.6. De plus, la métrique adélique image réciproque sur ' #() s'identie à la métrique adélique tordue par sur .

g

2.5. Nombres de Tamagawa Commençons par rappeler la dénition, due à Peyre (cf. [16] et [18]) des nombres de Tamagawa associés à une métrique adélique sur le faisceau anticanonique. 2.5.1. Hypothèses.  Soit X une variété propre, lisse et géométriquement intègre sur F telle que H1 (X; X ) = H2 (X; X ) = 0 et que X (F ) soit Zariski-dense dans X . Sous

O

ces conditions, Pic(XF )Q est un

O

Q-espace vectoriel de dimension nie.

!X d'une métrique adélique. Pour toute place v de F , une construction classique de Weil fournit une mesure X;v sur X (Fv ) à partir de la métrique v -adique sur !X . Notons Lv (s; Pic(XF )) le facteur local en v de la fonction L d'Artin de la représentation de Gal(F =F ) sur Pic(XF )Q . Le théorème de Weil sur la mesure de X (Fv ) pour X;v et le théorème de Deligne sur les 2.5.2. Dénition.  Munissons le bré canonique

conjectures de Weil concernant le nombre de points rationnels des variétés sur les corps nis ont la conséquence suivante : il existe un ensemble ni  de places de F , contenant les places archimédiennes, tel que Y

v

2

X;v 

Y

v

62

Lv 1 (1; Pic(XF ))X;v




dénisse une mesure X; Q sur X (AF ) pour laquelle X (AF ) a un volume ni. Soit L (s; Pic(XF )) = v62 Lv (s; Pic(XF )) la fonction L partielle de Pic(XF ). Le produit eulérien converge en eet pour 1 et L a un pôle en s = 1 d'ordre la dimension t des invariants sous Gal(F =F ) de Pic(XF )Q . Notons L (1; Pic(XF )) = slim (s 1)r L (s; Pic(XF )): !1 On dénit alors le nombre de Tamagawa de X (associé à la métrique adélique choisie sur !X ) par Z   (X ) = L (s; Pic(X ))  : 

F

( )

X F



X;

Il est facile de vérier qu'il ne dépend pas de l'ensemble ni de places  choisi. Nous aurons à utiliser le lemme suivant.

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS Lemme 2.5.3.

23

 Supposons réalisées les hypothèses 2.5.1. Soit U un ouvert non vide Q

de X . Notons U (F ) l'adhérence de U (F ) dans v U (Fv ) pour la topologie produit (qui Q est la topologie induite sur v U (Fv ) par la topologie adélique de X ( F )). Alors, on a l'égalité Z Z

A

( )

U F

X; =

Démonstration.  Tout point x = (xv ) 2

( )

X F

X; :

Q U (Fv ) possède par dénition un voisinage Qv (n)

(pour la topologie induite) contenu dans v U (Fv ). Par suite, si une suite (x ) de points de X (F ) converge vers x, à partir d'un certain rang, x(n) appartientQà U (Fv ) pour toute place v , et donc x(n) 2 U (F ). Cela montre que U (F ) = X (F ) \ v U (Fv ). Q Ainsi, le complémentaire de U (F ) dans X (F ) est contenu dans X (AF ) n v U (Fv ), donc dans la réunion [ Y

(X n U )(Fv )

v

6=

X (Fw ):

w v

La dénition de la mesure X;v implique que (X n U )(Fv ) est de mesure nulle pour X;v . On voit donc que X (F ) n U (F ) est réunion dénombrable d'ensembles de mesure nulle pour la mesure de Tamagawa sur X (AF ), donc est de mesure nulle. On se place maintenant dans la situation 2.3.4, S étant le spectre Spec oF de l'anneau des entiers d'un corps de nombres F . Lemme 2.5.4.

 Si

X

B

et F satisfont les hypothèses 2.5.1 nécessaires pour la dénition des nombres de Tamagawa, F les satisfait aussi. F

Démonstration.  Que

Y

Y

soit lisse, propre et géométrique intègre est clair. D'autre part, les points rationnels de YF sont denses dans chaque bre au-dessus d'un point rationnel de BF , lesquels sont supposés denses dans BF . Comme YF ! BF est propre, un argument élémentaire de platitude puis de dimension implique que les points rationnels de YF sont Zariski-denses. D'autre part, les hypothèses sur XF impliquent que R0  OYF = OBF et que F

R1  OYF = R2  OYF = 0:

La suite spectrale des foncteurs composés implique que Hq (OYF ) est un quotient de L i j i j i+j =q H (BF ; R  OYF ). Si j = 1 ou si j = 2, on a H (R  OYF ) = 0 puisque Rj  OYF = 0. Si j = 0 et i 2 f1; 2g, Hi (R0  OYF ) = Hi (OBF ) = 0 en vertu des hypothèses faites sur BF .

Supposons donc que XF et BF satisfont ces hypothèses 2.5.1. Le faisceau canonique sur Y admet d'après la proposition 2.1.8 une décomposition ! = #(! =S )   ! :

Y

X

B=S Choisissons une structure de bré en droite hermitien (G; K1) linéarisé sur !X =S compatible à la linéarisation canonique sur !X =S (autrement dit, pour toute place archimédienne  , une métrique hermitienne K -invariante sur X  C). Choisissons aussi

24

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

une métrique hermitienne sur !B=S . Il en résulte une métrique hermitienne canonique sur !Y =S par la construction 2.3.5. Le fait de disposer d'un modèle sur oF induit de plus des métriques v -adiques au places nies, d'où des métriques adéliques sur !XF , sur !BF et sur !YF . Théorème 2.5.5.

 Muni de ces métriques adéliques, on a l'égalité

 (YF ) =  (XF ) (BF ):

B

Démonstration.  Soit U un ouvert de Zariski non vide de F tel que Notons V =  1 (U )  F , de sorte que V est un ouvert non vide de

à XF

Y

 U , et que dans cette décomposition, la mesure (2.5.6) Y ;v j (U ) = X ;v B;v jU :

T j ' G U. Y j isomorphe U

S

F

1

Pour toute place v de F , il résulte du corollaire au théorème 2.2.4 la relation entre facteurs locaux (2.5.7)

Lv (s; Pic(YF )) = Lv (s; Pic(XF ))Lv (s; Pic(BF )):

Alors, les équations (2.5.6) et (2.5.7) impliquent que la restriction de la mesure de Q Tamagawa de X (AF ) à v V (Fv ) s'écrit comme le produit

Y ; jQv V (Fv ) = X ; B;jQv U (Fv ) : Q Or, si U (F ) est l'adhérence de U (F ) dans le produit v U (Fv ), l'adhérence de V (F ) Q dans v V (Fv ) s'identie à X (F )  U (F ). Intégrons Y ; sur V (F ) ; en utilisant le lemme 2.5.3, on obtient

Z

X ; 

Z

B;: B(F ) L'équation (2.5.7) implique aussi que pour 1, L (s; Pic(YF )) = L (s; Pic(XF ))L (s; Pic(BF )): Par suite, l'ordre du pôle en s = 1 pour la fonction L de Y est la somme des ordes Y (F )

Y ; =

Z

X (F )

des pôles pour X et B , et donc L (1; Pic(YF )) = L (1; Pic(XF ))L (1; Pic(BF )): Le théorème est donc démontré.

2.6. Torseurs trivialisants Le paragraphe 2.4 a montré que le phénomène de torsion des métriques adéliques intervient naturellement dans nos constructions. Cependant, la hauteur tordue n'est facile à calculer que lorsqu'il existe des sections propres pour l'action du groupe. L'existence de sections canoniques permet comme on l'a vu de disposer d'une fonction hauteur sur les points adéliques. Les torseurs trivialisants que nous introduisons ici ont pour fonction de fournir  au prix d'un changement de variété  d'une droite canonique de sections.

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS

25

Dans ce paragraphe, nous nous plaçons sur un corps F . Supposons que PicG (X ) ' Pic(X )  X  (G) est un groupe de type ni. Soit H un groupe algébrique sur F , X1 ! X un H -torseur qui induise par fonctorialité covariante des torseurs un isomorphisme X  (H ) ! Pic(X ). On suppose de plus que X1 est muni d'une action de G qui relève l'action de G sur X et qui commute à l'action de H . On peut construire un tel X1 en xant 1 ; : : : ; h des brés inversibles G -linéarisés sur X dont les classes forment une base de Pic(X ). On pose alors Qh X1 = i=1 (V(_i ) n f0g) et H = Ghm . Soit T le plus grand quotient de G tel que l'homomorphisme naturel X  (T ) ! X  (G) est un isomorphisme. (C'est le quotient de G par l'intersection des noyaux des caractères f= X f1 T et  : X f ! X la composition de la première projection de de G). On pose X f1 ! X . C'est un H  T -torseur muni d'une action de G (diagonale). de la projection X

X

= P nG est un espace de drapeaux généralisé pour un groupe algébrique simplement connexe semi-simple G sur F . On a Pic(X ) ' X  (P ) et G ! X est un P -torseur qui induit un isomorphisme X  (P ) ' Pic(X ). De plus, T = f1g. Ainsi, on peut prendre Xf = G. Exemple 2.6.1.  Supposons que

G est trivial, on retrouve les torseurs universels introduits dans le contexte des hauteurs par Salberger et Peyre (cf. [20], [18]). G Fait 2.6.3.  Si  2 Pic (X ),    admet une F -droite canonique de sections GExemple 2.6.2.  Lorsque le groupe

invariantes.

PicG (X )

' X  (H  T ) = X (H )   X (G) admet une réciproque qu'il est facile d'expliciter. En eet, soient H et G deux caractères de H et G respectivement. On dénit un bré inversible G-linéarisé sur X f  A1 = X f1  T  A1 par l'action de H donnée par comme suit : on quotiente X Remarque 2.6.4.  L'isomorphisme canonique

h
(xe; t; u) = (h
xe; t; H (h) 1 u); h 2 H; (xe; t; u) 2 Xf1  T  A1

f  A1 fournie par et la G-linéarisation provient de l'action de G sur X

(xe; t; u)
g = (g
xe; g
t; G1 (g )u); g 2 G; (xe; t; u) 2 Xf1  T  A1 :

Par la construction 2.1.2, on obtient ainsi un F -schéma Yf avec une projection Yf ! Y . Supposons que Y provient de la situation 2.3.4, on dispose en particulier de brés inversibles sur YF munis de métriques adéliques associés aux brés inversibles (G; K)linéarisés sur X . En particulier, on obtient sur Yf des brés inversibles avec métriques adéliques. Le fait nouveau est que l'on dispose d'une hauteur sur les points adéliques de Yf associée à ces brés inversibles. En eet, une fois remontés à Yf, ces brés inversibles possèdent une droite de sections F -rationnelle canonique.

2.7. Exemples 2.7.1. Action d'un tore.  Pour les applications auxquelles notre deuxième article

sera consacré, on considère l'action d'un tore T .

26

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

Un tel tore peut agir non seulement sur des variétés toriques, mais aussi sur des variétés de drapeaux généralisées P nG, via un morphisme T ! G. Dans le cas des variétés toriques sur un corps de nombres F , on dispose de modèles canoniques sur Spec oF (si le tore est déployé), et de métriques hermitiennes à l'inni canoniques sur les brés en droites. Pour tout plongement  de F dans C, les points complexes T (C) du tore admettent un unique sous-groupe compact maximal K , et les métriques hermitiennes introduites sont automatiquement K -invariantes. On obtient ainsi des brés hermitiens (T; K)-linéarisés (cf. par exemple [3]). Dans le cas des variétés de drapeaux P nG, une fois xé des sous-groupes compacts maximaux de G aux places à l'inni, il est aussi possible de munir les brés en droites P linéarisés de métriques hermitiennes invariantes pour ces sous-groupes compacts maximaux et donc pour le sous-groupe compact maximal de T (C). Aux places nies, les métriques v -adiques qu'on obtient admettent une description analogue en termes de la décomposition d'Iwasawa (cf. [12]). D'autre part, un T -torseur sur un F -schéma B , du moins quand le tore est déployé, à la donnée d'un morphisme X  (T ) ! Pic(B ), et donc, une fois xé une base de X  (T ), à des brés en droites 1 ; : : : ; t 2 Pic(B ). (On a noté t = dim T .) La trivialisation des T=K -torseurs correspond, ainsi qu'on l'a dit après la dénition 1.1.3 d'un T -torseur arithmétique, à une métrique hermitienne sur les brés en droites i . Dans le cas où T agit sur une variété torique, on obtient alors par la construction 2.3.4 une famille de variétés toriques sur B . On peut notamment compactier ainsi une variété semi-abélienne T ! B et construire sur la compactication Y des fonctions hauteurs canoniques. Dans ce cas, les i sont des brés en droites algébriquement équivalent à 0 sur une variété abélienne B . Si on a pris soin de les munir, ainsi que tous les brés en droites sur B , de leur métrique adélique canonique, pour laquelle le théorème du cube est une isométrie, on obtient sur Y les hauteurs canoniques, au sens de la hauteur de NéronTate. (Dans ce cas particulier, cf. [9] où l'on trouvera cette construction dans un esprit analogue, et [11], où est donnée une construction  à la Tate  de ces hauteurs canoniques, due à M. Waldschmidt). Dans le cas où T agit sur une variété de drapeaux généralisée, on obtient la variété de drapeaux (généralisée) d'un bré vectoriel sur B construit naturellement à partir des i . Ce cas était étudié (lorsque la base est aussi une variété de drapeaux) dans la thèse de M. Strauch ([22]).

B donne lieu à des variétés de drapeaux généralisées. Dans ce cas, le groupe G est le groupe linéaire GL(d), X est une variété P nG. On identie en eet un bré vectoriel de rang n sur B à un 2.7.2. Variétés de drapeaux.  Tout bré vectoriel sur

GL(d)-torseur. Si l'on choisit comme sous-groupe compact à l'inni le groupe unitaire U(d), la trivialisation à l'inni du G=K -bré correspond à une métrique hermitienne sur le bré vectoriel. Il est à noter que cette situation se retrouve, mais dans l'autre sens, dans le calcul du comportement de la fonction zêta des hauteurs d'une puissance symétrique d'une

TORSEURS ARITHMÉTIQUES ET ESPACES FIBRÉS

27

courbe C de genre g  2. Dans ce cas en eet, si d > 2g 2, Symd C est un bré projectif au-dessus de la jacobienne de C associé à un bré vectoriel de rang d + 1 g . 2.7.3. Action d'un groupe vectoriel.  Dans [10] et [9], on étudie des compactica-

tions d'extensions vectorielles de variétés abéliennes. Expliquons comment ce travail s'insère dans les constructions de cet article lorsque, pour simplier les notations, on prend G = Ga . Un Ga -torseur sur B correspond à une extension de OB par lui-même, soit un bré vectoriel E de rang 2 sur B . La trivialisation du Ga -torseur à l'inni correspond à un scindage C 1 de l'extension sur B (C). D'autre part, Ga agit naturellement sur P1 (via son plongement dans GL(2), a 7! ( 10 a1 )). On obtient ainsi une compactication du Ga-torseur en une famille de droites projectives sur B.

Références [1]

  Theory of intersections on the arithmetic surface , Proceedings (Vancouver, 1974), 1974, p. 405408. V. V. Batyrev & Yu. I. Manin   Sur le nombre de points rationnels de hauteur bornée des variétés algébriques , Math. Ann. 286 (1990), p. 2743. V. V. Batyrev & Yu. Tschinkel   Rational points on bounded height on compactications of anisotropic tori , Internat. Math. Res. Notices 12 (1995), p. 591635. ,  Height zeta functions of toric varieties , Journal Math. Sciences 82 (1996), no. 1, p. 32203239. ,  Manin's conjecture for toric varieties , J. Algebraic Geometry 7 (1998), no. 1, p. 1553. ,  Tamagawa numbers of polarized algebraic varieties , in Nombre et répartition des points de hauteur bornée [17], p. 299340. A. Borel   Sous-groupes compacts maximaux des groupes de Lie (d'après Cartan, Iwasawa et Mostow) , Séminaire Bourbaki 1950/51, Exp. 33. S. Bosch, W. Lütkebohmert & M. Raynaud  Néron models, Ergeb., no. 21, Springer Verlag, 1990. A. Chambert-Loir   Extensions vectorielles, périodes et hauteurs , Thèse, Univ. P. et M. Curie, Paris, 1995. ,  Extension universelle d'une variété abélienne et hauteurs des points de torsion , Compositio Math. 103 (1996), p. 243267. P. Cohen   Heights of torsion points on commutative group varieties , Proc. London Math. Soc. 52 (1986), p. 427444. J. Franke, Yu. I. Manin & Yu. Tschinkel   Rational points of bounded height on Fano varieties , Invent. Math. 95 (1989), no. 2, p. 421435. G. van der Geer & R. Schoof   Eectivity of Arakelov divisors and the Theta divisor of a number eld , Tech. report, math.AG/9802121, 1998. H. Gillet & C. Soulé   Arithmetic intersection theory , Publ. Math. Inst. Hautes Études Sci. 72 (1990), p. 94174. D. Mumford, J. Fogarty & F. Kirwan  Geometric invariant theory, Ergeb., no. 34, Springer Verlag, 1994. S. Ju. Arakelov

of the International Congress of Mathematicians

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

28

[16] [17] [18] [19] [20] [21] [22] [23]

  Hauteurs et mesures de Tamagawa sur les variétés de Fano , Duke Math. J. 79 (1995), p. 101218. (éd.)  Nombre et répartition des points de hauteur bornée, Astérisque, no. 251, 1998. ,  Terme principal de la fonction zêta des hauteurs et torseurs universels , in Nombre et répartition des points de hauteur bornée [17], p. 259298. D. Roy & J. L. Thunder   An absolute Siegel's lemma , J. Reine Angew. Math. 476 (1996), p. 126. P. Salberger   Tamagawa measures on universal torsors and points of bounded height on Fano varieties , in Nombre et répartition des points de hauteur bornée [17], p. 91258. S. Schanuel   Heights in number elds , Bull. Soc. Math. France 107 (1979), p. 433 449. M. Strauch   Thèse , Thèse, Universität Bonn, 1997. M. Strauch & Yu. Tschinkel   Height zeta functions of toric bundles over ag varieties , Tech. report, Universität Bonn, 1997. E. Peyre

alg-geom le 4 janvier 1999 Antoine Chambert-Loir, Institut de mathématiques de Jussieu, Boite 247, 4, place Jussieu, F75252 Paris Cedex 05  E-mail : [email protected] Yuri Tschinkel, Department of Mathematics, U.I.C., Chicago (IL) 60607 E-mail : [email protected] Soumis sur l'archive

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS par

Antoine Chambert-Loir & Yuri Tschinkel

Table des matières Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations et conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Fonctions holomorphes dans un tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Énoncé du théorème . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Démonstration du théorème . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Variétés toriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Préliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Transformations de Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Dénition d'une classe de contrôle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. La fonction zêta des hauteurs et la formule de Poisson . . . . . . . . 5. Application aux brations en variétés toriques . . . . . . . . . . . . . . . . . . . . 5.1. Holomorphie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Prolongement méromorphe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendice A. Un théorème taubérien . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendice B. Démonstration de quelques inégalités . . . . . . . . . . . . . . . . . . Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 4 4 9 13 14 17 19 19 25 25 27 30 32 36

Introduction Cet article est le deuxième d'une série consacrée à l'étude des hauteurs sur certaines variétés algébriques sur un corps de nombres, notamment en ce qui concerne la distribution des points rationnels de hauteur bornée. Précisément, soient X une variété algébrique projective lisse sur un corps de nombres F , L un bré en droites sur X et HL : X (F ) ! R+ une fonction hauteur (exponentielle) pour L . Si U est un ouvert de Zariski de X , on cherche à estimer le nombre

NU (L ; H ) = #fx 2 U (F ) ; HL (x)  H g

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

2

lorsque H tend vers +1. L'étude de nombreux exemples a montré que l'on peut s'attendre à un équivalent de la forme () N (L ; H ) = (L )H a(L ) (log H )b(L ) 1 (1 + o(1)); H ! +1 U

pour un ouvert U convenable et lorsque par exemple L et !X1 (bré anticanonique) sont amples. On a en eet un résultat de ce genre lorsque X est une variété de drapeaux [11], une intersection complète lisse de bas degré (méthode du cercle), une variété torique [4], une variété horosphérique [19], une compactication équivariante d'un groupe vectoriel [10], etc. On dispose de plus d'une description conjecturale assez précise des constantes a(L ) et b(L ) en termes du cône des diviseurs eectifs [1] ainsi que de la constante (L ) ([15], [5]). En fait, on étudie plutôt la fonction zêta des hauteurs, dénie par la série de Dirichlet

ZU (L ; s) =

X

x2U (F )

HL (x)

s

à laquelle on applique des théorèmes taubériens standard. Sur cette série, on peut se poser les questions suivantes : domaine de convergence, prolongement méromorphe, ordre du premier pôle, terme principal, sans oublier la croissance dans les bandes verticales à gauche du premier pôle. Cela permet de proposer des conjectures de précision variable. Dans cet article, nous considérons certaines brations localement triviales construites de la façon suivante. Soient G un groupe algébrique linéaire sur F agissant sur une variété projective lisse X , B une variété projective lisse sur F et T un G-torseur sur B localement trivial pour la topologie de Zariski. Ces données dénissent une variété algébrique projective Y munie d'un morphisme Y ! B dont les bres sont isomorphes à X . Le c÷ur du problème est de comprendre le comportement de la fonction hauteur lorsqu'on passe d'une bre à l'autre, comportement vraiment non trivial bien qu'elles soient toutes isomorphes. Dans notre premier article (Torseurs arithmétiques et espaces brés, [9]), nous avons exposé en détail la construction de hauteurs sur de telles variétés. Dans celui-ci, nous appliquons ces considérations générales au cas d'une bration en variétés toriques provenant d'un torseur sous un tore déployé, pour l'ouvert U déni par le tore. Nous avons construit les hauteurs à l'aide d'un prolongement du torseur géométrique en un torseur arithmétique, ce qui correspond en l'occurence au choix de métriques hermitiennes sur certains brés en droites. Écrivons la fonction zêta comme la somme des fonctions zêta des bres X X X ZU (L ; s) = HL (x) s = ZUb (L jUb ; s): b2B (F ) x2Ub (F ) b2B (F ) Chaque Ub est isomorphe au tore et on peut exprimer la fonction zêta des hauteurs de Ub à l'aide de la formule de Poisson adélique. De cette façon, la fonction zêta de U apparaît comme une intégrale sur certains caractères du tore adélique de la fonction L d'Arakelov du torseur arithmétique sur B .

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3

Ainsi, nous pouvons démontrer des théorèmes de montée : supposons que B vérie une conjecture, alors Y la vérie. Bien sûr, la méthode reprend les outils utilisés dans la démonstration de ces conjectures pour les variétés toriques ([4, 2, 3]). Par exemple, nous démontrons au Ÿ 5.1 l'holomorphie de la fonction ZU (L ; s) pour Re(s) > a(L ) sous la seule hypothèse de la convergence de la fonction zêta des hauteurs analogue sur B ; cela implique que pour tout " > 0, le nombre de points rationnels de hauteur HL inférieure à H est O(H a(L )+" ). Ensuite, sous des hypothèses raisonnables concernant B , nous établissons un prolongement méromorphe de cette fonction zêta à gauche de a(L ) et nous démontrons que l'ordre du pôle est inférieur ou égal à b(L ) ; cela précise la majoration du nombre de points en O(H a(L ) (log H )b(L ) 1 ). Enn, lorsque L = !Y 1 , nous démontrons que le pôle est eectivement d'ordre b(L ) d'où une estimation de la forme () et nous identions la constante (L ), établissant ainsi la conjecture de Manin ranée par Peyre. Pour un bré en droites quelconque, la preuve de la conjecture de BatyrevManin [1] avec son ranement par Batyrev Tschinkel [5] est ramenée à la détermination exacte de l'ordre du pôle, c'est-à-dire à la non-annulation d'une certaine constante. Dans le cas des variétés toriques ou des variétés horosphériques, l'utilisation de  brations L -primitives  dans [3] et [19] a permis d'établir cette conjecture. Moyennant des hypothèses sur B , cette méthode devrait s'étendre au sujet de notre étude. Notre méthode impose de disposer de majorations de la fonction zêta des hauteurs (pour B ) dans les bandes verticales à gauche du premier pôle ; nous avons ainsi tâché d'obtenir de telles majorations pour la variété Y . Il est en outre bien connu que cela entraîne un développement asymptotique assez précis pour le nombre de points de hauteur bornée, cf. le théorème taubérien donné en appendice. Quelques cas de variétés toriques sur Q avaient en eet attiré l'attention des spécialistes de théorie analytique des nombres (voir notamment les articles de É. Fouvry et R. de la Bretèche dans [16], ainsi que [6]). Notre méthode établit un tel développement pour les variétés toriques lisses, les variétés horosphériques, etc. sur tout corps de nombres. La démonstration de l'existence d'un prolongement méromorphe de la fonction zêta des hauteurs pour les variétés toriques ou pour les variétés horosphériques faisait intervenir un théorème technique d'analyse complexe à plusieurs variables dont la démonstration se trouve dans [4], [3] et [19]. En vue d'obtenir les majorations exigées dans les bandes verticales, nous sommes obligés d'en préciser la preuve ; ceci est l'objet du Ÿ 3. Dans les Ÿ 4 et Ÿ 5 se situe l'étude de la fonction zêta des hauteurs d'une variété torique et d'une bration en variétés toriques. Pour les variétés toriques, nous améliorons le terme d'erreur à la suite de [4, 18, 8]. Le théorème de montée pour les brations généralise le résultat principal de [19].

Notations et conventions

Si X est un schéma, on note Pic(X ) le groupe des classes d'isomorphisme de faisceaux inversibles sur X . Si F est un faisceau quasi-cohérent sur X , on note

4

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

V(F ) = Spec Sym F et P(F ) = ProjSym F les brés vectoriels et projectifs associés à F . c X ) le groupe des classes d'isomorphisme de brés en droites hermitiens On note Pic( sur X (c'est-à-dire des brés en droites munis d'une métrique hermitienne continue sur X (C) et invariante par la conjugaison complexe).. Si X est un S -schéma, et si  2 S (C), on désigne par X le C-schéma X  C. Cette notation servira lorsque S est le spectre d'un localisé de l'anneau des entiers d'un corps de nombres F , de sorte que  n'est autre qu'un plongement de F dans C. Si G est un schéma en groupes sur S , X  (G) désigne le groupe des S -homomorphismes G ! Gm (caractères algébriques). Si X =S est lisse, le faisceau canonique de X =S , noté !X =S , est la puissance extérieure maximale de 1X =S . Enn, cet article commence au paragraphe 3. Les références aux paragraphes 1 et 2 renvoient ainsi à l'article précédent [9].

3. Fonctions holomorphes dans un tube Le but de ce paragraphe est de prouver un théorème d'analyse sur le prolongement méromorphe de certaines intégrales et leur estimation dans des bandes verticales. Ce théorème généralise un énoncé analogue de [4, 19]. La présentation en est un peu diérente et le formalisme que nous introduisons permet de contrôler la croissance des fonctions obtenues. Ce contrôle est nécessaire pour utiliser des théorèmes taubériens précis et améliorer ainsi le développement asymptotique du nombre de points rationnels de hauteur bornée. Les résultats de ce paragraphe n'interviennent que dans la preuve des théorèmes 4.4.6 et 5.2.5.

3.1. Énoncé du théorème Soit V un R-espace vectoriel réel de dimension nie muni d'une mesure de Lebesgue dv et d'une norme k
k. On dispose alors d'une mesure canonique dv  sur le dual V  . Notons VC = V R C le complexié de V . On appelle tube toute partie connexe de VC de la forme + iV où est une partie connexe de V ; on le notera T( ). Soit enn M un sous-espace vectoriel de V muni d'une mesure de Lebesgue dm. Dénition 3.1.1.  Une classe de contrôle D est la donnée pour tout couple M  V de R-espaces vectoriels de dimension nie d'un ensemble D (M; V ) de fonctions mesurables  : V ! R+ dites D (M; V )-contrôlantes vériant les propriétés suivantes : (a) si 1 et 2 sont deux fonctions de D (M; V ), 1 et 2 deux réels positifs, et si  est une fonction mesurable V ! R+ telle que   1 1 + 2 2 , alors  2 D (M; V ) ; (b) Si  2 D (M; V ) et si K est un compact de V , la fonction v 7! supu2K (v + u) appartient à D (M; V ) ; (c) si  2 D (M; V ), pour tout v 2 M n 0, (tv ) tend vers 0 lorsque t tend vers +1 ;

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5

(d) si  2 D (M; V ), pour tout sous-espace M1  M , la fonction M1 -invariante

M1 : v 7!

Z

M1

(v + m1 ) dm1

est nie et appartient à D (M=M1 ; V=M1 ) ; (e) si  2 D (M; V ), pour tout sous-espace M1  M et tout projecteur p : V de noyau M1 , la fonction M1 -invariante  Æ p appartient à D (M=M1 ; V=M1 ). 3.1.2. Il existe une classe de contrôle

!V

D max contenant toutes les classes de contrôles :

l'ensemble D max (M; V ) est déni par récurrence sur la dimension de M par les trois conditions (a, c, e) dans la dénition 3.1.1. La dernière condition est alors automatique. Dans la suite, on xe une classe de contrôle D , et on abrège l'expression D (M; V )contrôlante en M -contrôlante.

f : T( ) ! C sur un tube est dite M -contrôlée s'il existe une fonction M -contrôlante  telle que pour tout compact K  T( ), il existe un réel c(K ) de sorte que l'inégalité jf (z + iv)j  c(K )(v) soit vériée pour tout z 2 K et tout v 2 V . 3.1.4. Considérons une fonction sur un tube, f : T( ) ! C. Soit M un sous-espace vectoriel de V , muni d'une mesure de Lebesgue dm. On considère la projection  : V ! V 0 = V=M et on munit V 0 de la mesure de Lebesgue quotient. On pose, quand Dénition 3.1.3.  Une fonction

cela a un sens,

Z 1 (3.1.5) SM (f )(z) = (2)dim M f (z + im) dm; z 2 T( ): M .  Soit  V et f : T( ) ! C une fonction holomorphe M -contrôlée. 0 Soit M un sous-espace vectoriel de M et 0 l'image de par la projection V ! V=M 0 . Alors, l'intégrale qui dénit SM 0 (f ) converge en tout z 2 T( ) et dénit une fonction holomorphe M=M0 -contrôlée sur T( 0 ). Démonstration.  Comme f est M -contrôlée, il existe une fonction  2 D (M; V ) et, pour tout compact K  T( ), un réel c(K ) > 0 de sorte que pour tout v 2 V et tout z 2 K , on ait jf (z + iv )j  c(K )(v ). La condition (3.1.1, d) des classes de contrôles Lemme 3.1.6

jointe au théorème de convergence dominée de Lebesgue implique que l'intégrale qui dénit SM 0 (f ) converge et que la somme est une fonction holomorphe sur T( ). Par construction, cette fonction est iM 0 -invariante. Comme elle est analytique, elle est donc invariante par M 0 et dénit ainsi une fonction holomorphe sur T( 0 ). De plus, si  désigne la projection V ! V=M 0 , pour tout z 2 K et tout v 2 V , on a Z

jSM 0 (f )((z) + i(v))j  c(K ) 0 (v + m0 ) dm0 = c(K )0 ((v)) M 0 0 où  appartient par dénition à D (M=M ; V=M 0). Tout compact de T( 0 ) étant de la forme  (K ) pour un compact K de

T( ), le lemme est ainsi démontré.

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

6

 un cône convexe polyédral ouvert de V . La fonction caractéristique de  est la fonction sur T() dénie par l'intégrale 3.1.7. Fonction caractéristique d'un cône.  Soit

convergente

X(z) =

(3.1.8)

Z



 e hz;v i dv  ;

où   V  est le cône dual de , V  étant muni de la mesure de Lebesgue dv  duale de la mesure dv . Si  est simplicial, c'est-à-dire qu'il existe n = dim V formes linéaires indépendantes `1 ; : : : ; `n telles que v 2  si et seulement si `j (v ) > 0 pour tout j , alors

X (z) = kd`1 ^


^ d`nk

1 : ` (z ) j =1 j

n Y

(On a noté kd`1 ^


^ d`n k le volume du parallélépipède fondamental dans V  de base les `j .) Dans le cas général, toute triangulation de  par des cônes simpliciaux permet d'exprimer X sous la forme d'une somme de fractions rationnelles de ce type. Elle se prolonge ainsi en une fonction rationnelle sur T(V ) dont les pôles sont exactement les hyperplans de VC dénis par les équations des faces de . Elle est de plus strictement positive sur . Une autre façon de construire un cône est de s'en donner des générateurs, autrement dit de l'écrire comme quotient d'un cône simplicial. À ce titre, on a la proposition suivante. .  Soit  un cône polyédral convexe ouvert de V dont l'adhérence

Proposition 3.1.9

 ne contient pas de droite. Soit M un sous-espace vectoriel de V tel que  \ M = f0g. On note  la projection V ! V 0 = V=M . La restriction à T() de la fonction X est M -contrôlée (pour la classe D max (M; V )). L'intégrale qui dénit SM (X ) converge donc absolument et pour tout z 2 T(), on a SM (X )(z) = X0 ((z)): 0 Remarque 3.1.10.  Les hypothèses impliquent que  ne contient pas de droite. En 0 0 eet, s'il existait un vecteur non nul de  \  , il existerait deux vecteurs v1 et v2 de  tels que v1 + v2 2 M mais v1 62 M . Comme  \ M = f0g, v1 = v2 ce qui contredit l'hypothèse que  ne contient pas de droite. Démonstration.  La preuve est une adaptation des paragraphes 7.1 et 7.2 de [19].

Soit (ei ) une famille minimale de générateurs de . Chaque face de  dont la dimension est dim V 1 engendre un sous-espace vectoriel qui est l'orthogonal d'un des ei . Comme M \  = f0g, il existe une forme linéaire ` 2 V  qui est nulle sur M mais qui n'appartient à aucune face de  ; posons H = ker `. Soit H 0 un supplémentaire de R` dans V  . Si ' 2 V  et t 2 R sont tels que ' + t` 2  , on doit avoir pour tout générateur ej de  l'inégalité '(ej ) + t`(ej ) > 0, soit (rappelons que `(ej ) n'est pas nul), t > '(ej )=`(ej ) quand `(ej ) > 0 et t < '(ej )=`(ej ) quand `(ej ) < 0. Soit

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alors I (') = ]h1 ('); h2 (')[ l'intervalle de R déni par ces inégalités. (Si tous les `(ej ) sont positifs, c'est-à-dire ` 2  , on a h1  1, tandis que s'ils sont tous négatifs, h2  +1.) Les fonctions h1 et h2 sont linéaires par morceaux par rapport à un éventail de H 0 qu'on peut supposer complet et régulier (voir par exemple [12] pour la dénition, ou [2]). Alors, si v 2 T() et m 2 H , on a

X (v + im) =

= = =

Z Z Z Z

V H0

1 (')e hv+im;'i d'

Z

R

1 (' + t`)e hv+im;'i e thv;`i dt d'

Z h2 (')

H 0 h1 (') H0

e hv+im;'i e thv;`i dt d'

e e hv+im;'i

h1 (')hv;`i

e hv; `i

h2 (')hv;`i

d'

de sorte que la fonction H ! C telle que m 7! X (v + im) est (à une constante multiplicative près) la diérence des transformées de Fourier des fonctions

H 0 ! C; ' 7! e hv;'+hj (')`i pour j = 1 et 2. Comme v 2 T() et ' + hj (')` appartient au bord de  , hv; ' + hj (')`i est de partie réelle strictement positive, à moins que ' = 0. Soit K un compact de T(). Il résulte alors des estimations des transformées de Fourier de fonctions linéaires par morceaux et positives (voir [2], proposition 2.3.2, p. 614, et aussi infra, prop. 4.2.4) une majoration de la fonction

f;K (m) := de la forme

f;K (m)  c(K )

X

X dim YH 

jX (v + im)j

v2K

j =1

1

(1 + jhm; `;j ij)1+1= dim H ;

où pour tout , la famille (`;j )j est une base de H  . D'après le lemme 3.1.11 ci-dessous, la fonction f;K appartient à D max (M; V ). La fonction m 7! X (v + im) est donc absolument intégrable sur M . C'est la transformée de Fourier de la fonction ' 7! 1 (')e hv;'i dont il est facile de voir qu'elle est intégrable sur tout sous-espace et donc aussi M ? . La formule de Poisson s'applique (après un léger argument de régularisation) et s'écrit Z

X(v + im) dm = (2)

dim M

M

Z

 \M ?

e hv;'i d':

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

8

Or, l'application V

! V 0 identie (V 0) à M ?, et  \ M ? à (0). Ainsi, on obtient Z SM (X )(v) = 0  e h(v);'i d' = X0 ((v)): ( )

.  Soit V un R-espace vectoriel de dimension d, (`1 ; : : : ; `d ) une base Q V  et f la fonction v 7! dj=1 (1 + j`j (v )j) 1 1=d . Alors, f 2 D max (V; V ). Démonstration.  Soit M un sous-espace vectoriel de V de dimension m. Quitte à réordonner les indices, on peut supposer que M est l'image d'une application linéaire Rm ! Rd = V de la forme t = (t1 ; : : : ; tm ) 7! (t1 ; : : : ; tm ; 'm+1 (t); : :R: ; 'd (t)). Si on réalise V=M par son supplémentaire f0gm  Rd m , la fonction fM : v 7! M f (v + m) dm Lemme 3.1.11

de

est donnée par l'intégrale Z

d Y 1 1 1 ::: 1+1 =d 1+1 =d (1 + jtm j) j=m+1 (1 + jvj + 'j (t)j)1+1=d dt1 : : : dtm : Rm (1 + jt1 j)

Elle est dominée par l'intégrale convergente Z

1 1 ::: 1+1 =d (1 + jtm j)1+1=d dt1 : : : dtm Rm (1 + jt1 j)

et le théorème de convergence dominée implique alors que pour tout vecteur v (0; : : : ; 0; vm+1; : : : ; vd) distinct de 0,

lim

f s!+1 M

=

(sv) = 0:

Le lemme est ainsi démontré.

C un ouvert convexe de V ayant 0 pour point adhérent et  un cône polyédral ouvert contenant C . Soit   V  une famille de formes linéaires deux à deux non proportionnelles déDénition 3.1.12.  Soient

nissant les faces de . On note HM (; C ) l'ensemble des fonctions holomorphes f : T(C ) ! C telles qu'il existe un voisinage convexe B de 0 dans V de sorte que la fonction g dénie par

g (z ) = f (z )

'(z ) 1 + '(z) '2 Y

admet un prolongement holomorphe M -contrôlé dans

T(B ).

Par le théorème d'extension de Bochner (voir par exemple [13]), une telle fonction s'étend en une fonction holomorphe sur le tube de base l'enveloppe convexe C 0 de B [ C . En particulier, il n'aurait pas été restrictif de prendre pour C l'intersection du cône  avec un voisinage convexe de 0 dans V . On constate aussi que f est nécessairement M -contrôlée dans T(C ). Enn, il est facile de vérier que HM (; C ) ne dépend pas du choix des formes linéaires qui dénissent les faces de .

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 est un cône polyédral et si M est un sous-espace vectoriel de V tel que l'image de  dans V=M ne contient pas de droite, la proposition 3.1.9 implique donc que la fonction X appartient à l'espace HM (; ) déni par la classe de contrôle D max . 3.1.13. Si

Le théorème principal de cette section est le suivant. .  Soit M

Théorème 3.1.14

V

un sous-espace vectoriel muni d'une mesure de

Lebesgue. Soit C l'intersection de avec un voisinage convexe de et soit f 2 M C. 0 0 0 0 Soit M un sous-espace vectoriel de M ,  la projection V ! V V=M ,  et 0 C C. Alors, la fonction M 0 f appartient à M=M 0 0 C 0 . Si de plus l'adhérence du cône 0 ne contient pas de droite et si pour tout z 2 ,



= ( )

S ()

alors pour tout z 0

0

( ; )

H



=

H (; )  = ()



f (sz ) = 1; lim s!0+ X (sz ) 

2 0,

SM 0 (f )(sz ) = 1: lim s!0+ X 0 (sz 0 ) 0

 Corollaire 3.1.15.  Supposons de plus que f est la restriction à \ C d'une 0 fonction holomorphe M -contrôlée sur . Alors, la fonction M f sur V est méromorphe dans un voisinage convexe de 0 , ses pôles étant simples dénis par les faces (de codimension ) de 0 .

1

 



S ()



3.2. Démonstration du théorème

D'après le lemme 3.1.6, la fonction SM 0 (f ) est holomorphe et M=M0 -contrôlée sur T(C 0). Le but est de montrer qu'elle y est la restriction d'une fonction méromorphe dont on contrôle les pôles et la croissance. La démonstration est fondée sur l'application successive du théorème des résidus pour obtenir le prolongement méromorphe. La dénition des classes de contrôle est faite pour assurer l'intégrabilité ultérieure de chacun des termes obtenus. Par récurrence, il sut de démontrer le résultat lorsque dim M 0 = 1. Soit m0 un générateur de M 0 . Munissons la droite Rm0 de la mesure de Lebesgue d. Soit   V  une famille de formes linéaires deux à deux non proportionnelles positives sur  et dont les noyaux sont les faces de . Soit B un ouvert convexe et symétrique par rapport à l'origine, assez petit de sorte que pour tout ' 2  et tout v 2 B , j'(v )j < 1 et que la fonction

g (z ) = f (z )

'(z ) 1 + '(z) '2 Y

admette un prolongement holomorphe M -contrôlé sur T(B ). L'intégrale à étudier est Z +1 Y 1 + '(z + itm )

1

g (z + itm0 )

'2

'(z + itm0 )

0

dt:

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

10

On veut déplacer la droite d'intégration vers la gauche. Fixons  > 0 tel que 2m0 2 B . Ainsi, si Re(z ) 2 12 B , z + (u + it)m0 appartient à T(B ) pour tout u 2 [  ; 0] et tout t 2 R. Notons + ,  et 0 les ensembles des ' 2  tels que respectivement '(m0 ) > 0, '(m0 ) < 0 et '(m0 ) = 0. Soit B1  12 B l'ensemble des v 2 12 B tels que pour tout ' 2 + , j'(v )j < 2 '(m0 ). Dans la bande   s  0, les pôles de la fonction holomorphe

s 7! g (z + sm0 ) sont ainsi donnés par

s' (z ) =

1 + '(z + sm0) '(z + sm0 ) '2 Y

'(z ) ; ' 2 + : '(m0 )

Le pôle s = s' (z ) est simple si et seulement si pour tout

'(z ) (m0 )

2 + tel que 6= ',

(z)'(m0 ) 6= 0: (m0 )' '(m0 )

Comme ' et sont non proportionnelles, est une forme linéaire non y nulle ; notons B1  B1 le complémentaire des hyperplans qu'elles dénissent lorsque ' 6= parcourent les éléments de + . Si z 2 T(B1y ) et si T > maxfjIm(s' (z ))j ; ' 2 + g, la formule des résidus pour le contour délimité par le rectangle   Re(s)  0, T  Im(s)  T s'écrit Z T T

g (z + itm0 )

1 + '(z + itm0 ) dt '(z + itm0 ) '2 X 2i Y 1 + (z + s' (z )m0 ) = g (z + s' (z )m0 ) '(m0 ) (z + s'(z)m0 ) 6=' '2+ Z T Y + g(z m0 + itm0 ) 1 +'('z(z mm+0 +itmitm) 0 ) dt 0 0 T '2 Z  Y + g(z + sm0 + iT m0 ) 1 +'('z(+z +smsm+0 +iTiTmm) 0 ) ds 0 0 0 '2 Z 0 Y + g(z + sm0 iT m0 ) 1 +'('z(+z +smsm0 iTiTmm) 0 ) ds: 0 0  '2 Y

Lorsque T ! +1, l'hypothèse que g est M -contrôlée et l'axiome (3.1.1,c) des classes de contrôles impliquent que ces deux dernières intégrales (sur les segments horizontaux du rectangle) tendent vers 0. De même, l'axiome (3.1.1,d) assure la convergence des deux premières intégrales vers les intégrales correspondantes de 1 à +1.

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11

Par suite, si z 2 T(B1y \ ), on a (3.2.1)

SRm0 (f )(z) =

+

X

'2+ Z

1 + '(z) '(z ) '20 Y

1 + (z + s'(z)m0 ) (z + s'(z)m0 ) 6=' Y 1 + '(z m0 + itm0 ) m0 + itm0 ) dt: ' ( z m + itm ) 0 0 0 '2n

g (z + s' (z )m0 ) 1 1

g (z

Y

Il résulte alors des axiomes (3.1.1,e) et (3.1.1,d) des classes de contrôles que la fonction (3.2.2)

z 7! SRm0 (f )(z )

'(z ) Y Y '(s + s (z )m0 ) 1 + '(z) '2+ 620 [f'g 1 + (s + s'(z)m0 ) '20 Y

dénie sur T(B1y \) s'étend en une fonction holomorphe M=M0 -contrôlée sur T( (B1y )). En particulier, SRm0 (f ) se prolonge méromorphiquement à T(B1y ) et les pôles de SRm0 (f ) sont donnés par une famille nie de formes linéaires. Le lemme suivant les interprète géométriquement. 0 Lemme 3.2.3.  Les faces de  sont les noyaux des formes linéaires deux à deux non

V=Rm0 ' 2 0 et ' De plus, si ' et 2 + , le noyau de '

proportionnelles sur

2

'(m0 ) + et (m0 ) pour ' '(m0 ) 0 (m0 ) rencontre .



2

.

x 2 V appartient à  si et seulement si '(x) > 0 pour 0 tout ' 2 . Par suite,  (x) 2  si et seulement si il existe  2 R tel que '(x m0 ) > 0 pour tout ' 2 . Si ' 2 0 , cette condition est exactement '(x) > 0. Pour les autres ', elle devient Démonstration.  Un vecteur

'(x) '(x) <  < min+ max '2 '(m ) '2 '(m ) 0

0

d'où la première partie du lemme. Pour la seconde, soit ' et deux éléments distincts de + . Si le noyau de ' ne recontre pas 0 , quitte à permuter ' et , on a

'(v ) > '(m0 )

'(m0 ) (m0 )

(v) (m0 )

pour tout v 2  et cela contredit le fait que ' et .

dénissent deux faces distinctes de

On sait que SRm0 (f ) est holomorphe sur T(0 ). Il résulte du lemme que les formes linéaires + s' (z )' avec ' 2 + et 62 0 [ f'g sont des pôles apparents dès que 2 + . Les autres correspondent aux faces de 0 ! Autrement dit, nous avons déjà prouvé que SRm0 (f ) est la restriction à T( (B1 )) d'une fonction méromorphe dont les pôles (simples) sont donnés par les faces de 0 . Montrons comment contrôler la croissance de SRm0 (f ) dans les bandes verticales.

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

12

.  Soit V un espace vectoriel, M un sous-espace vectoriel, B un voisi-

Lemme 3.2.4

0

( )

nage de dans V . Soit h une fonction holomorphe sur T B et soit ` une forme linéaire sur V . Si la fonction z 7! h z 1+`(`z()z ) est M -contrôlée, h est M -contrôlée.

()

Démonstration.  Il faut montrer que h est M -contrôlée dans un voisinage de tout point de B . Soit donc x0 2 B et K un voisinage compact de x0 contenu dans B . Soit

 2 D (M; V ) telle que pour tout x 2 K et tout y 2 V , h



(x + iy) 1 +`(x`(+x +iyiy) )  (y):

Supposons d'abord que `(x0 ) 6= 0. Si  = j`(x0 )j =2 > 0, il existe un voisiange compact K1  K de x0 où j`j  . Alors, pour tout x 2 K1 et tout y 2 V , on a

)j  1 +  (y); jh(x + iy)j  (y) 1 +`(jx`(+x +iyiy )  ce qui prouve que h est M -contrôlée dans K1 . Si `(x0 ) = 0, soit u 2 V tel que `(u) = 1, K1 un voisinage compact de x0 assez petit et  > 0 tels que pour tout t 2 [ 1; 1] et tout x 2 K1 , x + tu 2 K . La fonction s 7! h(x + iy + su) est une fonction holomorphe sur le disque unité fermé jsj  1. D'après le principe du maximum, on a donc pour tout x + iy 2 T(K1 ),

jh(x + iy)j  sup jh(x + iy + su)j = sup jh(x + iy + su)j  1 +  sup (y + su): jsj1

jsj=1

jsj1

L'axiome (3.1.1,b) assure alors l'existence d'une fonction 1 tout x + iy 2 T(K1 ),

2 D (M; V ) telle que pour

jh(x + iy)j  1(y): La fonction h est donc M -contrôlée dans un voisinage de x0 . Il reste à démontrer que si pour tout z 2 ,

lim f (tz)=X (tz) = 1, alors

t!0+

lim SRm0 (f )(tz0)=X0 (tz0 ) = 1:

t!0+

Comme X (tz ) = t dim V X (z ), l'hypothèse f (tz )  X (tz ) se récrit

lim tdim V t!0

# g

(tz) = X(z):

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

13

D'autre part, la formule (3.2.1) donne

t

1+dim V

SRm0 (f )(tz) = t 1+dim V

+t

X '2+

1+dim V Z 1

g (tz + s' (tz )m0 )

1 + '(tz)  '(tz ) '20 Y

1 + (tz + s'(tz)m0 ) (tz + s'(tz)m0 ) 6 ' =

Y

1 + '(tz m0 + itm0 ) dt '(tz m0 + itm0 ) 1 '2n0 X Y = tdim V #g(t(z + s'(z)m0 )) 1 + t(z (+z s+ (sz')(mz)m) 0 ) ' 0 6=' '2+ Y (tz)  + t 1+dim V #0 1 +'t' (z) '20 Z 1 Y 1 + '(tz m0 + itm0 )  g(tz m0 + itm0 ) dt: ' ( tz m + itm ) 0 0 1 0 '2n Un vecteur non nul de V ne peut appartenir qu'à au plus dim V 1 faces de  et seuls les générateurs de  appartiennent à dim V 1 faces. Comme m0 est supposé n'être pas un générateur de , #0  dim V 2. Lorsque t tend vers 0, on a donc X Y 1 lim t 1+dim V SRm0 (f )(tz) = X (z + s' (z)m0 ) (z + s (z)m ) 

g (tz

m0 + itm0 )

Y

'2+

6='

'

0

où le second membre ne dépend plus de f . Comme on peut appliquer cette formule à f = X , on obtient donc

lim t1

dim V

(SRm0 (f ))(tz) = lim t1 dim V (SRm0 (X))(tz) = lim t1 dim V X0 (tz) = X0 (z):

Le théorème est ainsi démontré. Remarque 3.2.5.  La démonstration s'adapte sans peine lorsque f dépend uniformé-

ment de paramètres supplémentaires.

4. Variétés toriques Dans ce paragraphe, nous montrons comment les ranements analytiques du paragraphe 3 permettent de préciser le développement asymptotique obtenu par Batyrev Tschinkel dans [4] pour la fonction zêta des hauteurs d'une variété torique. Les résultats techniques que nous rappelons à l'occasion seront réutilisés au paragraphe suivant, lorsque nous traiterons le cas d'une bration en variétés toriques.

14

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

4.1. Préliminaires

S = Spec oF le spectre de l'anneau des entiers de F . Si v est une place de F , on dénit la norme k
kv sur Fv de la manière habituelle, comme le module associé à une mesure de Haar additive sur Fv . En particulier, si v est une uniformisante en une place nie v , kv kv est l'inverse du cardinal du corps résiduel en v . Soit G un tore déployé de dimension d sur S . Désignons par K la collection de ses Q1 Q sous-groupes compacts maximaux aux places à l'inni et KG = v 1 G(ov ) vj1 Kv  G(AF ). Il nous faut faire quelques rappels sur la structure du groupe AGL des caractères de G(F )nG(AF )=KG . On a un homomorphisme de noyau ni AG ! vj1 X  (G)R ,  7! 1 obtenu en associant à un caractère adélique son type à l'inni, c'est-à-dire sa restriction au sous-groupe de G(A) dont les composantes aux places nies sont triviales. En choisissant une norme sur X  (G)R , on obtient ainsi une  norme   7! k1 k sur 4.1.1. Rappels adéliques.  Notons

-

AG.

Il existe enn un homomorphisme X  (G)R ! AG , tel que l'image du caractère algébrique  2 X  (G) est le caractère adélique g 7! j(g)ji dont le type à l'inni s'identie à  sur chaque composante. Le quotient AG =X  (G)R est un Z-module de type ni et de rang ( 1)d (où  = r1 + r2 , r1 et r2 désignant comme d'habitude les nombres de places réelles et complexes) et l'on peut xer une décomposition AG = X  (G)R  UG , par exemple à l'aide d'un scindage de la suite exacte j
j  1 ! Gm(AF )1 ! Gm (AF ) ! R ! 1: (Rappelons que G est supposé déployé.) 4.1.2. Rappels sur les variétés toriques.  Notons

M

=

X  (G)R , c'est un espace

vectoriel sur R de dimension nie d. Considérons une compactication équivariante X de G, lisse sur S . D'après la théorie des variétés toriques (cf. par exemple [14], [12]), X est dénie par un éventail complet et régulier  de N := Hom(M; R) formé de cônes convexes simpliciaux rationnels. Il existe ainsi une famille (minimale) (ej )j 2J de vecteurs de N telle que tout cône  2  soit engendré par une sous-famille (ej )j 2J de cardinal dimvect( ). On note (d) l'ensemble des cônes de  de dimension d. L'espace vectoriel PL() des fonctions continues N ! R dont la restriction à chaque cône de  est linéaire est un espace vectoriel de dimension nie sur R, d'ailleurs égale à #J ; munissons le d'une norme arbitraire. L'espace vectoriel PicG(XF )R est isomorphe à PL() ; il possède une base canonique formée des brés en droites G-linéarisés associés aux diviseurs G-invariantsP sur XF . À chaque ej correspond un tel diviseur Dj ; à un diviseur G-invariant D = j j Dj correspond l'unique fonction ' 2 PL() telle que '(ej ) = j . Dans cette description, le cône des diviseurs eectifs correspond simplement l'ensemble des éléments de PicG (XF ) dont les coordonnées (j ) vérient j  0 pour tout j . Plus généralement, on notera t l'ensemble des éléments de PicG (XF ) tels que j > t pour tout j ; le cône ouvert 0 est aussi noté PL+ () et encore 0e (XF ).

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

15

Cette base (Dj ) de PicG (XF ) et l'homomorphisme canonique  : X  (G) ! PicG (X ) induisent des sous-groupes à un paramètre Gm ! G, d'où, pour tout caractère  2 AG , des caractères j de Gm (F )nGm (AF )=KGm , autrement dit des caractères de Hecke. Les brés en droites sur XF seront systématiquement munis de leur métrique adélique canonique introduite notamment dans [2]. Cela nous fournit un homomorphisme c X ) qui induit un homomorphisme canonique Pic(XF ) ! Pic( c G;K (X ): PicG(XF ) ! Pic

(4.1.3)

On vérie aisément, par exemple sur les formules données dans [2], que les sous-groupes compacts maximaux aux places archimédiennes agissent de manière isométrique. De plus, le choix d'une G-linéarisation fournit une unique F -droite de sections ne s'annulant pas sur G, donc en particulier une fonction hauteur sur les points adéliques de XF comme dans la dénition 1.3.3. Cette fonction s'étend en une application  bilinéaire  H : PL()  G(A ) ! C :

C

(On a identié PicG (XF )C et

F

PL()C.) 2 X  (G) et notons m 2 AG

.  Soit m

Lemme 4.1.4

dénit. On a alors

le caractère adélique qu'il

m (g) = H ((m); g) i: Démonstration.  Par dénition, (m) est le bré en droite trivial sur X muni de la G-linéarisation dans laquelle G agit par multiplication par le caractère algébrique m. Ainsi, la droite de sections rationnelles G-invariante et ne s'annulant pas sur G est engendrée par le caractère m vu comme fonction rationnelle sur X . La dénition de H implique que Y H ((m); g) = km(gv )k 1 = km(g)k 1 : v

Or,

m (g) = km(g)ki = H ((m); g) i:

v de F , on xe une mesure de Haar dxv sur Fv . On suppose que pour presqueQtoute place nie v , la mesure du sous-groupe compact ov est égale à 1. Alors, dx = v dxv est une mesure de Haar sur le groupe localement compact AF . On en déduit pour tout v une mesure de Haar 0Gm ;v = kxv kv 1 dxv sur Fv . Pour presque toute place nie v , la mesure de ov est égale à 1 qv 1 ; dénissons ainsi, si v est une place nie, Gm ;v = (1 qv 1 ) 1 0Gm ;v . On munit alors AF de la 4.1.5. Mesures.  Pour toute place

mesure

Y v

Gm ;v =

Y v-1

(1

qv 1 )

Remarquons que F;v (1) = (1 qv ) fonction zêta de Dedekind du corps F . 1

1 kx

1 v k dxv 

Y vj1

kxv k 1 dxv :

1 est le facteur local en la place nie

v de la

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

16

Tout oF -isomorphisme G ' Gdm induit alors des mesures de Haar 0G;v et G;v = F;v (1)d 0 sur G(Fv ) pour toute place v de F , indépendantes de l'isomorphisme. On G;v

Q

en déduit aussi une mesure de Haar G;v sur G(AF ). D'autre part, le bré canonique sur X est métrisé. Peyre a montré dans [15] comment en déduire une mesure sur X (AF ). Pour toute place v , on dispose d'une mesure 0X ;v sur X (Fv ) dénie par la formule

0X ;v = kd1 ^


^ ddkv 1 d1 : : : dd si (1 ; : : : ; d ) est un système arbitraire de coordonnées locales sur restreint la mesure 0X ;v à G(Fv ), on obtient donc

X (Fv ).

Si l'on

Hv ( ; x)0G;v ;

(4.1.6)

 désignant la fonction de PL() telle que pour tout j , ej 7! 1 ( correspond à la classe anticanonique). Pour presque toute place nie v , on a alors

0X ;v (X (Fv )) = qv d #X (kv ): La décomposition cellulaire des variétés toriques (point n'est besoin ici d'invoquer le théorème de Deligne sur les conjectures de Weil) implique alors que

#X (kv ) = qvd + rang(Pic XF )qvd 1 + O(qvd 2): Par suite, le produit inni Y v-1

0v (X (Fv ))F;v (1)

est convergent. Dénissons une mesure X ;v sur

X ;v = F;v (1)

rang(Pic XF )

X (Fv ) par

rang Pic XF 0

X ;v

Q si v est nie et X ;v = 0X ;v si v est archimédienne. Ainsi, le produit inni v X ;v converge et dénit une mesure, dite mesure de Tamagawa sur X (AF ). Le nombre de Tamagawa de X (AF ) est alors dénie par

(4.1.7)

 (X ) = (AF =F )

d

ress=1 F (s)rang(Pic XF ) X (AF ):

Remarque 4.1.8.  La diérence de formulation avec la dénition que donne Peyre

dans [15] n'est qu'apparente. Peyre a choisi la mesure sur Fv de la façon suivante : si v est une place nie, dxv (ov ) = 1, si v est une place réelle, dxv est la mesure de Lebesgue usuelle sur R et si v est une place complexe, dxv est le double de la mesure usuelle sur C. Le volume de AF =F est alors égal à
1F=2 .

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

17

4.2. Transformations de Fourier

On s'intéresse à la transformée de Fourier de la fonction g 7! H ( ; g ) sur le groupe abélien localement compact G(AF ). Rappelons qu'on a noté 1 l'ensemble des  2 PL() tels que j > 1 pour tout j . Alors, si  2 T(1), la fonction g 7! H ( ; g) est intégrable (cf. [19], Ÿ 3.4), si bien que la transformée de Fourier existe pour tout  2 T(1 ). Elle se décompose par construction en un produit H = H f  H 1 , où

H f

= (ress=1 F (s))

d

Y v-1

(1

qv 1 ) d H v

 1 = vj1 H v sont les produits des intégrales locales (renormalisées) aux places et H nies et archimédiennes. (Les transformées de Fourier locales existent même dès que pour tout j , Re(j ) > 0.) Q

.  Soit 2=3

Lemme 4.2.1

 PL() la partie convexe dénie par j > 2=3 pour tout

j . Il existe une fonction cf : T(2=3 )  AG ! C; (; ) 7! cf (; ); holomorphe en  telle que log jcf j est bornée et telle que le produit des transformées de Fourier locales aux places non archimédiennes s'écrive, pour tout  2 AG et tout  2 T() Y H f ( ; ) = cf (; ) L(j ; j ): j

log

Démonstration.  Si  est xé, c'est la proposition 2.2.6 de [2]. Le fait que jcf j soit borné indépendamment de  se déduit immédiatement de la preuve dans loc. cit.

 f se prolonge .  La fonction H en une fonction méromorphe pour Q 

Corollaire 4.2.2

( )

1)Hf ( ; ) se prolonge en une j (j ( ) Y lim (j 1)H f ( ; ) = 0 !(1;:::;1)

2 T 2=3 . Plus précisément, le produit fonction holomorphe dans T 0 et 

si et seulement si

 6= 1.

j

Comme conséquence facile de l'estimation par Rademacher des valeurs des fonctions L de Hecke pour les caractères non ramiés, estimation qui repose sur le principe de PhragménLindelöf, on obtient la majoration suivante : .  Pour tout " > 0, il existe

Corollaire 4.2.3

Re(j ) > 1

Æ,

Y j

0 < Æ < 1=3 et un réel c" tels que si

jj 1j H ( ; )  c 1 + kIm()k
" 1 + kk
": " jj j f

Passons maintenant aux places archimédiennes. La proposition suivante précise la proposition 2.3.2 de [2].

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

18

 2=3  PL()R , il existe un réel cK

.  Pour tout compact K

Proposition 4.2.4

telle que pour tout

' 2 T(K ) et tout m 2 M , on ait la majoration

1 + k'k : (1 + jh e ; m ij ) j j 2 J  2(d) Q Q e l'éventail e = .  Désignons par   dans N vj1 vj1 N . Si ' 2 P PL(), désignons par 'e la fonction Ne ! R dénie par (nv )v 7! '(nv ). Pour tout compact K de PL() contenu dans 2=3 , il existe une constante cK telle que pour tout ' 2 T(K ) et tout  2 AG décomposé sous la forme  = im + u 2 iM  UG, on ait jF (m)j  cK 1 + 1kmk

X

Q

Corollaire 4.2.5

H 1('; ) 

cK

X

1 + kIm 'eke : eje + m e ij) e2e (1 + jhe; Im '

Q

1 + kk e2e Démonstration.  Si l'on note m e = (mv )v la décomposition de  à l'inni, on remarque que

H 1('; ) =

Y vj1

H v ('; ) =

Y vj1

F ('; mv ) = F (';e me ):

Il sut alors d'appliquer la proposition précédente. Preuve de la proposition 4.2.4.  Il faut estimer Z

F (m) =

N

exp( '(v) ihv; mi) dv:

Soit  2  un cône de base (e1 ; : : : ; ed ). Si jdet(ej )j désigne la mesure du parallélotope de base les ej , on a Z 

exp( '(v) ihv; mi) dv =

(4.2.6)

Z

d Y

Rd+ j =1

= c()

exp

1 : ' ( e ) + i h e ; m i j j j =1

d Y

Ainsi, on a (4.2.7)

F (m) =

X 

c( )

1 : ' ( e ) + i h e; m i e2

Y




tj ('(ej ) + ihej ; mi)

jdet(ej )j

Y

dtj

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

19

6= 0, on peut intégrer par parties et écrire   @' 1 F (m) = im @v exp( '(v) ihv; mi) dv

D'autre part, supposons que mj Z

j N  Z 

j

@' exp( '(v) ihv; mi) dv N @vj X @' Z = @v exp( '(v) ihv; mi) dv j    X @' Y = c() @v '(e) +1ihe; mi : j  e2 

imj F (m) =

(4.2.8)

En combinant les égalités (4.2.7) et (4.2.8) pour tous les indices j tels que mj obtient une majoration

jF (m)j  1 + 1kmk

X 

6= 0, on

1 + k'k : e2 j'(e) + ihe; mij

c( ) Q

Finalement, comme ' 2 T(K ), on a une estimation

j'(e) + ihe; mij  1 + jIm(')(e) + he; mij

et la proposition s'en déduit.

4.3. Dénition d'une classe de contrôle Soit un réel strictement positif. Si M et V sont deux R-espaces vectoriels de dimension nie avec M  V , notons D ;"(M; V ) le sous-monoïde de F (V; R+ ) engendré par les fonctions h : V ! R+ telles que pour tout " > 0, il existe c > 0, " 2 ]0; 1[ et une famille (`j ) de formes linéaires sur V vériant :  la famille (`j jM ) forme une base de M  ;  pour tout v 2 V et tout m 2 M , on a (1 + kvk) Q 1 (4.3.1) h(v + m)  c (1 + kmk)1 " (1 + j`j (v + m)j) :

Notons alors D

= T">0 D ;". .  Les D (M; V ) dénissent une classe de contrôle au sens de la

Proposition 4.3.2

dénition 3.1.1.

La preuve de cette proposition consiste en une série d'inégalités faciles mais techniques. Nous la repoussons à l'appendice B.

4.4. La fonction zêta des hauteurs et la formule de Poisson

On s'intéresse fonction zêta des hauteurs de X restreinte à l'ouvert dense formé par le tore G ; c'est par dénition la série génératrice

Z () =

X

x2G(F )

H ( ; x);

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

20

quand elle converge. Des théorèmes taubériens standard (voir l'appendice) permettront de déduire de résultats analytiques sur Z un développement asymptotique du nombre de points de hauteur bornée

N (; H ) = #fx 2 G(F ) ; H (; x)  H g: Lemme 4.4.1.  Lorsque Re() décrit un compact de 1 , la fonction zêta des hauteurs converge uniformémént en . Plus généralement, la série X H ( ; xg) x2G(F )

converge absolument uniformément lorsque compact de G AF .

( )

Re() décrit un compact de 1

Démonstration.  Compte tenu d'estimations pour

et g un

H ( ; xg)=H ( ; x) lorsque g

décrit un compact de G(AF ), x 2 G(F ) et  2 T(1 ), c'est en fait un corollaire de l'intégrabilité de la fonction H ( ;
) sur G(AF ). Voir [4], Th. 4.2 et aussi [19], Prop. 4.3. Par conséquent, on peut appliquer la formule sommatoire de Poisson sur le tore adélique G(AF ) pour le sous-groupe discret G(F ). Compte tenu de l'invariance de l'accouplement de hauteurs par le sous-groupe compact maximal KG de G(AF ), on en déduit la formule

Z () =

(4.4.2)

Z

H ( ; ) d

AG où d est la mesure de Haar sur le groupe AG des caractères unitaires continus sur le groupe G(F )nG(AF )=KG duale de la mesure de comptage sur G(F ). Rappelons que l'on a décomposé le groupe AG = M  UG , où UG est un groupe discret. De plus, si  = m  u ,

H ( ; ) = H (  im; u )

si bien que

Z () =

Z

X M

u 2UG

!

H (  im; u ) dm

où dm est la mesure de Lebesgue sur M telle que dm du de comptage sur UG.

= d, du étant la mesure

.  Si d0 m est la mesure de Lebesgue sur M dénie par le réseau M , on

Lemme 4.4.3

a

dm = (2 vol(AF =F ) ress=1 F (s))

d 0 d m:

=

Démonstration.  Par multiplicativité, il sut de traiter le cas G Gm et d 1  Notons AF le sous-groupe de AF formé des x tels que kxk . La suite exacte 1    logkxk

1 ! AF =F ! AF =F

=1 !R!0

= 1.

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

21

permet de munir A1F =F  de la mesure de Haar dx1 telle que d x = dx1 d0 n. La suite exacte duale 1 ! R ! (AF =F ) ! (A1F =F ) ! 1 et la discrétudePdu groupe des caractères de A1F =F  permet de munir (AF =F  ) de le mesure d0 m . Avec ces normalisations, la constante devant la formule de Poisson est (2 vol(A1F =F  )) 1 . Compte tenu des normalisations choisies, le théorème classique selon lequel  (Gm ) =  (Ga ) = 1, cf. par exemple [20], p. 116, devient vol(A1 =F ) = vol(A =F ) res  (s); s=1 F

F

F

d'où le lemme.

= (1; : : : ; 1) 2 PL(). On décale la fonction zêta des hauteurs  2 PL()+ , ! Z X Z ( + ) = H (   im; u ) dm Soit 

Soit F la fonction PL()

M

+

de  : si

u 2UG

! C dénie par la série

 7! (vol(AF =F ) ress=1 F (s))

de sorte que si  2 PL()+ ,

X

d

u 2UG

H (

1

; u );

Z 1 0 (4.4.4) Z ( + ) = (2)d M F ( + im) d m: .  Si > 1, la fonction F appartient à l'espace HM (PL()+ ) déni par la classe de contrôle D du paragraphe 4.3. De plus, pour tout  2 PL()+ , lim F (s) =  (X ); s!0 XPL()+ (s) le nombre de Tamagawa de X . Proposition 4.4.5

Démonstration.  On a vu que l'on pouvait écrire

H (  ; ) = cf ( + ; )H 1 ( ; )

Par suite, la fonction

 7! H (  ; )

Y

Y j

L(j + 1; j ):

j

j + 1 admet un prolongement holomorphe pour Re(j ) > 1. j

De plus, il résulte des corollaires 4.2.3 et 4.2.5 que pour tout " > 0, il existe Æ < 1=3 tel que si pour tout j on a Re(j ) > Æ , alors H

(

 ; )

Y j



(1 + kIm()k)  j + 1 (1 + k1k)1 " j

1+" X e d) e2(

1 ; e2e1 (1 + jhe; Im()je + 1 ij)

Q

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

22

formule dans laquelle 1 désigne L l'image de  par l'homomorphisme de noyau ni  type à l'inni  AG ! M1 = vj1 M . Ainsi, on obtient un prolongement holoQ morphe de la fonction  :  7! F () j j =(1 + j ) pour Re(j ) > Æ si l'on prouve e d), la série que pour tout e 2 (

1 (1 + ) (1 + jhe; Im()je + u;1ij) u 2UG e d). Alors, lorsque converge localement uniformément en  si Re(j ) > Æ . Fixons e 2 ( e 2 e1 , les formes linéaires he;
i forment une base de M1 . Il est facile de remplacer la sommation sur le sous-groupe discret UG;1 par une intégrale sur l'espace vectoriel 1

X

ku;1k 1 " Qe2e1

qu'il engendre, lequel est d'ailleurs un supplémentaire de M envoyé diagonalement dans M1 . La convergence est alors une conséquence de la proposition B.3. Pour obtenir l'assertion sur la croissance de F , il faut montrer que si > 1, K est un compact de PL()+ ,  2 T(K ) et m 2 M , on a une majoration

)k) j( + im)j  (1(1++kkIm( mk)1 "

1 1 + j`;k (Im() + m)j  k où  parcourt un ensemble ni et où pour tout , f`;k gk est une base de PL() . Il nous  obtenue ci-dessus en remarquant faut récrire un peu diéremment la majoration de H XY

que si la forme des transformées de Fourier aux places nies fournit le prolongement méromorphe, la convergence de la série provient, elle, des estimations archimédiennes. On écrit ainsi

H (   im; u ) = cf ( +  + im; u )

et donc H

X

j( + im)j 

e d) e2(

j

L(j + 1 + im; u;j )H 1 (  ; m u )



j + im  (   im; u ) 1 +  + im j j ) + mk)" (1 + ku k)"  (1 + kIm( 1 + km + u;1k Y

Par suite,

Y

X e d) e2(

1 + kIm()ke : e2e1 (1 + jhe; Im()je + m + 1 ij)

Q

(1 + kIm()ke )(1 + kIm() + mk)"Ge (Im(); m)

où e(';m) est déni par la série

e ('; m) = On a la majoration

(1 + kuk)" Y 1 : 1 + k m +  k 1 + jh e; ' eje + m + u;1 ij u; 1 u 2UG e2e1 X

1 + kuk  1 + km + u;1k + kmk  (1 + km + u;1k)(1 + kmk)

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

23

et comme précédemment, on remplace la sommation sur le sous-groupe discret UG par l'intégrale sur l'espace vectoriel qu'il engendre. La proposition B.3 fournit alors pour tout "0 > " une estimation

Ge ('; m) 

1

XY

(1 + kmk)

1 "0



k

1 1 + j`;k (m + 'je )j

où f`;k gk est une base de M  et 'je l'élément de M qui coïncide avec ('; : : : ; ') 2 L e . L'application ' 7! `;k ('je ) est une forme e de l'éventail  vj1 PL() sur le cône  linéaire `e;;k sur PL(). On a ainsi

1 (1 + k Im( )k)(1 + kIm() + mk)" X X Y jG( + im)j  0 1 " " (1 + kmk) 1 + j`e;;k (Im() + m)j e  k + kIm()k)1+" X X Y 1  (1 : 0 1 2 " " (1 + kmk) 1 + j ` (Im(  ) + m ) j  e ;;k  k e Comme on peut prendre " et "0 arbitrairement petits, la contrôlabilité est établie. Il reste à calculer la limite quand s ! 0 par valeurs supérieures de F (s)=XPL()+ (s). Le cône PL()+ est simplicial et

XPL()+ () = Q1 : j j

Ainsi,

F ()

= (vol(AF =F ) ress=1 F (s)) + ()

XPL()

d

Y j

j

X u 2UG

H (  ; u ):

D'après ce qui précède, la série qui dénit F converge uniformément pour Re(j ) > Æ ; cela permet de permuter sommation et limite, si bien que

F (s) lim + s!0 XPL()+ (s)

= (vol(AF =F ) ress=1 F (s))

d

X u 2UG

lim H (

s!0+

s ; u )

Y j

(sj )

!

:

En écrivant,

H ( s ; )

Y

Y

j

j

(sj ) = cf (s; )

sj L(sj + 1; j ) H 1 (s; );


on voit que la limite est nulle si l'un des j 6= 1 (car une des fonctions L(
; j ) n'a pas de pôle en 1, les autres ont au plus un pôle simple). Étudions maintenant le cas  = 1.

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

24

Utilisant la formule (4.1.6), il vient

H ( s ; 1)

=

Y

j Y v-1

F (1 + j s)

v (1)

d



=

Y v-1

Y

1

v (1 + j s)

j YZ

vj1 G(Fv ) Y d v j

v (1)

1

Z G(Fv )

H ( s ; x)0G;v

H ( s ; x)0G;v

(1 + j s)

1

Z G(Fv )

H(

C'est un produit eulérien absolument convergent pour ment par continuité en s = 0, de valeur Y v-1

v (1)d

X ;v (X (Fv ))

#J 0

Y vj1

Z

Y s)0X ;v  H ( s)0X ;v : vj1 G(Fv )

Re(s) >

", d'où un prolonge-

0X ;v (X (Fv ))

=  (X )(AF =F )d(ress=1 F (s)) rang(Pic XF ) en vertu de la dénition (4.1.7) de la mesure de Tamagawa de X (AF ). Ainsi,  ( s ; 1)(Y sj ) = (ress=1 F (s))#J lim H ( s ; 1) Y F (1 + sj ) 1 lim H s!0 s!0 j

=  (X )(AF =F ) (ress=1 F (s)) d

d:

j

Finalement, on a donc

lim F (s)XPL+() (s) 1 = (vol(AF =F ) ress=1 F (s)) d(AF =F )d(ress=1 F (s))d  (X ) =  (X );

s!0

ainsi qu'il fallait démontrer. L'équation (4.4.4) et le théorème 3.1.14 impliquent alors le théorème suivant. .  La fonction zêta des hauteurs (décalée)

Théorème 4.4.6

 7! Z ( + )

(PL() )

+ et dénit une fonction hoconverge localement uniformément sur le tube T lomorphe sur T 0e F . Si > et si désigne la classe de contrôle intro0 duite au sous-paragraphe 4.3, elle appartient à l'espace f0g 0e F e F (déni en 3.1.12) des fonctions méromorphes f g-contrôlées dont les pôles sont simples et donnés par les faces du cône 0e F . De plus, pour tout  2 0e F ,

( (X ))

1

D

0

 (X )  (X ) Z (s + ) lim =  (X ): s!0 X 0 (s) e

H ( (X );  (X ))

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

25

En spécialisant la fonction zêta des hauteurs à la droite C qui correspond au bré en droite anticanonique, on obtient le corollaire :

.  Si > 1, il existe " > 0, une fonction f holomorphe pour

Corollaire 4.4.7

Re(s)  1 " telle que (i) f (1) =  (X ) ; (ii) Pour tout  2 [1 "; 1 + "] et tout  2 R, jf (
+ i )j  (1 + j j) ; r (iii) Pour tout  > 1 et tout  2 R, Z (s! ) = s s 1 f (s). .  Si r désigne le rang de Pic(XF ), il existe un polynôme unitaire P de degré r 1 et un réel " > 0 tels que pour tout H > 0,  (X ) 1 " N (!X1 ; H ) = (r 1)! HP (log H ) + O(H ): Corollaire 4.4.8

Lorsque F = Q et lorsque la variété torique X est projective et telle que !X1 est engendré par ses sections globales, ce corollaire avait été démontré précédemment par R. de la Bretèche. Sa méthode est diérente ; elle est fondée sur le travail de P. Salberger [18] et une étude ne des sommes de fonctions arithmétiques en plusieurs variables (voir [7, 6] et [8] pour un cas particulier).

5. Application aux brations en variétés toriques 5.1. Holomorphie Soit B un S -schéma projectif et plat. Soit T ! B un G-torseur, et notons  :  X (G) ! Pic(B ) l'homomorphisme de fonctorialité des torseurs. Fixons un relèvec B ) de cet homomorphisme (c'est-à-dire, un choix de métriques ment b : X  (G) ! Pic( hermitiennes à l'inni sur les images d'une base de X  (G), prolongés par multiplicativité à l'image de  ). Donnons nous une S -variété torique lisse X , compactication équivariante de G. Soit Y le S -schéma obtenu par les constructions du Ÿ 2.1. On obtient alors un diagramme canonique, qui provient des propositions 2.1.11, 2.3.6, du théorème 2.2.4 et de l'oubli des métriques hermitiennes : (5.1.1)

/

X  (G) /

X  (G)

0

0 Le schéma Y

/

PicG(XF ) O Pic(BF ) / c G;K

c B) Pic (X )  Pic(

/

Pic(OYF ) Pic(Y )

/

0

/ c

contient T comme ouvert dense. On s'intéresse à la fonction zêta b l'image de  par l'homomordes hauteurs de T . Lorsque  2 PicG (XF )C, notons  c B ), on notera enn phisme (4.1.3). Si de plus  b 2 Pic( X b  b)    Z (; b) = Z (#( b; Y ) = H (#(b)   b; y ) 1: y2T (F )

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

26

b .  Soient 

c B )R une partie convexe telle que Z (  Pic( b; B ) c converge normalement si la partie réelle de  b 2 Pic(B )C appartient à . b  Alors, la fonction zêta des hauteurs de T converge absolument pour tout (; b) tel 0 que la partie réelle de  !X appartient à e (XF ) et la partie réelle de  appartient à . La convergence est de plus uniforme si la partie la partie réelle de  !X décrit un compact de 0e (XF ). Démonstration.  On peut décomposer la fonction zêta des hauteurs de T en écrivant X b  (5.1.3) Z (; b) = H (b; b) 1 Z (#(b); T jb ): Proposition 5.1.2

b2B (F )

D'après la remarque 2.4.6, le bré inversible  admet une section G-invariante s qui n'a ni pôles ni zéros sur l'ouvert G  X . En utilisant cette section, on obtient, en vertu du théorème 2.4.8 et de la proposition 2.4.3 une égalité (5.1.4)

Z (#(b); T jb ) =

X

x2G(F )

b s; gb
x) 1 ; H (;

où gb 2 G(AF ) représente la classe du G-torseur arithmétique Tcjb . On rappelle que si x 2 G(AF ), on a une expression de la hauteur en produit de hauteurs locales b s; x) = H (;

Y v

kskv (xv ) 1:

On peut appliquer la formule sommatoire de Poisson sur le tore adélique G(AF ), d'où, en utilisant l'invariance des hauteurs locales par les sous-groupes compacts maximaux, Z

); T jb) =  1 (gb)H ( b; ) d AG où l'intégration est sur le groupe AG des caractères (unitaires continus) du groupe localement compact G(F )nG(AF )=KG , muni de son unique mesure de Haar d qui (5.1.5)

Z (#(

b

permet cette formule. L'utilisation de la formule de Poisson est justiée par le fait que les deux membres convergent absolument. La série du membre de gauche est traitée dans [4], Theorem 4.2, lorsque gb = 1, c'est-à-dire lorsqu'il n'y a pas de torsion. Comme il existe une constante C (; gb ) ne dépendant que de gb et b  telle que b H ; s

1

b s ; gb H ;

1

( ; gb
x)  C ( ) ( ; x) b s; x) = H ( b; x), la convergence absolue du membre de gauche en résulte. et comme H (;

(Voir aussi le lemme 4.4.1.) Quant à l'intégrale du membre de droite, on peut négliger le caractère  dont la valeur absolue est 1 et on retrouve une intégrale dont la convergence absolue est prouvée dans [4] (preuve du théorème 4.4). Cela prouve aussi que lorsque Re() décrit un compact de !X1 +0e (XF ), la fonction zêta des hauteurs Z (#(b); T jb) de la bre en b 2 B (F ) est bornée indépendamment de b. En reportant cette majoration dans la décomposition (5.1.3), il en résulte la convergence absolue de la fonction zêta des hauteurs de T lorsque la partie réelle de  b

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

27

b et  !X appartient à 0 (XF ), uniformément lorsque  !X décrit appartient à  e un compact de ce cône.

Dans [9], dénition 1.4.1, on a déni la notion de fonction L d'Arakelov attachée à un torseur arithmétique et à une fonction sur un espace adélique. Appliquée au G  Gm torseur arithmétique sur B déni par TcB  b et à la fonction  1
k
k, la dénition devient X L(Tc b;  1  k
k) =  1 (gb )H (b; b) 1 : b2B (F ) (On a utilisé le fait que gb 2 G(F )nG(AF )=KG est la classe du G-torseur arithmétique T jb.) Un corollaire de la démonstration de la proposition précédente est alors le suivant : .  Sous les hypothèses de la proposition 5.1.2, on a la formule

Corollaire 5.1.6

Z(

)=

b  ; b

Z

H ( b; )L(Tc b; 

) d:

1  k
k

AG Démonstration.  Compte tenu de la majoration établie à la n de la preuve du théorème précédent et des rappels faits sur les fonctions L d'Arakelov, il sut de reporter l'équation (5.1.5) dans la formule (5.1.3) et d'échanger les signes somme et intégrale.

Cette dernière formule est le point de départ pour établir, moyennant des hypothèses supplémentaires sur B , un prolongement méromorphe de la fonction zêta des hauteurs de T .

5.2. Prolongement méromorphe

c B ) Z Q ! Pic(BF ) Q, Fixons une section de l'homomorphisme canonique Pic( autrement dit un choix de fonctions hauteurs compatible au produit tensoriel, ce que Peyre appelle système de hauteurs dans [17], 2.2. Concernant X , on utilise toujours les métriques adéliques canoniques utilisées au paragraphe 4. Ainsi, on écrira  et , c B ) est supposée être la les chapeaux devenant inutiles. L'application b : X  (G) ! Pic( composée de l'application  : X  (G) ! Pic(BF ) donnée par la restriction du torseur c B ) Q xée. à la bre générique, et de la section Pic(BF ) Q ! Pic( Ces restrictions ne sont pas vraiment essentielles mais simplient beaucoup les notations. Notons V1 = PicG (XF )R, M1 = X  (G)R , n1 = dim V1 et V2 = Pic(BF )R . Soient 1  V1 et 2  V2 les cônes ouverts, intérieurs des cônes eectifs dans PicG (XF )R et Pic(BF )R . L'espace vectoriel V1 possède une base naturelle, formée des brés en droites G-linéarisés associés aux diviseurs G-invariants sur X F . Dans cette base, le cône 1 est simplement l'ensemble des (s1 ; : : : ; sn1 ) strictement positifs. On note  : M1 ! V2 l'application linéaire déduite de b et M = (id;  )(M1 )  V1  V2 . Notons V = V1  V2 . Les théorèmes 2.2.4 et 2.2.9 identient Pic(YF )R à V=M , et l'intérieur du cône eectif de YF à l'image de 1  2 par la projection V ! V=M .

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

28

Si !X est muni de sa G-linéarisation canonique, la proposition 2.1.8 dit que !Y est l'image du couple (!X ; !B ) par cette même projection. .  Avec ces notations, la formule du corollaire du paragraphe précédent

Lemme 5.2.1

peut se récrire :

Z ( + !X

1; 

+ !B ) =

où la fonction

1

Z M1

f ( + im1 ;  i (m1 )) dm1 ;

f : T(1  2 ) ! C

est dénie par

f (; ) =

Z

UG

H ( ( + !X1 ); u )L(Tc ( + !B1 ); u 1  k
k) du

U = A = U On note que UG est un groupe discret et que la mesure du est donc proportionnelle

et dm1 , du sont des mesures de Haar sur M1 et G telles que d dm1 du dans la décomposition G M1  G du paragraphe 4.1.1 (cf. aussi le lemme 4.4.3).

à la mesure de comptage. Démonstration.  Si

les égalités

 2 AG s'écrit (m1 ; u ) dans M1  UG , on remarque que l'on a H ( ; ) = H (  (i m1 ); u)

et



( )H (b; b) 1 = u 1(gb )H (b

1 g b

 (m1 ); b)

1

car (lemme 4.1.4)

m1 (gb ) = exp(i k
k)([m1 ] Tcjb ) = exp(i kb(m1 )jbk) = H ( b(m1 ); b): On utilise ensuite le théorème de Fubini. On utilise enn les notations du Ÿ 3. .  On fait les hypothèses suivantes :

Hypothèses 5.2.2



( )

 le cône 2 est un cône polyédral (de type ni). Notons `j les formes linéaires dénissant ses faces ;  la fonction zêta des hauteurs de converge localement normalement pour  !B 2 2 ;  il existe un voisinage convexe B2 de l'origine dans V2 et pour tout caractère  2  !B 2 2 , G une fonction holomorphe g 
sur le tube T B2 tels que, si

+

B



( ;)

A

L(Tc ; 

)=

1  k
k

( )

Y j

Re( + ) 

`j () g (;  + !B ); `j ( + !B )

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS  il existe un réel strictement positif tel que pour tout vérient une majoration uniforme

29

" > 0, les fonctions g (;
)







jg(;  + !B )j  C" 1 + kIm()k 1 + kk ";

1

pour un réel " < et une constante C" ;  si  désigne le nombre de Tamagawa de Z s ! 1

(B)

B, pour tout  appartenant à 2, (B; + B ) =  (B) 6= 0: lim s!0+ X2 (s) Remarque 5.2.3.  Dans le cas où B est une variété de drapeaux généralisée, ces hypothèses correspondent à des énoncés sur les séries d'Eisenstein tordues par des caractères de Hecke. Ils sont établis dans [19]. Dans la suite, on travaille avec les classes de contrôle graphe 4.3.

D

introduites au para-

.  Sous les hypothèses précédentes, pour tout réel > 1, la fonction f

Lemme 5.2.4

appartient à

HM (1  2), pour la classe D + .

Démonstration.  Il sut de reprendre la démonstration de la proposition 4.4.5, d'y

insérer les majorations que nous avons supposées et de majorer

(1 + kIm k) (1 + kIm k)  (1 + kIm k + kIm k) + :

Grâce au théorème d'analyse 3.1.14, on en déduit un prolongement méromorphe pour la fonction zêta des hauteurs de T .

T admet un prolonge( (Y )) Pic(Y )C. Cette fonction a  (Y ). De plus, si  2 0e (Y ), (T ; + Y ) =  (Y ); lim + s!0 Xe (Y ) (s) le nombre de Tamagawa de Y . .  La fonction zêta des hauteurs décalée de

Théorème 5.2.5

ment méromorphe dans un voisinage de T 0e dans des pôles simples donnés par les équations des faces de 0e Z s ! 1

Démonstration.  Le seul point qui n'a pas été rappelé est que le nombre de Tamagawa

est

Y

est le produit de ceux de

X

et

B ([9], théorème 2.5.5).

.  Il existe " > 0 et un polynôme P tels que le nombre de points

Corollaire 5.2.6

T( )

de F dont la hauteur anticanonique est inférieure ou égale à pement asymptotique N H HP H O H1 "

+

( )=

(log ) + (

)

lorsque H tend vers 1. Le degré de P est égal au rang de coecient dominant vaut Xe (Y ) !Y 1  :

( ) (Y )

H vérie un dévelop-

Pic(YF ) moins 1 et son

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

30

Appendice A Un théorème taubérien Le but de ce paragraphe est de démontrer un théorème taubérien dont la preuve nous a été communiquée par P. Etingof. Ce théorème est certainement bien connu des experts mais que nous n'avons pu le trouver sous cette forme dans la littérature. .  Soient (n )n2N une suite croissante de réels strictement positifs, (cn )n2N

Théorème A.1

une suite de réels positifs et

f

la série de Dirichlet

f (s) =

1 X

n=0

cn

1:

sn

On fait les hypothèses suivantes :  la série dénissant f converge dans un demi-plan Re(s) > a > 0 ;  elle admet un prolongement méromorphe dans un demi-plan Re(s) > a Æ0 > 0 ;  dans ce domaine, elle possède un unique pôle en s = a, de multiplicité b 2 . On note  = lims!a f (s)(s a)b > 0 ;  enn, il existe un réel  > 0 de sorte que l'on ait pour Re(s) > a Æ0 l'estimation, b
( s a ) = O (1 + Im(s)) : f (s) b s

N

Alors il existe un polynôme unitaire X tend vers +1,

N (X ) def =

X

n X

P

de degré

cn =

a (b



b

1 tel que pour tout Æ < Æ0 , on ait, lorsque

a a Æ 1)! X P (log X ) + O(X ):

On introduit pour tout entier k  0 la fonction

'k (X ) =

de sorte que '0

= N.

X

n X

an (log(X=n ))k ;

.  Sous les hypothèses du théorème A.1, il existe pour tout entier k >  un polynôme Q de degré b 1 et de coecient dominant k !=(ak+1 (b 1)!) tel que pour tout Æ < Æ0 , on ait l'estimation, lorsque X tend vers +1, Lemme A.2

Démonstration.  Soit a0

'k (X ) = X a Q(log X ) + O(X a

Æ ):

> a arbitraire. On remarque, en vertu de l'intégrale classique 2i log+()
k ;  > 0 ds s k+1 = k! a0 +iR s

Z

que l'on a la formule

k! 2i

Z

ds f (s)X s k+1 ; s a0 +iR l'intégrale étant absolument convergente puisque  < k . '(X ) =

On veut décaler le coutour d'intégration vers la droite verticale Re(s) = a Æ, où Æ est un réel arbitraire tel que 0 < Æ < Æ0 . Dans le rectangle a Æ  Re(s)  a0 , jIm(s)j  T , il y a un

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

31

unique pôle en s = a. Le résidu y vaut

s Ress=a f (s) sXk+1 = ak+1 (b 1)! X a Q(log X ) où Q est un polynôme unitaire de degré b 1. Il en résulte que 1 Z a0 +iT f (s)X s ds 2i a0 iT sk+1 Z a Æ+iT 1 ds  f (s)X s k+1 + I+ I + k+1 X a Q(log X ); = 2i s a ( b 1)! a Æ iT

où I+ et I sont les intégrales sur les segments horizontaux (orientés de la gauche vers la 0 droite). Lorsque T tend vers +1, ces intégrales sont O (T  k 1 X a ) et tendent donc vers 0. Les hypothèses sur f et le fait que k >  montrent que f (s)X s =sk+1 est absolument intégrable sur la droite Re(s) = a Æ, l'intégrale étant majorée par O (X a Æ ). Par conséquent, on a

k! '(X ) =  k+1 a (b

Le lemme est ainsi démontré.

a a Æ 1)! X Q(log X ) + O(X ):

Preuve du théorème.  On va démontrer par récurrence descendante que la conclusion du

lemme précédent vaut en fait pour tout entier k  0. Arrivés à k prouvé. Montrons donc comment passer de k  1 à k 1. Pour tout  2 ]0; 1[, on a facilement l'inégalité

= 0,

le théorème sera

'k (X (1 )) 'k (X ) ' (X (1 + )) 'k (X )  k'k 1 (X )  k : log(1 ) log(1 + ) Fixons un réel Æ0 tel que 0 < Æ 0 < Æ < Æ0 . D'après le lemme précédent, il existe un réel C tel que

'k



0 (X ) ak+1k(! X a Q(log X )  CX a Æ : b 1)! On constate que l'on a alors, si 1 < u < 1, 'k (X (1 + u)) 'k (X ) log(1 + u) Q(log X + log(1 + u))(1 + u)a Q(log X ) = ak+1k(! Xa + R(X ); b 1)! log(1 + u)

où

jR(X )j  2CX a Æ0 = jlog(1 + u)j = O(X a Æ0 =u) si u tend vers 0 et X ! +1. Toujours lorsque X ! +1 et u ! 0, on a Q(log X + log(1 + u))(1 + u)a Q(log X ) log(1 + u) b 1 + u)a 1 + X 1 Q(n) (log X ) log(1 + u)n 1(1 + u)a = Q(log X ) (1log(1 + u) n=1 n! = Q(log X ) (a + O(u)) + Q0(log X ) (1 + O(u)) + O((log X )b 1 u) = (aQ + Q0)(log X ) + O((log X )b 1 u):

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

32

= 1=X " où " > 0 est choisi de sorte que Æ0 + " < Æ. Alors, lorsque X ! +1, jR(X )j = O(X a Æ ) et Q(log X + log(1 + u))(1 + u)a Q(log X ) = (aQ + Q0 )(log X ) + O(X Æ ): log(1 + u) Prenons u

On a alors un développement

1

'k 1 (X ) = X a (aQ + Q0 )(log X ) + O(X a k Le coecient dominant de (aQ + Q0 )=k est égal à (k 1)!=(ak (b

Æ)

récurrence descendante.

1)!) d'où le théorème par

Appendice B Démonstration de quelques inégalités Le but de cet appendice est de démontrer les inégalités sous-jacentes à la proposition 4.3.2 qui armait l'existence d'une classe de contrôle. Rappelons les notations. Soit un réel strictement positif. Si M et V sont deux -espaces vectoriels de dimension nie avec M  V , notons D ;" (M; V ) le sous-monoïde de F (V; + ) engendré par les fonctions h : V ! + telles qu'il existe c > 0 et une famille (`j ) de formes linéaires sur V vériant :  la famille (`j jM ) forme une base de M  ;  pour tout v 2 V et tout m 2 M , on a

R

R

R

1 (1 + kvk) Q (1 + kmk)1 " (1 + j`j (v + m)j) : T On dénit ensuite D (M; V ) = ">0 D ;" (M; V ). Théorème B.2.  Les D (M; V ) dénissent une classe de contrôle h(v + m)  c

(B.1)

tion 3.1.1.

au sens de la déni-

Démonstration.  Les points (3.1.1,a) et (3.1.1,c) sont clairs. L'axiome (3.1.1,e) est vrai car la famille (`j Æ pjM ) contient une base de (M=M1 ) . L'axiome (3.1.1,b) résulte de l'inégalité

1 (1 + j`(v + m)j min (1 + j `(v + tu + m)j)  1 + j `(u)j jtj1

valable pour tous v 2 V , u 2 V et m l'objet de la proposition suivante. .  Soient M

Proposition B.3

2 M . Enn, l'axiome (3.1.1,d), le plus délicat, fait

 V , V 0 un supplémentaire de M

dans

V , dm une mesure

de Lebesgue sur M , (`j ) une base de V  . Pour tout "0 > ", il existe une constante c"0 et un ensemble ((`j; )j ) de bases de (V 0 ) tels que pour tous v1 et v2 2 M 0 , Z

1 1 M (1 + kv1 + mk)

1 : 1 j (1 + j`j; (v2 )j)  Démonstration.  On raisonne par récurrence sur dim M . Soient u 2 M , M 0  M tels que M = M 0  Ru et xons une mesure de Lebesgue dm0 sur M 0 telle que dm0
dt = dm. Alors, dm

" Q(1 + j`j (v2 + m)j)

 (1 + kcv"0 k)1

"0

X

Q

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS

Z

Ru

Z

::: 

1

33

dt

Q dt R (1 + kv1 + m0 k + jtj)1 " j (1 + j`j (v2 + m0 )t`j (u)j)

1  1 + j ` ( v j 2 + m0 )j j ; `j (u)=0 Z  (1 + kv + m1 0k + jtj)1 1 R Y



1 1 + j`j (v2 + m0) + tj dt j ; `j (u)6=0 Y

"

et, en appliquant le lemme B.4 ci-dessous, 1 + kv1 + m0k) X Y  1 +(1log(1 0 1 " 0 + kv1 + m k)  j 1 + j`j; (v2 + m )j

"0 (1 + kv +1 m0k)1 1

"0

X



1 : (1 + j `j;(v2 )j) j

Q

.  On a une majoration, valable pour tous réels t1

Lemme B.4

Z

1

1 1 (1 + A + jtj)1

n Y "

j =1

1

1 + jt tj j

dt 




 tn et tout A  0,

1 + log(1 + A) X nY1 1 (1 + A)1 "  j=1 1 + j;j j

où pour tout  et tout j , ;j = ta(;j ) tb(;j ) de sorte que pour tout , notant base canonique de n , les familles (e;j = ea(;j ) eb(;j ) )j sont libres.

R

Démonstration.  On découpe l'intégrale en

Pour l'intégrale de

Z t1

1

1 à t1, on a n Y

R t1

R t2 R1 1, t1 , . . ., tn et on majore chaque terme.

Z t1

1 dt 1 " 1 + j t t j (1 + A + j t j) 1 + t1 t j 1 1 j =2 Z 1 n Y 1 dt  1 + jt1 t j 1 " j 1 0 (1 + A + jt t1 j) 1 + t j =2 n Y + A)  1 + jt1 t j 1 +(1log(1 1 + A) " j 1 j =2

::: 

1

d'après le lemme B.5. La dernière intégrale (de tn à Z tk+1

tk






Y

jk+1 1 + jtk+1 tj j  

Z tk+1

tk

1 dt (1 + A + jtj)1 " (1 + t tk )(1 + tk+1 t)

ANTOINE CHAMBERT-LOIR & YURI TSCHINKEL

34

et cette dernière intégrale s'estime comme suit :

1 dt 1 " (1 + A + jtj) (1 + t tk )(1 + tk+1 t) =   Z tk+1 1 1 1 1 = (1 + A + jtj)1 " 2 + tk+1 tk 1 + t tk + 1 + tk+1 t dt tk Z 1 1 dt 1  2+t 1 " 1 + t tk k+1 tk tk (1 + A + jtj) Z tk+1  1 dt + 1 Z(1 + A + jtj)1 " 1 + tk+1 t  1 1 dt  2+t 1 t 1 " (1 + A + j t + t j) 1 +t k+1 k 0 k Z 1  + (1 + A + jt1 t j)1 " 1 dt+ t k+1 0 + A)  1 + t 1 t 1 +(1log(1 1 " + A ) k+1 k en vertu du lemme B.5. .  On a une majoration, valable pour tout A  1 et tout a > 0,

Lemme B.5

Z

1

1

dt

(A + jt + aj) 1 + t

A  1 +Alog  :

0 Il reste à démontrer ce lemme. Pour cela, on a besoin de deux lemmes supplémentaires !

 1 et tous ; > 0 tels que  + > 1, 8 > >1 + log(B=A) si  = 1 et B > A ; Z 1 min( A; B ) < dt ; A B  >1 + log(A=B ) si = 1 et A > B ; 0 (A + t) (B + t) > :1 sinon. .  Pour tous A et B

Lemme B.6

A < B , l'autre étant symétrique et le cas A = B élémentaire. Faisons le changement de variables A + T = (B A)eu , d'où B + T = (B A)(1 + eu ). Pour t = 0, u = log A=(B A). Lorsque t ! +1, u ! +1. Ainsi, l'intégrale Démonstration.  On ne traite que le cas

vaut

I (A; B ; ; ) = Si A < B

Z

1

1 (B A)+

e(1 )u 1 log A=(B A) (1 + eu ) du:

A+ 1

A B

 2A, on majore l'intégrale par Z 1 1 (1  )u du I (A; B ; ; )  (B A)+ 1 log A=(B A) e
 (B A1)+ 1 1 1 B A A + 1  1  A

puisque 1=A  2=B .

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS Lorsque B  2A, log A=(B A) lorsque u  0, d'où les inégalités

 0. On minore 1 + eu

(B A)+ 1 I (A; B ; ; ) = 

Z 0

+

Z

log A=(B A) 1 e(1 )u

Z

0

par

1 lorsque u  0 et par eu

1 Z 0

(1 + eu) du + log A=(B 8 0, l'intégrale se majore par Z 1 1 dt 1 + log A 1+t  ( A + t ) A 0

d'après le lemme B.6. Si a < 0, on découpe l'intégrale de de 0 à a vaut Z

Z

1 dy 1 + log A   A 0 0 (A + u) (1 a) u en vertu du lemme B.7, tandis que l'intégrale de a à +1 s'estime ainsi : Z 1 Z 1 1 dt 1 du 1 + log A 1+t = 1 a+u  ( A + t + a )) ( A + u ) A a 0 en appliquant de nouveau le lemme B.6 et en distinguant suivant que A  1 a ou A  1 a. a

1

0 à a et de a à +1. L'intégrale

dt

(A t a) 1 + t =

a

Références [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

& Yu. I. Manin   Sur le nombre de points rationnels de hauteur bornée des variétés algébriques , Math. Ann. 286 (1990), p. 2743. V. V. Batyrev & Yu. Tschinkel   Rational points on bounded height on compactications of anisotropic tori , Internat. Math. Res. Notices 12 (1995), p. 591635. ,  Height zeta functions of toric varieties , Journal Math. Sciences 82 (1996), no. 1, p. 32203239. ,  Manin's conjecture for toric varieties , J. Algebraic Geometry 7 (1998), no. 1, p. 1553. ,  Tamagawa numbers of polarized algebraic varieties , in Nombre et répartition des points de hauteur bornée [16], p. 299340. R. de la Bretèche   Compter des points d'une variété torique rationnelle , Prépublication 41, Université Paris Sud (Orsay), 1998. ,  Estimations de sommes multiples de fonctions arithmétiques , Prépublication 42, Université Paris Sud (Orsay), 1998. ,  Sur le nombre de points de hauteur bornée d'une certaine surface cubique singulière , in Nombre et répartition des points de hauteur bornée [16], p. 5177. A. Chambert-Loir & Yu. Tschinkel   Torseurs arithmétiques et espaces brés , E-print, math.NT/9901006, 1999. ,  Points of bounded height on equivariant compactications of vector groups, III , Work in preparation, 2000. J. Franke, Yu. I. Manin & Yu. Tschinkel   Rational points of bounded height on Fano varieties , Invent. Math. 95 (1989), no. 2, p. 421435. W. Fulton  Introduction to toric varieties, Annals of Math. Studies, no. 131, Princeton Univ. Press, 1993. R. Narasimhan  Several complex variables, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995, Reprint of the 1971 original. T. Oda  Convex bodies and algebraic geometry, Ergeb., no. 15, Springer Verlag, 1988. E. Peyre   Hauteurs et mesures de Tamagawa sur les variétés de Fano , Duke Math. J. 79 (1995), p. 101218. V. V. Batyrev

FONCTIONS ZÊTA DES HAUTEURS DES ESPACES FIBRÉS [16] [17] [18]

1998.

37

(éd.)  Nombre et répartition des points de hauteur bornée, Astérisque, no. 251, ,  Terme principal de la fonction zêta des hauteurs et torseurs universels , in

Nombre et répartition des points de hauteur bornée [16], p. 259298.

  Tamagawa measures on universal torsors and points of bounded height on Fano varieties , in Nombre et répartition des points de hauteur bornée [16], p. 91258. [19] M. Strauch & Yu. Tschinkel   Height zeta functions of toric bundles over ag varieties , Selecta Math. (N.S.) 5 (1999), no. 3, p. 325396. [20] A. Weil  Adeles and algebraic groups, Progr. Math., no. 23, Birkhäuser, 1982. P. Salberger

Version du 4 décembre 2000, 16:35

Antoine Chambert-Loir, Institut de mathématiques de Jussieu, Boite 247, 4, place Jussieu, F-

75252 Paris Cedex 05



E-mail :

[email protected]

Yuri Tschinkel, Department of Mathematics, U.I.C., Chicago (IL) 60607 E-mail :

[email protected]

Hasse principle for pencils of curves of genus one whose Jacobians have a rational 2-division point (Close variation on a paper of Bender and Swinnerton-Dyer) J.-L. Colliot-Th´el`ene Let k be a number field and X/k a smooth projective surface equipped with a morphism f : X → P1k the generic fibre of which is a curve of genus one. Assume that each geometric fibre of f contains a multiplicity one component. If the set X(k) of rational points of X is not empty, one may ask whether it is then Zariski-dense in X. If no k-point is Q known, one may ask whether the Brauer-Manin condition (existence of an adelic point Mv ∈ v X(kv ) such that P inv (A(M v v )) = 0 for each element A of the Brauer group of X) ensures the existence of v (many) rational points. In [SwD1] Swinnerton-Dyer initiated a new technique for studying these problems. The technique was systematized in the joint work [CSS3]. Swinnerton-Dyer has since then written a number of papers on the topic: [SwD2], [SwD3], [BSwD] (joint with Bender), [SwD4]. Here are some common features of these papers. (i) They postulate the finiteness of Tate-Shafarevich groups of elliptic curves. (ii) With the notable exception of the very recent paper [SwD4], they build upon Schinzel’s Hypothesis (a very likely, but very unreachable conjecture). (iii) Some assumption is made on the torsion subgroup of the jacobian J/K of the generic fibre of f (here K denotes the function field k(P1 )). In the papers [SwD1], [CSS3], [SwD2], [SwD3], one assumes that the entire 2-torsion subgroup 2 J of J is rational, i.e. isomorphic to the constant K-group scheme (Z/2)2K . In [BSwD], the assumption is that 2 J contains a copy of the constant group (Z/2)K . (In the paper [SwD4] the 3-torsion satisfies a similar property.) The paper [BSwD] is written in the style of [SwD2]. The contribution of the present paper is merely to rewrite most of Sections 1 to 5 of [BSwD] in the style of [CSS3], i.e. in terms of the Brauer-Manin obstruction (I have not however produced the analogue of Section 4 of [CSS3]). Apart from gaining more practice in the new technique, my immediate motivation for writing the present paper was the hope to produce a more satisfactory statement than Theorem 3 in Section 6 of [BSwD]. Namely, under the Schinzel hypothesis and the finiteness of TateShafarevich group, one would hope to prove that the Brauer-Manin obstruction to the Hasse principle is the only obstruction for smooth complete intersections of two quadrics in P4k . It has been known for some time (Sansuc and the author, 1986; Harari [Ha1], Prop. 5.2.3) that this last statement implies the Hasse principle for smooth complete intersections of two quadrics in Pnk for n ≥ 5 (for n ≥ 8, the Hasse principle for such intersections of two quadrics is proved in [CTSaSwD]). Section 6 of [BSwD] contains an interesting geometric construction which one could have tried to combine with the techniques in Harari’s papers [Ha1] [Ha2]. But for the time being I still feel that serious difficulties are in the way. Theorem A below is a reformulation of Theorem 2 of [BSwD]. Theorem B below is a reformulation of Theorem 1 of [BSwD]. Conditions (3.a) and (3.b) in the statements of Theorems A and B are purely algebraic renditions of Conditions 7 and 8 of [BSwD]. The slightly technical Conditions 5 and 6 of [BSwD] (which involved the quadratic reciprocity law, hence had to do with the Brauer-Manin obstruction, but in a rather involved manner) are replaced by the sheer hypothesis that there is no vertical Brauer-Manin obstruction. Out of habit, I have used the one variable, inhomogeneous version of Schinzel’s hypothesis. In his papers, Swinnerton-Dyer uses the two variables, homogeneous version of that hypothesis. The latter hypothesis has now been proved in nontrivial cases (Heath-Brown for x3 + 2y 3 over the integers, Heath-Brown and Moroz for any binary cubic form ax3 + by 3 over the integers), hence the homogeneous version should be preferred in the long run. As Swinerton-Dyer points out, the homogeneous version

also has the advantage that the point at infinity on the projective line does not play a special rˆ ole. The reader of the present paper is supposed to be acquainted with Sections 1 and 2 of the paper [CSS3]. Plan of the paper 0. Statement of the Theorems 1. Selmer groups associated to a degree 2 isogeny 1.1. A statement from class field theory, and some linear algebra 1.2. Isogenies of degree 2 over an arbitrary field 1.3. Local computations 1.4. The Selmer groups as kernels of pairings 1.5. Small Selmer groups 2. Proof of theorem A 2.1. Fibres with points everywhere locally 2.2. The groups S0 , S00 and the maps δ 0 and δ 00 2.3. Independence of the choice of λ: the spaces 2.4. Independence of the choice of λ: the pairing 2.5. Completion of the proof of Theorem A 3. Proof of Theorem B 0. Statement of the Theorems Let k be a field, char(k)=0. Let c(t), d(t), m(t) ∈ k[t] be polynomials. We assume: (0.1) The polynomials c(t), d(t), m(t) are all of even degree, deg(d) = 2deg(c) > 0, c2 − d has even positive degree and d(c2 − d) has no square factor. Let E be the smooth projective surface over k which is the projective regular minimal model over P1k of the surface given in affine 3-space by the affine equation y 2 = (x − c(t))(x2 − d(t)), the fibration over the affine line A1k being given by (x, y, t) 7→ t. Given m as above, let π : X(m) → P1k denote the projective regular minimal model over P1k of the surface given in affine 4-space by the system of equations w12 = m(t)(x − c(t)), w22 = m(t)(x2 − d(t)), the fibration over the affine line A1k being given by (x, w1 , w2 , t) 7→ t. The restrictions on the degrees of d, c and m made above ensure that the fibres at t = ∞ of X(m)/P1k and of E/P1k have a geometrically integral component of multiplicity one; if moreover the degree of c2 − d is equal to the degree of d, then these fibres are smooth. Let U ⊂ A1k be the open set defined by d(c2 − d) 6= 0. Let U 0 ⊂ A1k be the open set defined by d 6= 0. Let S0 = k[U 0 ]∗ /k[U 0 ]∗2 . Let S00 ⊂ S0 be the subgroup of classes of functions whose divisor at infinity is even: these are exactly the classes (modulo squares) of polynomials m0 (t) ∈ k[t] which are of even degree and divide d. Let M0 be the set of closed points M of A1k such that d(M) = 0. Let U 00 ⊂ A1k be the open set defined by c2 − d 6= 0. Let S00 = k[U 00 ]∗ /k[U 00 ]∗2 . Let 00 S0 ⊂ S00 be the subgroup of classes of functions whose divisor at infinity is even: these are exactly the classes (modulo squares) of polynomials m00 (t) ∈ k[t] which are of even degree and divide c2 − d. Let M00 be the set of closed points M of A1k such that (c2 − d)(M) = 0.

For each M ∈ M = M0 ∪ M00 , let rM (t) be the monic irreducible polynomial defining M. Let ∗ ∗2 δ 0 : S0 → ⊕M ∈M00 kM /kM and ∗ ∗2 δ 00 : S00 → ⊕M ∈M0 kM /kM

be the map given by evaluation at the relevant points M. Since d(c2 − d) is squarefree, d and c2 − d are coprime hence these evaluation maps are well-defined. These maps induce maps ∗ ∗2 δ00 : S00 → ⊕M ∈M00 kM /kM

and ∗ ∗2 δ000 : S000 → ⊕M ∈M0 kM /kM ,

which will sometimes be denoted simply δ 0 and δ 00 . Clearly d is in the kernel of the first map, and c2 − d is in the kernel of the second map. We refer the reader to [CTSwD], Section 3, and [CT] for a detailed discussion of the BrauerManin obstruction. Given a morphism π : X → P1k with X/k smooth and projective, the vertical subgroup Brvert(X) of the Brauer group of X with respect to π is the subgroup of Br(X) consisting of classes whose restriction to the generic fibre of π come from Br(k(P1 )). When m ∈ S000 , each geometric fibre fibre of π : X(m) → P1 contains a component of multiplicity one. This implies that the quotient Brvert (X(m))/Br(k) is finite (the proof of this fact given in [Sk], ˆ is torsion free; Cor. 4.5 holds under the mere assumption that the Galois module denoted G such is the case when the above multiplicity one condition is fulfilled, as proved in [Sk] 3.2.4). Let Ω be the set of places of the numberQ field k. For v ∈ Ω, let kv be the completion of k at v. For any proper k-variety X, let X(Ak ) = v∈Ω X(kv ) be the space of ad`eles of X, equipped with Pthe product topology. For any element A ∈ Br(X), the map which sends the ad`ele {Pv } to v∈Ω invv (A(Pv )) ∈ Q/Z is a continuous function θA : X(Ak ) → Q/Z with finite image. Given B ⊂ Br(X), we let X(Ak )B ⊂ X(Ak ) denote the closed subset which is the intersection −1 (0) for A ∈ B. The diagonal inclusion X(k) ⊂ X(Ak )B of the set of k-rational of the kernels θA points of X in the set of ad`eles of X shows that X(Ak )B 6= ∅ is a necessary condition for the existence of a k-rational point on X. For π : X → P1k as above, the group Brvert(X)/Br(k) is finite, hence X(Ak )Brvert (X) is open in X(Ak ). For any commutative ring A, let H(A) = A∗ /A∗2 . Theorem A Let k be a number field. Let c, d, m ∈ k[t] satisfy (0.1) and let X = X(m). (1) Assume that the polynomial m divides c2 − d. The class of m ∈ H(k(t)) = k(t)∗ /k(t)∗2 then defines a class in S000 . (2) Assume that the class of m in S000 ⊂ H(k(t)) differs from that of 1 and of c2 − d. (3.a) Assume that the kernel of δ00 consists of the classes 1, d. (3.b) Assume that the kernel of the composite map δ000 ∗ ∗2 ∗ ∗2 00 S000 −→ ⊕M ∈M0 kM /kM → ⊕M ∈M0 kM /(kM , δM (m)), is spanned by c2 − d and m. (4) Assume that the Tate-Shafarevich group of each elliptic curve occurring as a fibre of 1 E/Pk is finite. (5) Assume Schinzel’s hypothesis.

Let R ⊂ P1 (k) be the set of points λ ∈ P1 (k) such that the fibre Xλ = π −1 (λ) is smooth and has infinitely many k-points. Let RE ⊂ P1 (k) be the set of points λ ∈ P1 (k) such that the fibre Eλ of E/P1k is an elliptic curve over k of rank (= dimQ (Eλ (k) ⊗ Q)) at least equal to one. We have R ⊂ RE . Then: (a) The closure of R in P1 (Ak ) coincides with π(X(Ak )Brvert (X)). (b) Assume X(Ak )Brvert (X) 6= ∅, i.e. assume that there is no vertical Brauer-Manin obstruction to the existence of a rational point (this is certainly the case if X(k) 6= ∅). Then for Q 1 any non-empty finite set S ⊂ Ω, the closure of R in v∈S P (kv ) contains a non-empty open set. The same therefore holds for RE . In particular R and RE are Zariski-dense in P1k , the set X(k) is Zariski-dense in X and the set E(k) is Zariski-dense in the surface E. (c) Assume X(Ak )Brvert (X) 6= ∅. Then there exists a finite set S0 ⊂ Ω such that Q for any finite set S ⊂ Ω with Q S ∩S0 = ∅, the closure of R under the diagonal embedding P1 (k) → v∈S P1 (kv ) coincides with v∈S π(X(kv )). Let us comment on the various hypotheses in the Theorem. Hypothesis (1) ensures in particular that all geometric fibres of π : X → P1k contain a component of multiplicity one. Some condition of that kind is required, if we hope to conclude that infinitely many k-fibres of π have rational points (see Cor. 2.2, Cor. 2.4, Prop. 4.1, Prop. 4.2 of [CSS1]). If Hypothesis (2) is not satisfied, then the generic fibre of X/P1k has a k(P1 )-rational point; it can be regarded as an elliptic curve isogenous to the generic fibre of E/P1k . In particular X has a k-rational point and the question of (Zariski) density of k-rational points on the surface X is equivalent to that question for the surface E. By analogy with the work done in Section 4 of [CSS3], Hypotheses (3.a) and (3.b) should be slightly stronger than the hypothesis that the 2-primary subgroup of the Brauer group of X is contained in the vertical part of the Brauer group. But we have not done the corresponding (algebraic) work here. As mentioned in [CSS3], this would fit in well with the obvious remark that the necessary condition X(Ak )Brvert (X) 6= ∅ for the existence of a rational point (assumption made in (b) and (c)) is a priori weaker than the equally necessary condition X(Ak )Br(X) 6= ∅ (a condition which does not appear in the Theorem). Let us now prepare for the statement of Theorem B. Let k be a field, char(k)=0. Let αi , βi , i = 0, · · · , 4 be elements of k[t]. Consider the variety given in A1k ×k P3k by the system of equations

(0.2)

α0 U02 + α1 U12 + α2 U22 + α3 U32 + 2α4 U2 U3 = 0, β0 U02 + β1 U12 + β2 U22 + β3 U32 + 2β4 U2 U3 = 0.

This defines a one-parameter family of intersections of two quadrics in P3k . Let dij = αi βj − αj βi . These dij satisfy a number of useful identities. For any subset {i, j, k, l} of {0, 1, 2, 3, 4}, we have the identity (0.3)

dij dk` + dik d`j + di` djk = 0.

We have the identities ([BSwD], (23))

(0.4)

4d224 (d204 − d02 d03 ) = (2d04 d24 − d02 d03 )2 − d202 (d223 + 4d24 d34 )

4d224 (d214 − d12 d13 ) = (2d14 d24 − d12 d23 )2 − d212 (d223 + 4d24 d34 ).

Let ∆ = d01 (d223 + 4d24 d34 )(d204 − d02 d03 )(d214 − d12 d13 ).

At any point t ∈ A1 where ∆ does not vanish, the fibre is a smooth, pure intersection of two quadrics in P3k . In particular, if ∆ does not identically vanish, the generic fibre is a smooth, pure intersection of two quadrics in P3k . We assume: (0.5) All the αi (t) are of the same degree r, all the βi (t) are of the same degree s, and ∆ is of degree r + s. This assumption, which one could slightly relax (see [BSwD]), ensures that the surface given by the system (*) admits a regular proper model over P1k whose fibre over ∞ ∈ P1k is smooth. We also assume: (0.6) ∆ has no square factor. (0.7) d204 − d02 d03 and d02 are coprime; d214 − d12 d13 and d12 are coprime. Let U ⊂ A1k be the open set defined by ∆ 6= 0. Let U 0 ⊂ A1k be the open set defined by ∆0 = d01 (d223 + 4d24 d34 ) 6= 0. Let S0 = k[U 0 ]∗ /k[U 0 ]∗2 . Let S00 ⊂ S0 be the subgroup of classes of functions whose divisor at infinity is even: these are exactly the classes (modulo squares) of polynomials r(t) ∈ k[t] which are of even degree and divide ∆0 . Let M0 be the set of closed points M of A1k such that ∆0 (M) = 0. This set breaks up into the set M01 consisting of closed points M such that d01 (M) = 0 and the set M02 consisting of closed points M such that (d223 + 4d24 d34 )(M) = 0. Let U 00 ⊂ A1k be the open set defined by ∆00 = (d204 − d02 d03 )(d214 − d12 d13 ) 6= 0. Let S00 = k[U 00 ]∗ /k[U 00 ]∗2 . Let S000 ⊂ S00 be the subgroup of classes of functions whose divisor at infinity is even: these are exactly the classes (modulo squares) of polynomials r(t) ∈ k[t] which are of even degree and divide ∆00 . Let M00 be the set of closed points M of A1k such that ∆00 (M) = 0. For each M ∈ M = M0 ∪ M00 , let rM (t) be the monic irreducible polynomial defining M. Let ∗ ∗2 δ 0 : S0 → ⊕M ∈M00 kM /kM and ∗ ∗2 δ 00 : S00 → ⊕M ∈M0 kM /kM

be the map given by evaluation at the relevant points M. Since ∆ = ∆0 .∆00 is squarefree, ∆0 and ∆00 are coprime hence these evaluation maps are well-defined. These maps induce maps ∗ ∗2 δ00 : S00 → ⊕M ∈M00 kM /kM

and ∗ ∗2 δ000 : S000 → ⊕M ∈M0 kM /kM

Using the formulas (0.4) and the various coprimality assumptions above, one checks that (the class of) d223 + 4d24 d34 ∈ S00 is in the kernel of δ00 and that (the classes of) d204 − d02 d03 and 00 d214 − d12 d13 in S000 are in the kernel of the maps δ0,M relative to the points M ∈ M02 . Let c = 4d04 d14 − 2d02 d13 − 2d03 d12 . Let d = 4d201 (d223 + 4d24 d34 ). We have

(0.8)

c2 − d = 16(d204 − d02 d03 )(d214 − d12 d13 ).

00 00 In particular, at any point M ∈ M01 , we have δM (d204 − d02 d03 ) = δM (d214 − d12 d13 ).

Theorem B Let k be a number field. Let αi , βi be as above, and let X be a smooth projective model of the surface defined by the system (0.2), which we may choose equipped with a k-morphism X → P1k extending the map to A1k given by the t-coordinate. Let E/P1k be the associated jacobian fibration. (1) Assume that all the αi (t) are of the same degree r, that all the βi (t) are of the same degree s, and that ∆ is of degree r + s, and assume that the product ∆ = d01 (d223 + 4d24 d34 )(d204 − d02 d03 )(d214 − d12 d13 ) has no square factor. (2) Assume that d204 −d02 d03 and d02 are coprime, and that d214 −d12d13 and d12 are coprime. (3.a) Assume that the kernel of the composite map δ00 ∗ ∗2 ∗ ∗2 S00 −→ ⊕M ∈M00 kM /kM → ⊕M ∈M00 kM /(kM , εM ), where εM is the class of (−d01 d02 )(M) if (d204 − d02 d03 )(M) = 0 and is the class of (−d01 d21 )(M) if (d214 − d12 d13 )(M) = 0, consists of the classes 1, d223 + 4d24 d34 . (3.b) Let c = 4d04 d14 − 2d02 d13 − 2d03 d12 . Assume that the kernel of the (composite) map δ000 ∗ ∗2 ∗ ∗2 00 00 S000 −→ (⊕M ∈M02 kM /kM ) ⊕ (⊕M ∈M01 kM /(kM , δM (d214 − d12 d13 ), δM (−c))) is spanned by the classes (d204 − d02 d03 ) and (d214 − d12 d13 ). (4) Assume that the Tate-Shafarevich group of each elliptic curve occurring as a fibre of 0 E /P1k is finite. (5) Assume Schinzel’s hypothesis. Then we have the same conclusions as in Theorem A, namely: Let R ⊂ P1 (k) be the set of points λ ∈ P1 (k) such that the fibre Xλ = π −1 (λ) is smooth and has infinitely many k-points. Let RE ⊂ P1 (k) be the set of points λ ∈ P1 (k) such that the fibre Eλ of E/P1k is an elliptic curve over k of rank (= dimQ (Eλ (k) ⊗ Q)) at least equal to one. We have R ⊂ RE . Then:

(a) The closure of R in P1 (Ak ) coincides with π(X(Ak )Brvert (X)). (b) Assume X(Ak )Brvert (X) 6= ∅, i.e. assume that there is no vertical Brauer-Manin obstruction to the existence of a rational point (this is certainly the case if X(k) 6= ∅). Then for Q any non-empty finite set S ⊂ Ω, the closure of R in v∈S P1 (kv ) contains a non-empty open set. The same therefore holds for RE . In particular R and RE are Zariski-dense in P1k , the set X(k) is Zariski-dense in X and the set E(k) is Zariski-dense in the surface E. (c) Assume X(Ak )Brvert (X) 6= ∅. Then there exists a finite set S0 ⊂ Ω such that Q for any finite 1 set S ⊂ Ω with Q S ∩S0 = ∅, the closure of R under the diagonal embedding P (k) → v∈S P1 (kv ) coincides with v∈S π(X(kv )).

Comment on (3.a). In their (more arithmetic) version (Condition 7 of [BSwD], as applied to ∗ ∗2 their Theorem 1), Bender and Swinnerton-Dyer only have ⊕M ∈M00 kM /kM as the target group (i.e. no εM ), but I think this is an oversight on their part. Comment on (3.b). To simplify matters, in their (more arithmetic) version (Condition 8 of [BSwD]), Bender and Swinnerton-Dyer replace this by the stronger condition where one only keeps the sum over points in M02 in the target group. But they write ∗ ∗2 00 ⊕M ∈M02 kM /(kM , δM (d214 − d12 d13 )) 00 whereas the formulas (0.4) above show that for such an M the class δM (d214 − d12 d13 ) is trivial.

1. Selmer groups associated to a degree 2 isogeny 1.1. A statement from global class field theory, and some linear algebra Let S be a finite subset of the set Ω of places of the number field k. Consider the vector space ⊕v∈S kv∗ /kv∗2 over the finite field F2 . We equip this space with the bilinear form ( , )S = ⊕v∈S ( , )v , where ( , )v is the bilinear pairing (the local symbol) H 1 (kv , Z/2) × H 1 (kv , Z/2) → H 2 (kv , Z/2) ,→ Z/2, where the right hand arrow is an isomorphism provided v is not a complex place. It is a standard fact of local class field theory that the pairing ( , )v , and hence the pairing ( , )S , is nondegenerate. This pairing is skewsymmetric: for any a, b ∈ kv∗ , we have (a, b)v + (b, a)v = 0. One also has (a, −a)v = 0 for any a ∈ kv∗ , but in general one need not have (a, a)v = 0, i.e. the pairing need not be alternating. Let oS be the ring of S-integers of k. Proposition 1.1.1 Let S be a finite set of places containing all the archimedean places and all the places over 2. Then the dimension of the F2 -vector space o∗S /o∗2 S is equal to half the ∗ ∗2 dimension of the F2 -vector space ⊕v∈S kv /kv . If moreover S contains a set of generators of the ∗ ∗2 ideal class group of k, then the natural map o∗S /o∗2 S → ⊕v∈S kv /kv is injective and its image is ∗ ∗2 a maximal isotropic subspace of ⊕v∈S kv /kv . This is [CSS3], Prop. 1.1.1. Let S0 denote a fixed family of places of k satisfying all the conditions of the proposition. Clearly, for any finite set S of places containing S0 , the statement of the proposition holds. For any place v of k, we let Vv = H 1 (kv , Z/2) = H(kv ). We equip this F2 -vector space with the nondegenerate skewsymmetric bilinear form (., .)v . For any finite set S of places containing S0 , and any family of objects {∗v }, v ∈ S, we shall consistently write ∗S = ⊕v∈S ∗v . The space VS comes equipped with the pairing eS : VS × VS → Z/2 sum of the pairings (., .)v . Clearly eS is skewsymmetric and nondegenerate (but unless −1 ∈ k ∗2 , it need not be alternating). Because S contains S0 , we have Pic(oS ) = 0. Hence the diagonal map M H 1 (oS , Z/2) → H 1 (kv , Z/2) v∈S

may be read o∗S /o∗2 S →

M v∈S

kv∗ /kv∗2

and it is injective by the previous proposition. By the global reciprocity law, the image I S of this map is isotropic with respect to the symplectic pairing eS defined above. By dimension count (using Proposition 1.1.1), we then see that it is maximal isotropic. For any S, the natural map H 1 (oS , Z/2) → H 1 (k, Z/2) is injective. For S containing S0 , we 1 ∗ ∗2 shall identify o∗S /o∗2 = H 1 (k, Z/2), S = H (oS , Z/2) with the corresponding subgroup of k /k and also with I S . The elements of I S may thus be identified with the elements of k ∗ /k ∗2 whose 1 image in kv∗ /kv∗2 belongs to o∗v /o∗2 / S. v = H (ov , Z/2) for each v ∈ We shall actually consider two copies of VS , call them VS0 and VS00 , and consider the bilinear pairing (., .)S : VS0 × VS00 → Z/2.

We have I 0S ⊂ VS0 and I 00S ⊂ VS00 . Note that I 0S is the exact orthogonal of I 00S under the pairing VS0 × VS00 → Z/2. The following Proposition is proved in [BSwD] (Lemma 8 and Corollary). It is linear algebra, but not entirely trivial. In the present context of degree 2 isogenies, it is the appropriate substitute for Prop. 1.1.2 of [CSS3]. Proposition 1.1.2 Let S0 be as above, and let S be a finite set of places containing S0 . There exist subspaces Kv0 ⊂ Vv0 , v ∈ S and Kv00 ⊂ Vv00 , v ∈ S such that: (a) for each place v ∈ S, Kv0 is the exact orthogonal of Kv00 under the pairing Vv0 ×Vv00 → Z/2; (b) VS0 = I 0S ⊕ KS0 , where KS0 = ⊕v∈S Kv0 ; and similarly VS00 = I 00S ⊕ KS00 , where KS00 = ⊕v∈S Kv00 ; (c) For v ∈ / S0 , one may choose Kv0 = H(ov ) and similarly Kv00 = H(ov ).

In the sequel, given S0, we shall take the Kv0 and Kv00 for v ∈ S0 as given by the Proposition, and we shall take Kv0 and Kv00 to be equal to H(ov ) for v ∈ / S0 . 1.2. Isogenies of degree 2 The object of this section, copied directly from Section 3 of [BSwD], is to recall the standard formulae for an elliptic curve with a rational 2-division point, for the associated isogeny, and for multiplication by 2 on such a curve. The notation introduced in this paragraph will be standard throughout the paper. Let k be an arbitrary field of characteristic not 2. An elliptic curve E 0 defined over k and having a primitive 2-division point P 0 defined over k can be written in the form E 0 : y 2 = (x − c)(x2 − d)

(1.2.1)

where c, d are in k and d, (c2 − d) non-zero and where P 0 is (c, 0). If O0 is the point at infinity on E 0 , which is the identity element under the group law, then there is an isogeny φ0 : E 0 → E 00 = E 0 /{O0 , P 0 } where E 00 , whose point at infinity will be denoted by O00 , is given by the equation E 00 : y12 = (x1 + 2c)(x21 + 4(d − c2 )).

(1.2.2) Explicitly, φ0 is given by

x1 = ((d − x2 )/(c − x)) − 2c,

y1 = y(x2 − 2cx + d)/(x − c)2 .

There is a dual isogeny φ00 : E 00 → E 0 , and φ00 ◦ φ0 and φ0 ◦ φ00 are the doubling maps on E 0 and E 00 respectively. The elements of H 1 (k, Z/2.P 0 ) ' k ∗ /k ∗2 classify the φ0 -coverings of E 00 ; the covering corresponding to the class of m0 is (1.2.3)

v12 = m0 (x1 + 2c),

v22 = m0 (x21 + 4d − 4c2 )

with the obvious two-to-one map to E 00 . The φ0 -covering corresponding to P 00 is given by m0 = d. Similarly the φ00 -coverings of E 0 are classified by the elements of H 1 (k, Z/2.P 00 ) ' k ∗ /k ∗2 , the covering corresponding to the class of m00 being w12 = m00 (x − c),

(1.2.4)

w22 = m00 (x2 − d).

The φ00 -covering corresponding to P 0 is given by m00 = c2 − d. The composite map φ00 ◦ φ0 : E 0 → E 00 → E 0 is multiplication by 2. There is an obvious commutative diagram

0

0

→

2E

0

  y

→ Z/2

→

2

E 0 −−−→   0 yφ

→ E 00

φ00

−−−→

E0 →   y=

0

E0

0.

→

This diagram induces a map H 1 (k, 2 E 0 ) → H 1 (k, Z/2) which may be interpreted as sending 2-coverings of E 0 to φ00 -coverings of E 0 . Let us now discuss multiplication by 2 on E 0 . Let K = k[t]/(t2 − d). The kernel 2 E 0 of multiplication by 2 on E 0 may be identified with the group RK/k µ2 ; the point P 0 defines a natural map Z/2 → RK/k µ2 . We have H 1 (k, 2 E 0 ) = H 1 (k, RK/k µ2 ) = K ∗ /K ∗2 . From [BSwD], we also extract the following pieces of information. √ √ Let a + b d ∈ K ∗ . The 2-covering of E 0 associated to the class of a + b d ∈ K ∗ /K ∗2 is given in homogeneous form by the intersection of two quadrics

(1.2.5)

Z22 + dZ32 = (a/(a2 − db2 ))Z12 + (ac + bd)Z02 , 2Z2 Z3 = (b/(a2 − db2 ))Z12 + (a + bc)Z02 .

in P3k . Let us denote this curve by Γ0 . The covering map Γ0 → E 0 can be factorized as Γ0 → C 00 → E 0 ; here C 00 is the φ00 -covering of E 0 given by (1.2.4), with m00 = a2 − db2 , and the map Γ0 → C 00 is W1 = Z1 /Z0 , W2 = (Z22 − dZ32 )/Z02 . From (1.2.5) it follows that if we are given an elliptic curve which has one nontrivial rational 2-division point then any 2-covering of it can be written as a smooth, complete intersection of two quadrics in P3k given by a system of homogeneous equations

(1.2.6)

α0 U02 + α1 U12 + α2 U22 + α3 U32 + 2α4 U2 U3 = 0, β0 U02 + β1 U12 + β2 U22 + β3 U32 + 2β4 U2 U3 = 0.

We refer to [BSwD] for the (simple) proof that conversely any curve of genus 1 defined over k by a system (1.2.6) is a 2-cover of an elliptic curve which has one nontrivial rational 2-division point.

As mentioned in the introduction, letting dij = αi βj − αj βi , for any subset {i, j, k, l} of {0, 1, 2, 3, 4} we have the identity (1.2.7)

dij dk` + dik d`j + di` djk = 0.

The system (1.2.6) implies

(1.2.8)

d10 U02 + d12 U22 + 2d14 U2 U3 + d13 U32 = 0, d01 U12 + d02 U22 + 2d04 U2 U3 + d03 U32 = 0.

and is in fact equivalent to it, since the assumption that (1.2.6) defines a smooth curve of genus one implies d01 6= 0 ∈ k. From [BSwD] we reproduce the formulas, some of which have already been mentioned: a = −2(2d14 d34 + d13 d23 )(d214 − d12 d13 ), 2 b = −d−1 01 d13 (d14 − d12 d13 ),

c = 4d04 d14 − 2d02 d13 − 2d03 d12 , d = 4d201 (d223 + 4d24 d34 ),

c2 − d = 16(d204 − d02 d03 )(d214 − d12 d13 ), m00 = a2 − db2 = 16d234 (d214 − d12 d13 )3 .

From these formulas we read the equation y 2 = (x − c)(x2 − d) of the curve E 0 , we read √ K = k[t]/(t2 − d), we read the class a + b d ∈ K ∗ /K ∗2 of the 2-covering Γ0 of E 0 which is isomorphic to the curve (1.2.5), and we also read the curve C 00 given by (1.2.4) through which the map Γ0 → E 0 factorizes. 1.3. Local computations Let E 0 , E 00 , φ0 , φ00 be as above. We let V 0 = H 1 (k, {O0 , P 0 }) = H 1 (k, Z/2) = k ∗ /k ∗2 and V 00 = H 1 (k, {O00 , P 00 }) = H 1 (k, Z/2) = k ∗ /k ∗2 . The isogeny φ0 gives rise to a natural embedding E 00 (k)/φ0 (E 0 (k)) ,→ V 0 and the isogeny φ00 gives rise to a natural embedding E 0 (k)/φ00 (E 00 (k)) ,→ V 00 . We let W 0 ⊂ V 0 denote the image of the first embedding and W 00 ⊂ V 00 be the image of the second. As mentioned above, the group V 0 classifies the set of φ0 -coverings of E 00 up to isomorphism; the subgroup W 0 classifies the set of isomorphism classes of such coverings which have a k-point. In this section k is a local field of characteristic different from 2. If k is nonarchimedean, we denote by A its ring of integers. As with any pair of dual isogenies, the (nondegenerate) Weil pairing φ0 E

0

× φ00 E 00 → µ2

(for a general pair of isogenies, µ2 should be replaced by a big enough group of roots of unity) induces a pairing V 0 × V 00 → H 2 (k, µ2 ) ⊂ Br(k). Since k is a local field, this pairing is nondegenerate.

Lemma 1.3.1

Assume that k is a local field. Under the nondegenerate pairing V 0 × V 00 → H 2 (k, µ2 ) ' Z/2

the subgroups W 0 ⊂ V 0 and W 00 ⊂ V 00 are each other’s orthogonal. A proof of this well-known lemma is provided in [BSwD] (Lemma 3). Note that once the diagrams written down there have been shown to commute, the result follows from Tate’s duality theorem over a nonarchimedean local field, and from Witt’s result over the reals. For the commutativity, one may refer to [CTSa], Lemme 5.2 et Remarque 5.3. We shall need an explicit description of the groups W 0 ⊂ V 0 and W 00 ⊂ V 00 . When k is a nonarchimedean nondyadic local field, and E 0 , hence also E 00 , has good reduction over the ring A of integers of k, the description is simple: W 0 ⊂ V 0 = k ∗ /k ∗2 coincides with the subgroup A∗ /A∗2 ⊂ k ∗ /k ∗2 and the same result holds for W 00 ⊂ V 00 . The following general lemma will enable us to compute W 0 and W 00 in the nondyadic case. Lemma 1.3.2 Let A be a discrete valuation ring, K its fraction field, v its valuation, π a uniformizing parameter of A and κ the residue field. Assume v(2) = 0. Let c, d, m ∈ A with d(c2 − d) 6= 0. Let X/Spec(A) be a regular proper integral scheme over Spec(A) whose generic fibre over Spec(K) is K-isomorphic to the smooth projective K-curve C given in P3K by the projective system of equations w12 = m(xt − ct2 ), w22 = m(x2 − dt2 ). Assume that either v(d) = 0 or v(c2 − d) = 0. Assume v(m) = 0 or v(m) = 1 (one may always reduce to this hypothesis). Then the special fibre of X/Spec(A), which is a κ-curve, has a geometrically integral component of multiplicity one if and only if c, d, m satisfy one of the conditions: (i) v(d) = 0, v(c2 − d) = 0 and v(m) = 0. (ii) v(m) = 0, v(d) > 0 is odd and m is a square modulo π. (iii) v(m) = 0, v(d) > 0 is even and either m is a square modulo π or −mc is a square modulo π. (iv) v(c2 − d) > 0 is odd. (v) v(c2 − d) > 0 is even, v(m) = 0. (vi) v(c2 − d) > 0 is even, v(m) = 1 and 2c is a square modulo π.

Proof Let us recall the following facts. The property that the special fibre of X/A has a multiplicity one component which is geometrically integral over κ does not depend on the regular proper model X/A of C/K. The genus one curve C actually admits a unique regular proper minimal model over A. If Y /A is some integral (not necessarily regular or proper) model of X/K such that Y → Spec(A) is surjective and the special fibre over κ has a geometrically integal component of multiplicity one, then a regular proper model X/A also has this property. In case (i), the curve X ⊂ P3A given by the equations in the Lemma is smooth over A, its special fibre is a geometrically integral smooth curve of genus one over κ. Suppose v(d) ≥ 0, v(c2 −d) = 0 and v(m) = 1. We claim that in that case any component of the special fibre of X/A has even multiplicity. Let V be a discrete valuation of rank one on the function field L of C, and assume e = V (π) > 0. From the affine equations w12 = m(x − c), w22 = m(x2 − d) viewed as identities in L we deduce that both e + V (x − c) and e + V (x2 − d) are even. Assume e odd. This now implies V (x − c) odd and V (x2 − d) odd. From V (x2 − d) odd, and V (d) = ev(d) ≥ 0 we deduce V (x) ≥ 0. We now have V (x − c) ≥ 1, V (x2 − d) ≥ 1 hence V (c2 − d) > 0, which implies v(c2 − d) > 0, contradicting our hypothesis.

Assume v(d) > 0, v(c2 − d) = 0 and v(m) = 0. If m is a square modulo π, then the special fibre of the model Y /A given by the system of affine equations w12 − m(x − c) = 0, w22 − m(x2 − d) = 0 breaks up as the union of two smooth irreducible conics defined over the residue field κ. Assume d = uπ 2n+1 with u a unit. Suppose a regular proper model X/A of the K-curve given by the affine system of equations w12 = m(x − c), w22 = m(x2 − d) has a multiplicity one component which is geometrically integral over κ. On the function field of the curve, let V denote the valuation associated to the codimension one point of the model defined by that component. The multiplicity one assumption ensures V (π) = 1. We have V (x2 − d) = inf(2V (x), 2n + 1). From the second equation, we obtain 2V (w2 ) = V (m(x2 − d)) = inf(2V (x), 2n + 1), hence 2V (w2 ) = 2V (x) < 2n + 1 = V (d). The same equation now implies that m is a square modulo π. For later use, let us consider the special case where v(d) = 1, v(c2 − d) = 0 and v(m) = 0. One easily checks that the subscheme of P3A defined by the homogeneous equations w12 = m(xt − ct2 ), w22 = m(x2 − dt2 ) is the regular proper minimal model X/A of C/K. The special fibre is the union of two smooth conics meeting transversely in two points. Each conic is defined over κ[t]/(t2 − m). If this extension is a field, they are conjugate. The two points are defined over κ[t]/(t2 + mc). Assume now that d = uπ 2n with u a unit, and that m is not a square modulo π. Suppose a regular minimal model (over A) of the K-curve given by the affine system of equations w12 = m(x−c), w22 = m(x2 −d) has a multiplicity one component which is geometrically integral over κ. On the function field of the curve, let V denote the valuation associated to the codimension one point of the model defined by that component. The multiplicity one assumption ensures V (π) = 1. If V (x) ≤ 0, then from the second equation we deduce V (w22 ) = v(x2 ) ≤ 0 and m is a square modulo π, which we have excluded. Assume V (x) > 0. Then from the first equation we deduce that −mc is a square modulo π. Conversely, assume that −mc is a square modulo π. Make the change of variables x = π n X. Then affine equations for C/K read w12 = m(π n X − c), w22 = m.π 2n .(X 2 − u). Another change of variables tranforms this system into w12 = m(π n X − c), w22 = m(X 2 − u). Now these equations over A reduce to w12 = −mc, w22 = m(X 2 − u) over κ, and provided −mc is a square, this decomposes into two smooth geometrically integral conics. Assume v(d) = 0, v(c2 − d) > 0 and v(m) = 0. In this case the special fibre of the model Y /A given by w12 = m(x − c), w22 = m(x2 − d) is clearly irreducible and of multiplicity one.

Assume v(d) = 0, v(c2 − d) > 0 and v(m) = 1. Assume v(c2 − d) > 0 is odd. We claim that here also the special fibre of a suitable model contains a geometrically integral fibre of multiplicity one. Let E be the elliptic curve over K given by the affine equation y 2 = (x − c)(x2 − d). The curve C is K-birational to the unramified double cover of E given by w2 = m(x − c). Let σ denote translation by the 2-torsion point P given by x = c, y = 0 on E. The divisor of x − c is a double, and the evaluation of the unramified double cover of E given by x − c = z 2 at the point P is the class (c2 − d) ∈ K ∗ /K ∗2 . This is well-known to imply that in the function field of E there exists a formula (x − c) ◦ σ = (c2 − d).(x − c).g 2 with g a suitable rational function. Thus C viewed as an unramified double cover of E after pulling-back by σ is birationally given by the set of equations (x − c) = uz 2 , y 2 = (x − c)(x2 − d), where u = m/(c2 − d) ∈ K ∗ has even valuation. Multiplying z by a suitable power of the uniformizing parameter reduces to the case where u is a unit, i.e. to the situation v(m) = 0.

Assume v(d) = 0, v(c2 − d) > 0 even, and v(m) = 1. Note that this implies v(c) = 0. We claim that the special fibre has a geometrically integral component of multiplicity one if and only if 2c is a square modulo π. Suppose that there exists such a component, let V be the discrete valuation associated to it. By hypothesis, π is a uniformizing parameter. In the function field of the K-curve, we have w12 = m(x − c) and w22 = m(x2 − d). We have V (π) = 1, hence V (m) = 1. From the second equality we deduce V (x) = 0. From the first equality we conclude x = c + m2r+1 u, with r ≥ 0, V (u) = 0 and u is a square modulo π. Hence x2 = c2 + 2cm2r+1 u + m4r+2 u2 . The second equality now gives w22 = m(c2 − d + 2cm2r+1 u + m4r+2 u2 ).

Since V (m) = 1 and V (c2 − d) is even, this equality implies V (2cu.m2r+2 ) < V (c2 − d), hence 2cu is a square modulo π, hence finally 2c is a square modulo π. Conversely, suppose that 2c is a square modulo π. By K-birational transformations we transform the system of equations defining C into mw12 = x − c, mw22 = x2 − d

which is K-isomorphic to mw12 = x − c, w22 = 2cw12 + (c2 − d)/m + mw14 . By assumption, v(m) = 1 and v((c2 − d)/m) > 0 (since it is odd). The reduction modulo π of the above system reads 0 = x − c, w22 = 2cw12 . Since 2c is a square, this has a geometrically integral component of multiplicity one. Remark 1.3.3 Lemma 4 in [BSwD] is an arithmetic consequence of the above geometric lemma. Suppose A is a complete (or henselian) discrete valuation ring with finite residue field of characteristic not 2. If X/A has a special fibre which contains a multiplicity one geometrically integral component, then one may find on it a zero-cycle of degree one whose support lies in the smooth locus of the special fibre. By Hensel’s lemma, one may lift this zero-cycle to a zero-cycle of degree one on C/K. Since C/K is a curve of genus one, this curve admits a K-rational point. Conversely, if C/K admits a rational point, this defines an A-point of the regular proper A-scheme X/A, hence the special fibre of the latter contains a geometrically integral component of multiplicity one with a smooth κ-point (which as a matter of fact does not belong to any other component). Let k be a p-adic local field, p 6= 2, let c, d be in the ring A of integers. Let v denote the valuation of k. The above computation enables us to determine W 00 ⊂ k ∗ /k ∗2 . By duality, this also determines W 0 . (i) If v(d) = 0 and v(c2 − d) = 0, then W 0 = A∗ /A∗2 and W 00 = A∗ /A∗2 . (ii) If v(d) = 1 and v(c2 − d) = 0, then W 0 = k ∗ /k ∗2 and W 00 = 1 ⊂ k ∗ /k ∗2 . (iii) If v(d) = 0 and v(c2 − d) = 1, then W 0 = 1 ⊂ k ∗ /k ∗2 and W 00 = k ∗ /k ∗2 . This will be enough to prove Theorem A. But when proving theorem B, we shall also need: (iv) If v(d) > 0 is even and v(c2 − d) = 0 (hence v(c) = 0), then W 00 = 1 or W 00 = A∗ /A∗2 depending on whether −c is or is not a square in A∗ . In the first case, W 0 = k ∗ /k ∗2 , in the second case, W 0 = A∗ /A∗2 . We shall now study, though in less details, the analogue of Lemma 1.3.2 for a curve (1.2.6). Lemma 1.3.4 Let A be a discrete valuation ring, K its fraction field, v its valuation, π be a uniformizing parameter of A and κ the residue field. Assume v(2) = 0. Let αi , βi , i = 0, · · · , 3

be elements of A, and consider the closed subscheme of P3A defined by the system of homogeneous equations

(1.2.6)

α0 U02 + α1 U12 + α2 U22 + α3 U32 + 2α4 U2 U3 = 0, β0 U02 + β1 U12 + β2 U22 + β3 U32 + 2β4 U2 U3 = 0.

Assume that the generic fibre of this A-scheme is a smooth intersection of two quadrics in P3K . Noting as above dij = αi βj − αj βi ∈ A, this amounts to the assumption that the product d01 (d223 + 4d24 d34 )(d204 − d02 d03 )(d214 − d12 d13 ) 6= 0 ∈ A. If the valuation of this product is zero, the above A-scheme is smooth over A, and the special fibre is a smooth intersection of two quadrics in P3κ . Assume that the valuation of this product is equal to one (we shall only be interested in this case). Assume moreover that the valuation of (d204 − d02 d03 )d02 is at most one, and that the valuation of (d214 − d12 d13 )d12 is at most one. Let X/A be a regular proper integral scheme over Spec(A) whose generic fibre over Spec(K) is K-isomorphic to the curve defined by (1.2.6) over K. Then the special fibre of X/Spec(A), which is a κ-curve, has a geometrically integral component of multiplicity one if and only if one of the following conditions is satisfied: (i) v(d01 ) = 1 and either d204 − d02 d03 or d214 − d12 d13 is a square modulo π (in which case both are squares); (ii) v(d204 − d02 d03 ) = 1 and −d01 d02 is a square modulo π; (iii) v(d214 − d12 d13 ) = 1 and d12 d01 is a square modulo π; (iv) v(d223 + 4d24 d34 ) = 1. Proof

By assumption, we have v(d01 (d223 + 4d24 d34 )(d204 − d02 d03 )(d214 − d12 d13 )) = 1.

Thus exactly one of v(d01 ), v(d204 − d02 d03 ), v(d214 − d12 d13 ), v(d223 + 4d24 d34 ) is equal to 1, the other ones are zero. (i) Assume v(d01 ) = v(α0 β1 − α1 β0 ) = 1. Then each of v(d223 + 4d24 d34 ), v(d204 − d02 d03 ), v(d214 − d12 d13 ) equals zero. One of α0 , β0 , α1 , β1 has valuation zero. By the symmetry of the equation, we may assume v(α1 ) = 0. If we had v(α0 ) > 0, then we would have v(β0 ) > 0. But v(α0 ) > 0 and v(β0 ) > 0 imply v(d204 − d02 d03 ) > 0, which is excluded. Thus we have v(α1 ) = 0 and v(α0 ) = 0. Another (possibly singular) model for (1.2.6) is given by the system

(1.3.1)

α0 U02 + α1 U12 + α2 U22 + α3 U32 + 2α4 U2 U3 = 0, d10 U02 + d12 U22 + d13 U32 + 2d14 U2 U3 = 0.

Suppose there exists a multiplicity one, geometrically integral component γ of the special fibre of X/A. Let V be the associated valuation on the function field of our curve. By hypothesis, V (π) = 1. Suppose d214 − d12 d13 is not a square mod π. Then it is not a square in the residue field of V either, because the κ-curve γ is geometrically integral. Since d214 − d12 d13 is not a

square mod π, certainly each of d12 , d13 is a unit with respect to v, hence also with respect to V . In the function field L of X, we have the equality V (−d10 (U0 /U2 )2 ) = V (d12 + d13 (U3 /U2 )2 + 2d14 (U3 /U2 )). The left hand side has odd valuation. The right hand side has even valuation, because it is equal to the valuation of a norm of an element in a nontrivial quadratic extension, which is unramified and nonsplit at V (indeed, d214 − d12 d13 is a unit, and not a square in the residue field). We conclude: if the special fibre of X/A contains a multiplicity one, geometrically integral component, then d214 − d12 d13 is a square mod π. Conversely, suppose that d214 − d12 d13 is a square mod π. Equations for the special fibre of (1.3.1) are

(1.3.2)

α ˜ 0 U02 + α ˜ 1 U12 + α ˜ 2 U22 + α ˜ 3 U32 + 2α ˜ 4 U2 U3 = 0, d˜12 U 2 + d˜13 U 2 + 2d˜14 U2 U3 = 0. 2

3

The discriminant of this last equation is a nonzero square. Hence it breaks up as the product of two nonproportional linear forms, defining two distinct planes. The 2 by 2 determinants associated to the matrix α ˜2 α ˜ 3 ˜2α4 d˜12 d˜13 2d˜14 are α ˜ 1 d23 , 2α ˜ 1 d24 , 2α ˜ 1 d34 . One of them is nonzero (recall v(d223 + 4d24 d34 ) = 0). Thus the form 2 2 α ˜ 2 U2 + α ˜ 3 U3 + 2α ˜ 4 U2 U3 does not vanish identically on at least one of the two planes. From v(α1 ) = 0 and v(α0 ) = 0 we conclude that the trace of α ˜ 0 U02 + α ˜ 1 U12 + α ˜ 2 U22 + α ˜ 3 U32 + 2α ˜ 4 U2 U3 = 0 on that plane is a smooth conic: this produces a geometrically integral component of multiplicity one. (iii) (the argument for (ii) is entirely similar). Assume v(d214 − d12 d13 ) = 1. By assumption this implies v(d01 ) = 0 and v(d12 ) = 0. Suppose there exists a multiplicity one, geometrically integral component γ of the special fibre of X/A. Let V be the associated valuation on the function field L of our curve. By hypothesis, V (π) = 1. Let ui = Ui /U3 ∈ L. From the second equation in (1.3.1), we deduce the following equation in the function field of X: u20 − (d12 /d01 )(u2 + d14 /d12 )2 = (d214 − d12 d13 )/d10 d12 . If d12 d01 were not a square in the residue field of v, hence of V , then the V -valuation of the left hand side would be even. But the valuation of the right hand side is odd. Conversely, assume that d12 d01 is a square in the residue field of v. As a possibly singular A-model for the K-curve defined by (1.2.6) we may take Y ⊂ P3A defined by the system of homogeneous equations d10 U02 + d12 U22 + 2d14 U2 u3 + d13 U32 = 0, d01 U12 + d02 U22 + 2d04 U2 u3 + d03 U32 = 0.

The fibre of this A-scheme over the residue field κ of A is the intersection of two quadrics in P3κ . The first of these quadrics decomposes as the product of two planes over κ, each of them being given by an equation U0 = ±l(U2 , U3 ) where l is a linear form. The second one is the cone over a nonsingular conic in the variables U1 , U2 , U3 : indeed, v(d01 ) = 0 and v(d204 − d02 d03 ) = 0. Thus the fibre Yκ is the union of two distinct smooth conics. (iv) One considers the possibly singular model Y ⊂ P3A defined by the system of homogeneous equations d10 U02 + d12 U22 + 2d14 U2 U3 + d13 U32 = 0, d01 U12 + d02 U22 + 2d04 U2 U3 + d03 U32 = 0. Under the assumption v(d223 + 4d24 d34 ) = 1, we have v(d01 ) = 0, v(d204 − d02 d03 ) = 0 and v(d214 − d12 d13 ) = 0. By (1.2.7), we have d12 d04 − d02 d14 = d01 d42 , d12 d03 − d02 d13 = d01 d32 , and d14 d03 − d04 d13 = d01 d34 . Since the valuation of d223 + 4d24 d34 is one, one at least of d42 , d34 , d32 is a unit in A, and we know that d01 is a unit. This implies that the reduction over κ of the two quadratic forms d12 U22 + 2d14U2 u3 + d13U32 and d01 U01 + d02 U22 + 2d04 U2 u3 + d03 U32 are nonproportional. As the reader will check, this implies that the variety in P3κ obtained by the reduction of the above system of equations is geometrically irreducible (with possibly one singular point). 1.4. The Selmer groups as kernels of pairings In this section, k is a number field. We follow the model of [CSS3], Section 1.2. We let E , E 00 , φ0 , φ00 be as in Section 1.2. Define Sφ0 , the φ0 -Selmer group of E 00 , to be the set of isomorphism classes of φ0 -coverings of E 00 which are everywhere locally soluble; and similarly for Sφ00 , the φ00 -Selmer group of E 0 . Let S0 be as in Section 1.1 and let S be any finite set of places containing S0 . From Section 1.1 we have the nondegenerate bilinear pairing (., .)S : VS0 × VS00 → Z/2. 0S 00S The group I S is the image of o∗S /o∗2 are its images in VS0 , VS00 respectively. S in VS , and I , I By Proposition 1.1.1, for any S, I 0S and I 00S are each other’s orthogonal under the pairing VS0 × VS00 → Z/2. Let Kv0 , Kv00 , KS0 , KS00 be as in Proposition 1.1.2. Recall that KS0 ⊂ VS0 and KS00 ⊂ VS00 are each other’s othogonal under the pairing VS0 ×VS00 → Z/2. For v a place of k, let Wv0 ⊂ Vv0 be the image of E 00 (kv )/φ0 (E 0 (kv )) ,→ Vv0 and similarly let Wv00 ⊂ Vv00 be the image of E 0 (kv )/φ00 (E 00 (kv )) ,→ Vv00 . Let WS0 = ⊕v∈S Wv0 ⊂ VS0 and let WS00 = ⊕v∈S Wv00 ⊂ VS00 . By Lemma 1.3.1, WS0 and WS00 are each other’s orthogonal under the pairing VS0 × VS00 → Z/2. 0

The following Lemma is the analogue of Prop. 1.2.1 of [CSS3]. Lemma 1.4.1 Let S0 be as above and suppose that S contains S0 and all the primes of bad reduction for E 0 , i.e. c, d and c2 − d are units at any v ∈ / S. Then Sφ0 is isomorphic to each of the following groups: (i) the intersection I 0S ∩ WS0 ; (ii) the left kernel of the map I 0S × WS00 → Z/2 induced by (. , .)S ; (iii) the left kernel of the map WS0 × I 00S → Z/2 induced by (. , .)S . A similar result holds for Sφ00 . Proof If v is not in S then the φ0 -covering (1.2.3) is soluble in kv if and only if m0 is in H(ov ); hence Sφ0 can be identified with the subgroup of I 0S for which (1.2.3) is soluble at every place of S. This proves (i). Since WS0 and WS00 are orthogonal complements with respect to (. , .)S , we deduce (ii). As for (iii), it follows from the corresponding fact for I 0S and I 00S .

For any finite set S of places containing S0, but not necessarily the primes of bad reduction for E 0 and E 00 , write I0S = I 0S ∩ (WS0 + KS0 ),

WS0 = WS0 /(WS0 ∩ KS0 ) = ⊕v∈S Wv0 /(Wv0 ∩ Kv0 )

and similarly for I00S and WS00 . Define t0S : VS0 → I 0S to be the projection along KS0 in VS0 , and similarly for t00S . The image of t0S lies in I 0S , and the image of 1 − t0S lies in KS0 , and similarly for t00S . For x ∈ VS0 and y ∈ VS00 , we have (t0S (x), t00S (y))S = 0 = (x − t0S (x), y − t00S (y))S , hence (1.4.1)

(x, y)S = (t0S (x), y)S + (x, t00S (y))S

The kernel of t0S is KS0 , so t0S induces a map WS0 → I 0S whose kernel is trivial and whose image is easily seen to be I0S ; in other words it induces an isomorphism τS0 : WS0 → I0S . There is an analogous isomorphism τS00 : WS00 → I00S . We shall denote the inverse isomorphisms by σS0 , σS00 respectively. Proposition 1.4.2

The pairing (. , .)S induces pairings I0S × WS00 → Z/2,

WS0 × I00S → Z/2.

The action of τS0 × σS00 takes the first pairing into the second.

Proof To prove the existence of the first pairing it is enough to show that I0S is orthogonal to WS00 ∩ KS00 with respect to (. , .)S . But I0S ⊂ WS0 + KS0 which is the orthogonal complement of WS00 ∩ KS00 . The argument for the second pairing is similar. To prove the last statement, let α0 + β 0 be any element of I0S , where α0 is in WS0 and β 0 in KS0 , and let α00 be any element of WS00 ; by abuse of language we also use α00 to denote the corresponding element of WS00 . Then σS0 (α0 + β 0 ) is the class of α0 , and τS00 α00 has the form α00 + β 00 for some β 00 in KS00 . Since we are in characteristic 2, what we need to prove is (α0 + β 0 , α00 )S + (α0 , α00 + β 00 )S = 0. But t0S (α0 ) = α0 + β 0 because β 0 is in KS0 and α0 + β 0 is in I 0S , and similarly t00S (α00 ) = α00 + β 00 ; the left hand side of the displayed equation is equal to (α0 + β 0 , α00 + β 00 )S by (1.4.1) and this vanishes because I 0S is orthogonal to I 00S . We shall denote the pairings in Prop. 1.4.2 by (. , .)0S and (. , .)00S respectively. Proposition 1.4.3 Suppose that S also contains all the primes of bad reduction for E 0 . Then the left kernel of either of the pairings in Proposition 1.4.2 is isomorphic to Sφ0 and the right kernel to Sφ00 . Proof By Lemma 1.4.1, Sφ0 can be identified with I 0S ∩ WS0 and is therefore contained in I ; also it is orthogonal to WS00 and therefore to WS00 . Conversely, anything in I0S which is orthogonal to WS00 must be orthogonal to WS00 and therefore lies in WS0 ; since I0S lies in I 0S , such an element lies in I 0S ∩ WS0 = Sφ0 . 0S

The left kernel of the second pairing in Prop. 1.4.2 is isomorphic through σS0 to the left kernel of the first pairing. The proof for the right kernels starts from the second pairing in Prop. 1.4.2 and is similar. What we shall actually use is the pairing WS0 × WS00 → Z/2 given by (x0 , x00 ) 7→ (τS0 (x0 ), x00 )0S = (x0 , τS00 (x00 ))00S . The equality here follows from the last sentence of Prop. 1.4.2, and Prop. 1.4.3 asserts that the left kernel of this map is isomorphic to Sφ0 and the right kernel to Sφ00 . 1.5. Small Selmer groups In [CSS3], one considers multiplication by 2 on an elliptic curve E/k whose 2-torsion is rational over k and whose 2-Selmer group S2 (E) is of order at most 8. On the Tate-Shafarevich group of E we have the alternating Cassels-Tate pairing. If the Tate-Shararevich group is finite, then the order of the 2-torsion subgroup of X(E) must be a square. Since it is of order at most 2, it has to be trivial and E(k)/2 ' S2 (E) which forces any 2-covering of E with points everywhere locally to have a k-rational point. Here we must give an argument taking into account the two isogenies E 0 → E 00 and E 00 → E 0 . This is Lemma 11 of [BSwD], for which I give a slightly revised proof. Let k be a number field. Let φ0 : E 0 → E 00 be a degree 2 isogeny with kernel Z/2.P 0 . Let φ00 : E 00 → E 0 be the dual isogeny, the kernel of which is Z/2.P 00 . The composite map φ0 ◦ φ00 is multiplication by 2 on E 00 . Similarly, the composite map φ00 ◦ φ0 is multiplication by 2 on E 0 . From the isogenies φ0 et φ00 one obtains exact sequences 0 → E 00 (k)/φ0 (E 0 (k)) → Sφ0 → φ0 X(k, E 0 ) → 0 and 0 → E 0 (k)/φ00 (E 00 (k)) → Sφ00 → φ00 X(k, E 00 ) → 0,

where Sφ0 is the Selmer group classifying φ0 -coverings of E 00 which have rational points everywhere locally, Sφ00 is the Selmer group classifying φ00 -coverings of E 0 which have rational points everywhere locally, and φ0 X(k, E 0 ) is the kernel of the map X(k, E 0 ) → X(k, E 00 ) induced by φ0 , and similarly for φ00 X(k, E 00 ). Lemma 1.5.1 If P 0 is the unique 2-torsion point of E 0 (k), then P 00 ∈ / φ0 (E 0 (k)). If P 00 is the unique 2-torsion point of E 00 (k), then P 0 ∈ / φ00 (E 00 (k)).

Proof Assume there exists M ∈ E 0 (k) such that φ0 (M) = P 00 . Then 2M = φ00 ◦ φ0 (M) = φ00 (P 00 ) = 0, hence M is a 2-torsion point on E 0 . Then M = 0 or M = P 0 , hence P 00 = φ0 (M) = 0, which contradicts our hypothesis. Proposition 1.5.2 Assume that P 0 is the only 2-torsion point of E 0 (k) and that P 00 is the only 2-torsion point of E 00 (k). Assume that Sφ0 has order 2 and that Sφ00 has order at most 4. Then φ0 X(E 0 ) = 0 and S2 (E 0 ) has order at most 4. If we moreover assume that the groups X(E 0 ) and X(E 00 ) are finite, then the groups φ00 X(E 00 ), 2 X(E 0 ) and 2 X(E 00 ) are all zero. In particular the curves corresponding to elements in the Selmer groups Sφ0 , Sφ00 , S2 (E 0 ), S2 (E 00 ) all have rational points. The finitely generated abelian groups E 0 (k) and E 00 (k) have (Mordell-Weil) rank r, where 0 ≤ r ≤ 1 and 2r+1 is the order of the group Sφ00 . Proof From Lemma 1.5.1 we have an injection Z/2.P 00 ,→ E 00 (k)/φ0 (E 0 (k)). The hypothesis on the order of Sφ0 thus implies Z/2.P 00 = E 00 (k)/φ0 (E 0 (k)) and φ0 X(E 0 ) = 0. From the first equality we deduce φ00 (E 00 (k)) = φ00 ◦ φ0 (E 0 (k)) = 2E 0 (k) ⊂ E 0 (k).

From the commutative diagram

0

0

2

0

→

E 0 −−−→   0 yφ

E0 → 0   y=

00

→

E 00

E0

→

2E

→

φ00 E

  y

φ00

−−−→

→ 0

we obtain the commutative diagram of exact sequences 0

0

→ →

E 0 (k)/2E 0 (k)   y

E 0 (k)/φ00 (E 00 (k))

→ S2 (E 0 ) →   y →

Sφ00

→

2 X(E

  y

0

φ00 X(E

)

→

0

00

) →

0,

where the middle and right vertical maps are induced by φ0 and the left vertical map is the natural projection map, which in the case in point is an isomorphism, as noticed earlier on. The kernel of the map 2 X(E 0 ) → φ00 X(E 00 ) is the group φ0 X(E 0 ), and we have shown that this group vanishes. Thus the left and right vertical maps are one-to-one, hence so is the map S2 (E 0 ) → Sφ00 , and the order of S2 (E 0 ) is at most 4. According to the lemma, the order of E 0 (k)/φ00 (E 00 (k)) is at least 2. The group Sφ00 has order at most 4. Thus φ00 X(E 00 ) has order at most 2. Since 2 X(E 0 ) → φ00 X(E 00 ) is one-toone, we conclude that 2 X(E 0 ) has order at most 2. If the group X(E 0 ) is finite, then the (alternating) Cassels-Tate pairing on that group is nondegenerate. This implies that the group X(E 0 ) is a direct sum of groups (Z/n)2 for various n’s, in particular the order of 2 X(E 0 ) is a square. Thus 2 X(E 0 ) = 0 and E 0 (k)/2E 0 (k) ' S2 (E 0 ), this last group being of order 2 or 4. The isogeny φ00 induces a homomorphism 2 X(E 00 ) → φ0 X(E 0 ) whose kernel is φ00 X(E 00 ), a group of order at most 2. From φ0 X(E 0 ) = 0 we now deduce that 2 X(E 00 ) is of order at most 2. The hypothesis that X(E 00 ) is finite implies that the order of 2 X(E 00 ) is a square. Hence 00 00 2 X(E ) = 0, and φ00 X(E ) = 0. From the commutative diagram above we then conclude that the groups E 0 (k)/2E 0 (k), E 0 (k)/φ00 (E 00 (k)) and Sφ00 are isomorphic. The last one has order 2r+1 , with 0 ≤ r ≤ 1. Thus E 0 (k), whose 2-torsion subgroup is of order 2, has Mordell-Weil rank r. Hence also E 00 (k). 2. Proof of Theorem A 2.1. Fibres with points everywhere locally Let k be a number field and let c(t), d(t), m(t) ∈ k[t] satisfy the hypothesis of the theorem. Thus all have even degree, deg(d) = 2deg(c) > 0, d(c2 − d) has no square factor and m divides c2 − d, hence is coprime with d. We let r(t) = d(t)(c2 (t) − d(t)) and we let rM (t) be the monic irreducible factors of r(t), where M denotes the associated closed point of A1k . The polynomial m(t) is the product of a constant ρm ∈ k ∗ by a product of some of the rM (t). We refer the reader to the introduction for the notation. The surface X = X(m) is equipped with the fibration π : X → P1k . Recall that M0 is the set of closed points M such that d(M) = 0 and that M00 is the set of closed points where c2 − d vanishes. We let r0 (t) be the product of the rM (t) for M ∈ M0

and similarly for r00 (t). Note that r0 (t) up to a scalar coincides with d(t) and that r00 (t) up to a scalar coincides with c2 (t) − d(t). The assumptions imply (Lemma 1.3.2 for M ∈ A1k , a direct computation for M = ∞ ∈ P1k ) that for M ∈ P1k , M ∈ / M0 , the fibre XM contains a component of multiplicity one which is geometrically integral over the residue field kM (the fibres need not be smooth, in contrast with what happened in [CSS3]; however, as is by now well known ([CSS2], Lemma 1.2), the above property is just as good for the method to be applied – see Lemma 2.1.1 hereafter). At a point M ∈ M0 , the fibre over kM when viewed over an algebraic closure of kM is the union of two smooth conics meeting transversally in two points; each conic is defined over the extension KM = kM [u]/(u2 − m(M)). We let Brvert (X) ⊂ Br(X) be the vertical Brauer group with respect to the map π, as defined in the introduction. Since the geometric fibres are reduced, Q the quotient Brvert (X)/Br(k) is finite ([Sk], Cor. 4.5). We start with an ad`ele {Pv } ∈ X(Ak ) = v∈Ω X(kv ). We assume that for all A ∈ Brvert(X), we have X invv (A(Pv )) = 0 ∈ Q/Z, v∈Ω

in other words we assume that {Pv } is in the kernel of all reciprocity maps θA : X(Ak ) → Q/Z associated to elements A ∈ Brvert(X) (see [CT]). Let S1 be a finite set of places of k containing all the archimedean places; for v ∈ S1, let Uv ⊂ P1 (kv ) be an open neighbourhood of π(Pv ). Under the five assumptions of Theorem A, we will find a point λ ∈ P1 (k), lying in Uv for v ∈ S1 , such that the fibre Xλ = π −1 (λ) is smooth and has infinitely many rational points. This will prove statement (a) in the Theorem. Statements (b) and (c) then follow easily. A standard procedure, which uses the finiteness of Brvert(X)/Br(k), and is described in full detail in [CSS2] (proof of Theorem 1.1) allows us to assume also that: (a) For each place v ∈ Ω, the point Pv lies on a smooth fibre of π and does not lie on the fibre at infinity (i.e. π(Pv ) lies in U(kv ), for U ⊂ A1k as in the introduction). (b) For each place v ∈ S1 , the projection X(kv ) → P1 (kv ) admits an analytic section σv : Uv → X(kv ) over Uv . (c) For each real place v, the neighbourhood Uv coincides with the open set tv > 0, and at any point of Uv the fibre of π is a smooth curve of genus one ; for v complex, Uv contains the point at infinity. The mentioned procedure involves a projective transformation of the projective line P1k sending a point of A1 (k), with smooth fibre, to the point at infinity. One checks that the procedure does not affect the hypotheses. This is clear for any transformation t → t − α, where α does not belong to the zeroes of d(c2 − d). Assume, as we may, that t = 0 is not a zero of d(c2 −d). Then the statement is also clear when one sets T = 1/t, and one replaces m(t), c(t), d(t) by M(T ) = T µ m(1/T ), D(T ) = T δ d(1/T ), C(T ) = T γ c(1/T ), where µ, δ, γ are the respective degrees of m, d, c, all even. Note that we then have deg(C 2 − D) = deg(D) = 2deg(C) > 0. From now on, we assume that conditions (a),(b),(c) above are fulfilled, as well as the improved condition (0.1), where the degree of c2 − d is equal to the degree of d. Given (the new) m and its associated regular minimal proper model X = X(m)/P1k , we shall define a finite set S0 of ‘bad’ places of k. As in Proposition 1.1.1, we want it to contain all the archimedean places and all the places above 2, and we want the class group of oS0 to be trivial. We also want it to contain the set S1 given above. For each v ∈ / S0, we want all polynomials c(t), d(t), m(t) to be ov -integral, we want ρm to be an ov -unit, and we want r(t) to define a closed subscheme of Spec(oS0 [t]) which is finite and ´etale over oS0 , the ring of S0 integers of k. This implies in particular that the leading coefficients of d(t) and of (c2 − d)(t) are units away from S0 . Given any monic irreducible factor rM (t) of r(t), the closed subscheme

˜ = Spec(oS0 [t]/rM (t)) of A1 , is finite and ´etale over Spec(oS0 ). For each M ∈ M0 , we M oS0 want the trivial or quadratic extension KM /kM (see above) to be unramified over the ring of S0 -integers of kM . For much later use, we want S0 to contain any finite place of k whose residue characteristic is less than or equal to the product of the degree [k : Q] and the degree of the polynomial r(t) (this will be used when applying Schinzel’s hypothesis). We also want S0 to be big enough so that the fibration π : X → P1k extends to a fibration X /P1oS0 with the following S ˜ is the set of points of P1 whose fibre property: the scheme Spec(oS0 [t]/r0 (t)) = M ∈M0 M oS0 does not contain a geometrically integral component of multiplicity one, and over any geometric point of this subscheme, the fibre consists of two smooth conics intersecting transversally in two points (that this can be achieved may be seen directly on projective equations, as in the introduction). Given a finite set S of primes containing S0 , we shall be interested in the affine line A1oS over oS and in the open set UoS = Spec(oS [t][1/r]), and the analogous Uo0 S and Uo00S . The group of units of oS [t][1/r] is the direct product of o∗S and the free group generated by the polynomials rM for M ∈ M, as one easily checks. Also, one easily checks that the Picard group of oS [t][1/r] vanishes (this uses the vanishing of the Picard group of oS ). The same results hold for oS [t][1/r0 ]) and oS [t][1/r00 ]). A point λ ∈ P1 (k) = P1 (oS0 ) (resp. λv ∈ P1 (kv ) = P1 (ov ) for v ∈ / S0 ) defines a section ˜ : Spec(oS ) → P1 (resp. a local section λ ˜v : Spec(ov ) → P1 ). To simplify the notation, we λ 0 oS0 ov shall sometimes write λ where it would be more correct to write λv . We shall call such a point ˜ (resp. local section λ ˜ v ) defined over Spec(oS ) (resp. transversal if the corresponding section λ 0 Spec(ov )) transversally intersects with the finite ´etale oS0 -scheme (resp. ov -scheme) which is the ˜ ’s (M ∈ M) and of the section at infinity of P1 union of all the restrictions over oS0 of the M oS0 (resp. which is the pull-back to ov of this ´etale scheme). In less geometric terms, this simply means that for each M ∈ M, the element rM (λ) ∈ k ∗ (resp. rM (λv ) ∈ kv∗ ) is either a unit or a uniformizing parameter at each v ∈ / S0 (resp. at v), and that if v(λ) < 0, then v(λ) = −1. Given any point λ ∈ A1 (k) such that d(c2 − d) does not vanish at λ, the fibre Xλ is a smooth curve of genus one over k. The analogue of Lemma 2.1.1 of [CSS3] (see [BSwD] Lemma 4) is here: Lemma 2.1.1 Let S0 be as above. Let v ∈ / S0 and let λ ∈ A1 (kv ) be a transversal point. ˜ ⊂ P1 (ov ) be its closure. Let λ ˜ does not intersect any of the M ˜ for M ∈ M0 , then Xλ (kv ) 6= ∅. (a) If λ ˜ intersects M ˜ for M ∈ M0 at some place w of kM (w unramified over v, of degree (b) If λ one over v), then Xλ (kv ) 6= ∅ if and only if w splits in KM . ˜ Proof Let Xλ/ov be the ov -scheme which is the inverse image of X /P1 under λ. oS0

(a) By the definition of S0 , the reduction mod v of the fibre Xλ/ov contains a geometrically integral component of multiplicity one. As noted in Remark 1.3.3, this implies Xλ (kv ) 6= ∅. (b) Since Xλ /ov is proper we have Xλ (kv ) = Xλ (ov ). Thus a point of Xλ (kv ) gives a local section over Spec(ov ) which intersects just one component of the closed fibre: that component must have multiplicity one and be geometrically integral. Thus the unramified extension KM /kM splits over w. Conversely, if this is so, the two conics which constitute the closed fibre of Xλ/ov at v are individually defined over Fw = Fv . We may then apply Remark 1.3.3. We may also argue directly: Any conic contains at least 3 points, thus we can always find a smooth rational point in such a fibre. By Hensel’s lemma this point can be lifted to a point over kv . Given M ∈ M, we may write rM (t) = NormkM /k (t − aM ), with aM ∈ kM the class of t in k[t]/rM (t). For M ∈ M0 , let (KM /kM , t − aM ) ∈ Br(kM (t))

be the class of the standard quaternion algebra associated to the element t − aM ∈ kM (t) and the quadratic extension KM (t)/kM (t). For such an M let aM (t) = CoreskM /k (KM /kM , t − aM ) ∈ Br(k(P1k )). The following Lemma is the analogue of Lemma 2.1.2 of [CSS3]. The hypotheses (a),(b) and (c) for the points {Pv }v∈S1 are in force.

Lemma 2.1.2 Let π : X → P1k and S0 (with S1 ⊂ S0 ) bePas above, let XU = π −1 (U), Q and let {Pv } ∈ v∈Ω X(kv ) be an adelic point of X such that v∈Ω invv (A(Pv )) = 0 for all A ∈ Brvert(X). There exists a finite set S of places of k, containing S0, and there exist points Qv ∈ XU (kv ) for v ∈ S, with Qv = Pv for v ∈ S0 (hence for v ∈ S1 ), such that X invv (aM (π(Qv ))) = 0 v∈S

for all M ∈ M0 . Proof This is just a special case of Harari’s ‘formal lemma’ ([Ha1], Cor. 2.6.1; [CTSwD], Theorem 3.2.1). We now have a certain finite set S of places containing S0 . The analogue of Prop. 2.1.3 of [CSS3] is the following proposition (the freedom allowed at the archimedean places in (6) below should also have been allowed in [CSS3].) Proposition 2.1.3 Let π : X → P1k , the set S and the points Qv ∈ X(kv ), v ∈ S, be as in the conclusion of the previous lemma. For each point M ∈ M ∪ ∞, let TM be a finite set of ˜ the places of k such that TM ∩ S = ∅, TM ∩ TN = ∅ for M 6= N . Let λ ∈ U(k) ⊂ P1 (k) and λ 1 associated point in P (oS ). Assume: (1) For each M ∈ M0 and each place v ∈ TM , there is an associated place w of kM (of p degree one over k) which splits in the quadratic extension KM = kM ( m(M))/kM (i.e. m(M) is a square in (kM )w ); p (2) each place v ∈ T∞ splits in each quadratic extension k( NormkM /k (m(M)))/k for each M ∈ M0 (i.e. for each such M, NormkM /k (m(M)) is a square in kv for each v ∈ T∞ ); ˜ is transversal over oS ; (3) λ ˜ and M ˜ , when viewed on M ˜ , consists (4) for any M ∈ M, the (transversal) intersection of λ of the places w ∈ TM and just one place wM of kM , of degree one over a place vM of k; moreover, all vM ’s are distinct from one another and none of them belongs to ∪M ∈M TM ; ˜ and the section at infinity of P1 consists exactly of (5) the transversal intersection of λ oS the places v ∈ T∞ (hence for any such v, we have v(λ) = −1); (6) λ is very close to λv = π(Qv ) ∈ U(kv ) for nonarchimedean places v ∈ S and lies in Uv for v ∈ S1 . S 00 0 0 0 TM and T 00 TM , let T (λ) = T ∪ ( Let T 0 = S ∪ = S ∪ M ∈M M ∈M M ∈M0 {vM }) and S T 00 (λ) = T 00 ∪ ( M ∈M00 {vM }). Let T = T 0 ∪ T 00 and T (λ) = T 0 (λ) ∪ T 00 (λ). Then one has: (i) Xλ (kv ) 6= ∅ for all places v of k; (ii.a) the evaluation map evλ : H(oT 0 ∪T∞ [U 0 ]) → H(oT 0 (λ)∪T∞ ) is an isomorphism, where 0 U = UoT 0 ∪T∞ ; (ii.b) the evaluation map evλ : H(oT 00 ∪T∞ [U 00 ]) → H(oT 00 (λ)∪T∞ ) is an isomorphism, where U 00 = UoT 00 ∪T∞ ; (iii.a) the image of the point P 0 (λ) of order 2 on Eλ0 under the Kummer map Eλ0 (k) → k ∗ /k ∗2 00 and the class of m(λ) in k ∗ /k ∗2 lie in the subgroup I 00T (λ) = H(oT 00 (λ)) and are independent;

(iii.b) the image of the point P 00 (λ) of order 2 on Eλ00 under the Kummer map Eλ00 (k) → 0 k ∗ /k ∗2 lies in the subgroup I 0T (λ) = H(oT 00 (λ) ) and is not trivial. Proof For (i), one uses Lemma 1.3.2 (or 2.1.1) together with Lemma 2.1.2, i.e. the hypothesis that there is no Brauer-Manin obstruction (this is what enables us to get the existence of a local solution at the ‘Schinzel primes’ vM ). For details, see [CSS3], p. 599/600. To prove (ii), one uses two analogues of diagram (2.1.1) of [CSS3], namely one over U 0 and one over U 00 . For the first one, the right hand side group is ⊕M ∈M0 Z/2, for the second one, it is ⊕M ∈M00 Z/2. These are commutative diagrams of exact sequences. 0

→

0

→

H(oT 0 ∪T∞ ) → ↓= H(oT 0 ∪T∞ ) →

H(oT 0 ∪T∞ [U 0 ]) → ⊕M ∈M0 Z/2 → ↓ evλ ↓= H(oT 0 (λ)∪T∞ ) → ⊕M ∈M0 Z/2 →

0 0

and 0 0

→ H(oT 00 ∪T∞ ) → H(oT 00 ∪T∞ [U 00 ]) → ↓= ↓ evλ → H(oT 00 ∪T∞ ) → H(oT 00 (λ)∪T∞ ) →

⊕M ∈M00 Z/2 ↓= ⊕M ∈M00 Z/2

→ 0 → 0

In these two diagrams, the bottom right hand side arrows are given by valuation at the places vM . To prove the independence statement in (iii.a), one then uses Assumption (2) in Theorem A, which is that m and c2 − d are independent (over Z/2) in S00 , that is to say that the class of m is not in the image of Z/2.P 0 under the Kummer map E 0 (k(t)) → k(t)∗ /k(t)∗2 coming form the isogeny E 00 → E 0 . (Recall that E 0 is given by y 2 = (x − c(t))(x2 − d(t)).) Similarly in (iii.b) one uses the fact that the class of d is not trivial in k(t)∗ /k(t)∗2 . That is automatic, since d is assumed nonconstant and squarefree. That the images lie in H(oT 00 (λ)), resp. in H(oT 0 (λ)), i.e. that they do not involve T∞ , is a consequence of the fact that c2 − d and m, resp. d, are of even degree, and that at any finite place v ∈ / S0 their coefficients are integral and their leading coefficient a unit; this implies that their evaluation at any point λ with v(λ) < 0 has even v-adic valuation. 2.2. The groups S0 , S00 and the maps δ 0 and δ 00 Lemma 2.2.1

The maps ∗ ∗2 δ 0 : S0 → ⊕M ∈M00 kM /kM

and ∗ ∗2 δ 00 : S00 → ⊕M ∈M0 kM /kM

given by evaluation at the relevant points M are homomorphisms. The kernels of these maps are finite. Proof The group S0 = k[U 0 ]∗ /k[U 0 ]∗2 is generated by the classes of the rM for M ∈ M0 and by k ∗ /k ∗2 . Since c2 − d is not constant, M00 is not empty. Now for any closed point M, ∗ ∗2 the map k ∗ /k ∗2 → kM /kM has finite kernel. Hence δ 0 has a finite kernel. The argument for the 00 kernel of δ is the same (here we use the fact that d is not constant).

2.3. Independence of the choice of λ: the spaces Given λ ∈ k as in Proposition 2.1.3, let us consider the curves Eλ0 and Eλ00 . They have good reduction at all places outside T (λ) (they indeed have good reduction at places in T∞ ). In Subsection 1.1 we defined I 0T (λ) = H(oT (λ) ) and in Subsection 1.4 we considered the subspace I0T (λ) ⊂ I 0T (λ) , and we defined similarly I00T (λ) ⊂ I 00T (λ) (since λ determines the curves Eλ0 and Eλ00 , I only use the subscript λ). By Lemma 1.4.1, the Selmer group Sλ0 = (Sφ0 )λ , resp. Sλ00 = (Sφ00 )λ (we here simplify the notation of Section 1.5), can be computed as a subgroup of I0T (λ), resp. I00T (λ) . The group I00T (λ) is defined by local conditions, namely, it consists of elements in H(oT (λ)) whose image under the map H(oT (λ) ) → H(kv ) belongs to Wv00 (λ) + Kv , this for all v ∈ T (λ). Similarly for the group I0T (λ). The spaces Kv do not depend on λ. We shall fix the approximation condition (6) in Proposition 2.1.3 so that for v ∈ S, with the notation of Section 1.3, the subspaces Wv00 (λ) and Wv00 (λv ) of Vv00 are identical. For a real place v, the independence of these subspaces follows from the fact that c(c2 − d) does not vanish on the interval Uv . Hence the conditions defining I00T (λ) at the places of S do not depend on λ. For such a λ, from Remark 1.3.3, for v ∈ T 0 (λ) \ S, one has Wv0 (λ)/(Wv0 (λ) ∩ Kv0 ) = Z/2, 00 Wv (λ)/(Wv00 (λ) ∩ Kv00 ) = 0, and Wv00 (λ) + Kv00 = Kv00 = H(ov ); for v ∈ T 00 (λ) \ S, one has Wv00 (λ)/(Wv00 (λ) ∩ Kv00 ) = Z/2, Wv0 (λ)/(Wv0 (λ) ∩ Kv0 ) = 0 and Wv0 (λ) + Kv0 = Kv0 = H(ov ) ; and for v ∈ / T (λ) = T 0 (λ) ∪ T 00 (λ), one has Wv0 (λ)/(Wv0 (λ) ∩ Kv0 ) = 0 and Wv0 (λ) + Kv0 = Kv0 = H(ov ), 00 Wv (λ)/(Wv00 (λ) ∩ Kv00 ) = 0 and Wv00 (λ) + Kv00 = Kv00 = H(ov ). 0 0 For such a λ, the natural embeddings I0T (λ) ⊂ I0T (λ) and I00T (λ) ⊂ I00T (λ) are isomorphisms, and the natural projection maps WT0 (λ) → WT0 0 (λ) and WT00 (λ) → WT00 00 (λ) are isomorphisms. 0

00

In particular we have inclusions Sλ0 ⊂ I0T (λ) ⊂ H(oT 0 (λ)) and Sλ00 ⊂ I00T (λ) ⊂ H(oT 00 (λ)). 0 00 From Section 1.4 recall the isomorphisms τλ0 : I0T (λ) ' WT 0 (λ) and τλ00 : I00T (λ) ' WT 00 (λ). From the above description of the sets Wv0 (λ), Wv00 (λ), Kv0 , Kv00 , one deduces the following lemma.

Lemma 2.3.1 Let T 0 , T 00 , T and λ be as in Proposition 2.1.3. (a) There are natural inclusions H(oS ) ⊂ H(oT 0 ) ⊂ H(oT 0 (λ)) ⊂ H(oT (λ) ). The intersections of the subgroup I0T (λ) ⊂ H(oT (λ) ) with these various subgroups are pre0 0 cisely I0S ⊂ I0T ⊂ I0T (λ) ⊂ I0T (λ) . 0 The group N00 := I0S depends neither on λ nor on the primes in T (λ). The group I0T only depends on T 0 = S ∪M ∈M0 TM , it depends neither on λ nor on the places vM . (b) We have the analogous statements for the natural inclusions H(oS ) ⊂ H(oT 00 ) ⊂ H(oT 00 (λ) ) ⊂ H(oT (λ) ) 00

for the group N000 := I00S , and for the group I00T . The proof to follow will involve the construction of ‘subdiagrams’ of the diagrams appearing in the proof of Prop. 2.1.3. Basically, we want to look at the inverse images under evλ of 0 0 00 00 I0T (λ) ⊂ H(oT 0 (λ)∪T∞ ) = I 0T (λ)∪T∞ and of I00T (λ) ⊂ H(oT 00 (λ)∪T∞ ) = I 00T (λ)∪T∞ . Before doing this, we shall fix T∞ . Consider the finite subgroup S00S of S00 spanned by I 00S and the elements rM (t) for M running through M00 . This subgroup contains m(t). For each M ∈ M0 , we have the finite 00 ∗ ∗2 subgroup δM (S00S ) ⊂ kM /kM , which defines a multiquadratic extension FM /kM . Consider also 0 the finite subgroup SS of S0 spanned by I 0S and the elements rM (t) for M running through 0 ∗ ∗2 M0 . For each M ∈ M00 , we have the finite subgroup δM (S0S ) ⊂ kM /kM , which defines a multiquadratic extension FM /kM .

Let F/k be the composite of all the extensions FM /k. It is a consequence of Tchebotarev’s theorem that there exist infinitely many prime principal ideals (πv ) in the ring of integers ok such that the (chosen) generator πv is a square (as a matter of fact, as close as one wishes to 1) in each completion p kw for w ∈ S, and such that moreover the place v splits in each quadratic extension k( NormkM /k (m(M))/k. Choose v∞ ∈ / S a finite place which satisfies these properties (required in Proposition 2.1.3), and which moreover is unramified in the extension F/k (note: v∞ is not an archimedean place, the index ∞ refers to the point at infinity on P1 ). Let µ = πv∞ ∈ ok be a generator of the corresponding principal ideal. For each w ∈ S, µ is a square in kw . For the rest of Section 2, we fix the set T∞ = {v∞ }, and we let λ∞ = λv∞ = 1/µ ∈ kv∞ . Lemma 2.3.2 Let notation be as in Proposition 2.1.3, with T∞ = {v∞ } and µ as above. (a) For each M ∈ M0 , there exists a uniquely defined a0M ∈ I 0S such that for any T 0 and λ 0 0 0 as in Proposition 2.1.3, a0M .µdeg(rM ) .rM (λ) ∈ I T (λ)∪T∞ is in the subspace of I0T (λ) ⊂ I T (λ)∪T∞ defined by the additional conditions that the local projection at v ∈ S belongs to Kv0 . (b) For each M ∈ M00 , there exists a uniquely defined a00M ∈ I 00S such that for any T 00 00 00 and λ as in Proposition 2.1.3, a00M .µdeg(rM ) .rM (λ) ∈ I T (λ)∪T∞ is in the subspace of I00T (λ) ⊂ 00 I T (λ)∪T∞ defined by the additional conditions that the local projection at v ∈ S belongs to Kv00 . Proof For any a0M ∈ I 0S , the valuation of a0M .µdeg(rM ) .rM (λ) at a place v ∈ / T 0 (λ) is even (for v = v∞ and rM of odd degree, this follows from the transversality conditions (3) 0 and (5) of Prop. 2.1.3). Thus any such a0M .µdeg(rM ) .rM (λ) belongs to I 0T (λ) . For M1 ∈ M0 with M1 6= M, and v ∈ TM1 ∪ {vM1 }, the local condition a0M .µdeg(rM ) .rM (λ) ∈ Wv0 (λ) + Kv0 is satisfied, since at any such place v the valuation v(a0M .µdeg(rM ) .rM (λ)) is even and for v ∈ / S, Kv0 consists of the classes in kv∗ /kv∗2 with even valuation. For any v ∈ TM ∪ {vM }, the condition a0M .µdeg(rM ) .rM (λ) ∈ Wv0 (λ) + Kv0 is also satisfied. Indeed at such a place we have Wv0 (λ) = kv∗ /kv∗2 (Remark 1.3.3). Since λ is v-adically very close to λv for v ∈ S (condition (6) in Prop. 2.1.3), the class of rM (λ) ∈ kv∗ /kv∗2 does not depend on λ. Since µ is a square in each kv for v ∈ S, the image of of µdeg(rM ) .rM (λ) in VS0 is independent of λ, T and T∞ . We have I 0S ⊕ KS0 = VS0 . There thus exists a uniquely defined a0M ∈ I 0S such that a0M .µdeg(rM ) .rM (λ) ∈ 0 0 0 I 0T (λ)∪T∞ is in the subspace of I0T (λ) ⊂ I 0T (λ)∪T∞ defined by the additional conditions that the local projection at each v ∈ S belongs to Kv0 . This completes the proof of (a). Mutatis mutandis, this also proves (b). For each M ∈ M0 , resp. M ∈ M00 , let a0M , resp. a00M , be the elements of I S determined in the previous lemma. We now define: a0M (t) = (a0M .µdeg(rM ) .rM (t)) ∈ H(oS∪T∞ [U 0 ]) ⊂ S0 for M ∈ M0 and similarly a00M (t) = (a00M .µdeg(rM ) .rM (t)) ∈ H(oS∪T∞ [U 00 ]) ⊂ S00 for M ∈ M00 .

Notation 2.3.3 Let A0 ⊂ H(oS∪T∞ [U 0 ]) ⊂ H(k[U 0 ]) = S0 be the subspace spanned by 0 0 the a0M (t) for M ∈ M0 . The subgroup A0 and I0T = I0T (λ) ∩ H(oT 0 ) of H(oT 0 ∪T∞ [U 0 ]) clearly 0 have trivial intersection. Let N 0 = A0 ⊕ I0T ⊂ H(oT 0 ∪T∞ [U 0 ]) be their direct sum. Let A00 and N 00 be defined similarly. The definition of N 0 depends on the choice of the family of subspaces Kv0 (fixed once and for all) as well as on the points (Lemma 2.1.2) Qv ∈ X(kv ), v ∈ S (points which determine the

spaces Wv (λ) = Wv (Qv ) for v ∈ S), as well as on T 0 and T∞ . But once these choices have been made, Lemma 2.3.1 shows that N 0 does not depend on the particular point λ chosen as in Proposition 2.1.3. More precisely, N 0 does not depend on the vM ’s (M ∈ M0 ). If we fix T∞ and µ as above, but still let the TM ’s and vM ’s for M ∈ M0 vary, there is a subspace of H(k[U 0 ]) which is contained in any N 0 , namely the subspace A0 ⊕N00 ⊂ H(oS∪T∞ [U 0 ]), where N00 = I0S is as above and may be defined as N00 = N 0 ∩ H(oS ) ⊂ H(oT 0 ∪T∞ [U 0 ]). Both N00 and A0 are independent of T , and they clearly do not intersect. The same considerations apply for N 00 , N000 , A00 . As we observed in the proof of part (ii) of Proposition 2.1.3, the evaluation maps evλ : H(oT 0 ∪T∞ [U 0 ]) → H(oT 0 (λ)∪T∞ ) and evλ : H(oT 00 ∪T∞ [U 00 ]) → H(oT 00 (λ)∪T∞ ) are isomorphisms as soon as properties (3) and (4) in that Proposition hold. We have the exact analogue of Prop. 2.3.4 of [CSS3], to which we refer for the proof. Proposition 2.3.4 Let T 0 , T 00 , T∞ and λ be as in Proposition 2.1.3, and let N 0 and N 00 be as above. (a’) The evaluation map evλ from H(oT 0 ∪T∞ [U 0 ]) to H(oT 0 (λ)∪T∞ ) induces an isomorphism 0 between N 0 and I0T (λ). (a”) The evaluation map evλ from H(oT 00 ∪T∞ [U 00 ]) to H(oT 00 (λ)∪T∞ ) induces an isomorphism 00 between N 00 and I00T (λ) . 0 ' (b’) Denote by ϕ0λ the composition of this isomorphism with the isomorphism I0T (λ) −→ WT0 0 (λ) in Section 1.4. There is a direct sum decomposition: 0 0 0 N 0 = N00 ⊕ A0 ⊕ ϕ−1 λ (⊕v∈T 0 \S (Wv (λ)/Wv (λ) ∩ Kv )).

(b”) We have the analogous statement for ϕ00λ . Q The class of d(t) ∈ H(oS [U 0 ]) may clearly be written as d(t) = ε. M ∈M0 a0M (t) with ε ∈ I 0S∪T∞ . Since the degree of d is even, one actually has ε ∈ I 0S . Since d(λ) ∈ k ∗ /k ∗2 belongs to the image Wλ0 of Eλ00 (k) → k ∗ /k ∗2 , ε belongs to I0S . From the above proposition we conclude that the class of d belongs to N 0 , more precisely that d belongs to N00 ⊕ A0 . The same argument shows that (c2 − d)(t) ∈ H(oS [U 00 ]) belongs to N000 ⊕ A0 ⊂ N 00 . The same argument also applies to m(t) viewed as an element in H(oS [U 00 ]). Indeed, the degree of m is even, m is the product of some of the rM (t) for M ∈ M00 by an element in o∗S , and the curve Xλ = X(m)λ has points in each kv , which implies that m(λ) belongs to Wv00 (λ) 00 for each place v, i.e. m(λ) ∈ I00T (λ). By the above proposition, the class of m(t) belongs to N 00 . Now m(t) divided by the appropriate product of elements a00M (t) is an element of I 00S , and it belongs to N 00 , thus it lies in N000 . 2.4. Independence of the choice of λ: the pairing At this point, for any λ as in Proposition 2.1.3, we have a pairing

0

e:N ×N

00

(ev0λ ,ev00 λ)

−−−−−→

I

0T 0 (λ)

×I

00T 00 (λ)

(id,τT−1 00 (λ) )

−−−−−−→ I0T

0

(λ)

× WT00 00 (λ)

eT (λ)

−−→

Z/2

which may also be read (ev0λ ,ev00 λ)

N 0 × N 00 (Recall that I0T

−−−−−→ 0

(λ)

I0T

0

(λ)

= I0T (λ), I00T

× I00T

00

(λ)

00

(λ)

→ WT0 0 (λ) × WT00 00 (λ)

eT (λ)

−−→

Z/2.

= I00T (λ), WT0 0 (λ) = WT0 (λ) and WT00 00 (λ) = WT00 (λ).)

Proposition 2.4.2 Let T 0 , T 00 , T∞ = {v∞ } and λ be as in Proposition 2.1.3, and let N 0 , N 00 be as above. Then provided λ is close enough to λ∞ = 1/µ at v∞ , the restriction of the pairing e to a pairing (N00 ⊕ A0 ) × (N000 ⊕ A00 ) depends neither on T 0 , T 00 nor on λ. Proof

It is identical to the one given in given in [CSS3], Section 2.4. The only differences

are: (1) We here have pairings kv∗ /kv∗2 ×kv∗ /kv∗2 → Z/2 rather than (kv∗ /kv∗2 )2 ×(kv∗ /kv∗2 )2 → Z/2. (2) We have to pair a0P and a00Q for P ∈ M0 and Q ∈ M00 , hence in particular P 6= Q : so the only computation one has to reproduce is the one given on pages 610, 611 and the first lines of page 612 of [CSS3]. 2.5. Completion of the proof of Theorem A We have reached Section 2.5 of [CSS3] (page 612); I now transcribe the end of Section 2 of [BSwD]. ∗ ∗2 For each r ∈ N 0 , if r 6= 1, d, there exists M ∈ M00 such that r(M) 6= 1 ∈ kM /kM . If r is of even degree, this is hypothesis (3.a) of the theorem. If r is of odd degree, one remarks that r is the product of an element of I 0S an odd power of µ and a product of rM (t)’s for M ∈ M0 . For 0 any M ∈ M00 , the class δM (r) is then nontrivial, since µ has been so chosen that it does not 0 belong to the image under δM of the group spanned by I 0S and the rM ’s (M ∈ M0 ). For such an M there exist infinitely many primes v M of kM which are of degree one over k (we pdenote by v the induced place on k) and which remain inert in the quadratic extension kM ( r(M))/kM . We shall apply this to each element different from 1, d in the left kernel of the restriction of e to N00 ⊕ A0 × N000 ⊕ A00 . In this way, we produce distinct primes of the above type (one for each element of N00 ⊕ A0 under consideration). Let v M be one such prime, let v ˜ at v M . To be the underlying prime of k and let λv ∈ ov be transversal to the corresponding M 0 00 0 00 such a λv are associated curves Eλv , Eλv and groups Wv (λv ), Wv (λv ). Assertion: there exists m00v ∈ Wv00 ⊂ kv∗ /kv∗2 which has a non-zero cup-product with r(λv ) ∈ ∗ kv /kv∗2 . This follows from the computations in Remark 1.3.3 (Lemma 4 in [BSwD]): for λv as above, one has v((c2 −d)(λv )) = 1, v(d(λv )) = 0 hence v(r(λv )) = 0 (all this modulo 2), the cupproduct is simply given by (r(λv ), m00v ), it is non-zero if v(m00v ) = 1 (possible since Wv00 = kv∗ /kv∗2 ) and p r(λv ) ∈ o∗v is not a square, which it is not since v M is inert in the quadratic extension kM ( r(M))/kM . For each r ∈ N 00 ⊂ H(k(t)), by hypothesis (3.b) of the theorem if the degree of r is even, or by the analogue of the above argument if the degree of r is odd, if r does not belong to the group spanned by c2 − d and m, there exists an M ∈ M0 such that r(M) does not belong to ∗ ∗2 (1, m(M)) ⊂ kM /kM . For this M ∈ M0 there exist infinitely many places v M of kM of degree one p over k (we denote by v the induced place on k), which are splitpin the quadratic extension kM ( m(M))/kM but remain inert in the quadratic extension kM p( r(M))/kM . M (Note: at this point we do want the places v to split in kM ( m(M))/kM , see Proposition 2.1.3 (1): this is to ensure that Xλ (kv ) 6= ∅ at such places.)

We apply this to each element of N000 ⊕A00 which does not belong to the subgroup {c2 −d, m} and which is in the right kernel of the restriction of e to N00 ⊕ A0 × N000 ⊕ A00 . In this way, we produce distinct primes of the above type (one for each element of N000 ⊕A00 under consideration). Let v M be one such prime, let v be the underlying prime of k and let λv ∈ ov be transversal to ˜ at v M . To such a λv are associated curves E 0 , E 00 and groups W 0 , W 00 . the corresponding M v v λv λv Assertion: there exists m0v ∈ Wv0 ⊂ kv∗ /kv∗2 which has a non-zero cup-product with r(λv ) ∈ kv∗ /kv∗2 . This follows from Remark 1.3.3: we have here v(d(λv )) = 1 hence v((c2 − d)(λv )) = 0 and v(r(λv )) = 0 (all this modulo 2), the cup-product is simply given by (m0v , r(λv )). This is ∗ nonzero if v(m0v ) = 1 (which occurs, by Remark 1.3.3) and r(λ v ) ∈ ov is not a square, which it p is not since v M remains inert in the quadratic extension kM ( r(M))/kM . Let us now consider the restriction of the pairing e to (N00 ⊕ A0 ) × (N000 ⊕ A00 ). Recall that this restriction does not depend on the (not yet made) choice of T 0 , T 00 , λ. The classes {d}, resp. {c2 − d}, {m} lie in N00 ⊕ A0 , resp N000 ⊕ A00 , as was pointed out after Prop. 2.3.4. We let N00 ⊕ A0 = F00 ⊕ F10 ⊕ F20 , where F00 = Z/2{d} and F00 ⊕ F10 is the left kernel of the pairing (N00 ⊕ A0 ) × (N000 ⊕ A00 ) → Z/2. We let N000 ⊕ A00 = F000 ⊕ F100 ⊕ F200 , where F000 = Z/2{c2 − d} ⊕ Z/2{m} and F000 ⊕ F100 is the right kernel of the pairing (N00 ⊕ A0 ) × (N000 ⊕ A00 ) → Z/2. Note that the pairing then induces a pairing F20 × F200 which has trivial kernel on both sides. We now proceed as on pages 615-616 of [CSS3]. To each nonzero element in r ∈ F10 we may by the above procedure associate an M ∈ M00 , a place vM in kM , of degree one over k, a transversal λv and an element m00v such that (r(λv ), m00v ) = 1 ∈ Z/2. Let S20 be the set of places of k thus produced. The bilinear pairing F10 ×(⊕v∈S20 Z/2) → Z/2 sending the pair (r, cv ) to the cup-product (r(λv ), cv ) ∈ Z/2 is thus nondegenerate on the left hand side, one may therefore extract from S20 a subset S2 of order exactly the rank of the Z/2-vector space F10 , such that the induced bilinear pairing F10 × (⊕v∈S2 Z/2) → Z/2 (sending the pair (r, 1v ) to r(λv ) ∈ o∗v /o∗2 v = Z/2) is nondegenerate on both sides. We now fix such a set S2 and the corresponding sets TM for M ∈ M00 . We also fix the λv ’s as above. We similarly proceed with F100 . We thus get sets TM ’s for suitable M ∈ M0 , consisting of places v M of degree one over the corresponding place of k (which we may assume distinct from the places appearing in S ∪S2 ) and associated λv ’s. This defines a set S3 of places of k such that the bilinear pairing (⊕v∈S3 Z/2) × F100 → Z/2 (sending the pair (1v , r) to r(λv ) ∈ o∗v /o∗2 v = Z/2) is nondegenerate on both sides. We now fix such a set S3 and the corresponding sets TM for M ∈ M0 . We also fix the λv ’s as above. If we now choose a λ as in Schinzel’s hypothesis (in the (H1 ) version, due to Serre, see [CTSwD], p. 71), close enough to the chosen λv ’s for v finite in S ∪ T∞ ∪ S2 ∪ S3 , and positive at the real places of k (at the real places, only the analogous condition should have been imposed on pages 598 and 617 of [CSS3]), then the associated pairing eλ , whose kernels by Prop. 1.4.3 determine the Selmer groups Sλ0 and Sλ00 , breaks up as (F00 ⊕ F10 ⊕ F20 ⊕ F30 ) × (F000 ⊕ F100 ⊕ F200 ⊕ F300 ) → Z/2, and we have the following properties. F00 is in the left kernel and F000 in the right kernel. F10 is orthogonal to F000 ⊕ F100 ⊕ F200 and F100 is orthogonal to F00 ⊕ F10 ⊕ F20 . The pairing F20 × F200 → Z/2 is nondegenerate. The pairing F10 × F300 → Z/2 is nondegenerate. The pairing F30 × F100 → Z/2 is nondegenerate. This implies that the left kernel of the total pairing is F00 = Z/2, spanned by the image d(λ) ∈ k ∗ /k ∗2 of P 00 (λ) ∈ Eλ00 (k) under the Kummer map, and that the right kernel of the total

pairing is F000 = Z/2 ⊕ Z/2, spanned by m(λ) and the class (c2 − d)(λ) ∈ k ∗ /k ∗2 of the image of P 0 (λ) ∈ Eλ0 (k) under the Kummer map. It then only remains to apply 1.5. Remark The whole discussion with T∞ and µ could be avoided if one made a stronger assumption in Theorem A, namely if one had hypothesis (3.a) for the map δ 0 on the whole group S0 and similarly for the hypothesis (3.b) for the map δ 00 on the whole group S00 . 3. Proof of Theorem B The proof is very close to the proof of Theorem A. Only the points which differ will be described. On the family X/P1k of Theorem B one first realizes a reduction analogous to the one described in 2.1. We have here several sets of closed points M ∈ A1k . The set M is defined by the vanishing of the separable polynomial ∆ = d01 (d223 + 4d24 d34 )(d204 − d02 d03 )(d214 − d12 d13 ). The set M01 is defined by the vanishing of d01 . The set M02 is defined by the vanishing of (d223 + 4d24 d34 ). Their union is the set M0 . The set M00 is defined by the vanishing of the product (d204 − d02 d03 )(d214 − d12 d13 ). To any closed point M ∈ M01 , with residue field kM , we attach two quadratic extensions. The first one, KM /kM is obtained by adding the square root of the value at M of (d204 − d02d03 ), or of (d214 − d12 d13 ) (at a point M ∈ M0 , the product of these two values is a square in kM ). The second extension, LM /kM , is the one obtained by adding the square root of the value of −c in kM (the value of c is given in the Introduction and repeated in Section 1.2). To a closed point M ∈ M00 , we associate a quadratic extension KM /kM . If (d204 − d02 d03 ) vanishes at M, then KM is obtained by adding the square root of the value of −d01 d02 . If (d214 − d12 d13 ) vanishes at M, then KM is obtained by adding the square root of the value of d01 d12 . We do not attach any quadratic extension to KM if M belongs to M02 . One defines a set S0 as in Section 2.1 (containing any preassigned finite set of places), with an associated nice model X /P1oS over the ring oS0 of S0 -integers. 0

˜ ⊂ P1 (ov ) be Lemma 3.1.1 Let v ∈ / S0 and let λ ∈ A1 (kv ) be a transversal point. Let λ its closure. ˜ does not intersect any of the M ˜ for M ∈ M0 ∪ M00 , then Xλ (kv ) 6= ∅. (a) If λ 1 ˜ intersects M ˜ for M ∈ M0 ∪ M00 at some place w of kM (w unramified over v, of (b) If λ 1 degree one over v), then Xλ (kv ) 6= ∅ if and only if w splits in KM . Note that the extension LM /kM does not play a rˆ ole at this stage. let

Given M ∈ M, we write rM (t) = NormkM /k (t − aM ), with aM ∈ kM . For M ∈ M01 ∪ M00 , (KM /kM , t − aM ) ∈ Br(kM (t))

be the class of the standard cyclic algebra associated to the element t − aM ∈ kM (t) and the cyclic extension KM (t)/kM (t). Let aM (t) = CoreskM /k (KM /kM , t − aM ) ∈ Br(k(P1k )).

For M ∈ M01 , let similarly bM (t) = CoreskM /k (LM /kM , t − aM ) ∈ Br(k(P1k )). −1 Lemma 3.1.2 Let π : X → P1k and S0 be Pas above, let XU = π (U), and let {Pv } ∈ v∈Ω X(kv ) be an adelic point of X such that v∈Ω invv (A(Pv )) = 0 for all A ∈ Brvert (X). There exists a finite set S of places of k, containing S0 , and there exist points Qv ∈ XU (kv ) for v ∈ S, with Qv = Pv for v ∈ S0 , such that X invv (aM (π(Qv ))) = 0

Q

v∈S

for all M ∈ M01 ∪ M00 and such that X

invv (bM (π(Qv ))) = 0

v∈S

for all M ∈ M01 .

Proposition 3.1.3. Let π : X → P1k , the set S of places and the Qv ∈ X(kv ), v ∈ S, be as in the conclusion of the previous lemma. For each point M ∈ M ∪ ∞, let TM be a finite set of places of k such that TM ∩ S = ∅, TM ∩ TN = ∅ for M 6= N . Let λ ∈ U(k) ⊂ P1 (k) and ˜ the associated point in P1 (oS ). Assume: λ (1) For each M ∈ M01 and each place v ∈ TM , there is an associated place w of kM (of degree one over k) which splits in the quadratic extension KM /kM and in the quadratic extension LM /kM ; for each M ∈ M00 and each place v ∈ TM , there is an associated place w of kM (of degree one over k) which splits in the quadratic √ extension KM /kM ; √ (2) Write KM = p kM ( αM ) and LM = kM (p βM ); each place v ∈ T∞ splits in each quadratic extension k( NormkM /k (αM ))/k and k( NormkM /k (βM ))/k; ˜ is transversal over oS ; (3) λ ˜ and M ˜ , when viewed on M ˜ , consists (4) for any M ∈ M, the (transversal) intersection of λ of the places w ∈ TM and just one place wM of kM , of degree one over a place vM of k; ˜ and the section at infinity of P1 consists exactly of (5) the transversal intersection of λ oS the places v ∈ T∞ (hence for any such v, we have v(λ) = −1); (6) λ is very close to λv = π(Qv ) ∈ U(kv ) for nonarchimedean places v ∈ S, and lies in Uv for v ∈ S1 . S Let T 0 = S ∪M ∈M0 TM and T 00 = S ∪M ∈M00 TM , let T 0 (λ) = T 0 ∪ ( M ∈M0 {vM }) and S T 00 (λ) = T 00 ∪ ( M ∈M00 {vM }). Let T = T 0 ∪ T 00 and T (λ) = T 0 (λ) ∪ T 00 (λ). Then one has: (i) Xλ (kv ) 6= ∅ for all places v of k; (ii.a) the evaluation map evλ : H(oT 0 ∪T∞ [U 0 ]) → H(oT 0 (λ)∪T∞ ) is an isomorphism, where U 0 = UoT 0 ∪T∞ ; (ii.b) the evaluation map evλ : H(oT 00 ∪T∞ [U 00 ]) → H(oT 00 (λ)∪T∞ ) is an isomorphism, where U 00 = UoT 00 ∪T∞ ; 00 (iii.a) the classes of (d204 − d02 d03 )(λ) and (d214 − d12 d13 )(λ) lie in the subgroup I 00T (λ) = H(oT 00 (λ)) ⊂ k ∗ /k ∗2 and are independent; 0 (iii.b) the class of (d223 + 4d24 d34 )(λ) lies in the subgroup I 0T (λ) = H(oT 0 (λ)) ⊂ k ∗ /k ∗2 and is not trivial. Just as in Section 2, one uses two analogues of diagram (2.1.1) of [CSS3], namely one over U and one over U 00 (same open sets as in Section 2). For the first one, the right hand side group 0

is ⊕M ∈M0 Z/2, for the second one, it is ⊕M ∈M00 Z/2. These are commutative diagrams of exact sequences.

0

→

0

→

H(oT 0 ∪T∞ ) → ↓= H(oT 0 ∪T∞ ) →

H(oT 0 ∪T∞ [U 0 ]) → ⊕M ∈M0 Z/2 → ↓ evλ ↓= H(oT 0 (λ)∪T∞ ) → ⊕M ∈M0 Z/2 →

0 0

and 0 0

→ H(oT 00 ∪T∞ ) → H(oT 00 ∪T∞ [U 00 ]) → ↓= ↓ evλ → H(oT 00 ∪T∞ ) → H(oT 00 (λ)∪T∞ ) →

⊕M ∈M00 Z/2 ↓= ⊕M ∈M00 Z/2

→ 0 → 0

In these two diagrams, the bottom right hand side arrows are given by valuation at the places vM . Statements (iii.a) and (iii.b) can be reformulated exactly as in Proposition 2.1.3, in terms of the curves Eλ0 and Eλ00 . One now looks for a λ ∈ A1 (k) such that 3.1.3 holds (this will be provided by Schinzel’s hypothesis) and such that the 2-torsion subgroup of the Tate-Shafarevich group of Eλ0 is trivial. Propositions 1.5.2 and 3.1.3 then enable one to conclude. We are nearly in the same situation as we were in Section 2.3, except that d = d201 .(d223 + 4d24 d34 ) ∈ k[t] contains a square factor. One first has to check the “independence of the spaces” on the choice of λ. A key point here is to check that for each M ∈ M and each v ∈ T (λ) \ S, the spaces Wv0 and Wv00 are exactly the same as they were in Section 2. Suppose M ∈ M01 , i.e. d01 (M) = 0. For each v ∈ TM ∪ {vM }, we have v(d(M)) = 2, and the extension LM /kM is split at v (for v = vM , the last statement follows from the reciprocity law.), i.e. the class of −c(M) is a square at the completion of kM at v. It then follows from ∗ ∗2 00 Remark 1.3.3 (iv) that Wv00 = 1 and Wv0 = kM,v /kM,v , hence, since Kv0 = o∗v /o∗2 / S, v = Kv for v ∈ 00 00 00 0 0 0 Wv /(Kv + Wv ) = 0 and Wv /(Wv + Kv ) = Z/2. Suppose M ∈ M02 , i.e. (d223 + 4d24 d34 )(M) = 0. For each v ∈ TM ∪ {vM }, we have v(d(M)) = 1. Remark 1.3.3 (ii) shows that we are exactly in the same situation as above. Suppose M ∈ M00 . For each v ∈ TM ∪ {vM }, we have v(d(M)) = 0 and v((c2 − d)(M)) = ∗2 ∗ /kM,v , hence finally v(∆00 (M)) = 1. By Remark 1.3.3 (iii), we have Wv0 = 1, Wv00 = kM,v 0 0 0 00 00 00 Wv /(Wv + Kv ) = 0 and Wv /(Kv + Wv ) = Z/2. Now the situation is essentially identical to the one considered at the begining of Section 2.3, and the rest of the proof of Theorem B is just the same as that of Theorem A, hence shall not be reproduced. Acknowledgements This paper is a close variation on the paper [BSwD] of Bender and Swinnerton-Dyer, some parts of which have been simply reproduced.

Work for the paper was started on a hike in the Gers. Most of the writing was done in January 2000, while the author was staying at the Tata Institute of Fundamental Research (Mumbai, India), under the auspices of the Centre franco-indien pour la promotion de la recherche avanc´ee (CEFIPRA/IFCPAR, Project 1601-2). The author conveys his thanks to the referee for numerous and insightful remarks.

References [BSwD] A. O. Bender and Sir Peter Swinnerton-Dyer, Solubility of certain pencils of curves of genus 1, and of the intersection of two quadrics in P4 . Preprint, 1999. To appear in Proc. of the London Math. Soc. [CT] J.-L. Colliot-Th´el`ene, The Hasse principle in a pencil of algebraic varieties, in Number Theory, Proceedings of a conference held at Tiruchirapalli, India, January 1996, K. Murty and M. Waldschmidt ed., Contemp. math. 210 (1998) 19-39. [CTSa] J.-L. Colliot-Th´el`ene et J.-J. Sansuc, Fibr´es quadratiques et composantes connexes r´eelles, Math. Ann. 244 (1979) 105-134. [CTSaSwD] J.-L. Colliot-Th´el`ene, J.-J. Sansuc and Sir Peter Swinnerton-Dyer, Intersections of two quadrics and Chˆ atelet surfaces, I, J. reine angew. Math. (Crelle) 373 (1987), 37-107; II, J. reine angew. Math. (Crelle) 374 (1987) 72-168. [CSS1] J.-L. Colliot-Th´el`ene, A. N. Skorobogatov and Sir Peter Swinnerton-Dyer, Double fibres and double covers: paucity of rational points, Acta Arithmetica LXXIX (1997) 113-135. [CSS2] J.-L. Colliot-Th´el`ene, A. N. Skorobogatov and Sir Peter Swinnerton-Dyer, Rational points and zero-cycles on fibred varieties: Schinzel’s hypothesis and Salberger’s device, J. reine angew. Math. (Crelle) 495 (1998) 1-28. [CSS3] J.-L. Colliot-Th´el`ene, A. N. Skorobogatov and Sir Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. math. 134 (1998) 579-650. [CTSwD] J.-L. Colliot-Th´el`ene and Sir Peter Swinnerton-Dyer, Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, J. reine angew. Math. (Crelle) 453 (1994) 49-112. [Ha1] D. Harari, M´ethode des fibrations et obstruction de Manin, Duke Math. J. 75 (1994) 221-260. [Ha2] D. Harari, Fl`eches de sp´ecialisations en cohomologie ´etale et applications arithm´etiques, Bull. Soc. math. France 125 (1997) 143-166. [Mi] J. S. Milne, Arithmetic Duality Theorems (Academic Press, 1986). [Sk], A. N. Skorobogatov, Descent on fibrations over the projective line, Amer. J. Math. 118 (1996) 905-923. [SwD1] Sir Peter Swinnerton-Dyer, Rational points on certain intersections of two quadrics, in Abelian Varieties, ed. Barth, Hulek and Lange, 273-292 (Walter de Gruyter, Berlin 1995). [SwD2] Sir Peter Swinnerton-Dyer, Some applications of Schinzel’s hypothesis to Diophantine equations. Number theory in progress, Vol. 1 (Zakopane-Ko´scielisko, 1997), 503–530 (Walter de Gruyter, Berlin, 1999). [SwD3] Sir Peter Swinnerton-Dyer, Arithmetic of diagonal quartic surfaces, II. Proc. London Math. Soc. (3) 80 (2000) 513-544. [SwD4] Sir Peter Swinnerton-Dyer, The solubility of diagonal cubic surfaces, a` paraˆıtre dans les ´ Annales scientifiques de l’Ecole Normale Sup´erieure.

J.-L. Colliot-Th´ el` ene C.N.R.S. UMR 8628, Math´ ematiques Bˆ atiment 425 Universit´ e de Paris-Sud F-91405 Orsay France e-mail : [email protected]

Enriques surfaces with a dense set of rational points, appendix to the paper by J.-L. Colliot-Thelene Alexei Skorobogatov The aim of this note is to deduce from the results of Bender and Swinnerton-Dyer [BS], as re ned by Colliot-Thelene [CT], that rational points are Zariski dense on certain Enriques surfaces de ned over a number eld k, conditionally on the Schinzel Hypothesis (H) and the niteness of Tate{Shafarevich groups of elliptic curves over k. It was shown by Bogomolov and Tschinkel [BT] that for any Enriques surface Y de ned over a number eld k there exists a nite extension K=k such that K -points are Zariski dense on Y (\potential density" of rational points). We intend to show that the results of [BS] and [CT] can be used to construct explicit families of Enriques surfaces over any number eld k with the property that already k-points are Zariski dense. Although the general construction is conditional on the above mentioned conjectures, once an equation is written it is often possible to give a direct (and unconditional) proof of the Zariski density of rational points. We check this for the explicit example in the end of this note using an idea from [BT]. It is not known whether or not there exists an Enriques or K 3-surface X over a number eld k such that X (k) is not empty but not Zariski dense in X . Proposition 1. Let k be a eld of characteristic zero. Let c; d 2 k[t] be polynomials, deg (c) = 2, deg (d) = 4, deg (c2 d) = 4, such that td(c2 d) is separable. Then there exists a regular minimal model Y of the aÆne surface y12 = t(x

c(t)); y22 = t(x2

d(t))

(1)

which is an Enriques surface (hence Y is unique). Any K 3 double covering of Y is given by t = mu2 , for some m 2 k . Proof. A standard argument (cf. [CSS97], Section 4) shows that for any discrete valuation v : k(Y ) ! Z, which is trivial on k, the valuation v (t) is even. Indeed, if v (t) > 0, then v (d) = v (c2 d) = 0 by our assumption. If v (x) 6= 0, then from the second equation (1) we see that v (t) is also even. The case v (x) = 0, v (x c) > 0, v (x2 d) > 0 leads to v (c2 d) > 0, which is a contradiction. Hence either v (x c) = 0 or v (x2 d) = 0, implying that v (t) is even. Now if v (t) = l < 0, then v (c) = 2l, v (d) = 4l, v (c2 d) = 4l. It is clear from the second equation (1) that if v (x) 6= 2l, then v (t) is even. Suppose v (x) = 2l. In the case when v (x2 d) > 4l and v (x c) > 2l, we have v (c2 d) > 4l, which is a contradiction. Hence at least one of x2 d and x c has even valuation, implying that v (t) is even in all cases. It follows that the divisor (t) on any regular model of (1), in particular, on Y , is a double, that is, (t) = 2D for some D 2 Div (Y ). Hence the double covering X (m) ! Y given by t = mu2 , m 2 k , is unrami ed. Let y1 = uw1 , y2 = uw2 , then X (m) is given by the aÆne equations

w12 = m(x

c(mu2 )); w22 = m(x2

d(mu2 ))

(2)

The aÆne surface (2) is isomorphic to

w22 = m 1 w14 + 2c(mu2 )w12 + m(c(mu2 )2 1

d(mu2 ))

(3)

Note that 0 is not a root of d(t) and c(t)2 d(t), hence c(mu2 ) and c(mu2 )2 d(mu2 )) are separable. Then it is easy to check that the aÆne surface (3) is regular. The projection to the coordinate u gives rise to a proper morphism  : X (m) ! P1k . It is an easy local calculation (cf. [CT], Lemma 1.3.2, as we are exactly in the situation of that paper) that the conditions imposed on c and d ensure that 1) each bre of  at a root of d(mu2 ) = 0 is a rational curve with one node ( bre of type I1 ); 2) each bre at a root of c(mu2 )2 d(mu2 ) = 0 is the union of two rational curves tranversally intersecting each other in two points ( bre of type I2 ); 3) all other bres of  , including the bre at in nity, are smooth. We shall write X = X (m), X = X k k, where k is an algebraic closure of k. The topological Euler characteristic e(X ) equals 8  2 + 8 = 24 (see, e.g. [BPV], Prop. 11.4, p. 97). The canonical class KX can be written as   (nP ), where P is a k-point on P1k , n = (OX ) 2, and (OX ) is the Euler characteristic of the structure sheaf on X ([BPV], Cor. 12.3, p. 162). In any case (KX )2 = 0, and hence the formula (KX )2 + e(X ) = 12(OX ) implies that (OX ) = 2. Now the above formula for the canonical class gives KX = 0. This implies H 2 (X; OX ) = 1, hence H 1 (X; OX ) = 0. This proves that X = X (m) is a K 3-surface. It is one of the equivalent de nitions of Enriques surfaces that these are the surfaces with a K 3 unrami ed double covering. In particular, Y is an Enriques surface. QED

Proposition 2. Suppose that m 2 k n k2 , and that d(mu2 ) and c(mu2 )2 d(mu2 ) are irreducible polynomials, such that the elds K1 = k[u]=(d(mu2 )) and K2 = k[u]=(c(mu2 )2 d(mu2 )) do not contain quadratic extensions of k. Then the elliptic K 3-surface X (m) satis es the assumptions (1), (2), (3.a) and (3.b) of Theorem A of [CT]. If m(c(mr2 )2 d(mr2 )) 2 k2 for some r 2 k, then X (m) has a k-point. Proof. Assumptions (1) and (2) clearly hold since m is a non-square constant. Assumption (3.a) says that the natural map k =k2 ! K1 =K12 is injective. Non-trivial elements of its kernel correspond to quadratic extensions of k contained in K1 , and we assumed that there are none. For the same reason k =k2 ! K2 =K22 is injective. Assumption (3.b) which asserts the injectivity of k =hk2 ; mi ! K2 =hK22 ; mi, is then obviously satis ed. If r; s 2 k are such that m(c(mr2 )2 d(mr2 )) = s2 , then w1 = 0, w2 = s, u = r, x = c(mr2 ) is a solution of (2). Since (2) de nes a smooth aÆne surface, this point gives rise to a k-point on X (m). QED

Lemma. Let f (t) be an irreducible polynomial of even degree n with Galois group G  Sn . Then kf = k[t]=(f (t)) contains a quadratic extension of k if and only if G \ Sn 1 is contained in a subgroup of G of index 2. Proof. Call K the splitting eld of f (t), then we have Gal(K=k) = G. Let H = G \ Sn 1 , then the eld kf is recovered as the eld of invariants kf ' K H . If k  L  kf  K , where [L : k] = 2, then the corresponding sequence of subgroups of G reads as G  F  H  1, for some index 2 subgroup F  G. QED

Corollary. Assume the Schinzel Hypothesis (H) and the niteness of the Tate{ Shafarevich groups of elliptic curves over Q. Suppose that c(t); d(t) 2 Q[t] satisfy the following conditions: 2

(a) deg (c(t)) = 2, deg (d(t)) = 4, deg (c(t)2 d(t)) = 4, and td(t)(c(t)2 d(t)) is separable; (b) d(t) and c(t)2 d(t) are irreducible with Galois group isomorphic to S4 ; (c) d( u2 ) and c( u2 )2 d( u2 ) are irreducible with Galois group isomorphic to the semi-direct product of (Z=2)4 and S4 , where S4 acts on (Z=2)4 by permutations of factors; (d) d(0) c(0)2 2 Q2 . Then Q-rational points are Zariski dense on the Enriques surface Y de ned by (1). Proof. Set m = 1. In the notation of the previous proof G is the semi-direct product of (Z=2)4 and S4 , where S4 acts on (Z=2)4 by permutations of factors. Hence H is the semi-direct product of (Z=2)3 and S3 . It is clear that H is not contained in a subgroup of G of index 2: such a subgroup is normal, but the conjugates of H generate G. Proposition 2 now shows that all the conditions of Theorem A of [CT] are satis ed for X ( 1). Our condition (d) implies that X ( 1) has a Q-point with u = 0. Hence Theorem A of [CT] implies that Q-points are Zariski dense on X ( 1), and hence also on Y . QED Conditions (a), (b) and (c) are satis ed for \generic" coeÆcients of d(t) and c(t). If one takes d(t) = t4 + t2 + t + 5; c(t) = 2(t2 + 1); then (a) and (d) hold. On the other hand, the Galois groups of d(t) and c(t)2 d(t) are of order 24, hence isomorphic to S4 . That of d( u2 ) and c( u2 )2 d( u2 ) are of order 384 = 24  16 (computed using MAGMA package). The Galois group of any polynomial f (u2 ), where deg (f ) = 4, is contained in the semi-direct product of (Z=2)4 and S4 . Hence the Galois groups of d( u2 ) and c( u2 )2 d( u2 ) are equal to this semi-direct product. Thus (b) and (c) also hold. Condition (d) is obviously satis ed. We conclude that the Enriques surface given by

y 2 = tx4 + 4t2 (t2 + 1)x2 + t3 (3t4 + 7t2

t

1)

(4)

should have a Zariski dense set of Q-rational points. In fact, this can be rigourously proved.

Proposition 3. The surface (4) has a Zariski dense set of rational points. Proof. It will be easier to work with the K 3-covering of (4) given by t = u2 . This surface X = X ( 1) is given by

w22 = w14 + 4(u4 + 1)w12

(3u8 + 7u4 + u2

1)

(5)

The surface (5) contains the curve C of genus one given by w1 = u2 . Its equation is

w22 = 3u4

u2 + 1

(6)

Let us prove that C (Q) is in nite. This curve contains the points P = (1=2; 3=4) and Q = (0; 1). Let us show that P Q has in nite order in the Jacobian of C . It is easier to work with the isogeneous curve C1 : z 2 = t( 3t2 t + 1), where the unrami ed covering of degree 2 is f : C ! C1 is t = u2 , z = uw2 (f will become an isogeny after an appropriate choice of base points). The standard procedure yields the coordinate change 3

t = 3 3 (x+3), z = 3 4 y , and the equation of C1 takes the standard form y 2 = x3 +Ax+B with A = 270, B = 27  29. Then M = f (P ) has coordinates (xM ; yM ) = (33; 6  27), and f (Q) is a point of order 2 (when the origin of the group law on C1 is the point at 2 (which is even) does not divide 4A3 + 27B 2 (which is in nity). Since yM 6= 0 and yM odd), we conclude that M = f (P ) has in nite order in C1 . Thus the same is true for f (P ) f (Q), and hence P Q has in nite order in the Jacobian of C . Consider the double covering  : C ! P1Q given by projection to the coordinate u; this is just the restriction of  : X ! P1Q to C . Let  : C ! C be the corresponding involution,  (u; w2 ) = (u; w2 ). We note that C (R) is connected, and since C (Q) is in nite, C (Q) is dense in C (R) in the real topology. Let R be one of two real points of C with coordinate w2 = 0 (rami cation points of ). There is a sequence Pn 2 C (Q) converging to R in the real topology. We observe that the bre of  through R is a smooth curve of genus one, as 3u4 u2 + 1 is coprime with the irreducible polynomials d( u2 ) and c( u2 )2 d( u2 ). Then  (Pn ) Pn has in nite order in the Jacobian of the bre of  through Pn , provided n > N for some positive N . Indeed, by Mazur's theorem the torsion of elliptic curves over Q is bounded. On the other hand,  (Pn ) Pn tends to zero, hence cannot be torsion for n > N . (The key idea that when a rami cation point lies on a smooth bre, rational points are Zariski dense, is taken from [BT].) We have found in nitely many Q- bres of  : X ! P1Q , each with in nitely many Qrational points. This implies the density of Q-rational points of X in the Zariski topology. QED Remarks. 1. We used Mazur's theorem to simplify the argument, without any doubt one can nd a proof not based on this theorem. 2. It should be possible to construct similar examples for Enriques surfaces given by

y12 = f (t)(x2

c(t)); y22 = f (t)(x2

d(t))

where f , c, d, c d are all of degree 2 such that fcd(c d) is a separable polynomial (cf. [CSS97], Example 4.1.2 and Remark 5.3.1). In this case the conditional result pointing to the Zariski density of rational points is proved in [CSS98]. 3. The morphism Y ! P1k has two double bres. It is proved in ([CSS97], Cor. 2.4) that if a pencil of curves of genus one de ned over a number eld k has at least 5 (geometric) double bres, then all k-points are contained in nitely many k- bres, and hence are not Zariski dense. I would like to thank Sir Peter Swinnerton-Dyer for showing me the curve (6), and Jean-Louis Colliot-Thelene for useful discussions.

References [BPV] W. Barth, C. Peters, A. Van de Ven. Compact complex surfaces. Springer{ Verlag, 1984. [BS] A.O. Bender and Sir Peter Swinnerton-Dyer. Solubility of certain pencils of curves of genus 1, and of the intersection of two quadrics in P4 . Proc. London Math. Soc., to appear. 4

[BT] F.A. Bogomolov and Yu. Tschinkel. Density of rational points on Enriques surfaces. Math. Res. Lett. 5 (1998) 623{628. [CT] J.-L. Colliot-Thelene. Hasse principle for pencils of curves of genus one whose Jacobians have a rational 2-division point, this volume. [CSS97] J.-L. Colliot-Thelene, A.N. Skorobogatov and Sir Peter Swinnerton-Dyer. Double bres and double covers: paucity of rational points. Acta Arithm. 79 (1997) 113{135. [CSS98] J.-L. Colliot-Thelene, A.N. Skorobogatov and Sir Peter Swinnerton-Dyer. Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points. Inv. Math. 134 (1998), 579{650. Department of Mathematics Imperial College 180 Queen's Gate London SW7 2BZ U.K.

5

Density of integral points on algebraic varieties Brendan Hassett and Yuri Tschinkel August 1, 2000

1 Introduction Let K be a number eld, S a nite set of valuations of K , including the archimedean valuations, and OS the ring of S -integers. Let X be an algebraic variety de ned over K and D a divisor on X . We will use X and D to denote models over Spec(OS ). We will say that integral points on (X; D) (see Section 2 for a precise de nition) are potentially dense if they are Zariski dense on some model (X ; D), after a nite extension of the ground eld and after enlarging S . A central problem in arithmetic geometry is to nd conditions insuring potential density (or nondensity) of integral points. This question motivates many interesting and concrete problems in classical number theory, transcendence theory and algebraic geometry, some of which will be presented below. If we think about general reasons for the density of points - the rst idea would be to look for the presence of a large automorphism group. There are many beautiful examples both for rational and integral points, like K3 surfaces given by a bihomogeneous (2; 2; 2) form in P1 P1 P1 or the classical Markov equation x2 + y2 + z2 = 3xyz. However, large automorphism groups are \sporadic" - they are hard to nd and usually, they are not well behaved in families. There is one notable exception - namely automorphisms of algebraic groups, like tori and abelian varieties. Thus it is not a surprise that the main geometric reason for the abundance of rational points on varieties treated in the recent papers [11], [3], [12] is the presence of elliptic or, more generally, abelian brations with multisections having a dense set of rational points and subject to some nondegeneracy conditions. Most of the eort goes into ensuring these conditions. 1

In this paper we focus on cases when D is nonempty. We give a systematic treatment of known approaches to potential density and present several new ideas for proofs. The analogs of elliptic brations in log geometry are conic bundles with a bisection removed. We develop the necessary techniques to translate the presence of such structures to statements about density of integral points and give a number of applications. The paper is organized as follows: in Section 2 we introduce the main definitions and notations. Section 3 is geometrical - we introduce the relevant concepts from the log minimal model program and formulate several geometric problems inspired by questions about integral points. In Section 4, we recall the bration method and nondegeneracy properties of multisections. We consider approximation methods in Section 5. Section 6 is devoted to the study of integral points on conic bundles with sections and bisections removed. In the nal section, we survey the known results concerning potential density for integral point on log K3 surfaces.

Acknowledgements. The rst author was partially supported by an NSF Postdoctoral Research Fellowship, NSF Continuing Grant 0070537, and the Institute of Mathematical Sciences of the Chinese University of Hong Kong. The second author was partially supported by the NSA. We bene tted from conversations with D. Abramovich, Y. Andre, F. Bogomolov, A. Chambert-Loir, J.-L. Colliot-Thelene, J. Kollar, D. McKinnon, and B. Mazur. We are grateful to P. Vojta for comments that improved the paper, especially Proposition 3.12, and to D.W. Masser for information on specialization of nondegenerate sections. Our approach in Section 6 is inspired by the work of F. Beukers (see [1] and [2]).

2 Generalities

2.1 Integral points

Let  : U ! Spec(OS ) be a at scheme over OS with generic ber U . An integral point on U is a section of ; the set of such points is denoted U (OS ). In the sequel, U will be the complement to a reduced eective Weil divisor D in a normal proper scheme X , both generally at over Spec(OS ). Hence an S -integral point P of (X ; D) is a section sP : Spec(OS ) ! X of , which does not intersect D, that is, for each prime ideal p 2 Spec(OS ) we have 2

sP (p) 2= Dp. We denote by X (resp. D) the corresponding generic ber. We generally assume that X is a variety (i.e., a geometrically integral scheme); frequently X is smooth and D is normal crossings. Potential density of integral points on (X ; D) does not depend on the choice of S or on the choices of models over Spec(OS ), so we will not always specify them. Hopefully, this will not create any confusion. If D is empty then every K -rational point of X is an S -integral point for (X ; D) (on some model). Every K -rational point of X , not contained in D is S -integral on (X ; D) for S large enough. Clearly, for any X and D there exists a nite extension K 0=K and a nite set S 0 of prime ideals in OK such that there is an S 0-integral point on (X 0; D0) (where X 0 is the base change of X to Spec(OS0 )). The de nition of integral points can be generalized as follows: let Z be any closed subscheme of X . An S -integral point for (X ; Z ) is an OS -valued point of X n Z . 0

2.2 Vojta's conjecture

A pair consists of a proper normal variety X and a reduced eective Weil divisor D  X . A morphism of pairs ' : (X1; D1 ) ! (X2 ; D2) is a morphism ' : X1 ! X2 such that (the support of) ' 1(D2) is a subset of D1. In particular, ' restricts to a morphism X1 n D1 ! X2 n D2. A morphism of pairs is dominant if ' : X1 ! X2 is dominant. If (X1 ; D1) dominates (X2; D2) then integral points are dense on (X2; D2) when they are dense on (X1; D1) (after choosing appropriate integral models). A morphism of pairs is proper if ' : X1 ! X2 is proper and the restriction X1 n D1 ! X2 n D2 is also proper; equivalently, we may assume that ' : X1 ! X2 is proper and D1 is a subset of ' 1(D2 ). A resolution of the pair (X; D) is a proper ~ D~ ) ! (X; D) such that  : X~ ! X is birational, morphism of pairs  : (X; X~ is smooth, and D~ is normal crossings. Let X be a normal proper variety of dimension d. Recall that a Cartier divisor D  X is big if h0 (OX (nD)) > Cnd for some C > 0 and all n suciently large and divisible.

De nition 2.1 A pair (X; D) is of log general type if it admits a resolution ~ D~ ) ! (X; D) with !X~ (D~ ) big.  : (X; Let us remark that the de nition does not depend on the resolution. 3

Conjecture 2.2 (Vojta, [30]) Let (X; D) be a pair of log general type. Then

integral points on (X; D) are not potentially dense. This conjecture is known for semiabelian varieties and their subvarieties ([9], [31], [16]). Vojta's conjecture implies that a pair with dense integral points cannot dominate a pair of log general type. We are interested in geometric conditions which would insure potential density of integral points. The most naive statement would be the direct converse to Vojta's conjecture. However this can't be true even when D = ;. Indeed, varieties which are not of general type may dominate varieties of general type, or more generally, admit nite etale covers which dominate varieties of general type (see the examples in [7]). In the next section we will analyze other types of covers with the same arithmetic property.

3 Geometry

3.1 Morphisms of pairs

De nition 3.1 We will say that a class of dominant morphisms of pairs ' : (X1; D1) ! (X2; D2) is arithmetically continuous if the density of integral

points on (X2 ; D2) implies potential density of integral points on (X1; D1). For example, assume that D = ;. Then any projective bundle in the Zariski topology P ! X is arithmetically continuous. In the following sections we present other examples of arithmetically continuous morphisms of pairs. De nition 3.2 A pseudo-etale cover of pairs ' : (X1; D1 ) ! (X2; D2) is a proper dominant morphism of pairs such that a) ' : X1 ! X2 is generically nite, and b) the map from the normalization X2norm of X2 (in the function eld of X1) onto X2 is etale away from D2 . Remark 3.3 For every pair (X; D) there exists a birational pseudo-etale ~ D~ ) ! (X; D) such that X~ is smooth and D~ is normal morphism ' : (X; crossings. The following theorem is a formal generalization of the well-known theorem of Chevalley-Weil. It shows that potential density is stable under pseudoetale covers of pairs. 4

Theorem 3.4 Let ' : (X1 ; D1) ! (X2; D2) be a pseudo-etale cover of pairs. Then ' is arithmetically continuous.

Remark 3.5 An elliptic bration E ! X , isotrivial on X n D, is arithmetically continuous. Indeed, it splits after a pseudo-etale morphism of pairs and we can apply Theorem 3.4. The following example is an integral analog of the example of Skorobogatov, Colliot-Thelene and Swinnerton-Dyer ([7]) of a variety which does not dominate a variety of general type but admits an etale cover which does.

Example 3.6 Consider P1  P1 with coordinates (x1 ; y1); (x2; y2) and involutions

j1 (x1 ; y1) = ( x1 ; y1) j2 (x2 ; y2) = (y2; x2) on the factors. Let j be the induced involution on the product; it has xed points x1 = 0 x2 = y2 x1 = 0 x2 = y2 : y1 = 0 x2 = y2 y1 = 0 x2 = y2 The rst projection induces a map of quotients (P1  P1 )= hj i ! P1 = hj1 i : We use X to denote the source; the target is just Proj(C [x21 ; y12]) ' P1 : Hence we obtain a bration f : X ! P1 . Note that f has two nonreduced bers, corresponding to x1 = 0 and y1 = 0 respectively. Let D be the image in X of (x1 = 0) [ (y1 = 0) [ (x2 = m1 y2) [ (x2 = m2y2) where m1 and m2 are distinct, m1 m2 6= 1, and m1 ; m2 6= 0; 1. Since D intersects the general ber of f in just two points, (X; D) is not of log general type. We can represent X as a degenerate quartic Del Pezzo surface with four A1 singularities (see gure 1). If we x invariants a = x21 x2 y2; b = x21 (x22 + y22); c = x1y1(x22 y22); d = y12(x22 + y22); e = y12x2 y2 then X is given as a complete intersection of two quadrics: ad = be; c2 = bd 4ae: 5

. .

X

D3

D4

. .

A1

A1

A1

A1

D1

D2

Figure 1: The log surface (X; D) The components of D satisfy the equations

D1 D2 D3 D4

= = = =

fa = b = c = 0g fc = d = e = 0g f(1 + m21 )a m1 b = (1 + m21 )e m1 d = 0g f(1 + m22 )a m2 b = (1 + m22 )e m2 d = 0g:

Our assumptions guarantee that D3 and D4 are distinct. We claim that (X; D) does not admit a dominant map onto a variety of log general type and that there exists a pseudo-etale cover of (X; D) which does. Indeed, the preimage of X n D in P1  P1 is (A 1 n 0)  (P1 n fm1; m2 ; 1=m1; 1=m2g); which dominates a curve of log general type, namely, P1 minus four points. However, (X; D) itself cannot dominate a curve of log general type. Any such curve must be rational, with at least three points removed; however, the boundary D contains at most two mutually disjoint irreducible components. The following was put forward as a possible converse to Vojta's conjecture.

Problem 3.7 (Strong converse to Vojta's conjecture) Assume that the pair (X2 ; D2) does not admit a pseudo-etale cover (X1; D1) ! (X2 ; D2) such that (X1 ; D1) dominates a pair of log general type. Are integral points for (X2; D2) potentially dense? 6

3.2 Projective bundles in the etale topology

We would like to produce further classes of dominant arithmetically continuous morphisms (X1 ; D1) ! (X2; D2).

Theorem 3.8 Let ' : (X1; D1) ! (X2; D2) be a projective morphism of pairs such that ' is a projective bundle (in the etale topology) over X2 n D2 . We also assume that ' 1(D2 ) = D1 . Then ' is arithmetically continuous. Proof. We are very grateful to Prof. Colliot-Thelene for suggesting this proof. Choose models (Xi; Di) (i = 1; 2) over some ring of integers OS , so that the morphism ' is well-de ned and a projective bundle. (We enlarge S as necessary.) We recall basic properties of the Brauer group Br(OS ). Let v denote a place for the quotient eld K and Kv the corresponding completion. Class eld theory gives the following exact sequences

0 ! Br(K ) ! v Br(Kv ) ! Q =Z ! 0 0 ! Br(OS ) ! Br(K ) ! v62S Br(Kv ): The Brauer groups of the local elds corresponding to nonarchimedean valuations are isomorphic to Q =Z. Given a nite extension of Kw =Kv of degree n, the induced map on Brauer groups is multiplication by n. Let p denote an S -integral point of (X2; D2). The ber ' 1 (p) is a BrauerSeveri variety over OS . If r 1 denotes the relative dimension of ' then the corresponding element (p) 2 Br(OS ) has order dividing r. Integral points in ' 1(p) are dense if rational points are dense, which is the case when (p) = 0. Our exact sequences imply that (p) yields elements of Br(Kv ) which are zero unless v 2 S , and are annihilated by r otherwise. It suces to nd an extension K 0=K inducing a cyclic extension of Kv of order divisible by r for all v 2 S . Indeed, such an extension necessarily kills (p) for each point p de ned over OS . If OS is the integral closure of OS in K 0 then ' 1(p) has dense S 0-integral points. 0 0

Remark 3.9 Let X be a smooth simply connected projective variety which

does not dominate a variety of general type. It may admit an projective bundle (in the etale topology) ' : P ! X , for example if X is a K3 surface. However, P cannot dominate a variety of general type. Indeed, given such a 7

dominant morphism  : P ! Y , the bers of ' are mapped to points by . In particular,  necessarily factors through '. (We are grateful to J. Kollar for emphasizing this point.)

Problem 3.10 (Geometric counterexamples to Problem 3.7) Are there pairs which do not admit pseudo-etale covers dominating pairs of log general type but which do admit arithmetically continuous covers dominating pairs of log general type?

3.3 Punctured varieties

In Section 3.1 we have seen that potential density of integral points is preserved under pseudo-etale covers. It is not an easy task, in general, to check whether or not some given variety (like an elliptic surface) admits a (pseudo-) etale cover dominating a variety of general type. What happens if we modify the variety (or pair) without changing the fundamental group?

Problem 3.11 (Geometric puncturing problem) Let X be a projective variety with canonical singularities and Z a subvariety of codimension  2. Assume that no (pseudo-) etale cover of (X; ;) dominates a variety of general

type. Then (X; Z ) admits no pseudo-etale covers dominating a pair of log general type. A weaker version would be to assume that X and Z are smooth. By de nition, a pseudo-etale cover of (X; Z ) is a pseudo-etale cover of a pair (X 0; D0), where X 0 is proper over X and X 0 n D0 ' X n Z .

Proposition 3.12 Assume X and Z are as in Problem 3.11, and that X is smooth. Then a) No pseudo-etale covers of (X; Z ) dominate a curve of log general type. b) No pseudo-etale covers of (X; Z ) dominate a variety of log general type of the same dimension. Proof. Suppose we have a pseudo-etale cover  : (X1 ; D1 ) ! (X; Z ) and a dominant morphism ' : (X1; D1) ! (X2 ; D2) to a variety of log general type. By Remark 3.3, we may take the Xi smooth and the Di normal crossings. Since D1 is exceptional with respect to , Iitaka's Covering Theorem ([13] Theorem 10.5) yields an equality of Kodaira dimensions

(KX ) = (KX1 + D1): 8

Assume rst that X2 is a curve. We claim it has genus zero or one. Let be the normalization of X in the function eld of X1. The induced morphism g : X norm ! X is nite, surjective, and branched only over Z , a codimension  2 subset of X . Since X is smooth, it follows that g is etale (see SGA II X x3.4 [10]). If X2 has genus  2 then ' : X1 ! X2 is constant along the bers of X1 ! X norm, and thus descends to a map X norm ! X2 . This would contradict our assumption that no etale cover of X dominates a variety of general type. Choose a point p 2 D2 and consider the divisor F = ' 1(p). Note that 2F moves because 2p moves on X2. However, 2F is supported in D1 , which lies in the exceptional locus for , and we obtain a contradiction. Now assume ' is generically nite. We apply the Logarithmic Rami cation Formula to ' (see [13] Theorem 11.5)

X norm

KX1 + D1 = '0(KX2 + D2 ) + R where R is the (eective) logarithmic rami cation divisor. Applying the Covering Theorem again, we nd that (KX1 +D1 R) = (KX2 +D2) = dim(X ). It follows that KX1 + D1 is also big, which contradicts the assumption that X is not of general type. 0

Problem 3.13 (Arithmetic puncturing problem) Let X be a projec-

tive variety with canonical singularities and Z a subvariety of codimension  2. Assume that rational points on X are potentially dense. Are integral points on (X; Z ) potentially dense? For simplicity, one might rst assume that X and Z are smooth.

Remark 3.14 Assume that Problem 3.13 has a positive solution. Then

potential density of rational points holds for all K3 surfaces. Indeed, if Y is a K3 surface of degree 2n then potential density of rational points holds for the symmetric product X = Y (n) (see [12]). Denote by Z the large diagonal in X and by
the large diagonal in Y n (the ordinary product). Assume that integral points on (X; Z ) are potentially dense. Then, by Theorem 3.4 integral points on (Y n;
) are potentially dense. This implies potential density for rational points on Y .

9

4 The bration method and nondegenerate multisections This section is included as motivation. Let B be an algebraic variety, de ned over a number eld K and  : G ! B be a group scheme over B . We will be mostly interested in the case when the generic ber is an abelian variety or a split torus G nm . Let s be a section of . Shrinking the base we may assume that all bers of G are smooth. We will say that s is nondegenerate if [n sn is Zariski dense in G.

Problem 4.1 (Specialization) Assume that G ! B has a nondegenerate section s. Describe the set of b 2 B (K ) such that s(b) is nondegenerate in

the ber Gb. For simple abelian varieties over a eld a point of in nite order is nondegenerate. If E ! B is a Jacobian elliptic bration with a section s of in nite order then this section is automatically nondegenerate, and s(b) is nondegenerate if it is nontorsion. By a result of Neron (see [26] 11.1), the set of b 2 B (K ) such that s(b) is not of in nite order is thin; this holds true for abelian brations of arbitrary dimension. For abelian brations A ! B with higher-dimensional bers, one must also understand how rings of endomorphisms specialize. The set of b 2 B (K ) for which the restriction End(A) ! End(A(b)) fails to be surjective is also thin; this is a result of Noot [21] Corollary 1.5. In particular, a nondegenerate section of a family of generically simple abelian varieties specializes to a nondegenerate point outside a thin set of bers. More generally, given an arbitrary abelian bration A ! B and a nondegenerate section s, the set of b 2 B (K ) such that s(b) is degenerate is thin in B . (We are grateful to Prof. Masser for pointing out the proof.) After replacing A by an isogenous abelian variety and taking a nite extension of the function eld K (B ), we obtain a family A0 ! B 0 with A0 ' Ar11  : : :  Armm , where the Aj are (geometrically) simple and mutually non-isogenous. By the Theorems of Neron and Noot, the Aj (b0) are simple and mutually nonisogenous away from some thin subset of B 0 . A section s0 of A0 ! B 0 is rj nondegenerate i its projection onto each factor Aj is nondegenerate; for b0 not contained in our thin subset, s0(b0 ) is nondegenerate i its projection 10

onto each Arjj (b0 ) is nondegenerate. Hence we are reduced to proving the claim for each Arjj . Since Aj is simple, a section sj of Arjj is nondegenerate i its projections sj;1; : : : ; sj;rj are linearly independent over End(Aj ). Away from a thin subset of B 0, the same statement holds for the specializations to b0 . However, Neron's theorem implies that sj;1(b0 ); : : : ; sj;rj (b0 ) are linearly independent away from a thin subset.

Remark 4.2 There are more precise versions of Neron's Theorem due to

Demyanenko, Manin and Silverman (see [28], for example). Masser has proposed another notion of what it means for a subset of B (K ) to be small, known as `sparcity'. For instance, the endomorphism ring of a family of abelian varieties changes only on a `sparse' set of rational points of the base (see [18]). For an analogue to Neron's Theorem, see [17]. Similar results hold for algebraic tori and are proved using a version of Neron's Theorem for G nm - brations (see [26] pp. 154). A sharper result (for one-dimensional bases B ) can be obtained from the following recent theorem:

Theorem 4.3 ([4]) Let C be an absolutely irreducible curve de ned over

a number eld K and x1 ; :::; xr rational functions in K (C ), multiplicatively independent modulo constants. Then the set of algebraic points p 2 C (Q ) such that x1 (p); :::; xr (p) are multiplicatively dependent has bounded height.

The main idea of the papers [11], [3], [12] can be summarized as follows. We work over a number eld K and we assume that all geometric data are de ned over K . Let  : E ! B be a Jacobian elliptic bration over a one dimensional base B . This means that we have a family of curves of genus one and a global zero section so that every ber is in fact an elliptic curve. Suppose that we have another section s which is of in nite order in the Mordell-Weil group of E (K (B )). The specialization results mentioned above show that for a Zariski dense set of b 2 B (K ) the restriction s(b) is of in nite order in the corresponding ber Eb . If K -rational points on B are Zariski dense then rational points on E are Zariski dense as well. Let us consider a situation when E does not have any sections but instead has a multisection M . By de nition, a multisection (resp. rational multisection) M is irreducible and the induced map M ! B is nite

at (resp. generically nite) of degree deg(M ). The base-changed family 11

E B M ! M has the identity section Id (i.e., the image of the diagonal under M B M ! E B M ) and a (rational) section M := deg(M )Id Tr(M B M ) where Tr(M B M ) is obtained (over the generic point) by summing all the points of M B M . By de nition, M is nondegenerate if M is nondegenerate. When we are concerned only with rational points, we will ignore the distinction between multisections and rational multisections, as every rational multisection is a multisection over an open subset of the base. However, this distinction is crucial when integral points are considered. If M is nondegenerate and if rational points on M are Zariski dense then rational points on E are Zariski dense (see [3]).

Example 4.4 ([11]) Let X be a quartic surface in P3 containing a line L.

Consider planes P2 passing through this line. The residual curve has degree 3. Thus we obtain an elliptic bration on X together with the trisection L. If L is rami ed in a smooth ber of this bration then the multisection is nondegenerate and rational points are Zariski dense. This argument generalizes to abelian brations  : A ! B . However, we do not know of any simple geometric conditions insuring nondegeneracy of a (multi)section in this case. We do know that for any abelian variety A over K there exists a nite extension K 0 =K with a nondegenerate point in A(K 0 ) (see [12]). This allows us to produce nondegenerate sections over function elds.

Proposition 4.5 Let Y be a Fano threefold of type W2 , that is a double

cover of P3 rami ed in a smooth surface of degree 6. Then rational points on the symmetric square Y (2) are potentially dense. Proof. Observe that the symmetric square Y (2) is birational to an abelian surface bration over the Grassmannian of lines in P3 . This bration is visualized as follows: consider two generic points in Y . Their images in P3 determine a line, which intersects the rami cation locus in 6 points and lifts to a (hyperelliptic) genus two curve on Y . On Y (2) we have an abelian surface bration corresponding to the degree two component of the relative Picard scheme. Now we need to produce a nondegenerate multisection. Pick two generic points b1 and b2 on the branch surface. The preimages in Y of the corresponding tangent planes are K3 surfaces `1 and `2 , of degree two with

12

ordinary double points at the points of tangency. The surfaces `1 and `2 therefore have potentially dense rational points (this was proved in [3]), as does `1  `2 . This is our multisection; we claim it is nondegenerate for generic b1 and b2. Indeed, it suces to show that given a (generic) point in Y (2) , there exist b1 and b2 so that `1  `2 contains the point. Observe that through a (generic) point of P3 , there pass many tangent planes to the branch surface. 0 Remark 4.6 Combining the above Proposition with the strong form of Problem 3.13 we obtain potential density of rational points on a Fano threefold of type W2 - the last family of smooth Fano threefolds for which potential density is not known. Here is a formulation of the bration method useful for the analysis of integral points: Proposition 4.7 Let B be a scheme over a number eld K , G ! B a at group scheme, T ! B an etale torsor for G, and M  T a nondegenerate multisection over B . If M has potentially dense integral points then T has potentially dense integral points. Proof. Without loss of generality, we may assume that B is geometrically connected and smooth. The base-changed family T B M dominates T , so it suces to prove density for T B M . Note that since M is nite and at over B , M is a well-de ned section over all of M (i.e., it is not just a rational section). Hence we may reduce to the case of a group scheme G ! B with a nondegenerate section  . We may choose models G and B over Spec(OS ) so that G ! B is a group scheme with section  . We may also assume that S -integral points of  are Zariski dense. The set of multiples  n of  , each a section of G ! B, is dense in G by the nondegeneracy assumption. Since each has dense S -integral points, it follows that S -integral points are Zariski dense. 0 A similar argument proves the following Proposition 4.8 Let ' : X ! P1 be a K3 surface with elliptic bration. Let M be a multisection over its image '(M ), nondegenerate and contained in the smooth locus of '. Let F1 ; : : : ; Fn be bers of ' and D a divisor supported in these bers and disjoint from M . If M has potentially dense integral points then (X; D) has potentially dense integral points. 13

Proof. We emphasize that X is automatically minimal and the bers of ' are reduced (see [3]). Our assumptions imply that M is nite and at over '(M ). After base-changing to M , we obtain a Jacobian elliptic bration X 0 := X P1 M with identity and a nondegenerate section M . Let G  X 0 be the open subset equal to the connected component of the identity. Since D0 := D P1 M is disjoint from the identity, it is disjoint from G. Hence it suces to show that G has potentially dense integral points. We assumed that M is contained in the smooth locus of ', so M is contained in the grouplike part of X 0, and some multiple of M is contained in G. Repeating the argument for Proposition 4.7 gives the result. 0

5 Approximation techniques In this section we prove potential density of integral points for certain pairs (X; D) using congruence conditions to control intersections with the boundary. Several of these examples are included as support for the statement of Problem 3.13.

Proposition 5.1 Let G = QNj Gj where Gj are algebraic tori G m or geo-

metrically simple abelian varieties. Let Z be a subvariety in G of codimension >  = maxj (dim(Gj )) and let U = G n Z be the complement. Then integral points on U are potentially dense. Proof. We are grateful to Prof. McKinnon for inspiring the following argument. The proof proceeds by induction on the number of components N . The base case N = 1 follows from the fact that rational points on tori and abelian varieties are potentially dense, so we proceed with the inductive step. ConQ 0 0 sider the projections  : G ! G = j6=N Gj and N : G ! GN . By assumption, generic bers of 0 are geometrically disjoint from Z . Choose a ring of integers OS and models Gj over Spec(OS ). We assume that each Gj is smooth over Spec(OS ) and that GN has a nondegenerate S integral point q (see [12], for example, for a proof of the existence of such points on abelian varieties). Let T be any subscheme of GN supported over a nite subset of Spec(OS ) such that GN has an S -integral point pN disjoint from T . We claim that such

14

integral points are Zariski dense. Indeed, for some m > 0 we have

mq  0 (mod p) for each p 2 Spec(OS ) over which T has support. Hence we may take the translations of pN by multiples of mq. After extending OS , we may assume U has at least one integral point p = 0 (p ; pN ) so that 0 1(p0) and N1(pN ) intersect Z in the expected dimensions. In particular, 0 1 (p0) is disjoint from Z . By the inductive hypothesis, we may extend OS so that (N1(pN ) ' G 0; N1(pN ) \ Z ) has dense integral points. In particular, almost all such integral points are not contained in 0(Z ), a closed proper subscheme of G 0. Let r be such a point, so that Fr = 0 1(r) ' GN intersects Z in a subscheme T supported over a nite number of primes. Since (r; pN ) 2 Fr is disjoint from T , the previous claim implies that the integral points of Fr disjoint from T are Zariski dense. As r varies, we obtain a Zariski dense set of integral points on G n Z. 0

Corollary 5.2 Let X be a toric variety and Z  X a subvariety of codimension  2, de ned over a number eld. Then integral points on (X; Z ) are potentially dense. Another special case of the Arithmetic puncturing problem 3.13 is the following:

Problem 5.3 Are integral points on punctured simple abelian varieties of dimension n > 1 potentially dense?

Example 5.4 Potential density of integral points holds for simple abelian varieties punctured in the origin, provided that their ring of endomorphisms contains units of in nite order.

6 Conic bundles and integral points Let K be a number eld, S a nite set of places for K (including all the in nite places), OS the corresponding ring of S -integers, and  2 Spec(OS ) the 15

generic point. For each place v of K , let Kv be the corresponding complete eld and ov the discrete valuation ring (if v is nonarchimedean). As before, we use calligraphic letters (e.g., X ) for schemes (usually at) over OS and roman letters (e.g., X ) for the ber over .

6.1 Results on linear algebraic groups

Consider a linear algebraic group G=K . Choose a model G for G over OS , i.e., a at group scheme of nite type G =OS restricting to G at the generic point. This may be obtained by xing a representation G ,! GLn(K ) (see also [32] x10-11). The S -rank of G (denoted rank(G; OS )) is de ned as the rank of the abelian group of sections of G (OS ) over OS . This does not depend on the choice of a model. Indeed, consider two models G1 and G2 with a birational map b : G1 9 9 KG2; of course, b is an isomorphism over the generic point and the proper transform of the identity section I1 is the identity. There is a subscheme Z  Spec(OS ) with nite support such that the indeterminacy of b is in the preimage of Z . It follows that the sections of G1 congruent to I1 modulo Z have proper transforms which are sections of G2 . Such sections form a nite-index subgroup of G1 (OS ). Let G m be the multiplicative group over Z, i.e., Spec(Z[x; y]= hxy 1i); it can be de ned over an arbitrary scheme by extension of scalars. There is a natural projection G m (Z) ! Spec(Z[x]) = A 1Z  P1Z so that P1Z n G m (Z) = f0; 1g. A form of G m over K is a group scheme G=K for which there exists a nite eld extension K 0=K and an isomorphism G K K 0 ' G m (K 0 ). These are classi ed as follows (see [23] for a complete account). Any group automorphism  : G m (K 0 ) ! G m (K 0) is either inversion or the identity, depending on whether it exchanges 0 and 1. The corresponding automorphism group is smooth, so we may work in the etale topology (see [19] Theorem 3.9). In particular, K forms of G m ' He1t(Spec(K ); Z=2Z): Each such form admits a natural open imbedding into a projective curve G ,! X , generalizing the imbedding of G m into P1 . The complement D = 16

X n G consists of two points. The Galois action on D is given by the cocycle in He1t (Spec(K ); Z=2Z) classifying G. There is a general formula for the rank of a torus T due to T. Ono and J.M. Shyr (see [22], Theorem 6 and [27]). Let T^ (resp. T^v ) the group of characters de ned over K (resp. Kv ), and (T ) (resp. (T; v)) the number of independent elements of T^ (resp. T^v ). The formula takes the form rank(T; OS ) =

X v2S

(T; v) (T ):

For forms of G m this is particularly simple. For split forms rank(G m ; OS ) = #fplaces v 2 S g 1: Now let G=K be a nonsplit form, corresponding to the quadratic extension K 0=K , and S 0 the places of K 0 lying over the places of S . Then we have rank(G; OS ) = #fplaces v 2 S completely splitting in S 0 g:

6.2 Group actions and integral points

Throughout this subsection, X is a normal, geometrically connected scheme and X ! Spec(OS ) a at projective morphism. Let D  X be an eective reduced Cartier divisor. Contrary to our previous conventions, we do not assume that D is at over OS . Assume that a linear algebraic group G acts on X so that X n D is a G-torsor.

Proposition 6.1 There exists a model G for G such that G acts on X and stabilizes D. Proof. Choose an imbedding X ,! PnOS and a compatible linearization G ,! GLn+1(K ) (see [20], Ch. 1, Cor. 1.6 and Prop. 1.7). Let G 0 ,! GLn+1(OS ) be the resulting integral model of G, so that G 0 stabilizes the ideal of X and therefore acts on it. Furthermore, G 0 evidently stabilizes the irreducible components of D dominating OS . The bral components of D are supported over a nite subset of Spec(OS ). We take G  G 0 to be the subgroup acting trivially over this subset; it has the desired properties. 0 Proposition 6.2 Assume (X ; D) has an S -integral point and that G has positive OS -rank. Then (X ; D) has an in nite number of S -integral points. 17

Proof. Consider the action of G (OS ) on the integral point  (which has trivial stabilizer). The orbit consists of S -integral points of (X ; D), an in nite collection because G has positive rank. 0 Now assume that X is a smooth rational curve. A rational section (resp. bisection) D  X is a reduced eective Cartier divisor such that the generic ber D is reduced of degree one (resp. two). Note that the open curve X n D is geometrically isomorphic to P1 f1g (resp. P1 f0; 1g), and thus is a torsor for some K -form G of G a (resp. G m ). This form is easily computed. Of course, G a has no nontrivial forms. In the G m case, we can regard D as an element of He1t (Spec(K ); Z=2Z), which gives the descent data for G. The following result is essentially due to Beukers (see [1], Theorem 2.3):

Proposition 6.3 Let (X ; D) ! Spec(OS ) be a rational curve with rational

bisection and G the corresponding form of G m (as described above). Assume that (X ; D) has an S -integral point and rank(G; OS ) > 0. Then S -integral points of (X ; D) are Zariski dense. Proof. This follows from Proposition 6.2. Given an S -integral point  of (X ; D), the orbit G (OS ) is in nite and thus Zariski dense. 0 Combining with the formula for the rank, we obtain the following:

Corollary 6.4 Let (X ; D) ! Spec(OS ) be a rational curve with rational bisection such that (X ; D) has an S -integral point. Assume that either a) D is reducible over Spec(K ) and jS j > 1; or b) D is irreducible over Spec(K ) and at least one place in S splits completely in K (D). Then S -integral points of (X ; D) are Zariski dense. When D is a rational section we obtain a similar result (also essentially due to Beukers [1], Theorem 2.1):

Proposition 6.5 Let (X ; D) ! Spec(OS ) be a rational curve with rational section such that (X ; D) has an S -integral point. Then S -integral points of (X ; D) are Zariski dense.

6.3

v

-adic geometry

For each place v 2 S , consider the projective space P1 (Kv ) as a manifold with respect to the topology induced by the v-adic absolute value on Kv . 18

For simplicity, this will be called the v-adic topology; we will use the same term for the induced subspace topology on P1 (K ). Given an etale morphism of curves f : U ! P1 de ned over Kv , we will say that f (U (Kv )) is a basic etale open subset. These are open in the v -adic topology, either by the open mapping theorem (in the archimedean case) or by Hensel's lemma (in the nonarchimedean case). Let f (B ) := #fz 2 Ofvg : jzjv  B and z 2 f (U (Kv ))g where B is a positive integer and

Ofvg := fz 2 K : jzjw  1 for each w 6= vg: We would like to estimate the quantity

f := lim inf  (B )=Id (B ) B!1 f i.e., the fraction of the integers contained in the image of the v-adic points of U .

Proposition 6.6 Let f : U ! P1 be an etale morphism de ned over Kv and f1 : C ! P1 a nite morphism of smooth curves extending f . If there exists a point q 2 f1 1 (1) \ C (Kv ) at which f1 is unrami ed then f = 1. U.

Proof. This follows from the fact that f (U (Kv )) is open if f is etale along

0

As an illustrative example, we take K = Q and Kv = R , so that Ofvg = Z. The set f (U (R )) is a nite union of open intervals (r; s) with r; s 2 R [f1g, where the ( nite) endpoints are branch points. We observe that

8 > 1=2 if f (U (R )) contains a one-sided neighborhood of 1; :1 if f (U (R )) contains a two-sided neighborhood of 1.

We can read o easily which alternative occurs in terms of the local behavior at in nity. Let f1 : C ! P1 be a nite morphism of smooth curves extending f . If f1 1(1) has no real points then f = 0. If f1 1(1) has unrami ed (resp. rami ed) real points then f = 1 (resp. f > 0.) We specialize to the case of double covers: 19

Proposition 6.7 Let U ! P1 be an etale morphism de ned over Kv and f1 : C ! P1 a nite morphism of smooth curves extending f . Assume that f1 has degree two and rami es at q 2 f1 1(1). Then f > 0. Proof. Of course, q is necessarily de ned over Kv . The archimedean case follows from the previous example, so we restrict to the nonarchimedean case. Assume f1 is given by

y2 = cnzn + cn 1zn 1 + : : : + c0; where z is a coordinate for the ane line in P1 (Kv ), cn 6= 0, and the ci 2 ov . Substituting z = 1=t and y = x=tdn=2e , we obtain the equation at in nity

(

x2 = cn + cn 1 t + : : : + c0tn for n even : x2 = cnt + cn 1t2 + : : : + c0 tn for n odd

If n is even then f1 1(1) consists of two non-rami ed points, so we may assume n odd. Then f1 1(1) consists of one rami cation point q, necessarily de ned over Kv . Write cn = u0 and z = u1 , where u0 and u1 are units and  is a uniformizer in ov . (We may assume that some power r is contained in OK .) Our equation takes the form

y2n  = u0un1 + cn 1un1 1  + : : : + c0 u1n  :

(1)

We review a property of the v-adic numbers, (proved in [25], Ch. XIV

x4). Consider the multiplicative group U (m) := fu 2 ov : u  1 (mod m)g: Then for m suciently large we have U (m)  Kv2 . In particular, to deter-

mine whether a unit u is a square, it suces to consider its representative mod m . Consequently, if is suciently large and has the same parity as , then we can solve Equation 1 for y 2 Kv precisely when u0u1 is a square. For example, choose any M 2 OK so that M  u0(r 1) (mod r ) and set z = M=r 2 Ofvg . Hence, of the z 2 Ofvg with jzjv  B (with B  0), the fraction satisfying our conditions is bounded from below. It follows that f > 0. 0 20

Now let f : U ! Consider the function

P1

be an etale morphism of curves de ned over K .

!f;S (B ) := #fz 2 OS : jzjv  B for each v 2 S and z 2 f (U (K ))g and the quantity

lim sup !f;fvg (B )=f (B ): B!1

We expect that this is zero provided that f does not admit a rational section. We shall prove this is the case when f has degree two. A key ingredient of our argument is a version of Hilbert's Irreducibility Theorem:

Proposition 6.8 Let f : U ! P1 be an etale morphism of curves, de ned over K and admitting no rational section. Then we have

lim sup !f;fvg (B )=Id (B ) = 0: B!1

Proof. We refer the reader to Serre's discussion of Hilbert's irreducibility theorem ([26], x9.6, 9.7). Essentially the same argument applies in our situation. 0 Combining Propositions 6.6, 6.7, and 6.8, we obtain:

Corollary 6.9 Let f : C !

P1

be a nite morphism of smooth curves de ned over K . Assume that f admits no rational section and that f 1(1) contains a Kv -rational point. We also assume that f has degree two. Then we have lim sup !f;fvg (B )=f (B ) = 0: B!1

In particular, the set fz 2 Ofvg : z 2 f (C (Kv )) n f (C (K ))g is in nite.

6.4 A density theorem for surfaces

Geometric assumptions: Let X and B be at and projective over Spec(OS ) and  : X ! B be a morphism. Let L  X be a closed irreducible subscheme, D  X a reduced eective Cartier divisor, and q := D \ L. We assume the generic bers satisfy the following: X is a geometrically connected surface, B a smooth curve,  : X ! B a at morphism such that the generic ber is a rational curve with bisection D. We also assume L ' P1K , jL is nite, 21

and L meets D at a single point q, at which D is nonsingular. Write X 0 for X B L, D0 for D B L, L0 for the image of the diagonal in X B L (now a section for 0 : X 0 ! L), and q0 for L0 \ D0. Finally, if C 0 denotes the normalization of the union of the irreducible components of D0 dominating L, we assume that C 0 ! L has no rational section over K (i.e., that C 0 is irreducible over K ).

Arithmetic assumptions: We assume that (L; q) has an S -integral point. Furthermore, we assume that for some v 2 S , C 0 has a Kv -rational point lying over 0(q0).

Remark 6.10 The second assumption is valid if any of the following are

satis ed: 1. D ! B is unrami ed at q. 2. D ! B is nite (but perhaps rami ed) at q and L ! B has rami cation at q of odd order. 3. D ! B is nite (but rami ed) at q and L ! B has rami cation at q of order two. Choose local uniformizers t; x; and y so that we have local analytic equations t + ax2 = 0 and t + by2 = 0 (with a; b 2 K ) for D ! B and L ! B . We assume that ab is a square in Kv . Note that in theplast case, D0 and C 0 have local analytic equations ax2 by2 = 0 and x=y = ` b=a respectively.

Theorem 6.11 Under the geometric and arithmetic assumptions made above, S -integral points of (X ; D) are Zariski dense. Proof. It suces to prove that S -integral points of (X 0 ; D0 ) are Zariski dense. These map to S -integral points (X ; D). Consider rst S -integral points of (L0; q0). These are dense by Proposition 6.5, and contain a nite index subgroup of G a (OS )  P1K . Corollary 6.9 and our geometric assumptions imply that in nitely many of these points lie in 0(C 0(Kv )) n 0(C 0(K )). Choose a generic S -integral point p of (L0; q0) as described above. Let 0 Xp = 0 1(p); Dp0 = Xp0 \ D0; and L0p = Xp0 \ L0, so that (Xp0 ; Dp0 ) is a rational curve with rational bisection and integral point L0p. Combining the results of the previous paragraph with Proposition 6.3, we obtain that S -integral 22

points of (Xp0 ; Dp0 ) are Zariski dense. As we vary p, we obtain a Zariski dense collection of integral points for (X 0; D0). 0

6.5 Cubic surfaces containing a line

Let X1 be a cubic surface in P3OS , D1  X1 a hyperplane section, and L1  X1 a line not contained in D1, all assumed to be at over Spec(OS ). Write q1 := D1 \ L1 , a rational section over Spec(OS ). Let P3OS 99 K B be the projection associated with L1, X = BlL1 X1 , and  : X ! B the induced projection (of course, B = P1OS if OS is a UFD). Let L  X be the proper transform of L1, D  X the total transform of D1, and q = L \ D. We shall apply Theorem 6.11 to obtain density results for S -integral points of (X1; D1). We will need to assume the following geometric conditions: GA1 D1 is reduced everywhere and nonsingular at q1; GA2 X1 has only rational double points as singularities, with at most one singularity along L1; GA3 D1 is not the union of a line and a conic containing q1 (de ned over K ). The rst assumption and the fact that D1 is Cartier imply that X1 is nonsingular at q1. We analyze the projection from the line L1 using the rst two assumptions. This induces a morphism  : X ! P1 : Of course, X = X1 if and only if L1 is Cartier in X1, which is the case exactly when X1 is smooth along L1 . We use L and D to denote the proper transforms of L1 and D1 . Our three assumptions imply that D equals the total transform of D1 and has a unique irreducible component C dominating P1 . We also have that the generic ber of  is nonsingular, intersects D in two points, and intersects L in two points (if X1 is smooth along L1 ) or in one point (if X1 has a singularity along L1 ). In particular, L is a bisection (resp. section) of  if X1 is nonsingular (resp. singular) along L1 . We emphasize that S -integral points of (X ; D) map to S -integral points of (X1; D1), and all the Geometric Assumptions of Theorem 6.11 are satis ed except for the last one. The last assumption is veri ed if any of the following hold: 23

GA4a The branch loci of C ! P1 and L ! P1 do not coincide. GA4b The curve C has genus one. GA4c X1 has a singularity along L1 . Clearly, either the second or the third condition implies the rst. We turn next to the Arithmetic Assumptions. AA1 (L1; q1) has an S -integral point. Note that S -integral points of (L1; q1) not lying in the singular locus of X1 ! Spec(OS ) lift naturally to S -integral points of (L; q). Our next task is to translate the conditions of Remark 6.10 to our situation. They are satis ed in any of the following contexts: AA2a D1 is irreducible over K and q1 is not a ex of D1; AA2b X1 has a singularity along L1 ; AA2c D1 is irreducible over K and q1 is a ex of D1. Let H be the hyperplane section containing L1 and the ex line. We assume that H \ X1 = L1 [ M , where M is a smooth conic. AA2d D1 is irreducible over K but q1 is a ex so that the hyperplane H containing L1 and the ex line F intersects X1 in three coincident lines, i.e., H \ X1 = L1 [ M1 [ M2 . Choose local coordinates x and y for H so that L1 = fx = 0g; F = fy = 0g; and M1 [ M2 = fax2 + cxy + by2 = 0g. Then we assume that ab is a square in Kv . AA2e D1 consists of a line and a conic C1 irreducible over K , intersecting in two distinct points, each de ned over Kv . In the rst case, the map D ! B is unrami ed at q. Note that in the second case L is a section for . In the third case, our assumption implies that L ! B is unrami ed at q. In the last case, we observe that the points of L lying over (q) are de ned over Kv , hence C 0 has a Kv -rational point over 0(q0). It remains to show that AA2d allows us to apply case 3 of Remark 6.10. We x projective coordinates on P3 compatibly with the coordinates already 24

chosen on H : y = 0 is the linear equation for the hyperplane containing D1 , z = 0 the equation for H , x = z = 0 the equations for L1 , and x = z = y = 0 the equations for q1 . Under our assumptions, the equations for D1 and X1 take the form

g := zw2 + ax3 + c1wxz + c2wz2 + c4 x2 z + c5 xz2 + c6z3 = 0 f := g + cx2 y + bxy2 + yz`(w; x; y; z) = 0 where ` is linear in the variables. The conic bundle structure  : X ! B is obtained by making the substitution z = tx

g0 = tw2 + x(wc1t + wc2t2) + x2 (a + c4 t + c5 t2 + c6 t3 ) = 0 f 0 = g0 + cxy + by2 + ty`(w; x; y; tx): We analyze the local behavior of D ! B at q using x as a coordinate for D. First dehomogenize

g00 = t + x(c1t + c2t2 ) + x2 (a + c4 t + c5t2 + c6 t3) = 0 and then take a suitable analytic change of coordinate on D to obtain t + aX 2 = 0. To analyze L ! B , we set x = 0 and use y as a coordinate

f 00 = t + by2 + ty`(1; 0; y; 0) = 0: After a suitable analytic change of coordinate on L, we obtain t + bY 2 = 0.

Remark 6.12 We further analyze condition AA2d when Kv = R . Then ab is a square if and only if ab  0. This is necessarily the case if c2 4ab < 0, i.e., if the lines M1 and M2 are de ned over an imaginary quadratic extension. We summarize our discussion in the following theorem:

Theorem 6.13 Let X1 be a cubic surface, D1  X1 a hyperplane section, and L1  X1 a line not contained in D1 , all assumed to be at over Spec(OS ). Write q1 := D1 \ L1 . Assume the following: 1. GA1,GA2,GA3, and AA1; 2. at least one of the assumptions GA4a,GA4b,or GA4c; 3. at least one of the assumptions AA2a,AA2b,AA2c,AA2d, or AA2e.

25

Then S -integral points of (X1 ; D1 ) are Zariski dense.

We recover the following result (essentially Theorem 2 of Beukers [2]):

Corollary 6.14 Let X1 be a cubic surface, D1  X1 a hyperplane section, and L1  X1 a line not contained in D1, all assumed to be at over Spec(Z). Write q1 := D1 \ L1. Assume that 1. X1 and D1 are smooth; 2. there exists an Z-integral point of (L1; q1); 3. if q is a ex of D1 , we assume that the hyperplane containing L1 and the ex line intersects X1 in a smooth conic and L1 . Then Z-integral points of (X1; D1) are Zariski dense. We also recover a weak version of Theorem 1 of [2]. (This theorem is asserted to be true but the proof is not quite complete; the problem occurs in the argument for the second part of Lemma 2.)

Corollary 6.15 Retain all the hypotheses of Corollary 6.14, except that we

allow the existence of a hyperplane H intersecting X1 in three lines L1 ; M1 ; and M2 and containing a ex line F for D1 at q. Let p be a place for Z (either in nite or nite). Choose local coordinates x and y for H so that L1 = fx = 0g; F = fy = 0g; and M1 [ M2 = fax2 + cxy + by2 = 0g, and assume that ab is a square in Q p . Then Z[1=p]-integral points of (X1 ; D1) are Zariski dense (where Z[1=1] = Z and Q 1 = R .) Of course, there are in nitely many primes p such that ab is a square in Q p . When p = 1, by Remark 6.12 it suces to verify that M1 and M2 are de ned over an imaginary quadratic extension. We also obtain results in cases where the boundary is reducible:

Corollary 6.16 Let X1 be a cubic surface, D1  X1 a hyperplane section, and L1  X1 a line not contained in D1, all assumed to be at over Spec(Z). Write q1 := D1 \ L1. Assume that 1. X1 is smooth; 2. there exists an S -integral point of (L1; q1); 26

3. D1 = E [C , where E is a line intersecting L1 and C is a conic irreducible over K ; 4. C intersects E in two points, de ned over Kv where v is some place in S; 5. there exists at most one conic in X1 tangent to both L1 and C . Then S -integral points of (X ; D) are Zariski dense. Note that the assumption on the conics tangent to L1 and C is used to verify GA4a.

6.6 Other applications

Theorem 6.11 can be applied in many situations.

Theorem 6.17 Let X = P1OS  P1OS , D  X a divisor of type (2; 2), and L  X a ruling of X , all at over OS . Assume that 1. D is nonsingular; 2. L is tangent to D at q ; 3. S -integral points of (L; q) are Zariski dense. Then S -integral points of (X ; D) are Zariski dense. Proof. Let  be the projection for which L is a section. Since C = D in this case, the second arithmetic assumption of Theorem 6.11 is easily satis ed. 0 We also obtain the following potential density result:

Theorem 6.18 Let X be a smooth Del Pezzo surface of degree KX2 and index one (i.e., ZKX is saturated in Pic(X )). Let D  X be a smooth anticanonical

divisor. If X and D are de ned over a number eld K then integral points of (X; D) are potentially dense. Proof. Applying the classi cation theory of surfaces (and enlarging the base eld), we may represent X as a conic bundle  : X ! P1 . First express X as a blow-up of P2 in 9 KX2 points. The pencil of lines in P2 containing

27

one of the points w gives the conic bundle structure. Let L denote a ( 1)curve not contained in a ber of , e.g., the exceptional curve lying over w. By adjunction, D is a bisection for  and intersects L in one point q. Finally, the irreducible components of the normalization of D P1 L all have positive genus, because D has positive genus. In particular, X; D; L; and q satisfy all the geometric assumptions of Theorem 6.11. On the other hand, its arithmetic assumptions may always be satis ed after judicious extensions of OS . It follows that integral points of (X; D) are potentially dense. 0

7 Potential density for log K3 surfaces We consider the following general situation:

Problem 7.1 (Integral points of log K3 surfaces) Let X be a surface

and D a reduced eective Weil divisor such that (X; D) has log terminal singularities and KX + D is trivial. Are integral points on (X; D) potentially dense? Problem 7.1 has been studied when D = ; (see, for example, [3]). In this case density holds if X has in nite automorphisms or an elliptic bration. The case X = P2 and D a plane cubic has also attracted signi cant attention. Silverman [29] proved potential density in the case where D is singular and raised the general case as an open question. Beukers [1] established this by considering the cubic surface X1 obtained as the triple cover of X totally branched over D. Implicit in [2] is a proof of potential density when X1 is a smooth cubic surface and D1 is a smooth hyperplane section. Note that this also follows from Theorem 6.13 (cf. also Corollaries 6.14 and 6.15.) After suitable extensions of K and additions to S , there exists a line L  X de ned over K and the relevant arithmetic assumptions are satis ed. Similarly, the case of X = P1  P1 and D a smooth divisor of type (2; 2) follows from Theorem 6.17. Finally, Theorem 6.18 gives potential density when X is an index-one Del Pezzo surface and D is a smooth anticanonical divisor. We summarize our results as follows:

Theorem 7.2 Let X be a smooth Del Pezzo surface and D a smooth anticanonical divisor. Then integral points for (X; D) are potentially dense.

28

We close this section with a list of open special cases of Problem 7.1. 1. Let X be a Del Pezzo surface and D a singular anticanonical cycle. Show that integral points for (X; D) potentially dense. 2. Let X be a Hirzebruch surface and D an anticanonical cycle. Find a smooth rational curve L, intersecting D in exactly one point p, so that the induced map ' : L ! P1 is nite surjective.

7.1 Appendix: some geometric remarks

The reader will observe that the methods employed to prove density for integral points on conic bundles (with bisection removed) are not quite analogous to the methods used for elliptic brations. The discrepancy can be seen in a number of ways. First, given a multisection M for a conic bundle (with bisection removed), we can pull-back the conic bundle to the multisection. The resulting bration has two rational sections, Id and M (see section 4). However, a priori one cannot control how M intersects the boundary divisor (clearly, this is irrelevant if the boundary is empty). A second explanation may be found in the lack of a good theory of ( nite type) Neron models for algebraic tori (see chapter 10 of [5]). We should remark that in some special cases these diculties can be overcome, so that integral points may be obtained by geometric methods completely analogous to those used for rational points. Consider the cubic surface x3 + y3 + z3 = 1 with distinguished hyperplane at in nity. This surface contains a line with equations x + y = z 1 = 0. Euler showed that the resulting conic bundle admits a multisection (x0; y0; z0 ) = (9t4; 3t 9t4 ; 1 9t3 ); which may be reparametrized as (x1 ; y1; z1) = (9t4 ; 3t 9t4; 1 + 9t3 ): Lehmer [15] showed that this is the rst in a sequence of multisections, given recursively by (xn+1; yn+1; zn+1 ) = 2(216t6 1)(xn; yn; zn) (xn 1; yn 1; zn 1) + ( 108t4 ; 108t4; 216t6 + 4) This is related to the fact that the norm group scheme u2 3(108t6 1)v2 = 1; 29

admits a section of in nite order (u; v) = (216t6 1; 12t3). Remarkably, this group acts regularly on the conic bundle, i.e., the coordinate transformations are integral polynomials in t. In general, one would only expect a rational action, de ned over the generic point of the t-line.

References [1] F. Beukers, Ternary forms equations, J. Number Theory 54 (1995), no. 1, 113{133. [2] F. Beukers, Integral points on cubic surfaces, Number theory (Ottawa, ON, 1996), 25{33, CRM Proc. Lecture Notes, 19, AMS, Providence, RI, (1999). [3] F. Bogomolov, Yu. Tschinkel, Density of rational points on elliptic K3 surfaces, alg-geom 9902092, to appear in Asian J. of Math. [4] E. Bombieri, D. Masser, U. Zannier, Intersecting a Curve with Algebraic Subgroups of Multiplicative Groups, Internat. Math. Res. Notices no. 20 (1999), 1119{1140. [5] S. Bosch, W. Lutkebohmert, M. Raynaud, Neron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21, Springer-Verlag, Berlin, 1990. [6] J.-L. Colliot-Thelene, A. N. Skorobogatov, P. Swinnerton-Dyer, Rational points and zero-cycles on bred varieties: Schinzel's hypothesis and Salberger's device, J. Reine Angew. Math. 495 (1998), 1{28. [7] J.-L. Colliot-Thelene, A. N. Skorobogatov, P. Swinnerton-Dyer, Double bres and double covers: paucity of rational points, Acta Arith. 79 (1997), no. 2, 113{135. [8] J.-L. Colliot-Thelene, P. Swinnerton-Dyer, Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, J. Reine Angew. Math. 453 (1994), 49{112. [9] G. Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549{576. 30

[10] A. Grothendieck, Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux (SGA 2), Advanced Studies in Pure Mathematics, Vol. 2. North-Holland Publishing Co., Amsterdam; Masson & Cie, E diteur, Paris, 1968. [11] J. Harris, Yu. Tschinkel, Rational points on quartics, alg-geom 9809015, to appear in Duke Math. J. [12] B. Hassett, Yu. Tschinkel, Abelian brations and rational points on symmetric products, alg-geom 9909074, to appear in the International J. of Mathematics. [13] S. Iitaka, Algebraic geometry: An introduction to birational geometry of algebraic varieties, Springer-Verlag, New York-Berlin, 1982. [14] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, 283{360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam-New York, 1987. [15] D. H. Lehmer, On the diophantine equation x3 + y3 + z3 = 1, J. Lond. Math. Soc. 31 (1956), 275{280. [16] M. McQuillan, Division points on semi-abelian varieties, Invent. Math. 120 (1995), no. 1, 143{159. [17] D. W. Masser, Specializations of nitely generated subgroups of abelian varieties, Trans. Amer. Math. Soc. 311 (1989), no. 1, 413{424. [18] D. W. Masser, Specializations of endomorphism rings of abelian varieties, Bull. Soc. Math. France 124 (1996), no. 3, 457{476.  [19] J.S. Milne, Etale cohomology, Princeton University Press, Princeton New Jersey, 1980. [20] D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory, Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34, Springer-Verlag, Berlin, 1994. [21] R. Noot, Abelian varieties|Galois representation and properties of ordinary reduction, Compositio Math. 97 (1995), no. 1, 161{171. 31

[22] T. Ono, On some arithmetic properties of linear algebraic groups, Ann. of Math. (2) 70 (1959), 266{290. [23] T. Ono, Arithmetic of algebraic tori, Ann. of Math. (2) 74 (1961), 101{ 139. [24] J. P. Serre, Algebraic groups and class elds, Graduate Texts in Mathematics, 117, Springer-Verlag, New York-Berlin, 1988. [25] J. P. Serre, Local elds, Graduate Texts in Mathematics, 67, SpringerVerlag, New York-Berlin, 1979. [26] J. P. Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15. Friedr. Vieweg & Sohn, Braunschweig, 1989. [27] J.M. Shyr, A generalization of Dirichlet's unit theorem, J. Number Theory 9 (1977), no. 2, 213{217. [28] J. Silverman, Heights and the specialization map for families of abelian varieties, J. reine und angew. Math. 342 (1983), 197{211. [29] J. Silverman, Integral points on curves and surfaces, Number theory (Ulm, 1987), 202{241, LNM, 1380, Springer, New York-Berlin, 1989. [30] P. Vojta, Diophantine approximations and value distribution theory, LNM, 1239, Springer-Verlag, Berlin, 1987. [31] P. Vojta, Integral points on subvarieties of semiabelian varieties. II, Amer. J. Math. 121 (1999), no. 2, 283{313. [32] V. E. Voskresenskii, Algebraic groups and their birational invariants, Translations of Mathematical Monographs, 179. American Mathematical Society, Providence, RI, 1998.

32

COMPOSITION OF POINTS AND MORDELL–WEIL PROBLEM FOR CUBIC SURFACES

arXiv:math/0011198v1 [math.AG] 23 Nov 2000

D. Kanevsky1,2 , Yu. Manin2 1

T.J. Watson Research Center, P.O. Box 218, 23-116A, Yorktown Heights, New York 10598, US 2 Max–Planck–Institut f¨ ur Mathematik, Bonn, Germany

Abstract. Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell–Weil theorem for V states that there is a finite subset P ⊂ V (k) such that the whole V (k) can be obtained from P by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. Equivalently, the group of birational transformations of V generated by reflections with respect to k–points is finitely generated. In this paper, elaborating an idea from [M3], we establish a Mordell–Weil type finite generation result for some birationally trivial cubic surfaces W . To the contrary, we prove that the birational automorphism group generated by reflections cannot be finitely generated if W (k) is infinite. §1. Introduction 1.1. Composition of points. Let V be a cubic hypersurface without multiple components over a field k in Pd , d ≥ 2. Three points x, y, z ∈ V (k) (possibly coinciding) are called collinear if either x + y + z is the intersection cycle of V with a line in Pd (with correct multiplicities), or x, y, z lie on a k–line belonging to V . If x, y, z are collinear, we write x = y ◦ z. Thus ◦ is a (partial and multivalued) composition law on V (k). We will also consider its restriction on subsets of V (k), e.g. that of smooth points. If x ∈ V (k) is smooth, and does not lie on a hyperplane component of V , the birational map tx : V → V, y 7→ x ◦ y, is well defined. It is called reflection with respect to x. Denote by Bir V the full group of birational automorphisms of V. The following two results summarize the properties of {tx } for curves and surfaces respectively. The first one is classical, and the second is proved in [M1], Chapter V. 1.2. Theorem. Let V be a smooth cubic curve. Then: (a) Bir V is a semidirect product of a finite group and the subgroup consisting of products of an even number of reflections {tx | x ∈ V (k)}. (b) We have identically t2x = (tx ty tz )2 = 1 (1.1) for all x, y, z ∈ V (k). 1

2

If in addition k is finitely generated over a prime field, then: (c) Bir V is finitely generated. (d) All points of V (k) can be obtained from a finite subset of them by drawing secants and tangents and adding the intersection points. 1.3. Theorem. Let V be a minimal smooth cubic surface over a perfect non– closed field k. Then: (a) Bir V is a semi–direct product of the group of projective automorphisms and the subgroup generated by {tx | x ∈ V (k)} and {su,v | u, v ∈ V (K); [K : k] = 2; u, v are conjugate over k} where su,v := tu tu◦v tv , and u, v do not lie on lines of V. (b) We have identically t2x = (tx tx◦y ty )2 = (su,v )2 = 1, stx s−1 = ts(x) ,

(1.2)

for all pairs u, v not lying on lines in V , and projective automorphisms s. (c) The relations (1.2) form a presentation of Bir V. We remind that V is called minimal if one cannot blow down some lines of V by a birational morphism defined over k. The opposite class consists of split surfaces upon which all lines are k–rational. 1.4. Main results of the paper. Although Theorems 1.2 and 1.3 look very similar, there is an important difference between finiteness properties in one– and two–dimensional cases. Basically, (1.1) means only that x + y := e ◦ (x ◦ y) is an Abelian group law with identity e: see [M1], Theorem I.2.1. The statements c) and d) of the Theorem 1.2 additionally assert that this group is finitely generated. Therefore, (1.1) generally is not a complete system of relations between {tx }. On the contrary, (1.2) is complete, and in §2 we will see that this prevents Bir V from being finitely generated if V (k) is infinite. This answers one of the questions raised in [M3]. Therefore, any reasonable analog of the Mordell–Weil problem must address the problem of finite generation for (V (k), ◦) or of quotients of V (k) with respect to various equivalence relations compatible with ◦. This is the subject of §§3–5.

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As in [M1], Chapter II, we can start with the universal equivalence relation U . By definition, this is the finest equivalence relation compatible with collinearity and such that ◦ induces a well defined operation on V (k)/U also denoted ◦. Then one of the Mordell–Weil type questions asks about finite generation (= finiteness) of the CH–quasigroup (V (k)/U, ◦) (see [M1], Chapter I.). In §3 and §4 we give a description of U refining earlier results of [M1]. Consider the set of intersections of V with tangent planes at points of V (k) and add to it all images of these curves with respect to the group generated by all tx , x ∈ V (k). Then one class of U consists of points that can be pairwise joined by a chain of curves belonging to this set of curves. This is the content of Theorem 3.3 below. We then discuss various versions of finite generation of (V (k), ◦). One essential choice is whether to allow to apply ◦ only to the different previously constructed points (for minimal surfaces, the result will then be uniquely defined). Another option giving more flexibility is to allow expressions x ◦ x and treat them as multivalued, thus adding at one step all the intersection points of V with a tangent plane at x. Finally, in §4 we extend the group–theoretic description of U given in [M1], II.13.10. The results of §3 and §4 are essentially algebraic and do not add any new cases of finite generation of (V (k), ◦) to the short list of locally compact local fields already treated in [M1]. (In fact, [M1] proves the finiteness of V (k)/U over such fields by establishing that V (k) is covered by a finite number of sets of the form (x ◦ x) ◦ (y ◦ y)). In §5 we study modified composition laws of points introduced in [Ma3]. The idea behind this development is to reinterpret the classical theorem on the structure of abstract projective planes as a finiteness result. Namely, let k be a finitely generated field. Start with a finite subset S ⊂ P2 (k) and add to it pairwise intersections of all lines passing through two points of S thus getting a new finite set S ′ . Apply the same procedure to S ′ , and so on. If S is large enough, in the limit we will get the whole P2 (k). This easily follows from the fact that if we start with S consisting of ≥ 4 points in general position, in the limit we will get an abstract projective plane satisfying the Desargues axiom and therefore coinciding with P2 (k ′ ) for k ′ ⊂ k up to a projective coordinate change. A trick, first introduced in [M3], allowed us to translate this remark into a finiteness theorem for V (k) assuming the existence of a birational morphism p : V → P2 defined over k. However, this required dealing with modified composition laws: roughly speaking, instead of looking at the collinearity relation induced by that in P3 , we now have to use the collinearity relations determined by the morphism p. In this paper we make some steps towards eliminating this complication. Although the final result falls short of what we would like to prove, we feel that

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the connection and analogies with the theory of abstract projective planes deserve further study. Acknowledgement. The first named author would like to thank V. Berkovich and J-L. Colliot-Th´el`ene for useful discussions. The work was partially supported by the Humboldt Foundation during the author’s stay at the Max–Planck–Institut f¨ ur Mathematik. §2. Cardinality of generators of subgroups in a reflection group 2.1. Notation. We shall call an abstract cubic a set S with a ternary relation L ⊂ S × S × S, satisfying the following axioms: (a) L is invariant with respect to permutations of factors. (b) If (x, y, z), (x, y, z ′) ∈ L and x 6= y, then z = z ′ . The reflection group GS of an abstract cubic S is generated by symbols tx , x ∈ S, subject to the following relations: t2x = 1 for all x ∈ S; (tx ty tz )2 = 1 for all (x, y, z) ∈ L. The following result is proved in [K1]. 2.2. Theorem. (a). Any element of finite order in GS is conjugate to either tx or to tx ty tz for appropriate x, y, z ∈ S. Let S be given effectively and L ⊂ S × S × S be decidable. Then: (b) The word problem in GS is decidable. (c) The conjugacy problem in GS is decidable. The proof is based on a direct description of GS as a limit of amalgamated sums. In [K2] it is shown that S can be sometimes reconstructed from GS . Moreover, under some additional assumptions it is proved that Aut GS is generated by GS and permutations of S preserving L. A different interesting description of GS and another proof of the Theorem 2.2 is given in [P]. For the purposes of our paper we need the following description of GS that is a special case of the general structure theorem 1.4 in [K1]. 2.3. Structural Theorem. Let x ∈ S be an arbitrary point and S ′ := {S \ x}. Then GS is canonically isomorphic to GS ′ ∗Π K (the free product of GS ′ and K with the amalgamated subgroup Π). The groups in this product can be described as follows. (a) GS ′ is the reflection group of the cubic S ′ with the ternary relation induced by L on S ′ .

5

(b) The amalgamated subgroup Π is a free group generated by free generators au,v = tu tv for all distinct pairs u, v ∈ S ′ such that (u, v, x) ∈ L and u < v (for some fixed ordering of S). ∼

(c) K → Z2 ∗ Z2 ∗ · · · ∗ Z2 . Generators of the subgroups Z2 in this free product are tx and tx au,v . (d) Π is of index 2 in K. The quotient group K/Π is generated by the class tx . This structural result leads to the following auxiliary statement, which we need to prove our main results in this section. 2.4. Definition–Lemma. (a) In the situation of Theorem 2.3 a family W = hR1 , tx , R2 , tx , . . . , tx , Rn i, where Ri ∈ GS ′ , is called a reduced tx –partition of g = R1 tx R2 tx . . . Rn if Ri ∈ / Π for 1 < i < n. (b) Let W be a reduced tx –partition of g ∈ GS . Let us define ordx (g) as the number of tx in W . This number depends on g and x and is the same for different reduced tx –partitions of g. (c) Let g ∈ GS be such that ordx (g) = 0. Then g ∈ GS ′ . (d) ordx (g1 g2 ) ≡ (ordx (g1 ) + ordx (g2 )) mod 2. (e) ordx (aga−1 ) ≡ ordx (g) mod 2 for any a, g ∈ GS . (f ) Let g ∈ GS . We put δ(g) := {x ∈ S | ordx (g) 6= 0}. The set δ(g) is finite. (g) δ(g1 g2 ) ⊂ δ(g1 ) ∪ δ(g2 ). (i) Let hh1 , h2 , . . . i be a family generating a subgroup H. Then ∪h∈H δ(h) = ∪i δ(hi ) . Now we can formulate the main theorem of this section. 2.5. Theorem. Consider a subgroup H = hg1 , g2 , . . . gn , . . . i ⊂ GS generated by an infinite family of elements such that δ = ∪i δ(gi ) is infinite. Then H is not finitely generated. Proof. Assume that H is finitely generated by h1 , . . . , hk . Then δ ′ = ∪i=1,..k δ(hi ) is finite. Therefore there exists some gr and x ∈ S such that the following holds: ordx (gr ) 6= 0 and x ∈ / δ ′ . Hence ordx (hi ) = 0 for all i = 1, . . . , k. By 2.4(i), H ⊂ GS ′ if H is generated by h1 , . . . , hk . Since ordx (gr ) 6= 0, gr ∈ / H. This contradiction proves the theorem. The following extension of Theorem 2.5 can be applied to various subgroups of Bir (V ).

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2.6. Corollary. Let GS be the reflection group of an abstract cubic S and let W be the group of permutations of S, preserving its ternary relation L. Let G ∼ = W ∗GS be the semi-direct product of W and GS , such that wtx w−1 = tw(x) for any w ∈ W and x ∈ S. Let a subgroup H ⊂ G be generated by a finite subgroup W ′ ⊂ W and an infinite family of elements gi ∈ GS such that δ = ∪i=1,2,.. δ(gi ) is infinite. Then H is not finitely generated. Proof. Let us assume that H is generated by a finite number of elements h1 , . . . , hk ∈ GS and a finite number of elements from W ′ . Let δ = ∪i δ(gi ) and let δ ′ = ∪w∈W ′ wδ be obtained by applications of all w ∈ W ′ to δ. Since δ and W ′ are finite, δ ′ is finite. Therefore there exists a generator gi ∈ H such that δ(gi ) * δ ′ . Therefore, gi ∈ / Gδ′ , i.e. it cannot be obtained as a product of elements from h1 , . . . , hk and w ∈ W ′ . This contradiction proves the corollary. 2.7. Examples. In the situation of Theorem 1.3 assume that S = V (k) is infinite. Then the following subgroups of Bir (V ) cannot be finitely generated: (a) Bir (V ), B(V ) := htx | x ∈ V (k)i and G := htx , su,v | x ∈ V (k), u, v ∈ V (K); [K : k] = 2; u, v are conjugate over ki. (b) The commutant of any of subgroups described in (a). (c) Let B0 (V ) denote the normal subgroup of B(V ) generated by elements of the form tx ty tz tx′ ty tz′ , where (x, y, z) and (x′ , y, z ′ ) run through triples of collinear points of V (k). This subgroup was introduced in [M1] (II.13.9; beware of a misprint there: the second y carries a superfluous prime). It is closely related to the universal equivalence on V (k) (see the section 4 below). (d) Let B1 (V ) denote the normal subgroup of B(V ) generated by elements of the form tx ty tz , where (x, y, z) run through all possible triples of collinear points of V (k). This subgroup was introduced in [M2] because it is closely related to some admissible equivalence relations on V (k) We will now show that Theorem 2.3 implies the statement for the case (b). Other cases will be discussed later and stronger statements will be proved. −1 Proof of (b). The commutant of B(V ) contains elements tx ty t−1 x ty = tx ty tx ty = a2x,y . Let us consider an infinite family of elements a2xi yi , xi , yi ∈ S. The statement for (b) will follow if we show that δ(a2x,y ) contains x, since it will follow that ∪i δ(axi yi ) is infinite. For this it is enough to show that tx ty tx ty has the following reduced tx –partition: (tx , ty , tx , ty ). Indeed, in the notation of 2.4 one has to check that ty ∈ / Π. But this fact follows immediately from 2.3 if one notes that Π is a free group (hence it contains no nontrivial elements of finite order) and t2y = 1. This implies that ordx (a2x,y ) > 0, i.e. x ∈ δ(a2x,y ). Q.E.D. Our next theorem provides a lower bound for the number of generators in the normal closure.

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˜ 2.8. Theorem. For any g ∈ GS let δ(g) = {x ∈ S | ordx (g) 6≡ 0 mod 2}. Let H be the normal closure in GS generated by a family of elements that contains a subfamily of elements h = (h1 , . . . , hi , . . . ) such that the following condition holds. ˜ i ) such that xi ∈ ˜ j ) if i 6= j. (J): For any i there exist xi ∈ δ(h / δ(h Then H cannot be the normal closure in GS of less than card h generators. This theorem immediately implies 2.9. Corollary. In the situation of Theorem 2.8, H cannot be the normal closure of a finite number of elements if there is an infinite subsystem h satisfying (J). Proof of Theorem 2.8. Define a map of GS into the vector space FS2 as follows: ψ : GS → V, ψ(g) = (. . . , ordx (g) mod 2, . . . ). It follows from 2.4(d) that ψ is a group homomorphism so that it maps conjugacy classes in GS into one element. The theorem will follow if one shows that the image ψ(H) cannot be generated by less than card h vectors. But this follows immediately from the condition (J) in the theorem that guarantees that each image ψ(hi ) has a non–zero xi –component while all other vectors ψ(hj ) have a zero xi –component. 2.10. Corollary. None of the subgroups that are defined in (a),(c) and (d) in 2.7 can be obtained as the normal closure of a finite number of generators. ˜ x ) = x. Since GS contains the infinite Proof. (a) follows from the fact that δ(t number of tx , GS cannot be obtained as the normal closure of a finite number of elements. ˜ x ty tz tx′ ty tz′ ) contains x for (c) will follow similarly to (a) if we show that δ(t ′ ′ x 6= y, z, x , z . This follows from the fact that the tx –partition of tx ty tz tx′ ty tz′ is (tx , ty tz tx′ ty tz′ ) (where ty tz tx′ ty tz′ ∈ GS ′ ). Since V (k) has infinitely many collinear triples (x, y, z), such that x 6= y 6= z 6= x, one can find infinitely many generators in B0 (V ) satisfying the condition (J). The case (d) can be treated similarly. §3. Structure of universal equivalence 3.1. Setup. Let P be an abstract cubic with the collinearity relation L ⊂ P × P × P , such that for any x, y ∈ P, there exists z ∈ P with (x, y, z) ∈ P. An equivalence relation R on P is called admissible if the relation L/R induced on P/R has the following property: for any X, Y ∈ P/R, there exists a unique Z

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with (X, Y, Z) ∈ L/R. An admissible equivalence relation is called universal if it is finer that any other admissible relation. In [M1] it was proved that the universal relation exists (and of course, is unique) by a simple argument: just take the intersection of all admissible relations. Here we will clarify its structure by representing it as a limit of a sequence of explicitly constructed equivalence relations of which every next one is less fine than the previous one. 3.2. Approximations. For every i ≥ 0, we will describe inductively a symmetric and reflexive binary relation ∼i on P and its transitive closure ≈i . By definition, ∼0 and ≈0 are simply identical relations x = x′ . 3.2.1. Definition. If ∼i and ≈i are already defined, we put x ∼i+1 x′ iff x = x′ or there exist u, v, u′ , v ′ ∈ P such that u ≈i u′ , v ≈i v ′ , and (u, v, x) ∈ L, (u′ , v ′ , x′ ) ∈ L. Furthermore, we put x ≈i+1 x′ iff there is a sequence of points x = y0 , y1 , . . . , yr = ′ x such that ya ∼i+1 ya+1 for all a < r. Let us consider the case i = 1. By definition, x ∼1 x′ iff there exist u, v ∈ P such that (u, v, x), (u, v, x′) ∈ L. Let P be the set of k–points of a cubic surface V and L the usual collinearity relation. Assume for simplicity that V does not contain lines defined over k. Then x ∼1 x′ means that x = x′ or x and x′ lie on the intersection of V with the tangent plane at some k–point u (with u deleted if the double tangent lines to u in this plane are not defined over k). So one equivalence class for ≈1 consists of one point or of a maximal connected union of such quasiprojective curves, two of them being connected if they have an intersection point defined over k. The case of general cubic surface allows a similar description, but points of k–lines in V must be added as subsets of equivalence classes. 3.3. Theorem.(a) If x ≈i x′ then x ≈i+1 x′ . (b) Denote by ≈ the equivalence relation x ≈ x′

⇐⇒

∃i, x ≈i x′ .

Then it is admissible and universal. Proof. (a) It suffices to prove that if x 6= x′ , x ∼i x′ then x ≈i+1 x′ . For i = 0 this is clear. Assume that we have proved that u ∼i−1 u′ implies u ≈i u′ . If x ∼i x′ then by definition (u, v, x) ∈ L and (u′ , v ′ , x′ ) ∈ L for some u ≈i−1 u′ and v ≈i−1 v ′ . From the inductive assumption it follows that u ≈i u′ and v ≈i v ′ . By definition, then x ≈i+1 x′ . (b) Let us first prove that ≈ is admissible, in other words, if (u, v, x) ∈ L, ′ ′ (u , v , x′ ) ∈ L, and u ≈ u′ , v ≈ v ′ , then x ≈ x′ . In fact, for some i we have u ≈i u′ , v ≈i v ′ , so that x ≈i+1 x′ and x ≈ x′ .

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Now denote temporarily the universal equivalence relation by ≈U . The previous argument shows that x ≈U x′ ⇒ x ≈ x′ . It remains to prove that x ≈i x′ ⇒ x ≈U x′ . We argue by induction. Again, it suffices to check that x ∼i x′ ⇒ x ≈U x′ assuming x 6= x′ . We can then find (u, v, x) ∈ L, (u′ , v ′ , x′ ) ∈ L such that u ≈i−1 u′ , v ≈i−1 v ′ . Therefore u ≈U u′ , v ≈U v ′ , and finally x ≈U x′ . 3.4. Types of finite generation. Let us say, as in [M3], that P is ◦–generated by (xα | α ∈ A) if for any y ∈ P there is a non–associative commutative word in xα ’s such that, informally, y is one of the values of this word. This means that when we calculate this word in the order determined by the brackets, every time that we have to calculate some u ◦ v, we may replace it by any x such that (u, v, x) ∈ L. 3.4.1. Claim. If P is ◦–generated by (xα | α ∈ A), then the CH–quasigroup P/U is generated by the classes Xα of xα . We consider the following different types of ◦-generation. 3.4.2. Values of nonassociative words. Let W be a non-associative commutative word in finite number of variables Xi , P as in 3.1, and xi a family of elements of P with the same set of indices. We define different rules of computing values of W on (xi ) in the order determined by the brackets inductively as follows for i = 0, 1, . . . ∞. We set x ≈∞ y if x ≈ y (i.e. x ≈j y for some j). Rule Ai . If the word W = X has length 1, then a value of W at any point x ∈ P is any y ∈ P such that x ≈i y. In particular, A0 means that the value of W at x coincides with x. The rule A1 means that the set of values of W consists of those y for which which there are points uj , yj , j = 0, . . . , r, y0 = x, yr = y such that the following holds: (uj , uj , yj−1 ) ∈ L, (uj , uj , yj ) ∈ L for j = 1, . . . , r. If the word W = X ◦ Y has length 2, its set P (x, y) of values of W at x, y ∈ P 2 is defined as follows. P (x, y) = {z ∈ P | z ≈i z ′ , (x, y, z ′) ∈ L}. If the word W has length more than two, it is a product of two non empty words W = W1 ◦ W2 . Let P (Wi ) be a set of values of Wi that is defined inductively. Then the set of values P (W ) is defined as ∪P (x, y) for all (x, y) ∈ P (W1 ) × P (W2 ). We say that P is ◦Ai generated by P ′ = (xα | α ∈ A) if it is generated by application of the rule Ai to points in P ′ . The inverse statement of 3.4.1 is valid for ◦A∞ by trivial reasons. 3.4.2. Claim. If CH-quasigroup P/U ≈ is ◦-generated by classes of (xα | α ∈ A), then P is ◦A∞ generated by xα . 3.4.3. Questions. Let us define the generation index i(P ) of P as the smallest i such that P is ◦Ai –generated by a finite number of points in P . Let P = V (k) for some cubic surface.

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(1) For which fields k and for which classes of cubic surfaces i(P ) is finite? In particular, is i(P ) = 0 for V defined over a number field (the original Mordell-Weil problem)? (2) If the the CH–quasigroup P/U is finite, is the index i(P ) finite? It would be worthwhile to study (2) for an abstract cubic P that has an additional property: every three points of it generate an Abelian group like points on a plane cubic curve. §4. A group–theoretic description of universal equivalence In [M1], II.13.10 a group–theoretic description of universal equivalence was given for a cubic surface that is defined over an infinite field and has a point of general type. In this section we extend this description of universal equivalence. We relate the sequence of explicitly constructed equivalence relations from §3 to a filtration by subgroups in the reflection group associated with a minimal cubic surface. Let B(V ) and B0 (V ) be the groups described in the examples 2.7. Here the field k over which the cubic surface V is defined can be finite and therefore we do not assume that V (k) is infinite. Define x ∼ y mod U if tx ty ∈ B0 (V ). It is clear that U is an equivalence relation on V (k). The proof of the following theorem differs from the proof of the corresponding theorem 13.10 in [M1] in the following respects. It uses the explicit description of the universal admissible equivalence from the section 3 and the structural description of the reflection group of S = V (k). 4.1. Theorem. U is the universal admissible equivalence relation. Proof. We will check in turn that each of the equivalence relations is finer than the other one. Assume first that z ′ and z are universally equivalent. We want to show that z ′ ∼ z mod U . According to Theorem 3.3, z ′ ≈i z for some i. Since U is an equivalence relation, it is sufficient to treat the case z ′ ∼i z. The following Lemmma does the job. 4.2. Lemma. Denote by B i (V ), i = 0, 1, . . . , the normal closure of the family {tx tx′ | x ∼i x′ } in B(V ). Let x ∼i x′ , y ∼i y ′ , (x, y, z) ∈ L and (x′ , y ′ , z ′ ) ∈ L . Then the following holds: tz tz′ ∈ tz tz′ B i (V ) = tx ty tz tx′ ty′ tz′ B i (V ) ⊂ B i+1 ⊂ B0 (V ) Proof of Lemma 4.2. Using relations t2x = 1 and tx ty tz = tz ty tx we get b = tz ty tx tx′ ty′ tz′ = tz tz′ b′ where b′ = tz′ ty tx tx′ ty′ tz′ . Next, b′ is conjugate to b′′ =

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ty tx tx′ ty′ . And, finally, b′′ is a product of ty ty′ ∈ B i (V ) and ty tx tx′ ty which is conjugate to tx tx′ ∈ B i (V ). This proves the equality tz tz′ B i (V ) = tx ty tz tx′ ty′ tz′ B i (V ). It remains to show the inclusion B i+1 (V ) ⊂ B0 (V ). We will prove this inductively. B 1 (V ) is generated by tz tz′ such that z ′ and z lie on the intersection of V with a tangent plane at some k–point u. In this case tz′ tz = tz′ tu tu tz tu tu ∈ B0 (V ). Assume that we already proved that B i (V ) ⊂ B0 (V ) and let us prove that tz tz′ ∈ B0 (V ). Let z ′′ ∈ V (k) be such that (x′ , y, z ′′ ) ∈ L. Then tz tz′ = tz tz′′ tz′′ tz′ and the following inclusions hold: tz tz′′ ∈ tz ty tx tx′ ty tz′′ B i (V ) ⊂ B0 (V )B i (V ) ⊂ B0 (V ), tz′′ tz′ ∈ tz′′ tx′ ty tz′ tx′ ty′ B i (V ) ⊂ B0 (V )B i (V ) ⊂ B0 (V ). Since tz tz′ ∈ B i+1 (V ), this proves the inductive statement, establishes the Lemma and the first part of the Theorem. We turn now to the second part. Let A be any admissible equivalence relation. We shall show that x ∼ y mod U implies x ∼ y mod A. Let X, Y, Z be the A– classes of x, y, z. Then Z = X ◦ Y in the sense of the composition law induced by collinearity relation on S = V (k). Denote by E = V (k)/A the set of classes with the induced structure of the symmetric quasigroup. Let tX : E → E be the map tX (Y ) = X ◦ Y . The map tx 7→ tX extends to an epimorphism of groups ϕ : B(V ) → T (E). We will show that its kernel contains B0 (V ). Therefore if tx ty ∈ B0 (V ) then ϕ(tx ty ) = tX tY = 1. This implies that tX = tY and that X = Y . To prove this property of ϕ we need to extend the Theorem 13.1 (ii),(iii) in [M1] to our case. Recall that the Theorem 13.1 uses assumptions for cubic hypersurfaces that implies the fact that every equivalence class is dense in the Zariski topology. This is not true any more in general in our case. 4.3. Lemma. (a) ϕ : B(V ) → T (E) is well defined and is an epimorphism of groups. (b) In T (E) the following equality holds: tX tY tZ = tY ◦Y . Proof. (a) Our proof is based on the representation of elements in B0 (V ) as “minimal” words in the group K S , the free product of groups Z2 generated by symbols Tx , one for each point x with the relations Tx2 = 1 (cf. [K1], 2.6 and §6). In order to construct the homomorphism B(V ) → T (E), we first define the action of B(V ) on E. Denote by Tx1 Tx2 . . . Txn a minimal representation in K S of some s ∈ B(V ). Choose Y ∈ E and put s(Y ) = X1 ◦ (X2 ◦ . . . (Xn ◦ Y ) . . . ) where Xi are classes of xi in E. One can show that this definition does not depend on the choice of a minimal representation of s in K S . This can be done inductively on the length of minimal

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words in K S . All minimal words of length one representing the same element in B(V ) coincide. Let us assume that the statement is proved for minimal words of the length i − 1. Consider now two different minimal words w = T1 . . . Ti , w′ = T1′ . . . Ti′ of the length i representing s ∈ B(V ). (Minimal words representing the same element have the same length). If Ti = Ti′ then the action of w (resp. w′ ) on E can be factored through the actions of Ti and w1 = T1 . . . Ti−1 (resp. ′ w1′ = T1′ . . . Ti−1 ) . Since w1 and w1′ represent the same element in B(V ) and have the length i − 1, the statement follows by the inductive assumption. Otherwise, if Ti 6= Ti′ , consider a Ti –partition of w′ (it is defined in the same way as tx –partition above): (R1 , Ti , . . . Rk−1 , Ti , Rk ). From [K1] it follows that Rk = Tu1 Tv1 Tu2 Tv2 . . . Tur Tvr and (uj , vj , u) ∈ L for all j = 1, . . . r and Tu = Ti . Moreover, if we replace Ti Rk in w′ with Rk′ Ti where Rk′ = Tv1 Tu1 Tv2 Tu2 . . . Tvr Tur , then we get a new word w′′ that is already a minimal representation of s. Since w′′ and w both end with the same element Ti = Tu , they act in the same way on T (E). In order to prove that w′ and w′′ also act identically on T (E) it is enough to check that Tu Rk and Rk′ Tu act in the same way on T (E). This can be shown using the fact that tuj tvj tu = tu tvj tuj . To complete (a) we need to show that for any two elements s1 , s2 ∈ B(V ) and Z ∈ E we have s1 (s2 (Z)) = (s1 s2 )(Z). We will prove this statement by induction on the sum of lengths of minimal representation of s1 and s2 . The statement is obvious if s1 has length 0. Assume now that s1 has a minimal representation w1 = Tx1 . . . Txi , i ≥ 1, and s2 has a minimal representation w2 = Ty1 . . . Tyk . If w = w1 w2 is the minimal representation of s = s1 s2 than the action of s on E is defined via the action of w by the rule X1 ◦ (. . . Xi ◦ (Y1 ◦ . . . (Yk ◦ Z) . . . ) where Xi (resp. Yj ) are the classes of xi (resp. yj ) and Z ∈ E. Therefore s1 (s2 (Z)) = (s1 s2 )(Z). Assume now that w1 w2 is not minimal. Consider first the case when there exists such minimal representation of w1 , w2 that Txi = Ty1 (i.e. the last element in w1 coincides with the first element in w2 ). Let s′1 ∈ B(V ) be represented by w1 = Tx1 . . . Txi−1 and s′2 ∈ B(V ) be represented by w2′ = Ty1 . . . Tyk−1 . Then s′1 (s′2 (Z)) = s1 (s2 (Z)) and one can apply the inductive statement to s′1 and s′2 . Otherwise, let us assume that the word w1 w2 has the following Tx –partition; R1 Tx R2 . . . Tx Rl Rl+1 Tx Rl+2 Tx . . . Tx Rm where R1 Tx R2 . . . Tx Rl (resp. Rl+1 Tx Rl+2 Tx . . . Tx Rm ) is a minimal partition of w1 (resp. w2 ). Since w1 w2 is not minimal, Tx can be chosen in such a way that Rl Rl+1 = Tu1 Tv1 Tu2 Tv2 . . . Tur Tvr , where (us , vs , x) ∈ L for s = 1, . . . , r. As in the case of minimal words above one can replace Tx Rl Rl+1 in w1 w2 with Tv1 Tu1 Tv2 Tu2 . . . Tvr Tur Tx

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and obtain a new word w′ that has the same action on E that w1 w2 . Since w′ has two subsequent elements Tx , we can split it into a product of w1′ that ends with Tx and w2′ that starts with Tx . This case was already considered in this proof. (b) follows from properties of the group law on plane cubic curves. This proves the Lemma 4.2. To finish the proof of Theorem 4.1, we use the following identity: ϕ(tx ty tz tx′ ty tz′ ) = tX tY tX◦Y tX ′ tY tX ′ ◦Y ′ = t2Y ◦Y = 1. Here X, Y, . . . are the classes of x, y, . . . mod A. As a consequence, B0 (V ) ⊂ Ker ϕ, proving the theorem. 4.4. Corollary. Let V be a minimal cubic surface over a finite field with q elements. Then B(V )/B0 (V ) = Z2 , except when all points of V (k) are Eckardt points. In the later case we have either q = 2, card V (k) = 3, or q = 4, card V (k) = 9. Proof. This follows from the description of the universal equivalence for V over finite fileds in [Sw–D]. 4.5. Remarks. (a) As it follows from the proof of Theorem 4.1, it can be extended to an abstract cubic for which every three points generate an abelian group, in the same sense as for a plane cubic curve. We believe that this theorem can be proved also for an abstract cubic using only a structural description of GS without this additional assumption. We plan to address this problem elsewhere. (b) Groups GS were studied in [P] using different methods. [P] asked whether the dependency problem DP (n) is decidable for reflection groups of an abstract cubic for n ≥ 3 or n = ∞. DP (n) can be formulated as follows. We will say that g0 is dependent on (g1 , . . . , gk ) if there is a family (gi1 , . . . , gip and elements u1 , . . . , up of G such that −1 g0 (u1 gi1 u−1 1 ) . . . (up gip up ) = 1.

If n is a positive number or infinity then the dependence problem DP (n) asks for an algorithm to decide for any sequence (g0 , . . . , gk ), 0 ≤ k < n, of elements of G whether or not g0 is dependent on (g1 , . . . , gk ). The problems D(1), D(2) are usually called the word problem and the conjugacy problem. A special case of the dependence problem for tx ty ∈ B0 (V ) can be related to the decidability of universal equivalence. Namely, if DP (∞) is decidable for gi = txi tyi tzi tx′i tyi tzi′ and g0 = tx ty than one can efficiently define whether x, y are universally equivalent.

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Since the decidability of the universal equivalence seems to be a very difficult problem in general, one can infer about the difficulty of the DP (∞) for B0 (V ). Question. Let an abstract cubic S be decidable. Is DP (n) decidable for arbitrary tx ty and generators of the subgroup B0 (V ) described in 2.7(c)? (c) Another construction of a filtration of the group of birational automorphism of V reflecting the structure of admissible equivalences is given in [M2]. One can apply the method from [M2] to the classes of universal equivalence. One can show that there exist classes of universal equivalence that are abstract cubics. One can consider universal equivalence on the set of points of such a class (considered as the abstract cubic). Applying this construction iteratively one can get a set of abstract cubics that corresponds to a filtration of subgroups in reflection groups. As in [M2] one can ask whether this sequence of subgroups stabilizes and what is its intersection. §5. Birationally trivial cubic surfaces: a finiteness theorem 5.1. Modified composition. Let V be a smooth cubic surface, and x, y ∈ V (k). Let C ⊂ V be a curve on V passing through x, y, and p : C → P2 an embedding of C into a projective plane such that p(C) is again a cubic, and p(x) ◦ p(y) is defined in p(C). We assume that C and p are defined over k. In this situation, following [M3], we will put x ◦(C,p) y := p−1 (p(x) ◦ p(y)). Example 1. Choose C = a plane section of V containing x, y. If p is the embedding of C into the secant plane, then x ◦(C,p) y = x ◦ y in the standard notation. Notice that the result does not depend on C if x 6= y. If x = y, then the choice of C determines a choice of one or two tangent lines to V at x so that the multivaluedness of ◦ is taken care of by the introduction of this new parameter. Example 2. Assume now that V admits a birational morphism p : V → P2 defined over k (e.g., V is split). We will choose and fix p once for all. Then any plane section C of V not containing one of the blown down lines as a component is embedded by p into P2 as a cubic curve. Therefore we can apply to (C, p) the previous construction. Notice that this time x ◦(C,p) y depends on C even if x 6= y. 5.2. Theorem. Assume that k is a finitely generated field. In the situation of Example 2, the complement to the blown down lines in V (k) is finitely generated with respect to operations ◦(C,p) with the additional restriction: (C) the operation x ◦(C,p) y is applied only to the different previously constructed points. Proof. This theorem was stated and proved in [M3] without the additional condition (C). It uses the following auxiliary construction. Choose a k–rational

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line l ⊂ P2 . Then Γ := p−1 (l) is a twisted rational cubic in V. The family of all such cubics reflects properties of that of lines: a) any two different points a, b of V (k) belong to a unique Γ(a, b); b) any two different Γ’s either have one common k–point, or intersect a common blown down line. The proof of this theorem is based on generation of points by adding intersections of lines l passing through pairs of previously constructed points in a projective plane. This induces generation of points on V that are intersections of p−1 (l). Analysis of this proof in [M3] shows that it considers only different points in pairs of previously constructed points hereby providing the statement of the theorem with the condition (C). If one drops the condition (C) one can prove the stronger statement. 5.3. Theorem. Let V be a smooth cubic surface over an arbitrary field k. Assume that V admits a birational morphism p : V → P2 . Then the complement P to all blown down lines in V (k) is generated by any single point from P (in the sense of the composition ◦(C,p) ). Proof. Let us choose a point x ∈ P . The theorem will follow if we prove that the set of points x ◦(C,p) x contains P (here C runs through all k–rational plane sections of V passing through x). Let us show that for any other point y in P there exists such C that y = x ◦(C,p) x. Indeed, following arguments of [M3], for y ∈ P there exists a twisted cubic curve G(x, y) := p−1 (l) where l is the line through p(x), p(y) in P2 . Let l1 be the tangent line to G(x, y) at x. Let a plane through points x, y and l1 cut a curve C on V . Then l1 is a tangent line to C at x, i.e. G(x, y) is tangent to C at x. Hence l in P2 is tangent to p(C) at p(x). Since this line l passes through p(y), on p(C) we have p(y) = p(x) ◦ p(x). This gives y ∈ x ◦(C,p) x proving the statement. One can apply this theorem to the proof of the triviality of the 3–component of the universal equivalence on P = V (k). 3–component of the universal equivalence can be defined as the finest admissible equivalence U3 for which the following condition holds: For any class X ∈ P/U3 , X ◦ X = X. Simillarly one can define the 2–component of the universal equivalence as the finest admissible equivalence for which the following condition holds: For any class X ∈ P/U2 , X ◦ X = O for some fixed class O ∈ P . It follows from [M1] that U = U3 ∩U2 , where U denotes the universal equivalence. 5.4. Corollary. Let V be a smooth cubic surface over an arbitrary field k. Assume that V admits a birational morphism p : V → P 2 . Then U3 is trivial on V (k). The corollary can be deduced from the following two lemmas.

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5.5. Lemma. Let C be a smooth plane cubic curve defined over a field k such that C(k) is non–empty. Let p be another plane embedding of C over k. Then x ◦(C,p) y := p−1 (p(x) ◦ p(y)) = t−1 ((t(x) ◦ t(y)) where t ∈ Bir C is some birational automorphism of C over k which can be represented as a product of reflections of C defined over k. Proof. The statement easily follows from the following fact: p can be decomposed into a product of reflections of C over k and a projective isomorphism of C and p(C). Indeed, let us choose a point 0 ∈ C(k). Isomorphism classes of invertible sheaves of degree 3 are parametrized by the jacobian of C of degree 3, say, T , and T is a principal homogeneous space over C. This means that C(k) acts transitively on T (k), i.e. any two sheaves L1 , L2 differ by a translation by a point a ∈ C(k). Any translation is a product of two reflections, whereas a projective isomorphism preserves collinearity. 5.6. Lemma. In the same notation, for any two points x, y ∈ C(k) the following holds: t−1 (t(x) ◦ t(y)) ∼ x ◦ y mod U3 . Proof. Let t = tx1 . . . txn where xi ∈ C(k). It is enough to check the statement for n = 1 since the general statement can be obtained by induction. Let t = tz . We have: t−1 (t(x)◦t(y)) = tz (tz (x)◦tz (y)) = z ◦((z ◦x)◦(z ◦y)) = z ◦((z ◦z)◦(x◦y)) ∼ z ◦ (z ◦ (x ◦ y)) mod U3 ∼ x ◦ y mod U3 . Here we used z ◦ z ∼ z mod U3 . Q.E.D. We can now deduce the Corollary 5.4. Fix some x ∈ P, where P is the complement to all blown down lines in V (k). By the Theorem 5.3, any point z ∈ P can be represented as x ◦(C,p) x. Let z = x ◦C,p x for some C. If C is singular then all points on C(k) are equivalent mod U3 (this is a general property of any singular plane cubic curve that does not have a line as a component). Otherwise, by lemmas 5.5 and 5.6 z = x ◦(C,p) x ∼ x ◦ x mod U3 ∼ x mod U3 . 5.7. Elimination of ◦(C,p) . The use of the modified operation ◦(C,p) is somewhat annoying, and we would like to replace it by the standard composition ◦. For example, in the setup of the Theorem 5.2 for any three points x, y, z on a plane smooth section C ⊂ V the following equality holds: (x ◦(C,p) y) ◦(C,p) z = (x ◦ y) ◦ z. This naturally leads to the question whether one can obtain the traditional Mordell– Weil statement for the composition ◦ using our finiteness results for ◦(C,p) and some tricks like the formula above.

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The remaining part of the paper is dedicated to the description of our, not altogether successful, attempts to eliminate ◦(C,p) . We reformulate the finiteness theorem above in terms that do not use explicitly compositions ◦(C,p) and a morphism p of a cubic surface into a projective plane. We only use the standard operation ◦ and implicitly use some intersections of planes with lines that belong to this cubic surface. Before we can state a new statement we need to define a new kind of operation on a cubic surface that involves lines belonging to this cubic surface. 5.7.1. Definition. Let V be a smooth cubic surface over an arbitrary field k. Let Λ = {l1 , l2 , m} be three (not necessary k-rational) lines belonging to V and such that the following properties hold: (A) l1 and l2 are skew lines (i.e. they do not have a common point) and m intersects l1 and l2 . Given a triple of lines Λ satisfying (A) and an arbitrary plane T not containing lines in V , let us define a new composition of points u, and w on T ∩ V as follows: (B) u ◦(T,Λ) w = (x ◦ y) ◦ [z ◦ (u ◦ w)], where x = l1 ∩ T , y = l2 ∩ T and z = m ∩ T . Of course, the point u ◦(T,Λ) w is not necessarily k–rational even u, w, and T are k–rational. But there is a special case when the composition ◦(T,Λ) produces rational points (over k) when u, w, and T are defined over k (whereas lines in Λ are not necessarily defined over k). This case is described in the following statement that reformulates the Theorem 5.2 in terms of the composition ◦(T,Λ) . 5.7.2. Theorem. Let V be a smooth cubic surface. Assume that V admits a birational morphism to a projective plane defined over k. Assume that k is finitely generated field. Then there exists a triplet of lines on V satisfying the property (A) such that the following statement holds: the complement to the blown down lines in V (k) is finitely generated with respect to operations ◦(T,Λ) with the additional restriction: (D) the operation x ◦(T,Λ) y is applied only to different previously constructed points. (Here Λ is fixed and T runs through some set of k-rational planes). Similarly, one can reformulate Theorem 5.3 in terms of new operations. 5.7.3. Theorem. Let V be a smooth cubic surface over an arbitrary field k. Assume that V admits a birational morphism to a projective plane defined over k. Then the complement P to all blown down lines in V (k) is generated by any single point from P in the sense of compositions ◦(T,Λ) for some fixed triple of lines Λ in V. Below we will show how to replace operations ◦(C,p) by operations ◦(T,Λ) .

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5.7.4. Lemma. Let V be a smooth cubic surface defined over a field k and k¯ be ¯ Then an algebraic closure of k. Let p : V → P2 be a birational morphism over k. there exists a triplet of lines Λ satisfying the property (A ) such that for any plane section C of V not containing one of the blown down lines as a component and for ¯ lying on C the following holds: any two points u, w ∈ V (k) u ◦(C,p) w = u ◦(T,Λ) w where T is a plane that cuts the curve C on V . 5.7.5. Corollary. Assume that the birational morphism p in Lemma 5.7.4 is defined over k. Then a triplet Λ can be chosen in such a way that the point u◦(T,Λ) w is k–rational if u, w and the plane T are k–rational. The proof of Lemma 5.7.4 is a consequence of the following claims which might be of independent interest. 5.7.6. Claim. In the conditions of Lemma 5.7.4, let x, y, u, w be some points on C. Then the following equality holds: u ◦(C,p) w = (x ◦ y) ◦ [(x ◦(C,p) y) ◦ (u ◦ w)]. In other words, if we know how to compute z = x ◦(C,p) y at least for some two points x, y in C then operation ◦(C,p) for all other points in C can be computed in terms of ◦ only. 5.7.7. Claim. In the conditions of Lemma 5.7.4, let Λ = {l1 , l2 , m} be a triplet of lines satisfying (A) and such that p(m) is a line on the plane P2 , and l1 , l2 are blown down lines. Let x = l1 ∩ T , y = l2 ∩ T and z = m ∩ T , where the plane T cuts a curve C on V . Then z = x ◦(C,p) y. In other words, one can easily compute an operation ◦(C,p) for intersection of lines l1 and l2 with a plane T . The result of this composition is an intersection of a third line m with T ! To show that the Lemma 5.7.4 follows from these claims, it is sufficient to note the following. By Claim 5.7.6, the operation u ◦(C,p) w can be replaced by (x ◦ y) ◦ [(x ◦(C,p) y) ◦ (u ◦ w)] where x, y are any points on C. There exists a triplet of lines Λ on V satisfying (A), such that p(m) is a line on the plane P2 , and l1 , l2 are the blown down lines. By the Claim 5.7.7, x, y can be chosen as intersections of lines l1 , l2 with a plane T that cuts C on V and in this case x ◦(C,p) y = m ∩ T . Now we prove our Claims. Proof of the Claim 5.7.6. Step 1: Since C and p are fixed, one can simplify our notation by putting x ∗ y =: x ◦(C,p) y. In this step we show that for any points x, y, u, w on C the following equality holds: u ∗ w = (x ∗ y)(x ◦ y)−1 (u ◦ w),

(5.1)

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where the expressions in brackets are multiplied by using an Abelian structure on C: xy = a ◦ (x ◦ y) for some point a in C(k). First, we consider the case when C is smooth. In this case by the Lemma 5.5 p in the formula p(p−1 (u) ◦ p−1 (w)) can be replaced by a product of reflections of C. Let us check (5.1) for the case when p can be replaced by one reflection tb : u ∗ w = p(p−1 (u) ◦ p−1 (w)) = b ◦ ((b ◦ u) ◦ (b ◦ w)) = b ◦ ((b ◦ b) ◦ (u ◦ w)). The general case can be obtained by iterating this argument. Using the identity u ◦ w = (a ◦ a)u−1 w−1 we get: u ∗ w = b ◦ ((b ◦ b) ◦ (u ◦ w)) = b−1 (b ◦ b)(u ◦ w) Similarly we have for other two points: x ∗ y = b−1 (b ◦ b)(x ◦ y). Replacing b−1 (b ◦ b) with (x ∗ y)(x ◦ y)−1 in b ◦ ((b ◦ b) ◦ (u ◦ w)) gives (5.1). Step 2: Replacing the Abelian multiplication operation in (5.1) by a ◦ (. . . ) we can rewrite (5.1) as as: u ∗ w = a ◦ (r ◦ (u ◦ w)), where r = a ◦ {(x ∗ y) ◦ [(a ◦ a) ◦ (x ◦ y)]}. Since the point a is arbitrary, one can choose a = x ◦ y. This gives r = x ∗ y and immediately implies the formula in the Claim 5.7.6. In order to complete the proof of the Claim we need to consider the case when C is a singular plane cubic curve that does not contain a line. This can be done by appealing to an obvious limiting construction in the case of topological field k, or to a similar argument using the Zariski topology in general. Proof of the Claim 5.7.7. Since l1 , l2 are blown down lines and p(m) is a line in P2 , the points p(x), p(y), p(z) are intersections of the line p(m) with the curve p(C) in P2 . This means that on p(C) we have p(x) ◦ p(y) = p(z). This is equivalent to the equality z = x ◦(C,p) y in the Claim. References [K1] D. S. Kanevski. Structure of groups, related to cubic surfaces, Mat. Sb. 103:2, (1977), 292–308 (in Russian); English. transl. in Mat. USSR Sbornik, Vol. 32:2 (1977), 252–264. [K2] D. S. Kanevsky, On cubic planes and groups connected with cubic surfaces. J. Algebra 80:2 (1983), 559–565. [M1] Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. North Holland, 1974 and 1986.

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[M2] Yu. I. Manin. On some groups related to cubic surfaces. In: Algebraic Geometry. Tata Press, Bombay, 1968, 255–263. [M3] Yu. I. Manin. Mordell–Weil problem for cubic surfaces. In: Advances in the Mathematical Sciences—CRM’s 25 Years (L. Vinet, ed.) CRM Proc. and Lecture Notes, vol. 11, Amer. Math. Soc., Providence, RI, 1997, pp. 313–318. [P] S. J. Pride. Involutary presentations, with applications to Coxeter groups, NEC-Groups, and groups of Kanevsky. J. of Algebra 120 (1989), 200–223. [Sw–D] H. P. F. Swinnerton–Dyer. Universal equivalence for cubic surfaces over finite and local fields. Symp. Math., Bologna 24 (1981), 111–143.

E-mail addresses: [email protected] manin@mpim–bonn.mpg.de

TORSEURS UNIVERSELS ET MÉTHODE DU CERCLE 24 Janvier 2001 par

Emmanuel Peyre

Résumé. — Ce texte décrit les premières étapes d’une généralisation de la méthode du cercle au cas d’une hypersurface lisse dans une variété presque de Fano. En effet, sous certaines conditions, il est possible d’exprimer dans ce cas les deux membres d’une version raffinée de la conjecture de Manin sur le comportement asymptotique du nombre de points de hauteur bornée de l’hypersurface en termes du torseur universel de la variété ambiante qui joue, dans ce cadre, le rôle de l’espace affine. Abstract. — This paper presents the first steps of a generalization of the circle method for smooth hypersurfaces in almost Fano varieties. Indeed it is possible, under some conditions, to express both sides of a refined version of Manin’s conjecture on the asymptotic behavior of the number of points with bounded height on the hypersurface in terms of the universal torsor of the variety, which plays here the rôle of the affine space.

Table des matières 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Une version raffinée d’une conjecture de Manin . . . . . . . . . . . . . . . . . . 2.1. Variétés presque de Fano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Hauteurs d’Arakelov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Mesure de Tamagawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Enoncé d’une question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Passage au torseur universel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Structures sur les torseurs universels . . . . . . . . . . . . . . . . . . . . . . 3.2. Fonctions de comptage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Fonctions de Möbius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Montée du nombre de points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Montée de la constante . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Intersections complètes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification mathématique par sujets (1991). — primaire 14G05.

2 3 3 5 9 10 11 12 18 21 24 25 29

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EMMANUEL PEYRE

4.1. Encerclement du nombre de points . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Encerclement de la constante : introduction . . . . . . . . . . . . . . . . 4.3. Aspect géométrique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Aspect analytique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Transformation de Fourier locale . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Mesures adéliques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Transformation adélique et encerclement de la constante . . . . 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 32 35 36 37 38 42 43

1. Introduction L’objet de ce texte est le comportement asymptotique du nombre de points de hauteur bornée sur des variétés dont le faisceau anticanonique vérifie certaines conditions de positivité. De nombreux progrès ont été réalisés dans la compréhension de ce comportement asymptotique. Une interprétation géométrique de la puissance et de la puissance du logarithme qui interviennent a été proposée dans les articles de Franke, Manin et Tschinkel [FMT] et de Batyrev et Manin [BM]. Des descriptions adéliques de la constante ont été proposées lorsque la hauteur est associée au faisceau anticanonique dans [Pe1] puis dans un cadre plus général par Batyrev et Tschinkel dans [BT4]. Plusieurs stratégies ont été développées pour attaquer ces conjectures. Une première famille de méthodes est basée sur des techniques d’analyse harmonique fine qui s’appliquent notamment lorsque la variété est équipée d’une action non triviale d’un groupe algébrique. Parmi les cas traités par ce type de méthodes, on peut citer celui des variétés de drapeaux généralisées étudiées dans [FMT] et [Pe1] à l’aide des travaux de Langlands sur les séries d’Eisenstein, le cas des variétés toriques considéré par Batyrev et Tschinkel dans [BT1], [BT2] et [BT3] et celui des fibrations en variétés toriques au-dessus de variétés de drapeaux généralisées par Strauch et Tschinkel [ST], ainsi que diverses compactifications de l’espace affine dues à Chambert-Loir et Tschinkel [CLT1], [CLT2]. Parallèlement des techniques de descente ont été mises au point dans ce cadre. Elles apparaissent de manière implicite dans le cas des intersections complètes lisses dans Pn et dans l’étude de quelques variétés toriques (cf. [Pe1] et [Ro]). Salberger les a rendues explicites dans [Sa], redémontrant ainsi en partie les résultats de Batyrev et Tschinkel sur les variétés toriques. La méthode introduite par Salberger fut ensuite exploitée par de la Bretèche qui put, à l’aide d’outils d’analyse complexe, améliorer les estimations pour les variétés toriques [Bre]. Une autre famille de méthodes, issue de la méthode du cercle, qui a depuis longtemps prouvée son efficacité pour les intersections complètes dans l’espace projectif, a été tout récemment utilisée par Robbiani pour l’étude d’un cas sortant de ce cadre, à savoir celui d’une hypersurface dans Pm  Pm définie par l’annulation d’une section de O(1; 1). Bien que la variété considérée par Robbiani soit une variété de drapeaux pour laquelle la conjecture de Manin avait été démontrée, le fait qu’il ait étendu la méthode du cercle à ce cas laisse espérer

TORSEURS ET MÉTHODE DU CERCLE

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que celle-ci puisse également s’appliquer à des cas où le rang du groupe de Picard n’est pas égal à un. Le but de ce texte est d’étendre à un cadre plus général quelques étapes de la méthode du cercle en exploitant un principe de descente présenté dans [Pe2]. Il reste toutefois un important et difficile travail à faire concernant le cœur même de la méthode du cercle, à savoir la majoration de sommes d’exponentielle. Le paragraphe 2 de ce texte rappelle la description conjecturale du comportement asymptotique du nombre de points de hauteur bornée. Le troisième a pour objet le passage aux torseurs universels au niveau desquels le problème se décrit naturellement comme passage d’une somme à une intégrale. Dans le quatrième nous décrivons comment, dans le cas d’une hypersurface vérifiant certaines conditions, on peut passer du torseur universel de la variété ambiante à celui de la sous-variété à l’aide de formules inspirées de la formule d’inversion de Fourier.

2. Une version raffinée d’une conjecture de Manin 2.1. Variétés presque de Fano. — Nous utiliserons dans ce texte les notations suivantes : Notations 2.1.1. — Si X est un schéma sur un anneau commutatif A et B une A-algèbre commutative, X (B ) désigne l’ensemble HomSpec A (Spec B; X ) et X B le produit de schémas X Spec A Spec B . Si C est un monoïde, alors A[C ] désigne la A-algèbre associée. Si X est une variété lisse sur un corps E , son groupe de Picard est noté Pic X , son groupe de Neron-Severi NS(X ) et son faisceau canonique !X . On désigne par Ceff (X ) le cône des classes de diviseurs effectifs dans NS(X ) R. On note E une clôture algébrique de E et E s sa clôture séparable dans E . On pose alors X = XE et X s = XE s . Le dual d’un module M est noté M _ . Définition 2.1.2. — Une variété V sur un corps k de caractéristique nulle sera dite presque de Fano si elle est projective, lisse et géométriquement intègre et si elle vérifie les conditions suivantes : (i) les groupes de cohomologie H i (V; OV ) sont nuls pour i = 1 ou 2, (ii) le groupe de Néron-Severi géométrique, qui sous l’hypothèse (i) coïncide avec Pic V , est sans torsion, (iii) la classe [!V 1 ] de !V 1 dans NS(V ) R appartient à l’intérieur du cône des diviseurs effectifs. Exemple 2.1.3. — Si V est une variété de Fano, alors V est presque de Fano. En effet, par le théorème de Kodaira, la condition (i) est vérifiée, la condition (ii) résulte de [Pe1, lemme 1.2.1] et (iii) découle du fait que, par définition, !V 1 est ample. Exemple 2.1.4. — Si V est une variété torique projective et lisse, alors par [Da, corollary 7.4], les groupes H i (V; OV ) sont nuls pour i > 0, et par [Oda, lemma 2.3] tout fibré en droites a une base de sections équivariantes sous l’action du tore et donc le cône des diviseurs effectifs dans Pic V R est engendrée par les [D] où D décrit l’ensemble des sous-variétés irréductibles invariantes de codimension 1 dans V . La classe [!V 1 ] étant la somme de ces [D]

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EMMANUEL PEYRE

par [Oda, page 70, example], il est à l’intérieur du cône et la condition (iii) est vérifiée. La variété V est donc presque de Fano. Proposition 2.1.5. — Soit X une compactification équivariante projective et lisse d’un tore sur C, L1 ; : : : ; Lm des faisceaux inversibles amples sur X et s1 ; : : : ; sm des sections non (1) nulles de ces faisceaux. On note XT l’ensemble des sous-variétés irréductibles invariantes de codimension un de X . On suppose que dim X > m + 3, que

T



X

D2XT(1)

D

m X i=1

Li



z Æ { 2 Ceff(X );

que les hypersurfaces définies par les si se coupent transversalement, que leurs intersections (1) successives sont connexes et qu’elles coupent proprement les diviseurs D de XT . Alors la sous-variété V définie par l’annulation des si est presque de Fano. En outre, la restriction induit un isomorphisme

Pic X

! Pic V

e

qui envoie Ceff (X ) dans Ceff (V ) et la classe de

P

D2XT(1) D

Pm 1 i=1 Li sur celle de !V .

Démonstration. — Nous allonsP démontrer par récurrence sur n que V vérifie les assertions de la proposition et que si L = m j =1 i Lj avec j 2 Z et j 6 0 pour 1 6 j 6 m, alors le groupe H i (V; L) est nul si 0 < i < dim V . Si m = 0, l’énoncé de la proposition résulte de l’exemple précédent, l’assertion de nullité pour OV résulte de [Da, corollary 7.4] et celle pour les sommes de fibrés Li de [Da, theorem 7.5.2] et du théorème de dualité de Serre (cf. [Ha, corollary III.7.7]). Supposons le résultat démontré pour m 1 et soit V 0 la sous-variété de X définie par l’annulation de s1 ; : : : sm 1 . La variété V 0 vérifie alors les assertions ci-dessus. La variété V est alors définie dans V 0 comme lieu des zéros de sm . Par l’hypothèse de transversalité, V est lisse et étant connexe, elle est intègre. Par définition elle est projective. Par ailleurs, on a une suite exacte de faisceaux de Zariski sur V 0 (2.1.1)

0 ! Lm1 OV 0

! OV 0 ! OV ! 0:

D’où une suite exacte longue de cohomologie (cf. [Ha, lemma III.2.10])

On obtient

H i (V 0 ; Lm1 ) ! H i (V 0 ; OV 0 ) ! H i (V; OV ) ! H i+1 (V 0 ; Lm1 ): donc que H i (V; OV ) est nul pour 0 < i < dim V = dim V 0 1. Comme

dim V > 3, cela entraîne l’assertion (i) de la définition. De même, on obtient l’annulation des groupes de cohomologie de l’hypothèse de récurrence. Par le théorème de Lefschetz classique [Bo, corollary, page 212] on a un isomorphisme :

H 2 (V (C); Z)

! H 2 (V 0 (C); Z):

e

En utilisant la suite exacte de faisceaux analytiques

0 ! Z ! OV exp ! OV  ! 0

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et des théorèmes de comparaison entre géométrie algébrique et géométrie algébrique, on obtient un diagramme commutatif

0 0

! H 1 (V?0 ; OV 0 ) e! H 2 (V 0?(C); Z) ! ? yo y  1 2 ! H (V; OV ) e! H (V (C); Z) !

0 0

et on obtient que la restriction de Pic V 0 à Pic V est un isomorphisme. Le cône des classes de diviseurs effectifs de X étant engendré par les classes des divi(1) seurs D de XT , l’assertion sur les cônes effectifs résulte de l’hypothèse sur la propreté des intersections avec P ces diviseurs. P m 1 Enfin [!V 01 ] = D2X (1) [D] i=1 [Li ] et l’assertion correspondante pour V résulte de T [Ha, proposition II.8.20]. Remarque 2.1.6. — A priori le cône des diviseurs effectifs de V pourrait être plus grand Qt que celui de X . Toutefois, si X est de la forme i=1 PnCi et si m < inf 16i6t ni , alors il y a égalité entre les cônes de diviseurs. En effet la formule de Künneth implique que

8L 2 Pic X; H i (X; L) = 0

si

On obtient alors par récurrence sur m que

8L 2 Pic V; H i (V; L) = 0

si

0 < i < inf ni : 16i6t

0 < i < inf ni 16i6t

m:

et la suite exacte (2.1.1) tensorisée par L fournit une suite exacte

H 0 (V 0 ; L) ! H 0 (V; L) ! H 1 (V 0 ; L Lm1 )

ce qui implique que les deux cônes coïncident. 2.2. Hauteurs d’Arakelov. — La donnée naturelle pour construire des fonctions de comptage sur l’ensemble des points rationnels de variétés propres est une hauteur d’Arakelov dont nous allons rappeler la définition. Notations 2.2.1. — Dans la suite, k désigne un corps de nombres, Ok son anneau des entiers, d son discriminant, Mk l’ensemble de ses places, Mf celui de ses places finies et M1 celui de ses places archimédiennes. Pour toute place v de k , on note kv le complété correspondant et j:jv la norme sur kv normalisée par



8vjp; 8x 2 kv ; jxjv = Nkv =Qp (x) p :

Si v est une place finie, Ov est l’anneau des entiers de kv et Fv le corps résiduel. Définition 2.2.2. — Soit V une variété projective lisse et géométriquement intègre sur k , L un faisceau inversible sur V . Si v est une place de k , une métrique v -adique sur L est une application associant à un point x de V (kv ) une norme k:kv sur L(x) = Lx OV;x kv de sorte que pour toute section s de L définie sur un ouvert W de V l’application

x 7! ks(x)kv

soit continue pour la topologie v -adique.

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EMMANUEL PEYRE

Si v est un place finie de k , V un modèle projectif et lisse de V sur Ov et L un modèle de L, alors on peut lui associer une métrique v-adique sur L de la manière suivante : tout point x de V (kv ) définit un point x~ de V (Ov ) et x~ (L ) fournit une Ov -structure sur L(x) dont on peut choisir un générateur y0 ; la norme d’un élément y de L est alors donnée par la formule kykv = yy : 0 v Une métrique adélique sur L est une famille de métriques (k:kv )v2Mk telle qu’il existe un ensemble fini de places finies S , un modèle projectif et lisse V de V sur l’anneau OS des S -entiers et un modèle L de L sur cet anneau tel que pour tout v de Mf S , k:kv soit la métrique définie par L OV Ov . Nous appellerons hauteur d’Arakelov sur V la donné d’une paire

h = (L; (k:kv )v2Mk )

où L est un faisceau inversible sur V et (k:kv )v2Mk une métrique adélique sur ce fibré. Pour toute hauteur h sur V et tout point rationnel x de V , la hauteur de x relativement à h est définie par

8y 2 L(x);

h(x) =

Y

v2Mk

kykv 1 :

Remarque 2.2.3. — La formule du produit assure que le produit ci-dessus est indépendant de y . Rappelons quelques exemples de hauteurs (cf. également [Sa, exemples 1.7]). Exemple 2.2.4. — Si hi = (Li ; (k:kiv )v2Mk ) pour i = 1 ou 2 sont deux hauteurs d’Arakelov, leur produit tensoriel h1 h2 est (L1 L2 ; (k:kv )v2Mk ) où

8v 2 Mk ; 8x 2 V (kv ); 8y 2 L1 (x); 8z 2 L2 (x); ky z kv = kyk1v kz k2v :

On en déduit immédiatement l’égalité

8x 2 V (k);

h1 h2 (x) = h1 (x)h2 (x):

f

Exemple 2.2.5. — Soit h = (L; (k:kv )v2Mk ) une hauteur sur V et = (fv )v2Mk une famille de fonctions strictement positives sur V (kv ) telle que pour presque toute place v de k la fonction fv soit constante et égale à 1, alors

f :h = (L; (fv k:kv )v2Mk )

est une hauteur sur V . Réciproquement, si h0 = (L; (k:k0v )v2Mk ) est une autre hauteur sur V relative au même faisceau, alors pour toute place v de k le quotient k:k0v =k:kv définit une fonction fv sur V (kv ) qui, pour presque toute place, est constante et égale à 1. On a bien sûr h0 = :h.

f

Exemple 2.2.6. — Si  : V ! W est un morphisme de variétés projectives lisses et géométriquement intègres et h = (L; (k:kv )v2Mk ) une hauteur sur W alors  (h) est la hauteur ( L; (k(:)kv )v2Mk ) où l’on note également  l’application induite  L(x) ! L((x)) pour tout x de V .

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En particulier si L est un faisceau inversible très ample, il définit un morphisme

:V

! P( (V; L)_ )

de sorte que L =  (O(1)) et tout système de métriques sur O(1) induit une hauteur sur V . Exemple 2.2.7. — Si K=k est une extension de corps de nombres, V une variété projective lisse et géométriquement intègre sur k , L un faisceau inversible sur V et hK = (L

K; (k:kv )v2MK ) une hauteur sur VK , alors la hauteur induite h = (L; (k:k0v )v2Mk ) est définie par Y [K :k] 1

8p 2 Mk ; 8x 2 V (kp ); 8y 2 L(x); kyk0p =

Pjp

kyP kP

:

Cela permet également d’associer à tout hauteur h relative à un faisceau inversible L sur

VK une hauteur NK=k h relative au faisceau NK=k L sur V .

Exemple 2.2.8. — Soit V un schéma plat projectif et régulier sur Ok et (L ; h) un fibré en droites hermitien sur V (cf. [BGS, §2.1.2]), L désigne donc un fibré inversible sur V Q et h une forme hermitienne C 1 sur le fibré en droites holomorphe LC sur :k!C V (C) invariante sous l’action de la conjugaison. On suppose que h s’écrit comme produit tensoriel de formes hermitiennes C 1 que l’on notera h et telles que h soit la conjuguée de h . Pour toute place finie v de k , L induit comme ci-dessus une métrique k:kv sur L = L k et pour toute place archimédienne v de k , on a un plongement  de k dans C et la forme hermitienne h définit une métrique v -adique k:kv sur L. Par définition, h = (L; (k:kv )v2Mk ) est une hauteur sur V , et la hauteur d’un point rationnel est donné par la formule

8x 2 V (k);

d c1 (L )jx) h(x) = exp deg(^




où x est l’adhérence de x dans V , c^1 (L ) le caractère de Chern arithmétique de L (cf. [BGS, page 932]), (:j:) l’accouplement

  d (V )  Z (V ) ! CH d (Spec Ok )Q CH

d l’application degré sur le groupe de défini par Bost, Gillet et Soulé (cf. [BGS, §2.3]) et deg  d Chow arithmétique CH (Spec Ok ). En effet par [BGS, §3.1.2.1 et (2.1.15)],


d c1 (L )jx) = deg d x deg(^ ~ (L ) = log(#(~x L =Ok y))

où x ~ : Spec Ok ! V est définie par x et définition on obtient

X

:k!C

log h (y; y)1=2

y un élément de x~ (L )

d c1 (L )jx) = deg(^

X

v2Mk

log kykv :

 L(x). En suivant les

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EMMANUEL PEYRE

Définition 2.2.9. — On note H (V ) l’ensemble des classes d’isomorphismes de hauteurs d’Arakelov quotienté par la relation d’équivalence définie par

(L; (k:kv )v2Mk )  (L; (v k:kv )v2Mk ) L Q pour toute famille de réels (v )v2Mk 2 v2Mk R>0 telle que v2Mk v = 1. L’ensemble H (V ) est un groupe pour le produit tensoriel des hauteurs, il est muni d’une

structure de R>0 -ensemble donnée par

:(L; (k:kv )v2Mk ) = (L; (v k:kv )v2Mk ) Q si (v )v2Mk 2 v2Mk R>0 vérifie v2Mk v = . On dispose d’un morphisme d’oubli o : H (V ) ! Pic V . Si  : V ! W est un morphimes de variétés projectives, lisses et géométriquement intègres sur k , alors  définit un morphisme H (W ) ! H (V ) qui L

s’insère dans un diagramme commutatif :

H (W )

!

H (V )

Pic(W )

!

Pic(V ):

? ? y

? ? y

Enfin si K=k est une extension de corps de nombres on dispose d’un morphisme de norme

NK=k : H (VK ) ! H (V ):

Remarque 2.2.10. — Si x est un point rationnel et h une hauteur, h(x) ne dépend que de la classe de h dans H (V ). On notera evx le morphisme H (V ) ! R>0 obtenu. Exemple 2.2.11. — Si V = Spec k , alors une hauteur d’Arakelov est la donnée d’un espace vectoriel L de dimension un sur k et d’une famille de normes (k:kv )v2Mk sur L telle qu’il existe une Ok -structure de L de L de sorte que pour tout place finie v de k en-dehors d’un ensemble fini S , on ait

8y 2 L ; kykv = (#(Ov y=L Ov )) 1 :

Cette description explicite montre que le morphisme evSpec k est un isomorphisme. Notons qu’en outre on a pour toute variété V projective lisse et géométriquement intègre sur k et tout point x de V (k ) un diagramme commutatif.

H (V )

evx

? ?  yx

H (Spec k )

!

R>0



! R : >0

Définition 2.2.12. — On appelle système de hauteurs une section de l’application composée

o Pic V H (V ) !

! NS(V ):

Un système de hauteurs H sur V induit un accouplement

H : NS(V ) C  V (k ) ! C

qui est l’exponentielle d’une fonction linéaire en la première variable et telle que

8L 2 NS(V ); 8x 2 V (k);

H(L; x) = H(L)(x):

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Comme l’ont souligné Batyrev et Manin [BM], l’existence de sous-variétés accumulatrices susceptibles d’occulter certains phénomènes globaux dans le comportement asymptotique du nombre de points de hauteur bornée amène à se restreindre à un ouvert non vide assez petit de la variété. On utilisera donc la définition qui suit. Définition 2.2.13. — Soit V une variété projective, lisse et géométriquement intègre sur k et W un sous-espace localement fermé de V . Alors pour toute hauteur h sur V et tout nombre réel H strictement positif

nW;h (H ) = #fx 2 W (k) j h(x) 6 H g: Si H est un système de hauteurs sur V alors la fonction zêta associée est définie par X 8s 2 NS(V ) Z C; H (s) = H(s; x) 1 : x2W (k) Remarque 2.2.14. — Si [o (h)] appartient à l’intérieur de Ceff (V ), alors il existe un ouvert U de V tel que nU;h(H ) soit fini pour tout H . 2.3. Mesure de Tamagawa. — Dans la suite V désigne une variété presque de Fano sur k. Dans ce cas toute métrique adélique sur le fibré anticanonique !V 1 définit une mesure

de Tamagawa qui permet de donner une interprétation conjecturale du terme principal du nombre de points de hauteur bornée.

A

Notations 2.3.1. — Si X est un variété sur k , X ( k ) désigne l’espace adélique qui lui est associé. (cf. [We, §1]). Pour toute place v de k , la mesure de Haar dxv sur kv est normalisée de la manière suivante : R - Si v est finie, alors Ov dxv = 1, - si kv e! R, alors dxv est la mesure de Lebesgue usuelle, - si kv e! C, alors dxv = idz ^ dz . Soit h = (!V 1 ; (k:kv )v2Mk ) une hauteur sur une variété presque de Fano V . En toute place v de k on lui associe la mesure borélienne ! h;v sur V (kv ) définie par la relation (cf. [We], [Pe1, §2.2.1])

!h;v =

@

@x



^


^ @x@

dx1;v : : : dxn;v 1 n v

où x1 ; : : : ; xn désignent des coordonnées locales analytiques au voisinage d’un point x de V (kv ) et @x@ 1 ^


^ @x@n est vu comme section locale de !V 1 . D’après [Pe1, lemme 2.1.1], on peut se donner un ensemble fini S de places finies et un modèle projectif et lisse V de V sur OS dont les fibres sont géométriquement intègres et tel que pour toute place finie en-dehors de S , le groupe de Picard géométrique Pic V Fp soit isomorphe à Pic V de façon compatible aux actions des groupes de Galois et la partie l-primaire du groupe de Brauer Br(V ) soit finie pour tout nombre premier l n’appartenant pas à . Pour tout de Mk S , le terme local de la fonction L associée à Pic V est défini par

p

p

p

Lp (s; Pic V ) =

Det(1 (#Fp )

1

s Frp

j Pic V Fp Q)

10

EMMANUEL PEYRE

p

où Frp est le Frobenius en . La fonction L globale est définie par le produit eulérien

LS (s; Pic V ) =

Y

p2Mf

S

Lp (s; Pic V )

qui par [Pe1, lemme 2.2.5] converge absolument pour Re s > 1 et s’étend en une fonction méromorphe sur C avec un pôle d’ordre t = rg Pic V en 1. Les facteurs de convergence (v )v2Mk pour la mesure de Tamagawa sont définis par (

v =

Lv (1; Pic V ) si v 2 Mf 1 sinon.

S

Les conjectures de Weil montrées par Deligne impliquent la convergence de la mesure adéQ lique v2Mk v 1 ! h;v (cf. [Pe1, proposition 2.2.2]). Définition 2.3.2. — Avec les notation qui précèdent, la mesure de Tamagawa associée à h est définie par

1 t !h = slim !1(s 1) LS (s; Pic V ) p dim V

d

Y

v2Mk

v 1 !h;v :

Remarque 2.3.3. — Par construction elle est indépendante du choix de S et ne dépend que de l’image de h dans H (V ). Exemple 2.3.4. — Si 2.2.5, alors

f

= (fv )v2Mk

est une famille de fonction comme dans l’exemple

!f :h = Notation 2.3.5. — On pose h (V ) rationnels de V dans V ( k ).

A

 Y

v2Mk

= !h V (k)



fv !h :


où

V (k) désigne l’adhérence des points

2.4. Enoncé d’une question. — Pour énoncer notre question qui est une version raffinée d’une conjecture de Manin [BM, conjecture C0 ], nous utiliserons la notion d’accumulation qui suit :

o

Définition 2.4.1. — Soit h une hauteur d’Arakelov sur V telle que [ (h)] appartienne à l’intérieur du cône effectif. Un fermé irréductible strict F de V est dit modérément accumulateur pour h si et seulement si pour tout ouvert non vide W de F , il existe un ouvert non vide U de V tel que

n (H ) lim W;h > 0: H !+1 nU;h (H )

Nous renvoyons à [BT4] et [Pe2, §2.4] pour des exemples de telles sous-variétés. Notation 2.4.2. — Si V est une variété presque de Fano, on considère l’hyperplan affine P de NS(V )_ R d’équation hy; !V 1 i = 1: Cet hyperplan est muni d’une mesure canonique  définie par !V 1 (cf. [Pe1, page 120]). On note Ceff (V )_ le cône dual de Ceff (V ) défini par

Ceff (V )_ = fy 2 NS(V )_ R j 8x 2 Ceff (V ); hx; yi > 0g

TORSEURS ET MÉTHODE DU CERCLE

11

et on pose

(V ) = (Ceff (V )_ \ P ): On note également

(V ) = #H 1 (k; Pic V ): Remarque 2.4.3. — La constante (V ) définie par Batyrev et Tschinkel [BT1] est obtenue en multipliant par (t 1)! celle considérée ici. Question 2.4.4. — Soit V une variété presque de Fano sur k et h une hauteur sur V définie par une métrique adélique sur !V 1 . On suppose que V (k ) est dense pour la topologie de Zariski et que le complémentaire U dans V des sous-variétés modérément accumulatrices est un ouvert de Zariski non vide de V . A quelle condition a-t-on l’équivalence (2.4.1)

nU;h(H )  (V ) (V )h (V )H (log H )t 1

lorsque H tend vers l’infini ? Remarques 2.4.5. — (i) L’introduction du facteur (V ) est due à Batyrev et Tschinkel [BT1]. (ii) L’équivalence (2.4.1) est compatible avec le produit de variétés [FMT, §1.2, proposition], [Pe1, corollaire 4.3]. (iii) Elle est vérifiée dans les cas suivants : - Si V est une intersection complète lisse dans PN Q définie par m équations homogènes de degré d > 2 si

N > 2d 1m(m + 1)(d 1) [Bi], [Pe1, proposition 5.5.3], - Si V est une variété de drapeaux généralisée [FMT], [Pe1, théorèmes 6.1.1 et 6.2.2], - Si V est une variété torique lisse [Pe1, §8-11], [BT1], [BT3], [Sa], - pour certains fibrés en variétés toriques au-dessus de variétés de drapeaux généralisées [ST]. (iv) Comme me l’a signalé Tschinkel, la question 2.6.1 dans [Pe2] est mal posée. En général on peut seulement espérer que la fonction

s 7! H (s!V 1 )=Ceff(V ) ((s 1)!V 1 ) s’étende en une fonction holomorphe au voisinage de 1 et prenne la valeur (V )H (V ) en ce point.

3. Passage au torseur universel L’objectif de ce paragraphe est de relever au torseur universel chaque coté de (2.4.1). C’est l’objet des propositions 3.4.2 et 3.5.2.

12

EMMANUEL PEYRE

3.1. Structures sur les torseurs universels. — Nous allons commencer par rappeler la définition des torseurs universels qui est due à Colliot-Thélène et Sansuc [CTS1] [CTS3]. Définition 3.1.1. — Soient G un groupe algébrique linéaire sur un corps E et Y une variété sur E . Un G-torseur au-dessus de Y est la donnée d’un morphisme fidèlement plat  : X ! Y au-dessus de E et d’une action  : X  G ! X de G sur X au-dessus de Y telle que l’application

(g; x) 7! (gx; x) définisse un isomorphisme de variétés de G E X sur X Y X . Par [Mi, théorème III.3.9 et corollaire III.4.7], si G est lisse et abélien, les classes d’isomorphismes de G-torseurs au-dessus de Y sont classifiées par le groupe de cohomologie étale Hét1 (Y; G) et par [CTS3, (2.0.2) et proposition 2.2.8], si T est un tore sur E , c’est-à-dire une E -forme de Gnm et si X est une variété propre, lisse et géométriquement intègre ayant un point rationnel sur E , alors on dispose d’une suite exacte naturelle  0 ! H 1 (E; T ) ! H 1 (X; T ) ! HomGal(Es =E) (X (T ); Pic XEs ) ! 0 ét

où X  (T ) désigne le groupe des caractères de T s et où pour tout torseur T et tout caractère  de T , (T )( ) est la classe du Gm -torseur  (T ) dans Pic XE s qui est isomorphe à Hét1 (XEs ; Gm ). Soit X une variété propre, lisse et géométriquement intègre sur un corps E . On suppose que le groupe de Picard géométrique Pic X s est de type fini et sans torsion. On note alors TNS le tore dont le groupe de caractères est le Gal(E s =E )-module Pic X s . Un torseur universel pour X est un TNS -torseur T au-dessus de X dont l’invariant (T ) coïncide avec IdPic(X s ) .

Remarque 3.1.2. — Nous renvoyons à [CTS3, §2.5, §2.6] et [Pe2, §3.3] pour des exemples de torseurs universels. Rappelons seulement qu’il résulte de [CTS1, proposition 6] et de [Sa, §8] qu’un torseur universel au-dessus d’une compactification équivariante lisse d’un tore T est un ouvert d’un espace affine. Si Y est une intersection complète lisse dans une variété presque de Fano X ayant un point rationnel et si la restriction de Pic X s à Pic Y s est un isomorphisme, alors on a un diagramme commutatif

0

! H 1 (E; TNS ) ! Hét1 (X;? TNS ) !

EndGal(Es =E) (Pic X s )

!

0

0

! H 1 (E; TNS ) ! Hét1 (Y; TNS ) !

EndGal(Es =E) (Pic Y s )

!

0



?  yj

? ? y

o

où j désigne le plongement de Y dans X . Il en résulte que les torseurs universels au-dessus de Y sont obtenus en prenant l’image inverse de Y dans les torseurs universels au-dessus de X . On dispose donc de diagrammes commutatifs de la forme :

TY

! TX

Y

! X

? ? y

? ? y

TORSEURS ET MÉTHODE DU CERCLE

13

où l’application du haut est une immersion fermée de TNS -ensembles. Si, en outre, X est une compactification équivariante lisse d’un tore, alors TX se plonge comme ouvert dans un espace affine AN E et l’action de TNS s’étend à cet espace affine. A chaque torseur universel au-dessus d’une variété presque de Fano sont associées deux structures canoniques, à savoir un espace d’adèles et une mesure sur cet espace. Ces structures ont été définies dans [Pe2, §4.2 et 4.4] mais nous allons maintenant en redonner une construction intrinsèque. Notation 3.1.3. — Si L appartient à Ceff (V ), on pose

 Æ(L) = inf hx; Li; x 2 Ceff (V )_ \ Pic V _ f0g et on note Æ (V ) = Æ (!V 1 ). Hypothèses 3.1.4. — Dans la suite V désigne une variété presque de Fano sur k dont le cône des diviseurs effectifs Ceff (V ) est un cône polyédral rationnel de Pic V R. On suppose en outre que Æ (V ) > 1. On note U une ouvert non vide de V .

Remarque 3.1.5. — La condition (iii) dans la définition 2.1.2 assure que pour toute variété presque de Fano Æ (V ) > 0 et donc Æ (V ) > 1. Exemple 3.1.6. — Si V est une intersection complète lisse dans PN définie par m équations f1 ; : : : ; fm de degrés respectifs d1 ; : : : ; dm , alors 

!V 1 = OV N + 1 Pm

m X i=1

di



et la condition s’écrit Æ (V ) 1 = N i=1 di > 0, qui est exactement l’hypothèse faite dans [Pe1, page 131]. La raison pour laquelle cette condition apparaît dans [Pe1] est exactement la même qu’ici : elle assure la convergence de sommations liées à la formule d’inversion de Möbius.

(1)

Exemple 3.1.7. — Si V est une compactification équivariante lisse d’un tore T sur k et V T désigne l’ensemble des sous-variétés irréductibles invariantes de codimension un deV , on a une suite exacte canonique j 0 ! X  (T ) !

(3.1.1)

M

D2V (1)

 ZD ! Pic V

!0

T

où X  (T ) désigne le groupe des k -caractères de T ; le cône Ceff (V ) est engendré par les (D) pour D 2 V (1) T et X !V 1 = (D): (1) D2V T _ _ Supposons qu’il existe  de Ceff (V ) \ Pic V f0g vérifiant h; !V 1 i = 1. On a alors D

;

X

D2V (1) T

E

(D) = 1

et

8D 2 V (1) T ; h;  (D)i > 0

14

EMMANUEL PEYRE

et donc il existe D0

2 V (1) D tel que h; (D)i =

(

1 0

si D = D0 ; sinon.

Si on considère la suite exacte duale de (3.1.1),

0 ! Pic V

_ _

!

M

P 2V (1) T

ZD_

j_

! X (T )_ ! 0;

on obtient que D0_ =  _ () et donc D0_ 2 Ker j _ . Mais il résulte de [Da, §6] que, par définition de j , l’application j _ est non nulle en D0 , ce qui est contradictoire. Par conséquent les variétés toriques projectives et lisses vérifient les conditions ci-dessus. Exemple 3.1.8. — Si V est le surface obtenue en éclatant quatre points en position générale sur P2k , alors

Pic V = Z 

4 M i=1

ZEi

où on note  le relevé strict d’une droite de P2k et Ei les diviseurs obtenus par éclatement. Le cône effectif est engendré par les diviseurs Fi;5 = Ei pour 1 6 i 6 4 et Fk;l =  Ei Ej pour fi; j; k; lg = f1; 2; 3; 4g et le faisceau canonique est donné par

!V 1 = 3

4 X i=1

Ei = 2F1;2 + F3;4 + F3;5 + F4;5 :

Comme le groupe des automorphismes de V agit transitivement sur les diviseurs Fi;j , on obtient que pour tout i; j avec 1 6 i < j 6 5, !V 1 2Fi;j appartient au cône effectif. Par conséquent cette surface vérifie également le condition précédente. Notation 3.1.9. — On note A Ceff (V );k le schéma affine

Spec(k[ Ceff (V ) \ X (TNS )]G ) où G désigne le groupe de Galois absolu de k . Pour tout torseur universel T

T

on note bCeff (V ) le produit contracté

au-dessus de V ,

T TNS A C (V );k : On dispose d’une immersion ouverte T ! TbC (V ) , l’action de TNS s’étend à TbC (V ) et on a une fibration TbC (V ) ! V en variétés toriques affines géométriquement isomorphes à la eff

eff

eff

eff

variété A Ceff (V );k . On appelle espace adélique associé à T et Ceff (V ) l’intersection

(Ak ) = eff (V )

TC

 Y

v2Mk



T (kv ) \ TbC

eff

A

(V ) ( k )

qui peut être explicitement décrit comme produit restreint des T (kv ) (cf. également [Pe2, §4.2]).

TORSEURS ET MÉTHODE DU CERCLE

15

Nous allons maintenant démontrer la trivialité de !T . Lemme 3.1.10. — Si Y est une variété lisse sur un corps algébriquement clos X ! Y est un T -torseur où T est un tore, alors il existe un isomorphisme

!X

E et si  :

!  (!Y ):

e

En outre cet isomorphisme est canonique au signe près. Remarque 3.1.11. — Ce lemme est en fait un généralisation facile de l’existence d’une forme volume naturelle, bien définie, au signe près, sur le tore T . En fait, on pourrait aussi le voir comme une conséquence de la description du fibré cotangent relatif 1X=Y pour les torseurs sous un groupe algébrique lisse (cf. [Sa, proposition 3.8]). Démonstration. — Soit (i )16i6t une base de X  (T ). Comme E est algébriquement clos, cette base induit un isomorphisme de T sur Gtm . Les classes d’isomorphismes de T -torseurs sur Y sont classifiés par

Hét1 (Y; T )

1 (Y; G )t : ! Hét1 (Y; Gm )t e! HZar m

e

Par conséquent  est localement triviale pour la topologie de Zariski. Soit U = Spec A un ouvert affine de Y sur lequel  se trivialise, c’est-à-dire sur lequel il existe une section s : U ! X de , l’isomorphisme correspondant  :  1 (U ) e! U  T étant caractérisé par

 Æ s(y) = (y; e):

Soit Xi = i Æ  :  1 (U ) ! Gm . On a donc que Xi appartient à ( 1 (U ); OY ) . Alors, par [Ha, remarque 8.9.2 et exemple 8.11.1] la famille ( dXXii )16i6t est une base de 1X=Y en tant que OX module et donc ^ti=1 dXXii fournit une trivialisation de det( 1X=Y ) sur  1 (U ). D’autre part on a une suite exacte de fibrés vectoriels (cf. [Ha, proposition 8.11])

0 !  1Y=E

j !

1X=E ! 1X=Y ! 0;

où l’injectivité résulte de l’hypothèse de lissité. Par conséquent on a un isomorphisme canonique

!X=E

!  (!Y=E ) det( 1X=Y ):

e

Donc ^ti=1 dXXii fournit un isomorphisme (3.1.2)

!X=E j 1 (U )

!  (!Y=E )j

e

Si s0 est une autre section trivialisante de  et Xi0 : dantes, on a alors Xi0 = ai Xi où ai est définie par

1 (U ) :

 1 (U ) ! Gm les fonctions correspon-

ai = Xi0 Æ s 2 (U; OU ) = A :

On obtient que

dXi0 d(ai Xi ) (dai )Xi + ai (dXi ) dXi = = = Xi0 ai Xi ai Xi Xi

16

EMMANUEL PEYRE

puisque dai = 0 dans 1B=A où B = ( 1 (U ); OX ). Donc l’isomorphisme (3.1.2) est indépendant de la section choisie et, par recollement, définit un isomorphisme

!X=E e!  (!Y=E ) Si (i0 )16i6t est une autre base de X  (T ), alors on note M 2 GLn (Z) la matrice de changement de base. La section ^ti=1 dXXii sera remplacée par det(M ) ^ti=1 dXXii ce qui montre qu’au

signe près l’isomorphisme est indépendant de la base choisie.

Lemme 3.1.12. — Avec les notations ci-dessus, le fibré canonique

!T

est trivial.

Démonstration. — On a une suite exacte

0 ! H 1 (k; k[T ] ) ! Pic(T ) ! Pic(T ):

Mais il découle de [CTS3, proposition 2.1.1] que

  (T ; O T ) = (V; OV ) = k : 90, Pic(T ) s’injecte dans Pic(T )

Et, par le théorème d’Hilbert et il suffit de montrer le résultat sur k . Mais par le lemme précédent, on a un isomorphisme

!T

!  (!V ):

e

En appliquant à nouveau [CTS3, proposition 2.1.1], l’application   de Pic(V ) à Pic(T ) est triviale et, par conséquent, !T est triviale.

Notation 3.1.13. — Par conséquent, il existe une section ! T de ! partout non nulle et,  , cette section est unique à une constante Tmultiplicative près. Par comme (T ; O ) = k T T définit pour toute place v de k une mesure !T ;v sur T (kv ). [We, §2] cette section ! Le résultat suivant est annoncé dans [Pe2, remarque 4.4.4]. Lemme 3.1.14. — Avec les hypothèses ci-dessus, si T a un point rationnel, le produit des mesures !T ;v converge et coïncide avec la mesure ! T définie dans [Pe2, définition 4.4.3].

Démonstration. — Il suffit de montrer que l’on peut choisir la section ! T de sorte que la mesure !T ;v coïncide avec celle définie dans [Pe2, notations 4.4.1]. Or, par définition, ! T ;v est localement donnée par la formule 

!T ;v = @x@ ^


^ @x@ ; !T 1 N



v

dx1;v : : : dxN;v

où x1 ; : : : ; xn désignent des coordonnés locales analytiques au voisinage d’un point x de V (kv ). D’un autre coté, la mesure ! 0T ;v définie par [Pe2, notations 4.4.1] est construite de la manière suivante : on note !TNS ;v la mesure définie par la forme différentielle canonique ! TNS sur TNS et on se donne un morphisme !V 1 de T dans !V 1 dont l’image ne rencontre pas la section nulle et qui est compatible avec le morphisme de tore de TNS ! Gm induit par l’injection Z ! Pic V envoyant 1 sur la classe de !V 1 . Pour tout point x de V (kv ), on considère sur la fibre Tx (kv ) la mesure ! Tx ;v donnée par Z

Tx(kv )

f (r)! Tx;v (r) =

Z

TNS (kv )

f (r:y)

1

!V 1 (r:y ) v

!TNS ;v (r)

TORSEURS ET MÉTHODE DU CERCLE

17

où y est un point arbitraire de Tx (kv ). La mesure ! 0T ;v est alors définie par la relation Z

T (kv )

f (y)!0T ;v (y) =

Z

V (kv )

!h;v (x)

Z

Tx (kv )

f (y)!Tx ;v (y):

Mais, par la démonstration du lemme 3.1.10, ! TNS fournit une trivialisation !~ TNS du faisceau det( 1T =V ) et donc un isomorphisme !T e!  !V . Par conséquent, !_ 1 : T ! !V V fournit une section partout non nulle de !T , qu’on peut supposer égale à ! T . D’autre part, on peut choisir des coordonnés locales x1 ; : : : ; xn sur un ouvert W de V (kv ) sur lequel T se trivialise et fixer cette trivialisation

T (kv )jW e! W  TNS(kv ):

Des coordonnées locales xn+1 ; : : : ; xN sur TNS (kv ) fournissent alors des coordonnées locales sur T (kv ). On a alors les relations

!0T ;v =

@x@ 1



^


^ @x@

k !V 1 (x1 ; : : : ; xN )kv 1 n v   @ @  @x ^


^ @x ; !TNS dx1;v : : : dxN;v n+1 N v   @ @ _ = ^


^ ; !V 1 (x1 ; : : : ; xN ) !~ TNS (x1 ; : : : ; xN ) @x1 @xN v  dx1;v : : : dxN;v = !T ;v :

Définition 3.1.15. — La mesure

!T =

Y

v2Mk

!T ;v

est, par la formule du produit, indépendante du choix de ! T . On l’appelle mesure canonique de TCeff (V ) ( k ).

A

Exemple 3.1.16. — Si V est une intersection complète dans une variété X , définie par l’annulation de sections s1 ; : : : ; sm de fibrés en droites L1 ; : : : ; Lm de sorte que la restriction donne un isomorphisme

Pic(X ) ! Pic(V ) et que X et V vérifient la convention 3.1.4 et si V a un point rationnel, alors par la remarque 3.1.2, un torseur universel TV est l’image inverse de V dans un torseur universel  : TX ! X . Comme les faisceaux inversibles  (Li ) sont triviaux pour 1 6 i 6 m, TV est donc défini dans TX par l’annulation de m fonctions f1 ; : : : ; fm qui vérifient 8y 2 TX (k ); 8t 2 TNS (k); fi (t:y) = [Li ](t)fi (y); où [Li ] 2 Pic V = X  (TNS ). Si ! TX est une trivialisation de !TX , On dispose alors d’une forme différentielle de Leray !  L;TV section de !TV et définie par la relation

8y 2 TV (k ); ! L;TV (y) ^ f 

m ^

i=1

!

dxi (y) = !TX (y):

18

EMMANUEL PEYRE

Cette forme différentielle est une section partout non nulle de!TV on peut donc poser !  TV

! L;TV .

=

Si, en outre, X est une compactification projective lisse d’un tore T , alors TX est un ouvert d’un espace affine AN k et on peut prendre

! TX = dx1

^


^ dxN :

La forme pour TV est alors donnée localement, au signe près, par l’expression explicite 



1 @fi (x) dx0 @xlj 16i;j 6m pour 1 6 l1 <


< lm 6 N .

!L;TV (x) = det

[

^


^ dd xl1 ^


^


^ dxlm ^


^ dxN

3.2. Fonctions de comptage. — Nous souhaitons maintenant expliciter et démontrer la description en termes des torseurs universels de la formule asymptotique (2.4.1) telle qu’elle est annoncée dans [Pe2, §5.4]. Le passage Q au torseur universel nécessite la construction d’un domaine fondamental dans le produit v2S T (kv ) sous l’action de TNS (OS ), qui permettra en fait de construire un domaine fondamental de TCeff (V ) ( k ) sous l’action de TNS (k ). Nous allons rappeler la construction d’un tel domaine donnée dans [Pe2]. On peut rapprocher cette construction du lien entre systèmes de métriques et sections des applications quotients

A

T (Ak )=KTNS ! V (Ak );

A

où KTNS est le sous-groupe maximal de TNS ( k ), indiquée par Salberger [Sa, page 94].

T sur k, on note X (T ) le Gal(k=k)-réseau dual de  X (T ) et pour tout place v de k, X (T )v le groupe X (T )Gal(kv =kv ) . En outre T (Ov ) désigne le sous-groupe compact maximal de T (kv ) et on pose Notations 3.2.1. — Pour tout tore

KT =

Y

v2Mk

T (Ov )

W (T ) = KT \ T (k):

et

Le groupe W (T ) est le groupe fini des éléments de torsion dans T (k ). On dispose d’une injection canonique

logv : T (kv )=T (Ov ) ! X (T )v R: Quitte à augmenter l’ensemble des mauvaises places S , on peut supposer qu’il contient les places archimédiennes et les places ramifiées dans une extension galoisienne fixée K=k qui déploie le tore TNS . Par [Ono1, theorem 4] et [Ono2, §3], on peut en outre supposer que l’application naturelle

TNS(k) !

M

v2Mk S

X (TNS )v

est surjective et qu’on a une suite exacte

0 ! W (TNS ) ! TNS (OS )

logS Y

!

v2S

X (TNS )v R

TORSEURS ET MÉTHODE DU CERCLE

où logS est induite par les applications logv pour v réseau dans le noyau du morphisme Y

v2S

19

2 S . En outre l’image M de logS est un

X  (TNS)_v R ! X (TNS )_k R

où X  (TNS )k = X  (TNS ) . On fixe une base de M et on note
le domaine fondamental correspondant de M dans ce noyau et pr une projection du groupe de gauche sur ce noyau. On se donne alors un système de hauteurs HK sur K et on note H le système de hauteurs défini par le diagramme commutatif

Gal(k=k)

NS(V ) ? ? y

[K :k]H

!

H (V )

x ?N ? K=k

NS(VK ) HK! H (VK ): On suppose en outre que h = H([!V 1 ]) et que (3.2.1) 8L 2 Ceff(V ) \ Pic V; 8x 2 V (k); H(L; x) > 1: Soit T un torseur universel au-dessus de V ayant un point rationnel y0 . Si L est un fibré en droites sur VK , L désigne le complémentaire de la section nulle dans L. Le morphisme Z ! Pic VK envoyant 1 sur la classe de L induit un morphisme L : TNS K ! Gm;K et L  (T ) est isomorphe à L. On note L : T ! L un morphisme partout non nul obtenu de cette manière. On fixe une place p0 de k , et on suppose que la hauteur (L; (k:kP )P2MK ) représente HK ([L]), on note alors 8 k (y)k P < L k L (y0 )kP si P 6 j p0 L 8P 2 MK ; 8y 2 T (KP ); kykP = : k L(y)kP [KP :kp0 ] k L (y0 )kP HK (L; (y0 )) [K:k] sinon. Les fonctions k:kL P ne dépendent que de HK ([L]), de y0 et de p0. Elles induisent des foncpour toute place v de k et tout L de Pic Vv . On obtient des fonctions tions k:kL v _ ~ log H T ;v : T (kv ) ! (Pic Vv ) R caractérisées par les relations

~ log

8y 2 T (kv ); 8L 2 Pic Vv ; kykLv = qv HT ;v (y)(L) où qv = #Fv si v 2 Mf , qv = e si kv est isomorphe à R et qv = e2 sinon. On considère alors

(3.2.2)

n


HK (T ) = y 2

Y

v2S






o

T (kv ) pr (H~ log T ;v (yv ))v2S 2


qui, par [Pe2, proposition 4.3.1] est, sous réserve d’une augmentation de S , un domaine Q fondamental de v2S T (kv ) sous TNS (OS )=W (TNS ): Nous pouvons maintenant définir les fonctions de comptage.

20

EMMANUEL PEYRE

c de Tb Notations 3.2.2. — Quitte à agrandir S , on peut fixer un modèle lisse T Ceff (V ) . Pour toute place de k en-dehors de S , on note

p

TC

(V ) (Op ) = T (Op ) \ T (kp ): c

eff

et on considère

TC Pour tout élément b de et on pose

A

( k) = eff (V );S

L

p2Mk

Y

v2S

T (kv ) 

Y

p62S

S X (TNS )p , on note

TC

eff

(V ) (Op ):

bp la composante de b dans X(TNS)p

TNS ( Ceff (V ); bp ) = ft 2 TNS(kp ) j 8y 2 Ceff (V ) \ Pic Vp ; vp (y(t)) 6 hy; bpig: Notons que

TNS( Ceff (V ); 0):TNS( Ceff (V ); bp) = TNS( Ceff (V ); bp ) et si bp

2 TNS(kp ) est tel que logp (bp ) = bp , alors TNS( Ceff (V ); bp) = bp TNS( Ceff (V ); 0)

et

TNS ( Ceff (V ); bp ):TCeff(V ) (Op ) = bp:TCeff (V ) (Op ): En réalité les TNS ( alors

Ceff (V ); bp ) vont jouer le rôle d’idéaux dans notre cadre. On considère

b:TCeff (V );S (Ak ) =

Y

v2S

T (kv ):

Y

p62S

TNS ( Ceff (V ); bp )TCeff(V ) (Op ):

La fonction de comptage sur le torseur universel L T associée au système de hauteurs HK , au nombre réel positif H et à l’élément b de v2Mk S X (TNS )v est alors la fonction H ( H; b ; : ) indicatrice de l’ensemble des = ( y ) v v 2Mk de TCeff(V ) ( k ) vérifiant les condiT tions qui suivent :

y

(3.2.3) (3.2.4) (3.2.5) (3.2.6) (3.2.7)

A

8v 2 S; (yv ) 2 U (kv ); (yv )v2S 2
HK (T ); Y 8L 2 Ceff(V ); kyv kLv 6 1; Y

v2S

v2S !V 1
1 yv v

k k

y 2 b:TC

eff

6 H;

A

(V );S ( k ):

TORSEURS ET MÉTHODE DU CERCLE

21

3.3. Fonctions de Möbius. — Nous aurons besoin dans le prochain paragraphe de fonctions de Möbius que nous allons maintenant définir et étudier. Notations 3.3.1. — Soit M un Z-module libre de type fini et C rationnel strictement convexe, c’est-à-dire de la forme N X i=0

 M R un cône polyédral

R>0 mi

avec mi 2 M et tel que C \ C = f0g. Si R est un anneau commutatif, on note R[[C ]] (respectivement R((C ))) l’ensemble des fonctions M ! R dont le support est contenu dans C (respectivement dans un translaté de C ). On dispose sur ces R-modules d’un produit (de convolution) défini par

8x 2 M; fg(x) =

X

y+z=x

f (y)g(z ):

En effet si Supp(f )  m + C et Supp(g )  n + C alors le support de fg est contenu dans m + n + C . La fonction Æ0 indicatrice de f0g est une unité pour ce produit. Si A est une partie de M , on note 1A sa fonction indicatrice. Exemple 3.3.2. — Si C est un cône régulier c’est-à-dire de la forme m X i=0

R>0 mi

où (mi )06i6m peut être complété en une base de M , alors on a des isomorphismes évidents

Z[[C ]] = Z[[T1 ; : : : ; Tm ]] et Z((C )) = Z[[T1 ; : : : ; Tm ]][T1 1 ; : : : ; Tm 1 ]

où T1 ; : : : ; Tm sont des indéterminées.

Remarque 3.3.3. — Géométriquement, Q[[C ]] peut être vu comme complété de l’anneau local à l’origine de la variété torique affine

Spec Q[C \ M ]

pour la topologie définie par l’idéal maximal, l’origine étant définie par l’annulation des fonctions correspondant aux éléments de C \ M f0g. Notations 3.3.4. — On a un plongement canonique R[M ]  R((C )) et on pose

R[C ] = R[M ] \ R[[C ]] qui coïncide en fait avec l’algèbre du monoïde C \ M . On note T une indéterminée. Si m 2 M , T m désigne l’élément correspondant de R[M ]. Si f 2 R[[C ]], on pose X f (m)T m = f: m2M 0 Si  : M ! M est un morphisme de Z-modules libres de type fini envoyant C dans un cône polyédral rationnel strictement convexeC 0 de M 0 R et tel que Ker  \ C = f0g, alors on dispose d’un morphisme de R-modules  : R((C )) ! R((C 0 ))

22

EMMANUEL PEYRE

envoyant R[[C ]] dans R[[C 0 ]] défini par

8x 2 M 0 ;  f (x) =

X

(y)=x

f (y):

Lemme 3.3.5. — Avec les notation ci-dessus,  est un morphisme d’anneau. Démonstration. — Si f , g

2 R((C )) et x 2 M 0 , on a les relations

 (fg)(x) = =

X

f (y)g(z )

(y)+(z)=x X  X

y+z=x (y0 )=y

f (y0 )

= ( f )( g)(x):

 X

(z0 )=z



g(z 0 )

Exemple 3.3.6. — Si  2 M _ appartient à l’intérieur du cône C _ défini par

C _ = fx 2 M _ R j 8y 2 C; hx; yi > 0g alors  : M ! Z envoie C dans R>0 et on dispose d’un morphisme  : R((C )) ! R((T )): Lemme 3.3.7. — Avec les notations qui précèdent, si R est intègre, alors R((C )) est un anneau intègre. Démonstration. — Soient f et g deux éléments non nuls de R((C )). On peut choisir  de M _ à l’intérieur de C _ , x0 2 Supp f et y0 2 Supp g de sorte que 8x 2 Supp f fx0 g; (x) > (x0 ) et 8y 2 Supp g fy0 g; (y) > (y0 ): On en déduit que  (f ) et  (g ) sont non nuls et le lemme découle de l’intégrité de R((T )). Lemme 3.3.8. — Avec les notations ci-dessus, si R est intègre et si f 2 R[[C ]] vérifie f (0) 2 R , alors f est inversible dans R[[C ]]. Démonstration. — La fonction g est un inverse de f si et seulement si elle vérifie la relation X 8y 2 M; g(y x)f (x) = Æ0 (y): x2C Soit C  = C f0g, alors cette équation s’écrit également (3.3.1)

8y 2 M; g(y) = f (0)



1 Æ (y ) 0

X

x2C  _ _ Or pour tout m de M à l’intérieur du cône C , on a 8x 2 C  ; hx; mi > 0:



g(y x)f (x) :

Un récurrence sur hx; mi montre alors que (3.3.1) défini une fonction g dont le support est contenu dans C . Notation 3.3.9. — On note C l’inverse de 1C dans R[[C ]].

TORSEURS ET MÉTHODE DU CERCLE

23

Lemme 3.3.10. — En conservant les notations qui précèdent, Il existe famille finie (mj )j 2J d’éléments de M tels que

1C =

Q

j 2J

1C =

n 2Zm >0

T

2 Z[C ] et une

P : (1 T mj )

Démonstration. — Si C est un cône régulier de la forme s’écrire X

P

Pm nimi

i=1

=

m Q i=1

Pm i=0 R>0 mi , la fonction 1C peut

1 (1 T mi )

:

Dans le cas général (cf. par exemple [Oda, page 23]), on écrit C comme support d’un éventail régulier , c’est-à-dire que  est un ensemble de cônes polyédraux rationnels strictement convexes de M R tel que (i) si  2  et  0 est un face de  , alors  0 2 , (ii) si ;  0 2  alors  \  0 est une face de  et de  0 , (iii) C = [2  , (iv) tout  de  est régulier. La fonction 1C s’écrit alors

1C = avec 

X

2

 1

2 Z et le résultat découle du cas précédent.

Proposition 3.3.11. — Pour tout élément  de M _ à l’intérieur de C _ , il existe une constante R telle que

8x 2 C; jC (x)j < Rh;xi:

Démonstration. — L’élément P de Z[C ] du lemme 3.3.10 peut s’écrire X

P =1+

P 1 + m2C 

On pose Q = de P 1 vérifient

m T m:

m2C  m m T . La relation (3.3.1) montre alors que les coefficients

j j



8x 2 M; P

1 (x) 6 Q 1 (x):

Mais par le lemme 3.3.10 la fonction de Möbius s’écrit

C = P et donc

1 Y (1 j 2J



8x 2 C; jC (x)j 6 Q

T mi ) 

1 Y (1 + T mi ) (x): j 2J

24

EMMANUEL PEYRE

on en déduit l’inégalité



8x 2 C; jC (x)j 6  Q



1 Y (1 + T mj ) ((x)): j 2J

Soit R0 l’inverse de la plus petite des valeurs absolues des racines de  Q. Dans C((T )) si Q  Q s’écrit u di=1 (1 i Ti ) alors 

n 





Y Y Y X  Q 1 (1 + T mj ) = u (1 + T (mj ) )  ni T n : i=1 n>0 j 2J j 2J Pour tout nombre réel  > 0, on obtient que les coefficients de la série vérifient 

 Q 1

Q



(1 + T mj ) (x)

j 2J (R0 + )x

! x !

en outre C (0) = 1 et le lemme est démontré.

0 +1

Remarque 3.3.12. — L’utilisation de fonction de Moebius dans des situations similaires apparaît dans [Sc], [Pe1] et [Sa, §11]. 3.4. Montée du nombre de points. — Notre but est maintenant d’exprimer le nombre

nU;h (H ) en termes des torseurs universels.

Notations 3.4.1. — Une famille de représentants des classes d’isomorphisme de torseurs universels ayant un point rationnel au-dessus de V , qui est finie par [CTS2, proposition 2], est notée (Ti )i2I . Pour toute place v de Mf S , on considère le cône On pose v

Cv = fx 2 X (TNS)v j 8y 2 Ceff (Vv ); hx; yi 6 0g:

= Cv et

=

Y

v2Mk S

v :

M

v2Mk S

X (TNS )v ! R:

Proposition 3.4.2. — Avec les notations qui précèdent, quitte à augmenter nombre réel positif H , on a la relation :

nU;H (H ) =

X 1 #W (TNS) i2I b2

L

X

X (TNS )v

v2Mk S

(b)

X

y2Ti (k)

S , pour tout

H Ti (H; b; y):

Remarques 3.4.3. — (i) Les sommations du terme de droite ne font intervenir qu’un nombre fini de termes non nuls. (ii) Si on remplace !V 1 par un autre fibré en droites L à l’intérieur de Ceff (V ) dans la condition (3.2.6) de la définition des fonctions de comptage, la démonstration reste valide et on obtient une expression de nU;H(L) en termes des torseurs universels.

TORSEURS ET MÉTHODE DU CERCLE

25

Démonstration. — Soit x un point rationnel de U et i l’unique élément de I pour lequel Ti a un point rationnel au-dessus de x. Il nous faut montrer que :

1 #W (TNS ) b2

(3.4.1)

L

(

1 si h(x) 6 H; (b) H Ti (H; b; y) = 0 sinon. X (TNS )v y2Ti (k) S

X

v2Mk

X

Mais, pour S assez gros, il résulte de [Pe2, proposition 4.2.2] que Ti Ceff (V ) (Op ) peut être décrit de la manière suivante :

Ti C

eff

(V ) (Op ) = fy 2 Ti (kp ) j 8P 2 fP 2 MK

Il en résulte que

X

b2X(TNS )p

p (b)1TNS (

j Pjpg; 8L 2 Ceff (V ); kykLP > 1g:

Ceff (V );b):TCeff (V ) (Op )

est la fonction indicatrice de

fy 2 Ti (kp ) j 8Pjp; 8L 2 Ceff (V ); kykLP = 1g: 1

Le terme de gauche de (3.4.1) est donc #W (TNS ) fois la somme des valeurs de la fonction caractéristique de l’ensemble des y de Tix (k ) vérifiant les conditions suivantes :

(yv )v2S 2
HK (Ti ); Y (3.4.3) 8L 2 Ceff(V ); kyv kLv 6 1; v2S Y !V 1 1 (3.4.4) (kyv kv ) 6 H; v2S (3.4.5) 8P 2 MK SK ; 8L 2 Ceff (V ); kyP kLP = 1; où SK désigne l’ensemble des places de K au-dessus de S . Comme, par (3.2.1) (3.4.2)

(3.4.6)

Y

v2Mk

kyv kLv = H(L; (y)) 1 6 1;

la condition (3.4.5) implique (3.4.3). Mais on a une suite exacte

0 ! TNS(OS ) ! TNS(k) !

Y

v2Mk S

X (TNS)v ! 0

et donc les conditions (3.4.2) et (3.4.5) définissent, d’après (3.2.2) un domaine fondamental pour l’ensemble TCeff (V ) ( k ) sous TNS (k )=W (TNS ). D’autre part la condition (3.4.4), compte tenu de (3.4.5) et de (3.4.6), peut être remplacée par h( (y )) 6 H .

A

3.5. Montée de la constante. — Nous allons maintenant montrer l’analogue intégral du résultat précédent. Hypothèses 3.5.1. — Dans la suite nous supposons également que les torseurs universels au-dessus de V vérifient le principe de Hasse et l’approximation faible.

26

EMMANUEL PEYRE

Proposition 3.5.2. — Avec les notations qui précèdent, sous les hypothèses 3.1.4 et 3.5.1 il existe S tel que pour tout nombre réel positif H on ait (3.5.1)

(V ) (V )h (V )

Z log H

0

ut 1 eu du

Z

X X 1 = (b) H Ti (H; b; y)!Ti (y): #W (TNS) i2I b2L X (T ) T ( A ) i k Ceff (V )  NS v v2Mk S

R log H

ut 1 eu du est H (log H )t 1 ; c’est Remarques 3.5.3. — (i) Le terme principal de 0 en fait le seul ayant une signification pour le comportement asymptotique. (ii) L’intégrale converge par le lemme 3.1.14. Il résultera de la démonstration que la sommation sur b converge absolument. (iii) En rapprochant la proposition 3.4.2 de la proposition précédente, on constate que la question 2.4.4 se ramène à des majorations de la forme X H Ti (H; b; y) y2Ti (k)

Z

H Ti (H; b; y)!Ti (y) TiCeff (V ) (Ak )

comme c’était le cas dans [Pe2] pour les fonctions zêta associées. Notons que l’équivalence entre ces deux termes lorsque H tend vers l’infini est l’analogue, dans notre cadre, de la notion de variété strictement d’Hardy-Littlewood introduite par Borovoi et Rudnick dans [BR]. Démonstration. — Remarquons tout d’abord que si v est une place de k , T un torseur audessus de V , b un élément de TNS (kv ) et U un ouvert de T (kv ), alors par le lemme 3.1.14 et [Pe2, (4.4.1)]



!T ;v (bU ) = [!V 1 ](b) v 1 !T ;v (U ):

On en déduit que Z

Ti Ceff(V ) (Ak )

H Ti (H; b; y)!Ti (y) =

1

Q

p2Mk

S

#Fp h!V

Z

1;

H bp i Ti C (V ) (Ak ) T (H; 0; y)!Ti (y): eff

Mais par l’hypothèse 3.1.4,

8x 2 Ceff(V )_ \ Pic V _ f0g; hx; !V 1 i > 1

et, Cv étant l’opposé du cône dual de Ceff (Vv ), il en résulte que (3.5.2)

8v 2 Mf S; 8b 2 Cv f0g; hb; !V i > 2:

Or il découle du lemme 3.3.7 et de la proposition 3.3.11 que pour toute place v supportée par Cv et il existe une constante Rv telle que

8b 2 Cv ; Cv (b) < Rvhb;!V i :



Cv

est

TORSEURS ET MÉTHODE DU CERCLE

27

En outre, l’ensemble décrit par les paires (X (TNS )v ; Cv ) étant fini, on peut choisir une constante R indépendante de v . Quitte à agrandir S , la série X hb;! 1 i

b2L X (TNS )v

v (b)(#Fv )

V

converge absolument et par (3.5.2), il existe une constante R0 telle que 1

X

b2X(TNS )v



1 ) (#Fv )hb;!V i
2. (By genus, we mean geometric genus, i.e., the genus of a smooth projective desingularization.) The Riemann-Hurwitz formula yields 2gS

2 = (deg )(2gR

2) +

Xi (e

1):

Let r be the degree of R in Pn. Then each coordinate hyperplane meets R transversely in exactly r points, each belonging to only one coordinate hyperplane. These are the only points on R which ramify, and above each there are pn 1 points of S , each with rami cation index p. Thus 2gS 2 = pn (2gR 2) + (n + 1)rpn 1(p 1)  2pn + 4(pn pn 1): If p > 2, then gS  2. Thus  1 (Z ) K K consists of isolated points and curves over K of genus  2. Applying Faltings' theorem to each curve (over a nite extension of K over which it is de ned) shows that  1 (Z )(K ) is nite. But X 0 (K )   1 (Z ), so X 0 (K ) is nite too. On the other hand, we are free to choose the ci so that at least one point in X (K ) has a K -rational preimage. Thus by replacing X by X 0 , we have reduced the problem to the case where X (K ) is nite (and nonempty). The next step is to reduce to the case where X (K ) consists of a single point. Suppose instead that there exist distinct points P1 ; P2 2 X (K ). We construct a new covering X 0 ! X exactly as before, choosing p large enough so that K has no nontrivial p-th roots of unity. Because we are allowed a generic change of variables on the projective space containing X , we may assume that P1 and P2 are outside all coordinate hyperplanes and that the values taken by the rational function x1 =x0 at P1 and P2 are distinct in K  =K p . Then we may

HASSE PRINCIPLE FOR COMPLETE INTERSECTIONS

3

pick the ci so that P1 has a K -rational preimage in X 0 but P2 does not. Moreover, the choice of p guarantees that each K -rational point on X has at most one K -rational preimage on 0 0 X , so we have succeeded in constructing a cover X with fewer K -rational points than X , but still with at least one K -rational point. Iterating such covers eventually leads to a cover with exactly one K -rational point. Therefore we now assume X (K ) consists of a single point P . Again perform a generic change of variables and de ne a covering X 0 as before, using ci = 1 for i 6= 0 (c0 2 K  will be chosen later). By Bertini's theorem, we may assume that the intersection Y of X 0 with the hyperplane x0 = 0 is smooth and geometrically integral, and we may also assume that P = (1 : 1 :


: 1). Over all but nitely many completions of K , Y will have a point, and hence so will X 0 . Note that Y does not depend on c0 , so we are free to apply weak approximation to choose c0 2 K  such that c0 62 K p but such that c0 2 Kvp for each of the nitely many completions Kv of K for which Y (Kv ) = ;. This ensures that X 0 has a point (namely a preimage of P ) over each of the nitely many completions remaining. On the other hand, X 0 (K ) is empty, since any K -rational point of X 0 would have to lie above P. If there exists a smooth geometrically integral complete intersection X of dimension  3 in Pn over a number eld K , such that the Zariski closure of X (K ) has codimension  2 in X , but X (K ) is not empty, then there exists also a 3-dimensional smooth geometrically integral complete intersection X 0 in projective space over K such that X 0 (K ) is empty, but such that the emptiness of X 0 (K ) cannot be explained by the Brauer-Manin obstruction or Skorobogatov's generalization thereof. Corollary 2.

Proof. Choose P 2 X (K ). If dim X > 3, then we may repeatedly use the Bertini-type theorems of [KA] to replace X by its intersection with a suÆciently general K -rational hypersurface of large degree m through P , in order to reduce to the case where dim X = 3. Now apply Theorem 1.

Acknowledgement I thank the referee for several useful comments.

References [Ha] Hartshorne, R., Algebraic geometry, Springer-Verlag, New York, 1977. [KA] Kleiman, S. and Altman, A., Bertini theorems for hypersurface sections containing a subscheme, Comm. Algebra 7 (1979), no. 8, 775{790. [SW] Sarnak, P. and Wang, L., Some hypersurfaces in P4 and the Hasse-principle, C. R. Acad. Sci. Paris 321 (1995), 319{322. [Sk] Skorobogatov, A. N., Beyond the Manin obstruction, Invent. Math. 135 (1999), no. 2, 399{424. Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

E-mail address :

[email protected]

Une construction de courbes

k -rationnelles

sur les

surfaces de Kummer d'un produit de courbes de genre 1.

Ph. Satge

Resume. k etant un corps de caracteristique dierente de 2, nous decrivons une methode permettant de construire des courbes k-rationnelles (i.e. k-birationnellement equivalentes a la droite projective) sur les surfaces de Kummer associees a un produit de courbes de genre 1 munies d'involutions k-hyperelliptiques. Nous ramenons ce probleme a un probleme de geometrie enumerative sur le produit P1;k  P1;k de la droite projective par elle m^eme. Bien que la resolution generale du probleme de geometrie enumerative auquel nous arrivons soit hors de portee des methodes que nous connaissons, la recherche de solutions particulieres dans des systemes lineaires convenablement choisis permet d'obtenir des exemples interessants. On constate par exemple que l'on retrouve ainsi, de maniere assez systematique, plusieurs resultats qui apparaissent de maniere isolee dans la litterature. INTRODUCTION. On xe un corps k de caracteristique dierente de 2. Le but de ce papier est la construction de courbes k -rationnelles (i.e. k -birationnellement equivalentes a la droite projective) sur la surface de Kummer X = X1  X2 =(1  2 ) ou X1 et X2 sont deux k-courbes de genre 1, et 1 et 2 deux involutions k-hyperelliptiques de X1 et X2 , c'est a dire deux k-involutions de X1 et X2 telles que les quotients X1 =1 et X2 =2 sont k -isomorphes a la droite projective. Il est facile de calculer des equations locales de la surface X (ce calcul est donne au debut du x2) et cela permet de reformuler le probleme qui nous interesse de la maniere tres explicite suivante : etant donnes deux polyn^omes P1 (x1 ) et P2 (x2 ) de degre 3 ou 4, a coeÆcients dans k et sans racines multiples, nous cherchons a construire des solutions, dans le corps k (u) des fractions rationnelles a coeÆcients dans k , a l'equation diophantienne P2 (x2 (u))y (u)2 = P1 (x1 (u)). Les raisons de s'interesser aux courbes k -rationnelles de la surface de Kummer X sont nombreuses et classiques ; rappelons en quelques unes. On montre facilement que l'image reciproque d'une courbe k -rationnelle de X par la projection canonique de X1  X2 sur X est une courbe k -hyperelliptique ; la construction de courbes k -rationnelles sur X est donc equivalente a la construction de courbes k -hyperelliptiques dont la jacobienne contient deux facteurs elliptiques. Lorsque le corps k est le corps des complexes, c'est le tres classique probleme de la reduction des integrales hyperelliptiques a des integrales 1

elliptiques. Le cas particulier des courbes hyperelliptiques de genre 2 est particulierement interessant et est discute dans deux des exemples que nous traitons ; on notera que nous ne supposons pas le corps de base algebriquement cl^os et que le contr^ole des corps de rationalite est un point important de notre approche. On sait aussi que la surface X possede plusieurs brations en courbes de genre 1 sur la droite projective ; on peut utiliser les courbes k -rationnelles de X qui ne sont pas contenues dans des bres pour construire des changements de base qui augmentent le rang generique des brations ; lorsque le corps k est un corps de nombres, ce procede est (au langage pres) le procede standard utilise pour construire par specialisation des familles de courbes dont le rang est strictement plus grand que le rang generique. Ce point de vue est developpe dans [M-S]. En n, toujours dans le cas ou k est un corps de nombres, rappelons que les conjectures de Batyrev-Manin ([B-M], x3.5) predisent que les points rationnels de X sont presque tous situes sur des courbes k -rationnelles, et qu'il semble important de comprendre comment ils s'accumulent sur ces courbes. McKinnon ([McK]) a montre que les courbes k -rationnelles qui forment le premier ensemble d'accumulation des points rationnels sur certaines surfaces du type de X sont, pour des hauteurs bien choisies, les courbes k -rationnelles dont la construction est evidente (ce sont les courbes appellees triviales dans le x1.1 de ce papier). On sait tres peu de choses sur les ensembles d'accumulations suivants. Bien que nous ne soyons rien capable de prouver dans cette direction a l'heure actuelle, il est raisonnable de penser que certaines des courbes construites dans ce papier appartiennent au second de ces ensembles. La production un peu systematique d'exemples de courbes k -rationnelles sur la surface X nous semble interessante de ce point de vue. Dans tout le papier k est un corps de caracteristique dierente de 2 et k une clture algebrique de k . On xe deux couples (X1 ; 1 ) et (X2 ; 2 ) ou X1 et X2 sont deux k-courbes lisses, geometriquement integres de genre 1, et ou 1 et 2 sont deux involutions k -hyperelliptiques sur X1 et X2 , c'est a dire des k-involutions telles que les quotients X1 =1 et X2 =2 sont k-isomorphes a la droite projective P1;k . Le quotient X = X1  X2 =(1  2 ) est une k -surface geometriquement integre, possedant 16 points singuliers, et dont le modele lisse est une surface de Kummer ; c'est sur cette surface que nous voulons construire des courbes k -rationnelles. Pour i = 1; 2, on note i : Xi ! Xi =i = P1;k la projection canonique ; le produit 1  2 : X1  X2 ! P1;k  P1;k se factorise naturellement par la projection canonique X1  X2 ! X1  X2 =(1  2 ) = X et on note  : X ! P1;k  P1;k le k -morphisme de factorisation. Le principe de notre construction consiste a construire des courbes k -rationnelles sur X en remontant des courbes k -rationnelles de P1;k  P1;k ; l'eÆcacite du procede repose sur le fait que la geometrie de P1;k  P1;k est plus simple que celle de X. Nous avons divise ce travail en deux paragraphes. Dans le premier, on etudie les composantes irreductibles de l'image reciproque d'une courbe de P1;k  P1;k par le morphisme canonique  : X ! P1;k  P1;k . On montre (corollaire 1.2.4) que les courbes de P1;k  P1;k dont les images reciproques par  ont 2

des composantes k -rationnelles sont les courbes k -rationnelles de P1;k  P1;k qui rencontrent la courbe de rami cation Z du k -morphisme  d'une maniere imposee ; on ramene ainsi le probleme de la recherche des courbes k -rationnelles de X a un probleme de geometrie enumerative dans P1;k P1;k . On sait que c'est la un probleme diÆcile en general ; on montre dans le second paragraphe qu'il est abordable pour des courbes dont le bidegre reste raisonable (par rapport aux moyens de calculs dont on dispose) et donne des resultats non triviaux. Dans le second paragraphe nous traitons quatres exemples. Dans le premier exemple nous traitons le cas le plus simple qui est celui des courbes de bidegre (1; 1). On constate que l'on retrouve de cette maniere des formules classiques attribuees a Legendre et Jacobi. Le deuxieme exemple semble moins classique ; il illustre le fait que les couples (X1 ; 1 ) et (X2 ; 2 ) ne sont pas necessairement formes d'une courbe elliptique et de la multiplication par 1 ; dans le point de vue des equations diophantiennes, cela se traduit par le fait que les polyn^omes P1 (x1 ) et P2 (x2 ) sont de degre 4 et n'ont pas necessairement de racines dans le corps de base k . Le troisieme exemple est choisi de maniere a retrouver un resultat etabli, dans un autre contexte, par J.-F. Mestre. En n, dans le quatrieme exemple, on s'interesse aux courbes k -rationnelles de X qui correspondent aux courbes hyperelliptique de genre 2 dont la jacobienne est decomposee. Cela nous amene a rechercher dans P1;k  P1;k des courbes k -rationnelles de bidegre (n; n), tres singulieres, et rencontrant la courbe de rami cation Z de  de maniere imposee. Le cas n = 3 est facile a traiter et redonne des formules classiques. Peu de choses sont connus pour les valeurs de n > 3. La methode presentee ici permet d'aborder les cas n = 4 et n = 5 ; chaque fois que l'on a trouve un exemple dans la litterature (on en signale un certains nombre a la n du papier) il nous a ete relativement facile de le retrouver par notre procede. Une etude plus generale necessiterait d'avoir des informations complementaires sur les singularites des courbes de bidegre (n; n) que l'on cherche a contruire. Cette etude est abordee dans [Sat] et merite sans doute d'^etre poussee plus loin.

1

Relevement des courbes de P1;k  P1;k sur la surface de Kummer X.

Dans ce premier paragraphe nous expliquons comment calculer le genre geometrique des composantes irreductibles des images reciproques par  des courbes de P1;k  P1;k et nous en deduisons la caracterisation des courbes de P1;k  P1;k dont l'image reciproque par  contient des courbes k -rationnelles en utilisant le fait que les courbes k -rationnelles sont les courbes de genre geometrique 0 qui possedent des points rationnels. 1.1

Notations.

Pour i = 1; 2, on designe par R0(i) ; : : : ; R3(i) les quatre k -points xes de l'involution i , et par r0(i) ; : : : ; r3(i) leurs images respectives par la projection canonique 3

i : Xi ! Xi =i = P1;k ; on note P l'ensemble des seize k-points de P1;k  P1;k (1) (2) forme des rm;n = (rm ; rn ) ou m; n = 0; : : : ; 3. Pour tout k-point r de P1;k , on note L1 (r) (resp. L2 (r)) la k -courbe P1;k frg (resp. frg P1;k ) de P1;k  P1;k . (1) Pour n; m 2 f0; : : : ; 3g, on pose L1 (n) = L1 (rn(2) ) et L2 (m) = L2 (rm ) ; les courbes L1 (n) et L2 (m) se coupent donc en rm;n ; on note L l'ensemble des huit courbes fLi (n); i = 1; 2; n = 0; : : : ; 3g. On designe par Z la courbe (reduite) de P1;k  P1;k dont le support est la reunion des huit courbes de L ; c'est la courbe de rami cation du morphisme  ; ses points singuliers sont les seize points de P et sont aussi les images par  des seize points singuliers de X . Toute courbe Li (n) 2 L est une courbe k -rationnelle de P1;k  P1;k ; elle est k-rationnelle si et seulement si le point rn(i) est de ni sur k ; comme Li (n) est dans le lieu de rami cation de  , son image reciproque, munie de sa structure de schema reduit, est une courbe k -rationnelle de X qui est k -rationnelle si et seulement si Li (n) est k -rationnelle, c'est a dire si et seulement si rn(i) est de ni sur k . Dans la suite ces courbes k -rationnelles seront appelees les courbes k-rationnelles triviales de X . Si D est une k courbe geometriquement integre, on note k (D) le corps des k-fonctions rationnelles sur D ; les valuations discretes de k(D) qui sont triviales sur k et normalisees par le fait que leurs groupes des valeurs est Z tout entier seront appelees les branches de D. On note VD l'ensemble des branches de D ; le corps de de nition de la branche v 2 VD est le corps residuel de l'anneau de la restriction de v au sous corps k (D) de k(D) forme des k -fonctions rationnelles. S'il existe un k -point de D dont l'anneau local dans k (D) est domine par l'anneau de la valuation v , ce k -point est unique (les k -courbes sont, par hypothese, des k-schemas separes) et est appele le centre de v. Toutes les courbes considerees dans ce papier sont des courbes projectives, donc toutes les branches ont un centre. 1.2

Le calcul du genre.

Pour i 2 f1; 2g, choisissons des coordonnees (ui;0 : ui;1 ) sur la droite projective P1;k = Xi =i et notons Ui et Ui0 les k-ouverts P1;k n f(0 : 1)g et P1;k n f(1 : 0)g de P1;k . On sait que le couple (Xi ; i ) se decrit de la maniere suivante : il existe une forme Fi (ui;0 ; ui;1 ) de degre 4, bien de nie a multiplication par le carre d'un element non nul de k pres, telle que les ouverts  1 (Ui ) et  1 (Ui0 ) de Xi qui recouvrent X sont les k -courbes aÆnes d'equations respectives vi2 = Pi (ui ) et v0 2i = Pi0 (u0i ) ou Pi (ui ) = Fi (1; ui ) et Pi0 (u0i ) = Fi (1; u0i ) ; ces deux ouverts se recollent par u0i = 1=ui , vi0 = vi =u2i , et i envoie le point de coordonnees (ui ; vi ) (resp. (u0i ; vi0 )) sur le point de coordonnees (ui ; vi ) (resp. (u0i ; vi0 )). Un calcul elementaire montre que l'ouvert  1 (U1  U2 ) de X est la k -surface aÆne de l'espace aÆne A3k d'equation v 2 = P1 (u1 )P2 (u2 ) ; notons que P1 (u1 )P2 (u2 ) = 0 est une equation de la courbe Z sur U1  U2 , donc on obtient une equation de l'ouvert  1 (U1  U2 ) de X en extrayant la racine carre d'une equation locale de Z bien normalisee. On decrit bien s^ur de maniere analogue  1 (U1  U20 ),  1 (U10  U2 ), et  1 (U10  U20 ) qui, avec  1 (U1  U2 ), recouvrent X ; les 4

formules de recollement sont immediates a ecrire. On introduit la de nition suivante :

De nition 1.2.1 Une bonne k-equation locale de Z est une k-equation locale (U; f ) de Z (i.e. un k -ouvert aÆne U  P1;k  P1;k et une k -fonction reguliere f sur U dont le schema des zeros est Zp\ U ) telle que l'anneau des coordonnees du k-ouvert aÆne  1 (U ) de X est A[ f ] ou A designe l'anneau des coordonnees de U . On considere maintenant une k -courbe D geometriquement integre de P1;k  P1;k qui n'est pas contenue dans Z . Soit v une branche de D et (U; f ) une bonne k -equation locale de Z telle que le centre de v appartient a U (on peut par exemple prendre pour U l'un des quatres ouverts U1  U10 , U1  U20 , U2  U10 ou U2  U20 qui recouvrent P1;k  P1;k et pour f l'equation locale que l'on a explicitee plus haut). Comme D n'est pas inclue dans Z , la restriction de f a D n'est pas nulle et de nit donc un element non nul du corps k (D) que l'on note fD . L'entier v(fD ) ne depend pas du choix de (U; f ) et est, par de nition, la multiplicite d'intersection de la branche v avec Z . Si P est un k -point de D, la multiplicite d'intersection de D et Z en P est donc la somme des multiplicites des branches de D centrees en P avec Z . On pose :

De nition 1.2.2 Soit D une k courbe de P1;k  P1;k qui est geometriquement integre et qui n'est pas contenue dans la courbe Z . Pour toute branche v de D, on note m(v) la multiplicite d'intersection de la branche v avec Z . On note RD l'ensemble des branches de D pour lesquelles m(v) est impair. On rappelle que le genre geometrique d'une k -courbe geometriquement integre est, par de nition, le genre d'un k modele propre et lisse de cette courbe ; on a :

Proposition 1.2.3 Soit D une k courbe de P1;k P1;k qui est geometriquement integre et qui n'est pas contenue dans Z . Si gD est le genre geometrique de D et si rD est le cardinal de l'ensemble RD de ni en 1.2.2, on a : i) Si RD est non vide, l'image reciproque  1 (D) de D par  est une k courbe de X qui est geometriquement integre et dont le genre geometrique est g = 1 + 2(gD 1) + r2D (et donc rD est pair). ii) Si RD est vide, si (U; f ) est une bonne k-equation locale de Z et si fD 2 k(D) est la restriction de f a D, alors : ii)1 si fD n'est pas un carre dans le corps k(D), l'image reciproque  1 (D) de D par  est une k courbe de X qui est geometriquement integre et dont le genre geometrique est g = 1 + 2(gD 1) ; ii)2 si fD est un carre dans le corps k(D), l'image reciproque  1 (D) de D par  est la reunion de deux k-courbes de X qui sont k-birationnellement equivalentes a D ; ces deux k-courbes sont des k-courbes de X si et seulement si fD est un carre dans le corps k(D) et, dans ce cas, elles sont k-birationnellement equivalentes a D. 5

Demonstration: Fixons une famille nie (U (i) ; fi )i2I de bonnes k-equations locales de Z dans laquelle (U (i) )i2I est un recouvrement ouvert de P1;k  P1;k et les U (i) sont des k -ouverts aÆnes d'anneau de coordonnees Ai . La famille (U (i) \ D)i2I est un recouvrement ouvert de D et les U (i) \ D sont des k -ouverts aÆnes dont les anneaux de coordonnees sont notes Ai;D ; pour i 2 I , on note fi;D 2 Ai;D la restriction de fi a D. Parpde nition  1 (U (i) ) est un k-ouvert aÆne de X d'anneau de coordonnees Ai [ fi ], donc  1 (U (i) ) \  1 (D) est un p le k -schema aÆne d'anneau des coordonnees Ai [ fi ] Ai Ai;D = Ai;D [ fi;D ]. Ainsi,  1 (D) est geometriquement integre si et seulement si, quelque soit i 2 I , fi;D n'est pas un carre dans le corps des fractions de Ai;D k k qui est k(D). Soit maintenant (U; f ) une bonne k -equation locale de Z ; il resulte immediatement des de nitions que, pour tout i 2 I , il existe un gi 2 k (P1;k  P1;k ) tel que 2 f = fi gi2 ; en consequence la restriction fD de f a D est fD = fi;D gi;D ou gi;D est la restriction de gi a D. Ainsi fD n'est pas un carre dans k(D) si et seulement si aucun des fi;D n'est un carre dans k (D), et donc si et seulement si la courbe  1 (D) est geometriquement integre. Montrons i) : Par hypothese RD n'est pas vide, c'est a dire qu'il existe au moins une valuation discrete v de k (D) triviale sur k telle que v (fD ) est impair, donc fD n'est pas un carre dans k (D) ; on vient de voir que cela implique que  1 (D) est geometriquement integre. D'autre part, la theorie de Kummer montre qu'une valuation discrete w de k(D) triviale sur k est rami ee dans p l'extension k (D)( fD ) si et seulement si w(fD ) est impair, c'est a dire si et seulement si w 2 RD . Ilpresulte alors de la formule de Riemann-Hurwitz que le genre du corps k (D)( fD ), qui est le genre geometrique de  1 (D), est 1 + 2(gD 1) + r2D , ce qu'on voulait. Montrons ii) : Si fD n'est pas un carre dans k(D), on raisonne comme dans le cas i). Sinon, les assertions resultent immediatement de la description locale de  1 (D) que l'on a explicite dans la premiere partie de cette demonstration.

p

Remarque:

Les equivalences birationnelles du cas ii)2 de la proposition precedente ne sont pas necessairement des isomorphismes (on fabrique facilement des exemples ou D est une courbe singuliere, et ou ces equivalences birationnelles sont des desingularisations de D k k . Dans ce papier nous utiliserons le corollaire suivant de la proposition 1.2.3 :

Corollaire 1.2.4 Soit D une courbe k-rationnelle de P1;k  P1;k qui n'est pas contenue dans Z , et soit rD le cardinal de RD (de nition 1.2.2). Alors 1) Les deux assertions suivantes sont equivalentes : (i) l'image reciproque  1 (D) de D dans X est une k-courbe geometriquement integre de genre geometrique zero ; (ii) rD = 2. 2) Les deux assertions suivantes sont equivalentes : (i) l'image reciproque  1 (D) de D dans X possede deux k-composantes irreductibles C1 et C2 qui, si on les munit de leurs structures de schemas reduits, sont de genre geometrique zero ; 6

(ii) rD = 0. 3) Si rD 6= 0; 2, alors aucune composante de  1 (D) n'est k-rationnelle. Demonstration: La courbe D etant de genre geometrique 0, elle n'admet pas

de rev^etement non rami e de degre strictement positif, donc si RD est vide i.e. si rD = 0, l'image reciproque  1 (D) de D n'est pas geometriquement integre. Compte tenu de cette remarque, nos assertions resultent immediatement de la proposition 1.2.3. Comme toute courbe k -rationnelle de X est une composante de l'image reciproque par  de son image dans P1;k  P1;k , et que l'image d'une courbe k-rationnelle par un k-morphisme est une courbe k-rationnelle, le corollaire precedent ramene la recherche des courbes k -rationnelles sur la surface de Kummer X aux deux problemes suivants : d'une part le probleme geometrique de trouver les courbes k -rationnelles D de P1;k  P1;k avec rD = 0 ou 2, et d'autre part le probleme arithmetique de decider, dans le cas rD = 0 si les deux k -composantes de  1 (D) sont de nies sur k (ce qui implique qu'elles sont k-rationnelles puisqu'elles sont alors k-birationnellement equivalentes a D), et dans le cas rD = 2 si la k -courbe geometriquement integre  1 (D) qui est de genre geometrique 0 possede des points rationnels sur k .

2

Exemples.

Dans la presentation des exemples nous utiliserons les notations introduites au debut du x1.2. Les coordonnees (ui;0 : ui;1 ) sur la droite projective P1;k = Xi =i seront choisis de maniere adaptee a l'exemple traite. Rappelons que le k -ouvert aÆne  1 (U1  U2 ) de X est k -isomorphe a la k -surface aÆne de A3k d'equation v2 = P1 (u1 )P2 (u2 ). Les points singuliers de cet ouvert sont les points dont la coordonnee v est 0 ; le morphisme d'eclatement de ces points singuliers est le morphisme de la k -surface A2k  P1;k d'equation P2 (x2 )y12 = P1 (x1 )y02 sur la k-surface de l'espace aÆne A3k d'equation v2 = P1 (u1 )P2 (u2 ) qui envoie le point de coordonnees (x1 ; x2 ; (y0 : y1 )) sur (u1 = x1 ; u2 = x2 ; v = P2 (x2 )y1 =y0 ) si y0 6= 0 et sur (u1 = x1 ; u2 = x2 ; v = P1 (x1 )y0 =y1) si si y1 6= 0. La partie y0 6= 0 de cet eclate est donc un k -ouvert aÆne du modele lisse X lisse de X qui est k-isomorphe a la k-surface aÆne de A3k d'equation P2 (x2 )y2 = P1 (x1 ). Nous explicitons les parametrisations des courbes k -rationnelles que nous construisons dans cet ouvert aÆne de X lisse ; on trouve ainsi les solutions de l'equation diophantienne qui a ete mentionnee dans l'introduction. Pour toute k -courbe D de P1;k  P1;k non contenue dans Z , toute branche v de D, et tout couple (i; n) avec i = 1; 2 et n = 0; : : : ; 3, on note m(i; n; v) la multiplicite d'intersection de la branche v avec la droite Li (n) ; on a donc v 2 RD si et seulement si (i;n) m(i; n; v) est impair.

P

2.1

Exemple 1.

Relevement des courbes de bidegre (1; 1) 7

Une k -courbe D integre et de bidegre (1; 1) est propre, lisse et k -rationnelle ; l'application qui envoie une branche de D sur son centre permet donc d'identi er l'ensemble des branches de D avec l'ensemble des k -points de D. Pour tout couple (j; m), la courbe D coupe Lj (m) en un unique point pj;m et avec multiplicite 1 ; ainsi, si v (j;m) est la branche de D dont le centre est pj;m , on a m(j; m; v(j;m) ) = 1. Si pj;m n'est sur aucune droite de L distincte de Lj (m), on a m(i; n; v (j;m) ) = 0 pour tout (i; n) 6= (j; m), et donc v (j;m) 2 RD . Sinon il existe un n 2 f0; : : : 3g et un seul tel que pj;m est le point d'intersection de la droite Lj (m) avec la droite Li (n) ou i est de ni par fi; j g = f1; 2g ; on a alors pj;m = rm;n (resp. rn;m ) si j = 2 donc i = 1 (resp. j = 1 donc i = 2). Dans ce cas on a m(j; m; v(j;m) ) = m(i; n; v(j;m) ) = 1 et , pour tout (i0 ; n0 ) 6= (j; m); (i; n), on a m(i0 ; n0 ; v (j;m) ) = 0 ; on a donc v (j;m) 62 RD .

Le cas rD = 0. Supposons qu'il existe une k-courbe D integre et de bidegre (1; 1) avec rD = 0 ; alors, pour tout m = 0; : : : ; 3, il existe un et un seul n = (m) 2 f0; : : : ; 3g tel que p2;m = p1;(m) est le point d'intersection rm;(m) de L2 (m) avec L1 ((m). L'application  est une bijection de l'ensemble f0; : : : ; 3g sur lui m^eme (si (m) = (m0 ) avec m 6= m0 , la courbe D contient L1((m)), donc n'est pas integre) et le k-isomorphisme de P1;k dont le graphe est D envoie les quatre k -points r0(1) ; : : : ; r3(1) respectivement sur r(2)(0) ; : : : ; r(2)(3) ; notons  cet isomorphisme. Le k -isomorphisme  : P1;k ! P1;k se releve en un k isomorphisme ~ : X1 ! X2 qui est compatible avec les involutions. Il est alors facile de veri er que les deux k-composantes de  1 (D) sont les images par  : X1  X2 ! X des deux courbes images de (id; ~) : X1 ! X1  X2 et de (1 ; ~) : X1 ! X1  X2 . Ces composante sont de nies sur k si et seulement si ~ est de ni sur k . Bien entendu les courbes k -rationnelles ainsi produites sont evidentes a trouver directement, et nous n'avons traite ce cas que pour ^etre complet.

Le cas rD = 2. Supposons qu'il existe une k-courbe D integre et de bidegre (1; 1) avec rD = 2 ; alors, il existe au moins un m(D) 2 f0; 1; 2; 3g tel que p2;m(D) n'est sur aucune des quatre droites L1 (n) pour n = 0; : : : ; 3 puisque sinon, comme on vient de le voir, on aurait rD = 0. Pour la m^eme raison, il existe au moins un n(D) 2 f0; 1; 2; 3g tel que p1;n(D) n'est sur aucune des quatre droites L2(m) pour m = 0; : : : ; 3. Ainsi les deux branches de D centrees respectivement en p2;m(D) et en p1;n(D) sont dans RD , et donc aucune autre branche de D n'est dans RD . Comme dans le cas precedent, on en deduit que si m 2 f0; 1; 2; 3g n fm(D)g, il existe un et un seul n = (m) 2 f0; : : : ; 3g n fn1 (D)g tel que p2;m = p1;(m) est le point d'intersection rm;(m) de L2(m) avec L1 ((m)), et que  est une bijection de f0; 1; 2; 3gnfm(D)g sur f0; 1; 2; 3gnfn(D)g. Quitte a changer les indices nous pouvons supposer que m(D) = n(D) = 0, de sorte que la situation se decrit par la gure suivante ou les obliques representent les intersections de la courbe D avec les Li (n) (dans l'exemple ci-dessous, on respecte le choix m(D) = n(D) = 0, et on a (1) = 1, (2) = 3 et (3) = 2) :

8

L2 (1)

L2 (3)

r2;3

L1 (3)

r3;2

L1 (2) L1 (1)

r1;1

P ;k  fr g = L1 (0) 1

(2) 0

fr0(1) g  P1;k =L2 (0)

L2 (2)

Comme la courbe D est de nie sur k , les deux droites L2 (0) et L1 (0) sont de nies sur k , donc les deux points r0(1) et r0(2) sont de nis sur k et le zero cycle (r1;(1) ) + (r2;(2) ) + (r3;(3) ) de P1;k  P1;k est rationnel sur k . Notons (*) la condition suivante : (*) Il existe une bijection  de l'ensemble f1; 2; 3g sur lui m^eme telle que le zero cycle (r1;(1) ) + (r2;(2) ) + (r3;(3) ) de P1;k  P1;k est rationnel sur k . La condition (*) est une partie des conditions necessaires que l'on vient de mettre en evidence, a l'existence d'une k -courbe D integre de bidegre (1; 1) avec rD = 2. Notons qu'elle implique que chacun des deux points r0(1) et r0(2) est de ni sur k , et donc que chacune des deux droites L2 (0) et L1 (0) est aussi de nie sur k. Supposons la condition (*) veri ee ; notons qu'elle implique que les trois k-points r1;(1) , r2;(2) , et r3;(3) ne sont pas situes sur une reunion de courbes de bidegre (0; 1) ou (1; 0)). Il en resulte qu'il existe une et une seule courbe D de bidegre (1; 1) passant par les trois k-points r1;(1) , r2;(2) , et r3;(3) de P1;k  P1;k , et que cette courbe est integre ; de plus elle est de nie sur k puisque le zero cycle (r1;(1) ) + (r2;(2) ) + (r3;(3) ) de P1;k  P1;k est, par hypothese, rationnel sur k . On a rD = 2 si et seulement si les points d'intersection p1;0 et p2;0 de D avec L1 (0) et L2 (0) sont distincts, c'est a dire si D ne passe pas par le point r0;0 . Notons en n que, lorsque D veri e toute ces conditions, la partie arithmetique est automatiquement resolue : en eet, les images reciproques des deux points rationnels p1;0 et p2;0 par  sont deux points de  1 (D ) qui sont aussi rationnels puisque ce sont des points de rami cation de  . La courbe  1 (D ) est donc k -rationnelle. Placons nous dans des coordonnees adaptees a notre situation : pour i = 1; 2, on choisit les coordonnees (ui;0 : ui;1 ) sur la droite projective de sorte que r0(i) est le point de coordonnees (0 : 1) et que la ieme projection du point rationnel pi;0 est le point de coordonnees (0 : 1). Avec ces choix, les points p1;0 et p2;0 sont les points de P1;k  P1;k de coordonnees ((1 : 0); (0 : 1)) et ((0 : 1); (1 : 0)) ; la courbe D passe par ces deux points, est integre, et est de nie sur k , 9

donc elle a pour equation u1;1 u2;1 u1;0 u2;0 ou  est un element non nul de k . D'autre part, le point r0(i) etant de ni sur k , le point R0(i) est aussi de ni sur k ; on munit Xi de la structure de k -courbe elliptique pour laquelle R0(i) est l'origine ; l'involution i est alors la multiplication par 1 et on pose Xi = Ei pour rappeller ce choix. On prend ui = ui;1 =ui;0 comme coordonnee dans l'ouvert aÆne Ui de ni par ui;0 6= 0 et on note Pi (ui ) le polyn^ome unitaire dont les racines sont les coordonnees des trois points r1(i) , r2(i) et r3(i) de Ui . Ainsi vi2 = Pi (ui ) est un modele de Weierstrass de Ei ; comme la ieme projection de pi;0 est le point de coordonnees ui = 0 dans Ui , et n'est ni r1(i) , ni r2(i) , ni r3(i) , on (1) (2) a Pi (0) 6= 0. En n, pour m = 1; 2; 3, la courbe D passe par le point (rm ; r(m) ) de U1  U2 , donc le polynme P2 (u2 ) et la fonction rationnelle P1 (=u2 ) ont les m^emes zeros ; si l'on pose Pi (ui ) = u3i + ai u2i + bi ui + ci cela se traduit par a2 = b1=c1 ; b2 = a1 2 =c1 et c2 = 3 =c1 . Si toutes ces conditions sont veri ees, la courbe D d'equation u1;1 u2;1 u1;0 u2;0 est une k -courbe integre de bidegre (1; 1) pour laquelle rD = 2. Il est facile de calculer et de parametriser l'image reciproque de D ; en posant w = =c1 on trouve :

Proposition 2.1.1 La surface X contient une courbe k-rationnelle C dont l'image dans P1;k  P1;k est une courbe D de bidegre (1; 1) avec rD =2 si et seulement si X1 et X2 sont deux k-courbes elliptiques E1 et E2 de modele de Weirstrass v12 = u31 + au21 + bu1 + c et v22 = u32 + bwu22 + acw2 u2 + c2 w3 ou a, b, c et w sont dans k, et ou c et w non nuls, les involutions 1 et 2 etant la multiplication par 1. Dans l'ouvert aÆne de X lisse d'equation (x32 + bwx22 + acw2 x2 + c2 w3 )y2 = (x31 + ax21 + bx1 + c) dans A3k , la courbe k -rationnelle C admet la parametrisation z ! (x1 = 1 (z ); x2 = 2 (z ); y = 3 (z )) ou 1 (z ) = wz 2 ; 2 (z ) = zc2 ; 3 et 3 (z ) = zc . On peut interpreter la condition (*) en introduisant les modules galoisiens

E1 [2] et E2 [2] des k-points tues par 2 des courbes elliptiques E1 et E2 . La bijection ensembliste de E1 [2] = fR0(1) ; : : : ; R3(1) g sur E2 [2] = fR0(2) ; : : : ; R3(2) g (1) qui envoie R0(1) sur R0(2) et Rm sur R(2)(m) pour m = 1; 2; 3 est un isomorphisme

de groupe et la condition (*) est equivalente au fait que cet isomorphisme de groupe est un isomorphisme de Gal(k=k ) module. Il est immediat de veri er que la condition pour que la courbe D ne passe pas par le point r0;0 , c'est a dire pour que rD = 2, est que ce k -isomorphisme de Gal(k=k )-module n'est pas induit par un isomorphisme de courbes elliptiques. En utilisant, dans le cas particulier qui nous interesse ici, les techniques de [F-K] ou [Sat], on peut veri er que cette condition est la condition pour qu'il existe une k -courbe C de genre 2 et deux k -morphismes '1 : C ! E1 et '2 : C ! E2 qui sont de degre 2 10

et independants. On retrouve ce resultat en calculant l'image reciproque de la courbe k -rationnelle C de X par le morphisme  : E1  E2 ! X ; on trouve, avec les notations introduites dans la proposition 2.1.1, que cette image reciproque est la courbe plane d'equation y 2 = w3 x6 + aw2 x4 + bwx2 + c, et que la restriction a cette courbe des deux projections q1 et q2 de E1  E2 sur E1 et E2 sont les morphismes '1 et '2 qui envoient le point de coordonnees (x; y ) sur le point de E1 de coordonnees (u1 ; v1 ) = (wx2 ; y ) et sur le point de E2 de coordonnees (u2 ; v2 ) = (c=x2 ; cy=x3 ). A la n du dix neuvieme siecle, la forme generale de l'integrale hyperelliptique de genre 2 qui se ramene a des integrales elliptiques par des transformations rationnelles de degre 2 a ete donnee par Legendre ([Leg]) et Jacobi ([Jac]), et beaucoup discutee ([Kra], Chapitre X1, [Pic] par exemple). Comme c'etait l'habitude au XIX ieme siecle, Legendre et Jacobi enoncent leurs resultats a partir d'une forme de Rosenhain de la courbe hyperelliptique de genre 2, c'est a dire pour la courbe mise sous la forme Y 2 = X (X 1)(X )(X )(X ). Ils montrent que les integrales des formes holomorphes de cette courbe se ramenent a des integrales elliptiques par des transformations rationnelles de degre 2 si et seulement si =  . On retrouve ce resultat a partir du n^otre de la maniere suivante : on note t1 ; t2 ; t3 les trois racines de l'equation polyn^omiale w3 T 3 + aw2 T 2 + bwT + c = 0 et on calcule la forme de Rosenhain de la courbe d'equation y 2 = w3 x6 + aw2 x4 + bwx2 + c pour laquelle les trois points de Weierstrass de coordonnees (t1 ; 0); ( t1 ; 0) et (t2 ; 0) sont envoyes respectivement sur les points d'abscisses 0; 1 et 1. Les trois points de Weierstrass restants qui sont les points de coordonnees ( t2 ; 0); (t3 ; 0) et ( t3 ; 0) sont alors respectivement envoyes sur les points d'abscisses = ( tt22 +tt11 )2 ,  = tt22 +tt11 tt33 +tt11 et = tt22 +tt11 tt33 +tt11 ; on a bien =  . Bien entendu, la forme de Rosenhain n'est pas adaptee aux questions de rationalite puisque les points de Weierstrass ne sont pas en general de nis sur le corps de base de la courbe, ni m^eme d'ailleurs sur le corps de de nition des quotients elliptiques de la Jacobienne de cette courbe. 2.2

Exemple 2.

Relevement non rami e de certaines courbes de bidegre (2; 2). Dans cet exemple on cherche les courbes k -rationnelles de X dont l'image dans P1;k  P1;k est une courbe D de bidegre (2; 2) qui rencontre les huit droites Li (n) de L de la maniere decrite dans la gure ci-dessous :

11

L2 (1) L1 (3)

r2;3 r3;3

L1 (2)

r2;2 r3;2

L1 (1)

r0;1 r1;1

P ;k  fr g = L1 (0)

r0;0 r1;0

1

(2) 0

L2 (3)

fr0(1) g  P1;k =L2 (0)

L2 (2)

ou les obliques representent les intersections de la courbe D avec les Li (n). Autrement dit, la courbe D de bidegre (2; 2) passe par le sous-ensemble P0 de P forme des huit points r0;0 ; r0;1 ; r1;0 ; r1;1 ; r2;2 ; r2;3 ; r3;2 ; et r3;3 . Les seules branches d'une telle courbe qui rencontrent les Li (n) sont les huit v (n;m) centrees en rn;m 2 P0 . Pour ces huit branches, on a m(1; n; v (n;m) ) = m(2; m; v (n;m) ) = 1 et tous les autres m(i0 ; n0 ; v (n;m) ) sont nuls, donc v (n;m) 62 RD , et donc rD = 0. Supposons qu'il existe une courbe k -rationnelle D de bidegre (2; 2) qui coupe les Li (n) comme indique. Le fait que la courbe D est de nie sur k implique que les deux droites L1 (0) et L1 (1) (resp. L1 (2) et L1 (3), resp. L2 (0) et L2 (1), resp. L2 (2) et L2 (3)) sont soit de nies sur k , soit de nies sur une extension quadratique de k et conjuguees ; les deux points r0(2) et r1(2) (resp. r2(2) et r3(2) , resp. r0(1) et r1(1) , resp. r2(1) et r3(1) ) ont donc la mme propriete. D'autre part, comme une courbe de bidegre (2; 2) est de genre arithmetique 1, la courbe D a un point singulier et un seul, et donc ce point singulier est un point de P1;k  P1;k rationnel sur k. En n, on suppose que la courbe k-rationnelle D a des points lisses rationnels en dehors de l'ensemble P (ce qui est toujours le cas si le cardinal de k est plus grand ou egal a 11) et on xe un tel point. Pour i = 1; 2, on choisit alors les coordonnees (ui;0 : ui;1 ) de sorte que les points de P1;k  P1;k de coordonnees ((0 : 1); (0 : 1)) et ((1 : 0); (1 : 0)) sont respectivement le point rationnel lisse que l'on a xe et le point singulier de D. Prenons ui = ui;1 =ui;0 comme coordonnee dans l'ouvert aÆne Ui de ni par ui;0 6= 0. L'ouvert aÆne i 1 (Ui ) de Xi a un modele plan d'equation vi2 = Pi (ui ) ou Pi est un polyn^ome de degre 4 dont les racines sont les coordonnees des quatre points r0(i) ; : : : ; r3(i) de Ui . Comme les deux points r0(i) et r1(i) (resp. r2(i) et r3(i) ) sont soit de nis sur k , soit de nis sur une extension quadratique de k et conjugues, on a Pi (ui ) = ei (u2i + ai ui + bi )(u2i + ci ui + di ) ou ai , bi , ci , di et ei sont dans k ; on peut noter que bi et di sont non nuls puisque le point singulier de D est le point de coordonnees (0; 0) dans U1  U2 et que ce point singulier n'est sur aucune des Li (n) par hypothese. La courbe D appartient au systeme lineaire des courbes de P1;k  P1;k de bidegre (2; 2) passant par les huit points de P0 . Ce systeme lineaire est de dimension 1 puisque le systeme lineaire de toutes les courbes de bidegre (2; 2) 12

est de dimension 9 ; il est forme des courbes D(:) dont trace dans U1  U2 a pour equation ((a2 c2 )u21 u22 + ((a1 a2 c1 c2 ) + (c1 a1 ))u1 u22 + (a2 c2 )u21 u2 + ((a2 b1 d1 c2 ) + (d1 b1 ))u22 + ((a2 d2 b2 c2 ) + (b2 d2 ))u21 + (a2 c2 (a1 c1 ) + (a2 c1 a1 c2 ))u1 u2 + (a2 c2 (b1 d1 ) + (d1 a2 b1 c2 ))u2 + ((d2 a2 a1 c2 b2 c1 ) + (b2 c1 d2 a1 ))u1 + (b1 a2 d2 b2c2 d1 ) + (d1 b2 b1 d2 ) = 0.

Conformement a l'habitude, lorsque nous travaillons dans U1  U2 nous notons (1; 1) le point de P1;k  P1;k de coordonnees ((0 : 1); (0 : 1)). La seule courbe de cette famille qui peut tre geometriquement integre et passer par le point de coordonnees (u1 ; u2 ) = (0; 0) avec multiplicite 2 et par le point (1; 1) avec multiplicite 1, est la courbe D = D(0:1) . Cette courbe passe par (0; 0) et (1; 1) avec les multiplicites voulues si et seulement si d1 a2 b1 c2 = 0, c1 b2 d2 a1 = 0 et d1 b2 b1 d2 = 0 ; elle est geometriquement integre si et seulement si d2 c21 d1 c22 6= 0 comme on le voit sur l'eqution de sa trace dans U1  U2 qui est : c2 u21 u2 c1 u22 u1 + d2 u21 d1 u22 = 0:

p

En remontant cette courbe D, on trouve que les deux composantes de  1 (D) sont de nies sur l'extension quadratique k ( e1 =e2 ) et un calcul facile donne :

Proposition 2.2.1 Pour i = 1; 2 on designe par Xi la completee de la courbe aÆne plane lisse d'equation vi2 = ei (u2i + ai ui + bi )(u2i + ci ui + di ) ou ai , bi , ci , di et ei sont dans k, et ou ei , bi et di sont non nuls ; on munit Xi de l'involution i qui envoie le point de coordonnees (ui ; vi ) sur le point de coordonnees (ui ; vi ). On suppose que d1 a2 b1 c2 = 0, c1 b2 d2 a1 = 0, d1 b2 b1 d2 = 0, que d2 c21 d1 c22 6= 0, et que e1 =e2 = 2 pour un  2 k. Alors la surface de Kummer X contient deux courbes k-rationnelles C1 et C2 dont les images dans P1;k  P1;k sont de bidegre (2; 2) et rencontrent les Li (n) comme indique sur la gure. Dans l'ouvert aÆne de X lisse d'equation e2 (x22 +a2 x2 +b2)(x22 +c2 x2 +d2 )y2 = e1 (x21 + a1 x1 + b1 )(x21 + c1 x1 + d1 ) dans A3k , les courbes k-rationnelle C1 et C2 admettent respectivement les parametrisations z ! (x1 = 1 (z ); x2 = 2 (z ); y = 3 (z )) et z ! (x1 = 1 (z ); x2 = 2 (z ); y = 3 (z )) ou 2 1 (z ) = dc22zz cd11 2 2 (z ) = zd(2cz2 z dc11) 3 (z ) = z 2 . 2.3

Exemple 3.

L'exemple de Mestre. Deux k -courbes elliptiques E1 et E2 etant xees, J.-F. Mestre ([Mes]) montre qu'on peut toujours construire un k -revtement C de degre 2 de P1;k (c'est a dire une courbe k -hyperelliptique) et deux k -morphismes '1 et '2 de C vers E1 et E2 qui veri ent '1 Æ C = '1 et '2 Æ C = '2 ou C est l'involution hyperelliptique de C , et qui sont independants. La construction du triplet (C ; '1 ; '2 ) 13

est essentiellement equivalente a la construction de la courbe k -rationnelle de la surface de Kummer X = E1  E2 =fidg qui est l'image de C par le compose de ('1 ; '2 ) : C ! E1  E2 avec la projection canonique de E1  E2 sur X . On retrouve le resultat de Mestre en prenant (X1 ; 1 ) = (E1 ; id), (X2 ; 2 ) = (E2 ; id), et en relevant une courbe D de P1;k  P1;k de bidegre (3; 3) dont les intersections avec les Li (n) sont decrites par le diagramme suivant :

L2 (1) L1 (3)

r1;3 r2;3 r3;3

L1 (2)

r1;2 r2;2 r3;2

L1 (1)

r1;1 r2;1 r3;1

P ;k  fr g = L1 (0) 1

(2) 0

L2 (3)

r0;0

@

fr0(1) g  P1;k =L2 (0)

L2 (2)

ou, comme plus haut, les obliques representent les intersections de la courbe D avec les Li (n). Autrement dit, la courbe D de bidegre (3; 3) passe par les neuf points r1;1 ; r1;2 ; r1;3 ; r2;1 ; r2;2 ; r2;3 ; r3;1 ; r3;2 ; et r3;3 avec multiplicite 1 et par le point r0;0 avec multiplicite 2 ; notons que cela implique que le troisieme point d'intersection de D avec L1 (0) (resp. L2 (0)) n'est ni r1;0 , ni r2;0 , ni r3;0 (resp. ni r0;1 , ni r0;2 , ni r0;3 ). Comme dans le travail de Mestre nous obtenons comme cela le resultat cherche generiquement ; les cas particuliers restants se traitent apres coup. On veri e (en reprenant les arguments explicites dans les exemples precedents) que rD = 2 donc, si une telle courbe existe, son image reciproque  1 (D) est une k -courbe geometriquement integre de genre geometrique 0 de X . Notons p1;0 et p2;0 les points d'intersection de D avec L1(0) et L2(0) distincts de r0;0 ; comme dans l'exemple 1, ces points sont des points de D qui sont lisses et rationnels sur k , et au dessus desquels  est rami e ; leurs images reciproques sont donc deux points rationnels de  1 (D), et donc  1 (D) est une courbe k-rationnelle. Le genre arithmetique d'une courbe de bidegre (3; 3) etant egal a 4, il faut imposer d'autres singularites que le point double en r0;0 pour assurer que le genre geometrique de D est 0. Pour retrouver les equations de [Mes], nous imposons dans la suite que D poss ede un point triple ; cela implique que le genre arithmetique est 0, donc aussi que le genre geometrique est 0 (mais ce n'est pas la seule maniere, par exemple imposer trois points doubles ordinaires en plus de r0;0 marcherait aussi). Notons qu'alors le point double en r0;0 et le point triple sont les seuls points singuliers de D, donc le point triple est un point de P1;k  P1;k qui est rationnel sur k ; de plus il n'est sur aucune des Li (n) comme on le voit sur la gure representant l'intersection de D avec les Li (n). Pour prouver l'existence d'une courbe k -rationnelle D rencontrant les Li (n) comme sur la gure,nous choisissons des coordonnees adaptes a la situation. 14

Pour i = 1; 2, on choisit les coordonnees (ui;0 : ui:1 ) sur la droite projective de sorte que r0(i) est le point a l'in ni i.e. le point de coordonnees (0 : 1), et que la ieme projection du point triple de D est le point zero de coordonnees (1 : 0). On prend ui = ui;1 =ui;0 comme coordonnee sur l'ouvert Ui de ni par ui;0 6= 0, et on note Pi (ui ) le polynme unitaire de degre 3 a coeÆcients dans k dont les racines sont les coordonnees des trois points r1(i) , r2(i) et r3(i) dans Ui . Ainsi vi2 = Pi (ui ) est un modele de Weierstrass de Ei ; notons que Pi (0) 6= 0 puisque le point triple de D est le point de U1  U2 de coordonnees (0; 0) et n'est sur aucune des Li (n). Une courbe D qui rencontre les Li (n) comme indique sur la gure, appartient au systeme lineeaire des courbes de P1;k  P1;k de bidegre (3; 3) qui passent par les neufs points rn;m pour n; m = 1; 2; 3 avec multiplicite 1 et par le point r0;0 avec multiplicite 2. Ce systeme lineaire est de dimension 3 puisque le systeme lineaire de toutes les courbes de bidegre (3; 3) est de dimension 14 ; il est forme des courbes D(0 :1 :2 :3 ) dont la trace dans l'ouvert U1  U2 de P1;k  P1;k a pour equation

P1 (u1 )(0 u2 + 1 ) + P2 (u2 )(2 u1 + 3 ) = 0: Pour i = 1; 2 posons Pi (ui ) = u3i + ai u2i + bi ui + ci de sorte que ai , bi et ci sont des elements de k avec ci 6= 0 puisque Pi (0) 6= 0. L'equation de la courbe D(0 :1 :2 :3 ) est (u31 +a1 u21 +b1 u1 +c1 )(0 u2 +1 )+(u32 +a2 u22 +b2 u2 +c2 )(2 u1 + 3 ) = 0 ; le point (u1 ; u2 ) = (0; 0) est un point de multiplicite au moins egale a trois si et seulement si on a a1 1 = 0; b2 3 = 0; c1 1 + c2 3 = 0; c1 0 + b2 3 = 0; b1 1 + c2 2 = 0; et b1 0 + b2 2 = 0 ; comme a1 c2 6= 0; on a necessairement a1 = a2 = 0 et (0 : 1 : 2 : 3 ) = (b2 : c2 : b1 : c1 ). L'equation de la trace de D(b2 :c2 : b1 : c1 ) sur U1  U2 est b2 u31 u2 b1 u1 u32 + c2 u31 c1 u32 . Le point r0;0 = (1; 1) est un point double sur cette courbe si l'un au moins des coeÆcients b1 ou b2 est non nul. En n, si b1 ou b2 est non nul, on voit en tenant compte du fait que c1 et c2 sont non nuls, que D(b2 :c2 : b1 : c1 ) est geometriquement integre si et seulement si b31 c22 b32 c21 6= 0. Sous ces hypotheses, on calcule et on parametrise facilement l'image reciproque de D(b2 :c2 : b1 : c1 ) dans X et on trouve :

Proposition 2.3.1 Soient E1 et E2 deux k-courbes elliptiques qui admettent des modeles de Weierstrass v12 = u31 + b1 u1 + c1 et v22 = u32 + b2 u2 + c2 . On suppose c1 et c2 non nuls, b1 ou b2 non nul, et b31 c22 b32c21 6= 0. Alors la surface de Kummer X = E1  E2 =(id) contient une courbe k-rationnelle C dont l'image dans P1;k  P1;k est de bidegre (3; 3) et rencontre les Li (n) comme indiquee sur la gure. Dans l'ouvert aÆne de X lisse d'equation (x32 + b2x2 + c2 )y2 = (x31 + b1 x1 + 6 c1 ) dans A3k la courbe C admet la parametrisation z ! (x1 = bc12 z b2 zc41 ; x2 = c2 z6 c1 3 z2 (b1 b2 z4 ) ; y = z ). On reconnait bien dans cette proposition les formules de Mestre. Les cas exclus dans la proposition precedente se traitent par une etude cas par cas. 15

Traitons par exemple le cas suivant : on conserve les hypotheses c1 et c2 non nuls, b1 ou b2 non nul, mais on suppose b31 c22 b32 c21 = 0. On a alors b1 et b2 non nuls et on pose  = b1 c2 =b2 c1 de sorte que b1 = 2 b2 et c2 = 3 c1 (l'hypothese est donc p que les deux courbes elliptiques E1 et E2 sont isomorphes sur le corps k ( ) et que leurs invariants modulaires sont dierents de 0 et 1728). L'equation de la trace de D(b2 :c2 : b1 : c1 ) dans U1  U2 devient (u1 u2 )(b1 u21 u2 + b1 u1 u22 + c1 2 u21 + c1 u1 u2 + c1 u22 ) ; ainsi D(b2 :c2 : b1 : c1 ) est la reunion d'une courbe de bidegre (1; 1) et d'une courbe de bidegre (2; 2) ; on veri e que l'image reciproque de la courbe de bidegre (2; 2) est une courbe k -rationnelle sur X qui est non triviale, ce qui regle ce cas. Dans l'ouvert aÆne de X lisse d'equation (x32 + 2 b1 x2 + 3 c1 )y2 = (x31 + b1 x1 + c1 ) dans A3k cette courbe k-rationnelle admet 2 4 z +z2 +1) ; x = c1 (2 z4 +z2 +1) ; y = z 2 ). la parametrisation z ! (x1 = c1 (b1 (z 2 +1) 2 b1 z2 (z2 +1) Notons que l'image reciproque de la courbe de bidegre (1; 1) se calcule comme on l'a fait dans l'exemple 1 ; elle se compose de deux composantes k -rationnelles p conjuguees sur k ( ) et donc ne permet de construire une courbe k -rationnelle sur X que dans le cas particulier ou  est un carre dans k , i.e. lorsque E1 et E2 sont isomorphes sur k . Les cas restants sont essentiellement ceux pour lesquels les invariants modulaires des courbes E1 et E2 sont 0 ou 1728. Les choix de coordonnees que nous avons fait ne sont alors pas adaptes a la situation (essentiellement on ne peut plus demander dans ces cas que le point triple soit le point de coordonnees ((0 : 1); (0 : 1) dans P1;k  P1;k ). Les changements a faire pour traiter ces cas ne posent pas de diÆcultes. 2.4

Exemple 4.

Courbes k-rationnelles associees aux courbes de genre 2. 0n suppose encore que (X1 ; 1 ) = (E1 ; id) et (X2 ; 2 ) = (E2 ; id) o E1 et E2 sont deux k-courbes elliptiques. On xe un entier naturel n. Si l'entier n est impair et strictement plus grand que 1, on considere les courbes k -rationnelles D de P1;k  P1;k de bidegre (n; n) qui rencontrent les Li (n) de la maniere suivante : pour n = 1; 2; 3 (resp. m = 1; 2; 3) la courbe D rencontre la droite L1 (n) (resp. L2 (m)) en le point r0;n (resp. rm;0 ) avec multiplicite 1, et en (n 1)=2 autres points qui sont des points lisses de D, qui ne sont situes sur aucune des L2 (m) (resp. L1 (n)), avec multiplicite 2 ; la courbe D rencontre L1 (0) (resp. L2 (0)) en les trois points r1;0 , r2;0 et r3;0 (resp. r0;1 , r0;2 et r0;3 ) avec multiplicite 1 , et en (n 3)=2 autres points qui sont des points lisses de D, qui ne sont situes sur aucune des L2 (m) (resp. L1 (n)), avec multiplicite 2. Par exemple, pour n = 3, cette situation se represente par la gure suivante :

16

L2 (1)

  L1 (3)

r0;3

L1 (2)

r0;2

L1 (1)

r0;1

P ;k  fr g =L1 (0) 1

(2) 0

fr0(1) g  P1;k =L2 (0)

L2 (3)

   



 

@r ; @r ; @r ; 1 0

2 0

3 0

L2 (2)

o les intersections de D avec les Li (n) sont representees par des obliques lorsqu'elles sont de multiplicite 1 et par des ovales lorsqu'elles sont de multiplicite 2. Une branche d'une courbe D qui veri e ces conditions rencontre soit deux des droites Li (n) avec multiplicite 1, soit une des Li (n) avec multiplicite 2 de sorte que rD = 0. L'image reciproque  1 (D) de D sur X est donc formee de deux composantes k -rationnelles. Supposons trouvee une telle courbe D, et notons ' une parametrisation de D, i.e. un k-morphisme ' : P1;k ! P1;k  P1;k dont l'image est D ; notons en n C1 et C2 les deux composantes geometriques de  1 (D). Choisissons des coordonnees (z0 : z1 ) sur la droite projective source de ', posons z = z1 =z0 , et notons z1;0 , z2;0 , z3;0 , z0;1 , z2;0 et z0;3 les paramtres des six points r1;0 ; : : : ; r0;3 , c'est a dire les six valeurs telles que '(z1;0 ) = r1;0 ; : : : ; '(z0;3 ) = r0;3 . Un calcul facile montre que, pour i = 1; 2, l'image reciproque Ci =  1 (Ci ) de Ci par le morphisme canonique  : E1  E2 ! X est une courbe de genre 2 sur E1  E2 qui est k birationnellement equivalentes au revtement de degre 2 de P1;k rami e en les six points z1;0 , z2;0 , z3;0 , z0;1 , z2;0 et z0;3 ; si C1 et C2 sont k -rationnelles, les courbes C1 et C2 sont de nies sur k et sont k -birationnellement equivalentes au revtement de degre 2 de P1;k rami e en ces six points. Le k -isomorphisme ( idE1 )  idE2 de E1  E2 echange ces deux courbes. Si l'entier n est pair, on considere les courbes k -rationnelles D de P1;k  P1;k de bidegre (n; n) qui rencontrent les Li (n) de la maniere suivante : la courbe D rencontre L1 (0) (resp. L2 (0)) en n=2 points qui sont des points lisses de D, qui ne sont situes sur aucune des L2 (m) (resp. L1 (n)), avec multiplicite 2 ; les trois points r1;1 , r2;2 et r3;3 sont trois points doubles ordinaires de D et chacune des six branches de D centrees en l'un de ces trois points rencontre les Li (n) passant par ce point avec multiplicite 1 ; en n, pour m = 1; 2; 3 (resp. n = 1; 2; 3) la courbe D rencontre la droite L2 (m) (resp. L1 (n)) en (n 2)=2 points qui sont des points lisses de D, qui ne sont situes sur aucune des L1 (n) (resp. L2 (m)), avec multiplicite 2, et en le point double ordinaire rm;m (resp. rn;n ) comme on vient de le decrire. Pour n = 4, cette situation se represente par la gure suivante : 17

L2 (1)

L1 (3) L1 (2) L1 (1)

P ;k  fr g =L1 (0) 1

(2) 0

L2 (3)

         r ;  @ r; @  r;  @    3 3

2 2

1 1

fr0(1) g  P1;k =L2 (0)

L2 (2)

o, comme plus haut, les obliques et les ovales representent respectivement les branches de D qui coupent les Li (n) avec multiplicites 1 et 2 ; on veri e immediatement que, dans ce cas encore, on a rD = 0. Supposons trouvee une telle courbe D, et notons ' une parametrisation de D, i.e. un k -morphisme ' : P1;k ! P1;k  P1;k dont l'image est D ; notons en n C1 et C2 les deux composantes geometriques de  1 (D). Choisissons des coordonnees (z0 : z1 ) sur la droite projective source de ', posons z = z1 =z0 et, pour n = 1; 2; 3, notons zn;1, zn;2 les deux paramtres du point double ordinaire rn;n de D, c'est a dire les deux valeurs telles que '(zn;1 ) = '(zn;2 ) = rn;n . Un calcul facile montre que, pour i = 1; 2, l'image reciproque Ci =  1 (Ci ) de Ci par le morphisme canonique  : E1  E2 ! X est une courbe de genre 2 sur E1  E2 qui est k-birationnellement equivalente au revtement de degre 2 de P1;k rami e en les six points z1;1 , z1;2 , z2;1 , z2;2 , z3;1 et z3;2 ; si C1 et C2 sont k -rationnelles, les courbes C1 et C2 sont de nies sur k et sont k -birationnellement equivalentes au revtement de degre 2 de P1;k rami e en ces six points. Le k -isomorphisme ( idE1 )  idE2 de E1  E2 echange ces deux courbes. Chaque fois que l'on trouve un courbe k -rationnelle D dans P1;k  P1;k qui veri e les conditions que l'on vient de decrire, on note C l'une des deux courbes C1 ou C2. On veri e immediatement que les deux projections q1 et q2 du produit E1  E2 sur E1 et E2 induisent deux morphismes de degre n de C sur E1 et E2 . On montre dans [Sat] que toute courbe k-hyperelliptique de genre 2 dont la jacobienne est k -isogene a un produit de k -courbes elliptiques est obtenue de cette maniere avec un bon choix de n. Comme on l'a rappele dans l'introduction, lorsque le corps k est le corps de complexes, la recherche des courbes hyperelliptiques dont la jacobienne contient des courbes elliptiques est le probleme classique de la recherche d'integrales hyperelliptiques qui se reduisent a des integrales elliptiques. Dans le cas des integrales hyperelliptiques de genre 2 qui nous interessent dans cet exemple, c'est un execice amusant de retrouver les formules classiques que l'on trouve par exemple dans [Bol], dans le chapitre XI de [Kra], ou encore dans [Pic] en construisant les courbes D correspondantes. Terminons ce travail en revenant 18

au cas ou le corps de base est quelconque, et en signalant deux travaux qui, au point de vue pres, traitent des cas particuliers de la situation que l'on decrit ici. Tout d'abord, les calculs du x6 de [Kuh] s'interprtent immediatement comme l'explicitation de la parametrisation de ' dans le cas n = 3. D'autre part, notons E1 [n] et E2 [n] les Gal(k=k )-modules galoisiens des points de E1 et E2 qui sont tues par n. On sait ([F-K] ou [Sat] par exemple) que les courbes k hyperelliptiques de genre 2 tracees sur E1  E2 dont l'image dans P1;k  P1;k est une courbe k -rationnelle de type (n; n) qui coupent les Li (n) comme indique plus haut correspondent aux Gal(k=k )-isomorphismes de E1 [n] sur E2 [n] qui sont anti-symplectique (pour le pairing de Weil). Ainsi, les calculs de [R-S] donnent des exemples de la situation decrite ici avec n = 3 et n = 5. Notons que les calculs presentes dans les deux travaux que nous venons de citer necessitent l'emploi d'un logiciel de calcul formel. Nous avons recalcule ces exemples sans diÆculte dans le cadre expose ici en utilisant Maple. .

References [B-M]

Batyrev V.-V., Manin Yu. I., Sur le nombre de points rationnels de hauteur borne des varietes algebriques, Math. Ann., 286, (1990), 27-43.

[Bol]

Bolza 0., Ueber die Reduction hyperelliptischer Integrale erster ord-

[F-K]

Frey G., Kani E., Curves of genus 2 covering elliptic curves, Arithmetic Algebraic Geometry, Prog. in Math. 89, Birkhauser, (1991), 153-175.

[Jac]

Jacobi C.G., Anzeige von Legendre, Theorie des fonctions elliptiques, troisieme supplement (1832), Jour. fur die Reine und Angew. Math., Bd 8, reproduit dans les Gesammelte Werke, Bd 1, Berlin (1881).

[Kra]

Krazer A.,Lehrbuch der Thetafunctionen, Teubner, Leipzig (1903) (reedite par Chelsea, New-York (1970)).

[Kuh]

Kuhn R.-M., Curves of genus 2 with split Jacobian, Trans. of the Amer. Math. Society, Vol. 307, (1988), 41-49.

[Leg]

Legendre A.-M., Traite des fonctions elliptiques et des integrales euleriennes, 3ieme supplement, (1832).

[M-S]

Maer N., Satge Ph., Fibrations en courbes de genre 1 sur la surface de Kummer associee au produit de deux courbes de genre 1, prepublications de l'Universite de Caen.

nung und erster Gattung auf elliptische durch eine transformation vierten Grades, Math. Ann., XXVIII, (1886), 447-456.

19

[McK]

McKinnon D., Counting rational points of bounded height on K3 surfaces, a paraitre au J. Number Theory, Vol. 89,(2000).

[Mes]

Mestre, J.-F., Rang de courbes elliptiques d'invariant donne, C.R. Acad. Sci. Paris Ser. I Math., 314, (1992), 919-922.

[Pic]

Picard E., Sur quelques exemples de reduction d'integrales abeliennes aux integrales elliptiques, C.R.,93, Seance du 16 decembre 1881, 1126-11288; Sur la reduction des integrales abeliennes aux integrales elliptiques, C.R.,94, Seance du 26 juin 1882, 1704-1707 ; reproduits dans le volume III des oeuvres completes, Editions du CNRS, (1980).

[R-S]

Rubin K., Silverberg A., Families of elliptic curves with constant mod. p representations, Proceedings of the conference on elliptic curves and modular forms, Hong-Kong 1993, Cambridge University Press (1995).

[Sat]

Satge Ph., Le morphisme complementaire d'un morphisme d'une courbe de genre 2 vers une courbe de genre 1, soumis aux actes des journees rev^etements de St Etienne (Mars 2000).

20

Arithmetic Strati cations and Partial Eisenstein Series

Matthias Strauch Mathematisches Institut der Westfalischen Wilhelms-Universitat Munster (e-mail address: [email protected]) November 1999 Let P nG and QnH be generalized ag varieties over a number eld F . In this paper we study certain locally trivial bre bundles Y over P nG having QnH as general bre, and determine the arithmetic strati cation of Y with respect to a line bundle. The arithmetic strati cation is de ned in terms of height zeta functions and the height zeta function of a stratum is of the form 1 ehs;HP ( )i EQQw Q0 (s;  (p )) ; Abstract.

X

2P (F )nG(F )

where EQQw 1Q0 is a \partial Eisenstein series" associated to the Schubert cell QnQw 1Q0. The computation of the constant term of these gives estimates that allow one to determine the abcissa of convergence of the height zeta function of the stratum. Soient P nG et H nQ des varietes de drapeaux generalisees sur un corps de nombres F . Dans cet article nous etudions certaines bres localement triviales Y sur P nG ayant QnH comme bre generale, et nous determinons la strati cation arithmetique de Y relative a un faisceau inversible. La strati cation arithmetique est de nie par des fonctions zeta associees a un faisceau inversible metrise et la fonction zeta d'une strate prend la forme 1 ehs;HP ( )i EQQw Q0 (s;  (p )) ; Resume.

X

1 EQQw Q0

2P (F )nG(F )

etant la serie d'Eisenstein partielle associee a la cellule de Schubert QnQw 1Q0. Le calcul du terme constant de celles-ci fournit des estimations permettent de determiner l'abcisse de convergence de la fonction zeta d'une strate. 1

Contents

Introduction 1 The bre bundles: geometric-arithmetic preliminaries 2 Height zeta functions 3 Arithmetic strati cation References

2 4 9 18 23

Introduction

Consider a projective variety Y over a number eld F and a metrized line bundle L = (L; (k kv )v ) on Y . For any subvariety U  Y there is a counting function NU (L; H ) = #fy 2 U (F ) j HL(y )  H g : Suppose NU (L; H ) is nite for all H . Then one would like to understand the asymptotic behaviour of the counting function NU (L; H ) as H tends to in nity. A useful tool when studying this question is the (formal) Dirichlet series 1 ; ZU (L; s) = H (y )s y2U (F ) L called a height zeta function. Whenever this series converges for all s  0 let U (L) be its abcissa of convergence: U (L) = inf fs 2 R j ZU (L; s) convergesg : U (L) depends only on the line bundle L underlying L and not on the metric. U (L) = 1 if and only if U (F ) is a nite; otherwise log NU (L; H ) : U (L) = limsup log H Sometimes one can go much further: namely, establish analytic continuation of the height zeta function. Then one even gets an asymptotic formula for the counting function, using a Tauberian Theorem (provided an assumption on the poles is satis ed). But before going into these questions, one encounters some features that should be considered beforehand. A general phenomenon which may occur is this: there is a proper subvariety Y 0 in Y , such that Y (L) = Y 0 (L) > Y Y 0 (L). Such a subvariety is called accumulating. In

X

2

this case, counting rational points on Y 0 is asymptotically the same as counting rational points on Y , and therefore one should study Y 0 and Y Y 0 separately. It may happen that Y Y 0 contains again an accumulating subvariety. This leads us to the notion of the arithmetic strati cation of Y with respect to L, introduced by V.V.Batyrev and Yu.I.Manin in [BM]. Suppose that Y (F ) is dense in Y and that there is an open subset Y 0 such that ZY 0 (F )(L; s) converges for all s  0. The arithmetic strati cation of Y with respect to L is the descending sequence of Zariski open subvarieties Y  Y0  Y1  Y2 : : : where Y0 is the maximal Zariski open subset of Y for which the height zeta function ZY0 (L; s) converges for all s  0, and Yi+1  Yi is the maximal open subset of Yi such that Yi+1 (L) is strictly smaller than Yi (L). Note that the arithmetic strati cation depends on the line bundle L. As an example, consider the variety obtained by blowing up the projective plane over F in a rational point. The projective line replacing this point will be an accumulating subvariety with respect to the anticanonical line bundle. For a further discussion cf. the introduction of [FMT] and [BM]. In this paper we study a class of smooth projective varieties, give a description of their Picard groups, and determine the arithmetic strati cation with respect to an arbitrary line bundle. Here is a brief description of the varieties and the methods employed. The varieties are locally trivial bre bundles Y ! W = P nG over a generalized ag variety P nG. The bres are isomorphic to another ag variety QnH , and Y is constructed via a homomorphism  : P ! H which factors through a split torus. The Picard group is up to a nite cokernel given by the direct sum of character groups X (Q)  X (P ). In P ic(Y )R there is the closed cone (Y )e generated by the line bundles having non-zero global sections, and we denote the interior of this cone by (Y )Æe. From now on we assume that  satis es assumption 1.4 below. Let L be a line bundle on Y corresponding to the pair (; ) 2 X (Q)  X (P ). Suppose L lies in (Y )Æe , and consider the arithmetic strati cation Y  Y0  Y1  Y2! : : : of Y with respect to L. It turns out that the strata Yi Yi+1 are always unions of twisted products Y w = X w P G over P nG. X w = QnQw 1 Q0 is the Schubert cell in X corresponding to the Weyl group element w. The height zeta function of Y w (with respect to a certain metric on L) is a double sum 1 ehs;HP ( )i EQQw Q0 (s;  (p )) :

X

Qw 1 Q0

2P (F )nG(F )

is what we call a \partial Eisenstein series", because it is a subseries of a Langlands-Eisenstein series: the summation runs only over Q(F )nQ(F )w 1Q0 (F ). The EQ

3

partial1 Eisenstein series can be bounded from above and from below by its constant term Qw Q0 EQ;Q 0 . The constant term in turn is equal to Qw 1 Q0 ehs(w) (w0 0 );HP ( )i EQ;Q 0 (s; 1H ) : Therefore, replacing the partial Eisenstein series by its constant term, one gets Qw 1 Q0 G EQ;Q 0 (s; 1H )
EP (s( + (w)) (w0 0 ); 1G ) : This suÆces to determine the abcissa of convergence of the height zeta functions of the strata. The arithmetic strati cation with respect to L can be described purely in terms of a nite number of explicitely computable constants involving the parameters ;  and the Weyl group elements. The results in this paper are part of the authors Ph.D. thesis, whose second part concerning asymptotic formulas shall appear in [St]. It is a pleasure to thank Jens Franke for introducing me to the methods used in this paper and for his constant support while working on these questions. 1 The bre bundles: geometric-arithmetic preliminaries (1.1) Let G and H

be two semi-simple linear algebraic groups over the number eld F , and x minimal F -rational parabolic subgroups P0  G and Q0  H with (F -rational) Levi decompositions P0 = L0U0 , Q0 = M0 V0 . The quotients of G, respectively H , by parabolic subgroups P  P0 , Q  Q0 are (by de nition) generalized ag varieties: W = P nG and X = QnH: Consider a homomorphism  : P ! T0 , where T0 is the maximal split torus in the center of M0 . P acts via  on X  G from the right: (x; g)
p = (x(p); p 1g), and there is a canonical morphism  from Y = Y = (X  G)=P to W , giving Y the structure of a locally trivial bre bundle over W with general bre X , cf. [J], 5.14. (1.2) Let  2 X  (Q) = HomF -gps (Q; Gm;F ) be a character of Q and de ne an action of Q on H  A1 by q
(h; a) = (qh; (q ) 1a). The quotient of H  A1 by Q is a line bundle over X = QnH , to be denoted by L : 4

This gives an injection

L

= Qn(H  A1);

X  (Q)

! P ic(X )

with nite cokernel, cf. for instance [Sa], Prop. 6.10, Lemma 6.9. The same applies to X  (P ) and P ic(W ). The action of P on H  A1 , given by (h; a)
p = (h (p); a), descends to an action of P on L , and the quotient LY = (L  G)=P ; P acting on L  G by (l; g )
p = (l
p; p 1g ), is a line bundle over Y . Using the exact sequence (6.10.2) in [Sa], it is not diÆcult to show that (1.2.1) (; ) 7 ! class of LY L is a monomorphism with nite cokernel X  (Q)  X  (P ) ! P ic(Y ) : (1.3) The choice of Q0 determines a basis
H0 of the root system of H with respect to T0 . The roots of T0 occuring in the Lie algebra of the unipotent radical of Q0 are de ned to be positive and
H0 is the basis of simple positive roots with respect to this set of positive

roots. Let

WH = NH (F ) (T0 (F ))=ZH (F )(T0(F ))

be the Weyl group of H (NH (F ) and ZH (F ) denoting the normalizer, respectively centralizer in H (F )), and let (:; :) be a WH -invariant positive de nite form on X (T0)R. De ne the set X  (T0 )+R = fx 2 X  (T0 )R j for all 2
H 0 : (x; ) > 0 g : This is the positive Weyl chamber. Restriction of characters gives an injection and an isomorphism X  (Q)R ,! X  (Q0 )R ~!X  (T0 )R : 5

In the sequel we will always consider X  (Q)R as a subspace of X (Q0 )R which we will identify with X (T0). Let M denote the unique Levi component of Q which contains M0. There is a subset
M0 of
H0 such that X  (Q)R = fx 2 X  (T0 )R j for all 2
M 0 : (x; ) = 0 g : Denote by+X (Q)+R the interior (in X (Q)R) of the intersection of X  (Q)R with the closure of X (T0 )R. Let
HQ be the complement of
M0 in
H0 . Then we have X  (Q)+R = fx 2 X  (Q)R j for all 2
H Q : (x; ) > 0 g : It is well known, that under the isomorphism X  (Q)R ! P ic(X )R the closure of X (Q)+R is mapped onto the cone of eective divisors (X )e , i.e., the closed cone that is generated by those line bundles having non-zero global sections (cf. [J], II, 2.6). The de nitions depend on the once and for +all chosen minimal parabolic subgroup and the xed Levi component M0 . De ne X (P )R by analogy (depending on P0 and L0 ). The following assumption will be used to describe (Y )e and to determine the abcissa of convergence of the height zeta functions. (1.4) Assumption. For all 2
H0 the element Æ  of X  (P ) lies in the closure of X  (P )+R. Geometrically speaking, we consider those  which map the cone in X  (T0)R generated by
H0 into the cone of eective divisors in X (P )R. This subset of HomF -gps(P; T0) generates a sublattice of nite index. Moreover, it is easy to see that if one replaces  by a conjugate ww 1, w being an element of the Weyl group WH , the variety Yww 1 is isomorphic to Y . Sometimes such a conjugate of  satis es (1.4). This is always the case if P is a maximal proper parabolic subgroup of G. Indeed, the group X (P ) is in this case of rank one, and if  is not trivial, the kernel of the induced homomorphism  : X (T0)R ! X (P )R is a hyperplane, consisting of elements which are orthogonal to some  2 X  (T0)R, orthogonal with respect to the W -invariant positive de nite form (:; :) on X (T0 )R. Replacing  by  if necessary, we can assume that the half-space of  2 X  (T0)R with (; ) < 0 is mapped to X (P )+R by . Consider the (closure of the) chamber that contains  and the base
of the root system that corresp! onds to that chamber. For 2
one has Æ  2 X (P )+R, and this shows that a conjugate of  by an element of the Weyl group ful lls (1.4). 6

Identify WH with its image in Aut(X (T0 )R). The re ections along the elements of
H0 generate WH , and de ne the Bruhat ordering on WH and a length function. Let wH be the longest element of W . De ne  = f(; ) 2 X (Q)R  X (P )R j  2 X (Q)+R and (wH ) +  2 X (P )+R g : This is a closed cone in X (Q)R  X (P )R. (1.5) Proposition. Suppose  ful lls assumption 1.4. Then the line bundle LY   L has non-zero global sections if and only if (; ) lies in  . Proof. Suppose H 0 (Y; LY   L ) 6= 0. The pull-back of LY   L by the quotient map X  G ! Y is isomorphic to the pull-back of L by the projection onto the rst factor X  G ! X . It follows that L has a non-zero global section, hence  is an element of X (Q)+R (cf. [J], II, 2.6). Let T0 act on L = Qn(H  A1) by [h; a]t = [ht; a] and on H 0 (X; L ) by (t
s)(x) = s(xt)t 1 . Because T0 is split, H 0 (X; L ) decomposes as a direct sum of one-dimensional representations. For each  2 X (T0 ) let m() be the multiplicity of the corresponding weight space. Then one has ()  (LY   L ) ' Lm Æ+ :

M

2X  (Q)

Hence, there is a  with m() > 0 and () +  2 X (P )+R. This implies (wH ) +  = ( wH  ) () +  2 X  (P )+ R; because wH   2 2
H0 Z0 . Now suppose that (; ) lies in  . Then wH  is a weight of the representation of T0 on H 0(X; L ). Therefore, m( wH ) ()  (LY   L ) ' L Lm Æ+ (wH )Æ+ 

P

M

6= wH 

has a non-zero global section, and so does LY  L. 2 By Proposition 1.5, under the isomorphism X (Q)R  X (P )R ! P ic(Y )R induced by (1.2.1), the cone of eective divisors (Y )e is the image of  , and the interior Æ of  is mapped onto (Y )Æe . 7

Q

(1.6) Next we discuss metrics on the line bundles introduced above. a maximal compact subgroup KG = v KG;v  G(A), such that G(A) = P0 (A)KG ;

To do this, choose

where v runs over all places of F and A denotes the ring of adeles of F . For a place v and w 2 W (Fv ) choose k 2 KG;v which is mapped to w. If [g; a] 2 L (Fv ) is an Fv -valued point of the line bundle L over w, de ne its norm by k[g; a]kw = j(p)ajv ; where p 2 P (Fv ) is determined by g = pk and j
jv denotes the local absolute value that is the Haar multiplier. The family k
kv = (k
kw )w2W (Fv ) is a v-adic metric on L and L = (L ; (k
kv )v ) is a metrization of L , cf. [P], Lemme 6.2.3. De ne a map HP = HP;KG : G(A) ! HomR (X  (P ) R; R) by h; HP (pk)i = log( v j(pv )jv ), for  2 X (P ), p = (pv )v 2 P (A), k 2 KG and extend linearly. An easy computation shows (cf. [FMT], p. 428) that for the (exponential) height function we have (1.6.1) HL (w) = e h;HP ( )i ; whenever 2 G(F ) maps to w 2 W (F ) = P (F )nG(F ). Similarly, we x a maximal compact subgroup KH = v KH;v  H (A) such that H (A) = Q0 (A)KH , and provide all line bundles L ,  2 X  (Q), with the metrics as de ned above. Furthermore, KH is supposed to induce maximal compact subgroups of all the standard Levi subgroups. More precisely, we assume that for each parabolic subgroup Q0  Q0 of H with Levi decomposition Q0 = M 0 V 0 , M 0  M0, one has  Q0(A) \ KH = (M 0 (A) \ KH )(V 0 (A) \ KH ) ;  M 0 (A) \ KH is a maximal compact subgroup of M 0 (A) : By [MW], I.1.4., such KH exist. This implies in particular that KH contains the maximal compact subgroup of T0(A). Finally, we de ne metrics on the line bundles LY = (L  G)=P . Let k
kv be the v-adic metric on L de ned by means of KH;v . Recall the action L  P ! L = Qn(H  A1),

Q

Q

8

given by l
p = l
(p) = [h(p); a] if l = [h; a]. For an element [l; g] 2 L (Fv ) mapping to y = [x; g ] 2 Y (Fv ) de ne k [l; g] ky = k l
(p) kx(p) ; where g = pk, p 2 P (Fv ), k 2 KH;v . It follows from the remark above, that this is wellde ned (because P (Fv ) \ KG;v is mapped into KH;v ). The family k
kv = (k
ky )y2Y (Fv ) is a v-adic metric on LY and LY = (LY ; (k
kv )v ) is a metrization of LY . For 2 G(F ) and Æ 2 H (F ) mapping to x 2 X (F ), the pair (x; ) maps to an element y 2 Y (F ), and (1.6.2) HLY (y ) = e h;HQ (Æ(p))i , with g = pk, p 2 P (A), k 2 KG. 2 Height zeta functions

Denote by s a variable (taking values in R). By (1.6.1) the height zeta function of the generalized ag variety X = QnH corresponding to the metrized line bundle L is equal to (the value of) an Langlands-Eisenstein series, whenever the series converges: ZX (L ; s) = HL (x) s = ehs;HQ (Æ)i = EQH (s; 1H ) :

X

x2X (F )

X

Æ2Q(F )nH (F )

This crucial observation is due to J.Franke. In [FMT] all analytic properties of these Eisenstein series that are needed for applications concerning rational points on X are proved. In this paper we only need results on the domain of convergence, and so we found it appropriate to prove these more accessible statements here again instead of deducing them from much deeper theorems. Hence our rst task is to determine the domain of convergence of these Eisenstein series. Let Q be half the sum of the roots of T0 that occur in the Lie algebra of V (= unipotent radical of Q), counted with multiplicity. For convenience, in the de nition of EQH , we shifted the coordinate on X (Q) R by Q, compared with the usual convention. The following elementary lemma will turn out to be very useful. (2.1) Lemma. Let F  H (A) and C  X  (Q)R be relatively compact subsets. Then there is a constant c > 1 such that for all v 2 F ,  2 C and h 2 H (A) 9

c 1 eh;HQ (h)i  eh;HQ (hv)i  ceh;HQ (h)i :

Put h = qk, hv = q1 k1, q; q1 2 Q(A), k; k1 2 KH . Note that q 1q1 = kvk1 1 2 KH F KH \ Q(A) and the latter is a compact subset of Q(A). Hence there is c > 1 such that for all v1 2 KH F KH and  2 C c 1  eh;HQ (v1 )i  c : Hence we have for any triple (; v; h) 2 C  F  H (A) 1 c 1 eh;HQ (h)i  eh;HQ (q q1 )i eh;HQ (q)i = eh;HQ (hv)i ; 1 eh;HQ (hv)i = eh;HQ (q q1 )i eh;HQ (q)i  ceh;HQ (hv)i : 2 (2.2) The most important tool for us to investigate the Eisenstein series are Siegel domains. To introduce these, consider the homomorphism  = ( ) 2
H0 : T0 ! Gdm;F ; t 7! ( (t)) 2
H0 : It is surjective with nite kernel (d = dim(T )). Embed R>0 into A by mapping x 2 R to the idele (xv )v with archimedean components xv = x and non-archimedean components xv = 1. This way we consider Rd>0 as a subgroup of Gdm;F (A). Let AM0 the connected component of the preimage of Rd>0 under , which contains the trivial element. For all w 2 W we have wAM0 w 1 = AM0 . One has M0 (A) = AM0 M0 (A)1 with M0 (A)1 = ker(jj : M0 (A) ! R>0 ) ; Proof.

) ) := Q j (

where jj((mv v

\

2X  (M0 )

For t0 2 AM0 we de ne AM0 (t0 ) = ft 2 AM0 j for all 2
H 0 : (t) > (t0 ) g : Recall the Levi decomposition Q = MV , M being the unique Levi component containing M0 . For a compact subset  Q0 (A) put SQ; = AM AM0 (t0 )KH ; where AM = AM0 \ Z (M (A)), Z (M (A)) denoting the center of M (A). If is suÆciently big and t0 is suÆciently small (i.e., (t0) is suÆciently small for all 2
H0 ) then the following holds, c.f. [MW], I.2.1: v

 mv )jv .

10

H (A) = Q(F )SQ; :

In this case SQ; is called a Siegel domain. With these preparations we can determine the abcissa of convergence of the Eisenstein series. (2.3) Proposition. a) The series EQH ( + 2Q ; h) =

X

Æ2Q(F )nH (F )

eh+2Q ;HQ (Æh)i ;

 2 X  (Q) C, h 2 H (A), converges uniformly for (0  : R= Proof.

X

2
H 0

Attached to  there is an ample line bundle L on Q nH . Hence, there exists c > 0 such that for all Æ 2 H (F ) eh ;HQ (Æ)i = HL (Q (F )Æ ) 1  c : Now let U  H (A) be an open and relatively compact subset. By Lemma (2.1) there is a constant c > 0 such that for all 2
HQ , Æ 2 H (F ) and h 2 U eh ;HQ0 (Æh)i  c : Therefore, H (F )U is contained in V0(A)M0(A)1AM0 (t0 )KH for some t0 2 AM0 , where AM0 (t0 ) = ft 2 AM0 j for all 2
H 0 :  (t) <  (t0 )g : Next we use the fact that H (A) = Q(F )SQ; with some Siegel domain SQ; . By the de nition of SQ; we see that there is a fundamental domain for the action of Q(F ) on H (F )U which is contained in

0 (AM \ AM0 (t0 ))KH 11

for some compact subset 0  Q0 (A) which contains . By Lemma 2.1, for h in a compact subset of H (A), we can bound EQH ( + 2Q; h) by the integral

Z

Q(F )nH (F )U

eh+2Q ;HQ(h1 )i dh1 ;

with some Haar measure dh1 on H (A). Now, dh1 = eh 2Q ;HQ(m)i dvdmdk with Haar measures dv on V (A), dm on M (A), dk on KH (recall Q = MV and H (A) = Q(A)KH ). Therefore, the integral above can be bounded by

Z

AM \AM0 (t0 )

eh;HQ (a)i da ;

with some Haar measure da on AM . Writing  = that this integral is equal to

P

s  2
H Q

(constant depending on t0 and da)


Y1

2


H Q

with s 2 R>0, we see

s

:

Hence we have shown that EQH ( + 2Q; h) converges uniformly for (; h) in compact +  subsets of X (Q)R  H (A). b) To prove the second assertion, it is enough +to show that EQH ( + 2Q; 1H ) tends to in nity as  approaches the boundary of X (Q)R. Let Q0  Q be a parabolic subgroup with Levi component M 0  M such that Q \ M 0 is a maximal proper parabolic subgroup of M 0 . Decompose X (Q)R as follows: X  (Q)R = X  (Q0 )R  X  (TM =TM 0 )R ; where TM (resp. TM 0 ) is the maximal split torus in the 0 center of M (resp. M 0 ). With 0 M M respect to this decomposition we have 2Q = 2Q0 + 2Q\M 0 , where 2Q\M 0 is the sum of roots of T0 (counted with multiplicity) which occur in the Lie algebra of the unipotent radical+of Q \ M 0 . X (TM =TM 0 )R is a one dimensional R-vector space and the image0 of X  (Q)R under the projection onto this space is the open half-line generated by M Q\M 0 . + +   0 Now let  2 X (Q)R tend to the boundary of X (Q)R. Call  the image of  under the projection onto X (TM =TM 0 )R, and assume that 0 tends to zero. Next we note that 0 0 0 EQH ( + 2Q ; 1H ) = eh  +2Q0 ;HQ0 (Æ)i EQQ (0 + 2M Q\M 0 ; Æ ) :

X

Æ2Q0 (F )nH (F )

12

Hence it suÆces to show that EQM\0M 0 (0 +2MQ\0 M 0 ; 1H ) tends to in nity as 0 tends to zero. In other words, it suÆces to consider the case when Q is a maximal proper parabolic subgroup. Let Q be the opposite parabolic subgroup with Levi component V . The map from H to QnH maps V isomorphically onto an open subset of QnH . On V (A) the function h 7! eh2Q ;HQ(h)i is bounded from above but assumes every suÆciently small positive value. Let F  V (A) be an open and relatively compact subset such that V (F )F = V (A). Then, with the notations from part a) of the proof, there is some open and relatively compact subset !  Q0 (A) such that a fundamental domain for the action of Q(F ) on H (F )F KH contains ! (AM \ AM0 (t0 ))KH : Here we used the fact that AM is one-dimensional. By Lemma 2.1, EQH ( + 2Q; 1H ) is bounded from below by the integral

Z

Q(F )nH (F )F KH

eh+2Q ;HQ (h)i dh ;

Z

with some Haar measure dh on H (A), and this in turn can be bounded from below by AM \AM0 (t0 )

eh;HQ (a)i da ;

with some appropriate Haar measure da on AM . This integral tends to in nity as  tends to zero, and this completes the proof. 2 (2.4) Of course, the Proposition 2.3 applies equally to the Eisenstein series EPG (; G) = eh;HP ( g)i :

X

2P (F )nG(F )

For  2 X (Q)+R denote by a the abcissa of convergence of EQH (s; h). By Prop. (2.3) we have a = inffa j a 2 2Q + X  (Q)+ R g: For  2= X (Q)+R we put a = 1. The height zeta function of the variety Y with respect to the metrized line bundle LY L has, by (1.6.1) and (1.6.2), the following formal expression: 13

P

y2Y (F ) HLY  L

(y)

s

P =P =

2P (F )nG(F ) e

2P (F )nG(F ) e

hs;HP ( )i

P

Æ2Q(F )nH (F ) e

hs;HQ (Æ(p ))i

hs;HP ( )i E H (s;  (p )) Q

where = p k with p 2 P (A), k 2 KG. Even if LY L lies in the interior of the cone of eective divisors (Y )e, this series will in general not converge for any s. The reason for this is the occurrence of accumulating subvarieties. A posteriori it turns out that the right approach to nd these is to decompose the bre X into locally closed subvarieties, the images of the Bruhat cells: X= Xw ;

a

w2W M

where W M  W consists of the elements of minimal length in the classes wWM , w running through W , and WM is the Weyl group of M with respect to T0 . X w is the image of the generalized BruhatM cell Qw 1Q0  H . Denote by wHM the element of minimal length in wH WM . Then X wH is the open stratum, which we denote by X Æ . The partial Eisenstein series are de ned by summation over the rational points of X w : 1 EQQw Q0 (; h) = eh;HQ (Æh)i :

X

Æ2Q(F )nQ(F )w 1 Q0 (F )

The behaviour of the partial Eisenstein series on Siegel domains is of importance for us. To understand how a partial Eisenstein series decreases when one goes to in nity on such a Siegel domain, we will study its constant term. The constant term Q0 of a function  : H (A) ! C, which is left-invariant under V0(F ) and lies in L1;loc (H (A)), is de ned by

R

Z ( )=

Q0 h

V0 (F )nV0 (A)

(vh)dv ;

where the measure dv on V0(F )nV0(A) is the quotient of the Haar measures and is normalized by V0 (F )nV0 (A) dv = 1. The following statement is the key to determine the abcissa of convergence of the height zeta functions of the strata. We put 0 = Q0 . (2.5) Proposition. Let F  H (A) and C  2Q + X  (Q)+R be relatively compact subsets (C relatively compact in X  (Q)R ), and x t0 2 AM0 . 14

a) There is a constant c > 1, such that for all  2 C , w 2 W M , t 2 AM0 (t0 ) and v 2 F Qw 1 Q0 hw( 0 )+0 ;HQ0 (t)i E Qw 1 Q0 (; tv )  cE Qw 1Q0 (; 1 ) : c 1 EQ;Q H Q Q;Q0 0 (; 1H )  e

b) Now let w = wHM . Then there is a constant c > 1, such that for all  2 2Q + X  (Q)+ R EQH (s; 1H )  cEQQw

1 Q0

(s; 1H ) :

a) Let V0 be the unipotent radical of Q0 , and let F0  V0 (A) be a compact subset such that V0 (F )F0 = V0(A). Because ft 1 vt j t 2 AM0 (t0 ); v 2 F0g is relatively compact, Lemma 2.1 ensures the existence of a c > 1 such that for all h 2 H (A), t 2 AM0 , v 2 F , v0 2 F0 one has 1 1 c 1 eh;HQ (ht(t v0 t))i  eh;HQ (htv)i  ceh;HQ (ht(t v0 t))i : Therefore Proof.

c 1

Z

V0 (F )nV0 (A)

1 EQQw Q0

(; v0t)dv0 

1 EQQw Q0

Moreover,

R

Qw 1 Q0 V0 (F )nV0 (A) EQ

(; v0t)dv0 = =

R

(; tv

Z ) c

V0 (F )nV0 (A)

E Qw

1 Q0

(; v0t)dv0 :

h;HQ0 (w 1 v0 t))i dv0

(V0 \wV0 w 1 )(A)nV0 (A) e

ehw+0

w0 ;HQ (t)i E Qw 1 Q0 Q;Q0

(; 1H ) :

And this proves assertion a). b) Because H (F ) is dense in H and Qw 1Q0 is open (and dense) in H , there are elements Æ1 ; : : : ; Æn 2 H (F ) such that H (F ) = [ni=1 Q(F )w 1Q0 (F )Æi : Therefore, 1 (2.5.1) EQH (; 1H )  ni=1 EQQw Q0 (; Æi) . By Lemma 2.1, there exists c1 > 1 such that for all  2 C , Æ 2 H (F ) and i = 1; : : : ; n

P

15

eh;HQ (ÆÆi )i

 c1eh;HQ (Æ))i :

The sum on the right of (2.5.1) can be bounded by nc1EQQw

1 Q0

(s; 1H ).

2

The following lemma explains the signi cance of assumption 1.4 : the geometric assertion that  maps the cone in X (T0 )R generated by
H0 into the cone X  (P )+R  X  (P )R, has as analytic consequence that there exists a xed Siegel domain such that  maps the \P -part" of each rational element 2 G(F ) into this xed Siegel domain. To be more precise, put T0 (A)1 = M0 (A)1 \ T0 (A) ; cf. (2.2). By the product formula we have T0 (F )  T0 (A)1, and the quotient T0(F )nT0(A)1 is compact. Let 0  T0(A)1 be a compact subset such that T0(A)1 = T (F ) 0 . (2.7) Lemma. There is t0 2 AM0 , such that for all 2 G(F ) one has  (p ) 2 T0 (A)1AM0 (t0 ) ; where = p k , p 2 P (A), k 2 KG. Proof. For 2
H 0 we have eh ;HQ0 ((p ))i = eh Æ;HP ( )i = HL Æ (P (F ) ) : By hypothesis, the line bundle L Æ lies in the cone of eective divisors, hence the height function associated to L Æ is bounded from below. 2 (2.8) Before proving our result about the abcissa of convergence we have to introduce the following constants. For (; ) 2 X  (Q)R  X (P )R and w 2 W M put (2.6)

aw;

= inf fa j for all a0  a : a0 ( + (w)) (w0 0) 2 2P + X (P )+R g ; aw; = max fa ; aw; g : In general, aw; may not be nite (by convention, the in mum of the empty set is 1). For w = wHM put aÆ; = aw; . The strati cation of X gives rise to a strati cation of Y because all strata X w = QnQw 1Q0 are stable under the action of T0 from the right: 16

Y

=

a w2W M

Yw

with Y w = (X w  G)=P :

The open stratum, corresponding to w = wHM will be denoted by Y Æ. The corresponding height zeta functions take the following form: 1 ZY w (LY   L ; s) = ehs;HP ( )i EQQw Q0 (s;  (p )) ;

X

2P (F )nG(F )

with the notations from Lemma 2.7.

(2.9) Proposition. a) If  + (w) is not contained in X + (P )+R, then ZY w (LY   L ; s)

does not converge for any s > 0. b) If  + (w) is in X  (P )+R, then the abcissa of convergence of ZY w (LY   L ; s) is in the interval [aw; ; aw; ]. c) Suppose that LY   L lies in (Y )Æe . Then the abcissa of convergence of ZY Æ (LY

  L ; s) is aÆ; . Proof. a) Let us x t0 such that the assertion of Lemma 2.7 holds. Then we can write  (p ) = # o a with # 2 T (F ), o 2 0 and a 2 AM0 (t0 ). Since all partial Eisenstein series are left-invariant under T (F ) one has

(s; (p )) = EQQw 1Q0 (s; a o ) : For each real number a > a there is by Proposition 2.5 a c1 > 0 such that for all w 2 W M and all s 2 (a ; a] EQQw

1 Q0

Qw 1 Q0 hw(s 0 )+0 ;HQ0 (a )i E Qw 1 Q0 (s;  (p ))  c 1 E Qw 1 Q0 (s; 1 ) ; c1 EQ;Q

H 1 Q;Q0 Q 0 (s; 1H )  e Qw 1 Q0 where EQ;Q is the constant term of the partial Eisenstein series: 0 Qw 1 Q0 EQ;Q 0

(s;

Z 1 )= H

V0 (F )nV0 (A)

EQQw

1 Q0

(s; v0)dv0 :

For s 2 (a ; a] this function is bounded from below by a positive number c2. For such s we have then 17

Qw 1 Q0 ZY w (LY   L ; s)  c1 EQ;Q 0 (s; 1H )

P

2P (F )nG(F ) e

 c1 c2EPG(s( + (w)) (w0

hs;HP (a )i ehsw

w0 +0 ;HQ0 ((p ))i

0 ); 1G) :

It is well-known that 0 w0 lies in the closed cone generated by the positive simple roots, and by assumption 1.4 we conclude that (w 0 0 ) lies in the closure of X  (P )+R. +  Therefore, if  + (w) is not contained in X (P )R, the same is true for s( + (w)) (w0 0 ), for all s > 0. In this case, ZY w (L   L ; s) does not converge for any s > 0. b) Suppose now that  + (w) is an element of X  (P )+R. The above estimate shows that the abcissa of convergence of ZY w (L L ; s) is greater or equal to aw; . On the other hand, using the estimate above, we get that Qw 1 Q0 G ZY w (LY   L ; s)  c1 1 EQ;Q 0 (s; 1H )EP (s( + (w)) (w0 0 ); 1G ) : Hence, this function converges for s > maxfa; aw; g = aw;, by Proposition 2.3 a). c) Let w = wHM . By Proposition 2.5 b) and the estimate above, we have for s 2 (a ; a] Qw 1 Q0 Qw 1 Q0 EQ;Q (s; 1H )  c3EQH (s; 1H ) ; 0 (s; 1H )  c1 EQ for some c3 > 0. This shows that ZY Æ (LY L ; s) converges only for s > aÆ; . By part b) we conclude that the abcissa of convergence is exactly aÆ; . 2 3 Arithmetic strati cation

In this section we determine the arithmetic strati cation of Y = (QnH  G)=P with respect to line bundles LY  L which lie in the interior of the cone of eective divisors (Y )Æe . Otherwise, there is no open non-empty subset for which the corresponding height zeta function converges for all s  0. First we need a lemma. (3.1) Lemma. If X w0  X w , then aw;0  aw; . Proof. The assertion follows if we can show that for all a > a one has a( + (w)) (w0 0 ) a( + (w0 )) + (w0 0 ) 2 X  (P )+ R: Assumption 1.4 on  ensures us that this is true if we have 18

aw0 

w0 0

aw + w0 2

XR

2
H 0

0

for all a > a . Note that aw0  w0 0 aw + w0 = w0(a 2Q ) w(a 2Q) + w0 (2Q 0 ) w(2Q 0 ) : Now let wM be the longest element in WM . Then we have wH = wHM wM and 2Q 0 = wM 0 . Therefore, aw0  w0 0 aw + w0 = w0 (a 2Q) w(a 2Q ) + w0 wM 0 wwM 0 : By [J], II, 13.8 (4), the assumption X w0  X w implies that w0  w with respect to the Bruhat ordering on WH . It follows from the description of the Bruhat ordering given in [J], II, Proposition 13.7, together with l(w0 wM ) = l(w0 ) + l(wM ); l(wwM ) = l(w) + l(wM ) that moreover w0wM  wwM : Hence we are done, if we can show that whenever w1  w2 and x 2 X (Q0 )+R one has w1 x w2 x 2 R0 :

X

2
H 0

For a root of H with respect to T0 we denote by s the re ection along . By [BGG], Proposition 2.8, there exist positive roots 1 ; : : : ; k such that with si = s i w2 = sk
sk 1
: : :
s1
w1 ; for all i = 1; :::; k : l(si
si 1
: : :
s1
w1 ) = l(si 1
: : :
s1
w1) + 1 : Because w1 x

w2 x

X = k

i=1

si 1
: : :
s1
w1 x

si
si 1
: : :
s1
w1 x ;

we can assume that w2 = s w1 with a positive root and l(w2 ) = l(w1) + 1. By [Bou], ch. VI, x1, Proposition+ 17, this implies w2 1 < 0, i.e. w1 1 > 0. Now suppose x lies in the closure of X (Q0 )R. Then one has 19

w1 x

w2 x = w1 x

(w1x

and this proves the assertion.

( x; w1 1 ) ( w1 x; ) 2 ( ; ) ) = 2 ( ; ) ;

2

To describe the arithmetic strati cation, consider those aw; which are nite and order them: faw; j w 2 W M ; aw; < 1g = fa0; a1 ; : : : ; ar; g ; with a0 > a1 > : : : > ar; and put formally a 1 = 1. (3.2) Theorem. Let Y = Y be the bre bundle over W = P nG de ned in (1.1), and suppose  ful lls the assumption of (1.4).

a) If LY   L is not an element of (Y )Æe , then there is no open non-empty subset such that the corresponding height zeta function converges for s  0. b) Suppose LY

L lies in (Y )Æe . For i = 0; 1; : : : ; r = r; put Yi = Y

[

w2W M ; aw ; =ai 1

Yw:

Then the arithmetic strati cation of Y with respect to LY Y Proof.

L is

 Y0  Y1  : : :  Yr :

First we prove the second assertion. Lemma 3.1 implies that Yi = Yw:

[

w2W M ; aw ; ai

Hence we conclude that Yi+1 is properly contained in Yi (i = 0; :::; r 1). Let Yi (L) be the abcissa of convergence of ZYi (L; s) with L = LY  L. We shall show that Yi (L) = ai. Obviously, ZYi (L; s) = ZY w (L; s) :

X

w2W M ; aw ; ai

20

By Proposition 2.9 b), ZY w (L; s) converges for s > aw;. This shows that ZYi (L; s) converges for s > ai . On the other hand, there is a w 2 W M such that aw; = ai and in order for ZY w (L; s) to converge it is necessary that s > aw;. Because the closure of X Æ is X , we have aÆ; = ar , by Lemma 3.1, and ZY Æ (L; s) is a summand of ZYi (L; s). By Proposition 2.5 c), we know that for ZY Æ (L; s) to converge it is necessary that s > a. Hence, for ZYi (L; s) to converge it is necessary that s > maxfaw; ; a g = aw; = ai and we conclude that the abcissa of convergence of ZYi (L; s) is ai. Fix i 2 f 1; : : : ; r g and consider an open subset Y 0  Yi which contains Yi+1 properly (Y 1 = Y , Yr+1 = ;). The case i = 1 can only occur if ZY (L; s) does not converge for any s > 0. We have to show that ZY 0 (L; s) converges only for s > ai (and then this sum converges for all s > ai). By hypothesis, Y 0 properly contains the open subset Yi+1, hence there is a w 2 W M such that aw; = ai und Y 0 \ Y w 6= ;. Y 0 \ Y w is an open and dense subset of Y w . Therefore, there are open and dense subsets X 0  X w , G0  G, such that the image of X 0  G0 under the canonical projection X  G ! Y is contained in Y 0 \ Y w . Moreover, we may and we will assume that P G0  G0. Let H 0  H be the preimage of X 0 under! the projection H ! X . For h 2 H (A) put 0 EQH (s; h) = ehs;HQ (Æt)i :

X

Æ2Q(F )nH 0 (F )

Then we have 0 (3.2.1) ZY 0 (L; s)  2P (F )nG0 (F ) ehs;HP ( )i EQH (s;  (p )) . For each # 2 T0 (F ) the set H 0# is open in Qw 1Q0 , hence H~ = [#2T0 (F )H 0# is an open subset of Qw 1Q0 and there exist #1; : : : ; #m 2 T0 (F ) such that H~ = [1jmH 0#j . For h 2 H (A) we de ne as above ~ EQH (s; h) = ehs;HQ (Æt)i :

P

X

~ (F ) Æ2Q(F )nH

Fix a > a . By Lemma 2.1 there is a c1 > 0 such that for all s 2 (a ; a] and t 2 T0 (A) 0 (3.2.2) EQH (s; t)  c1 EQH~ (s; t) . Note that for all # 2 T0(F ) and h 2 H (A) one has EQH~ (s; #h) = EQH~ (s; h). For each ~ is open in Qw 1Q0 . Therefore, there are 1; : : : ; n 2 Q0 (F ) such  2 Q0 (F ) the set H ~ k . Write (p ) = # a o with # 2 T0(F ), o 2 0 and a 2 that Qw 1Q0 = [1knH AM0 (t0 ), where t0 is as in Lemma 2.7. The set fa 1 j a o j 2 P (F )nG0(F ); j = 1; : : : n g 21

is relatively compact in H (A). So we can nd c2 > 0 such that for all 2 P (F )nG0(F ) and j = 1; : : : ; n we have ~ ~ ~ EQH (s; j a o )  c2 EQH (s; a o ) = c2 EQH (s;  (p )) : Replacing nc2 by c2 we even get the following estimate 1 (3.2.3) EQH~ (s;  (p ))  c2 EQQw Q0 (s;  (p )) . Putting (3.2.1), (3.2.2) and (3.2.3) together and using Proposition 2.5 we have 1 ZY 0 (L; s)  c1 c2 2P (F )nG0 (F ) ehs;HP ( )i EQQw Q0 (s;  (p ))  c3

P P

hs(+ (w)) (w0 0 );HP ( )i ;

2P (F )nG0 (F ) e

with a suitable c3 > 0. Because G0 is dense in G there is c4 > 0 such that ehs(+ (w)) (w0 0 );HP ( )i  c4 EPG (s( + (w)) (w0

X

2P (F )nG0 (F )

0 )) :

(The argument is the same as in the proof of Prop. 2.5 b).) This shows that the abcissa of convergence of ZY 0 (L; s) is not less than aw;. Because Y 0 is dense in Y , the intersection Y 0 \ Y Æ is not empty. By the same reasoning as above we nd c5 > 0 such that Q(wM ) 1 Q0 ZY 0 (L; s)  c5 ehs;HP ( )i EQ H (s; (p )) :

X

2P (F )nG0 (F )

Again by Proposition 2.5 b), EQQ(wHM ) 1 Q0 (s;
) converges only for s > a , so we can conclude that ZY 0 (L; s) converges only for s > maxfaw;; a g = aw; = ai, and this proves the second statement. To prove the rst assertion, observe that we have just seen that the height zeta function of any open non-empty subset can be bounded from below by the height zeta function of Y Æ . But this series does not converge for any s > 0 if LY   L is not in the interior of the positive cone, by proposition 2.9. 2

22

References

[BGG] I.N.Bernstein, I.M.Gelfand, S.I.Gelfand - Schubert cells and the cohomology of the spaces G/P, Russian Math. Surveys 28, no. 3 (1973), 1-26 [BM] V.V.Batyrev, Yu.I.Manin - Sur le nombre des points rationnels de hauteur bornee des varietes algebriques, Math. Annalen 268 (1990), 27-43 [Bou] N.Bourbaki - Groupes et algebres de Lie, chapitres 4,5 et 6, Hermenn, Paris, 1968 [FMT] J.Franke, Yu.I.Manin, Yu.Tschinkel - Rational points of bounded height on Fano varieties, Invent. math. 95 (1989), 421-435 [Go] R.Godement - Introduction a la theorie de Langlands, Seminaire Bourbaki, expose 321 (1966/67) [J] J.C.Jantzen - Representations of algebraic groups, Academic press, Orlando, Florida, 1987 [MW] C.Mglin, J.-L.Waldspurger - Decomposition spectrale et series d'Eisenstein, Birkhauser, Basel, 1994 [P] E.Peyre - Hauteurs et mesures de Tamagawa sur les varietes de Fano, Duke Math. Journal 79 (1995), 101-218 [Sa] J.-J.Sansuc - Groupe de Brauer et arithmetique des groupes algebriques lineaires sur un corps de nombres, Journal f. d. reine u. angewandte math. 327 (1981), 12-80 [St] M.Strauch - Rational points on twisted products of ag varieties, in preparation

23


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HUA'S LEMMA AND EXPONENTIAL SUMS OVER BINARY FORMS 

Trevor D. Wooley

We establish mean value estimates for exponential sums over binary forms of strength comparable with the bounds attainable via classical, single variable estimates for diagonal forms. These new mean value estimates strengthen earlier bounds of the author when the degree d of the form satis es 5 6 d 6 10, the improvements stemming from a basic lemma which provides uniform estimates for the number of integral points on aÆne plane curves in mean square. Exploited by means of the Hardy-Littlewood method, these estimates permit one to establish asymptotic formulae for the number of integral zeros of equations de ned as sums of binary forms 17 2d , improving of the same degree d, provided that the number of variables exceeds 16 signi cantly on what is attainable either by classical additive methods, or indeed the general methods of Birch and Schmidt. Abstract.

1. Introduction. Rather general versions of the Hardy-Littlewood method due to Birch [2] and Schmidt [13] oer remarkably successful approaches to estimating the number of integral zeros of prescribed height satisfying a given homogeneous polynomial with integral coeÆcients. Both approaches require the polynomial under investigation to possess many variables in terms of its degree, and there are further hypotheses to be negotiated involving, directly or indirectly, the singular locus of the associated hypersurface. These unfortunate de ciencies of the method are signi cantly less pronounced whend the polynomial under investigation is diagod nal, which is to say, of the shape a1x1 +


+ as xs (see Chapter 9 of Vaughan [19]), and such is also the case when the polynomial diagonalises over C (see Birch and Davenport [3]). The availability of superior analytic methods for the diagonal situation motivates investigation of polynomials intermediate in complexity between the diagonal ones, and the quite general homogeneous polynomials investigated by Birch and Schmidt, the hope being that insight will be obtained relevant to the general situation. One such intermediate situation is that in which the polynomial splits as a sum of binary homogeneous polynomials, and such has been investigated with some success for cubic forms by Chowla and Davenport [7], and more recently by Brudern and Wooley [6]. The author [22] has rather recently obtained analogues 1991 Mathematics Subject Classi cation. 11D72, 11L07, 11E76, 11P55. Key words and phrases. Exponential sums, binary forms, diophantine equations.  Packard Fellow and supported in part by NSF grant DMS-9970440. 1

Typeset by AMS-TEX

2

TREVOR D. WOOLEY

of Weyl's inequality and Hua's lemma for exponential sums over binary forms of higher degree, and thereby has made progress on problems involving sums of binary forms of arbitrary degree. This work was hindered by our lack of good uniform estimates for the number of integral points on aÆne plane curves. The object of this paper is to sharpen our earlier conclusions, and this we achieve by developing useful mean square estimates for the number of integral points on certain families of aÆne plane curves. It is to be hoped that progress will be stimulated in problems involving higher degree forms in many variables. Before proceeding to the main thrust of this paper, it seems worthwhile to recall the conclusions stemming from the classical additive theory, and the work of Birch and Schmidt, so far as the density of integer points on hypersurfaces is concerned. First, on combining estimates of Weyl and Hua, one obtains the following classical conclusion (see Chapter 9 of Vaughan [19]). Theorem A (Classical). Let a1 ; : : : ; as 2 Z n f0g and write F (x) = a1 xd1 +


+ as xds : Then whenever s > 2d , one has card(fx 2 [ B; B ]s \ Zs : F (x) = 0g)  CB s d ; where C denotes the \product of local densities" within the box [ B; B ]s. In order to save space at this point, we avoid explaining what is meant by the term \product of local densities", and instead note merely that this number is positive and uniformly bounded away from zero whenever the equation F (x) = 0 possesses non-singular real and p-adic solutions for every prime p. We refer the reader to Vaughan [17], [18], Heath-Brown [10] and Boklan [4] for the theory underlying the latest developments concerning the asymptotic formula in the diagonal situation. In order to describe Birch's theorem (see [2]), we recall that the singular locus of the hypersurfaces de ned by the homogeneous equation F (x1; : : : ; xs) = 0 is the set of points y 2 C satisfying the equations @F @F (y) =


= (y) = 0: @x1 @xs Theorem B (Birch). Let F (x) 2 Z[x1; : : : ; xs] be homogeneous of degree d, and suppose that the variety de ned by the equation F (x) = 0 has a singular locus of dimension at most D. Then whenever s D > (d 1)2d , one has card(fx 2 [ B; B ]s \ Zs : F (x) = 0g)  CB s d ; where C denotes the \product of local densities" within the box [ B; B ]s. Mention of the singular locus is removed by Schmidt [13] at the cost of introducing an invariant h associated with the polynomial under consideration. When F (x) 2 Q [x1 ; : : : ; xs ] is a form of degree d > 1, write h(F ) for the least number h such that F may be written in the form F = A1 B1 + A2 B2 +


+ Ah Bh ; with Ai ; Bi forms in Q [x] of positive degree for 1 6 i 6 h.

EXPONENTIAL SUMS OVER BINARY FORMS

3

Theorem C (Schmidt). Let d be an integer exceeding 1, and write (d) = d2 24d d!. Let F (x) 2 Z[x1; : : : ; xs] be homogeneous of degree d, and suppose that h(F ) > (d). Then one has card(fx 2 [ B; B ]s \ Zs

: F (x) = 0g)  CB s

d;

[ B; B ]s. We reiterate that the relative simplicity and strength of Theorem A over Theorems B and C seems to us to justify the investment of further eort in investigations which carry successful elements of the classical methods over to more general situations. We are now at liberty to focus on the topics central to this paper. Over sixty years ago, Hua [11] greatly simpli ed the analysis of the asymptotic formula in Waring's problem and allied additive problems with the introduction of a new mean value estimate which, to this day, remains central to the theory of exponential sums of small degree in a single variable. Roughly speaking, Hua observed that by Weyl dierencing half of the exponential sums in a suitable mean value, and interpreting the result in terms of the underlying diophantine equation, one obtains a recursive estimate for successive mean values in terms of divisor sum estimates of particularly simple type. The author has recently obtained a version of Hua's lemma for exponential sums of the type where C denotes the \product of local densities" within the box

X

06x;y6P

e((x; y ));

in which (x; y) is a non-degenerate binary form with integral coeÆcients, and as usual, we write e(z) to denote e2iz (see [22]). By means of a carefully orchestrated dierencing procedure, we are able to engineer a recursion similar to that of Hua in the situation of a single variable. Unfortunately, however, the divisor sum estimates are complicated by the presence of estimates for the number of integral points on aÆne plane curves, and our relative ignorance of such matters somewhat weakens the ensuing mean value estimates. In this paper we sharpen our analogue of Hua's lemma by means of an enhanced treatment of the aÆne curves that arise from the dierencing process at the heart of our treatment. In order to describe our version of Hua's lemma, we require some notation. Suppose that (x; y) 2 Z[x; y] is a binary form of degree d exceeding 1. Then we say that  is degenerate if there exist complex numbers  and such that (x; y) is identically equal to (x + y)d . It is easily veri ed that when ( x; y ) is degenerate, d then there exist integers a, b and c with (x; y) = a(bx + cy) . Finally, de ne the exponential sum X f (; P ) = e((x; y )): (1.1) 06x;y6P

Theorem 1. Suppose that (x; y) 2 Z[x; y] is a non-degenerate form of degree d > 3. Then the following estimates hold.

4

TREVOR D. WOOLEY

(i) When d = 3 or 4 and j is an integer with 1 6 j 6 d, or when d > 5 and j or 2, one has for each positive number " the bound

1

Z

0

jf(; P )j2j 1 d  P 2j

=1

j +" :

(ii) When d = 5, one has for each positive number " the bounds

(iii)

Z

1

Z

0 1

jf (; P )j4d  P 21=4+"; jf(; P )j10d  P 127=8+";

1

Z

0

Z

jf(; P )j8d  P 49=4+"; 1

jf (; P )j17d  P 29+": 0 0 When 6 6 d 6 10 and j is an integer with 3 6 j 6 d 2, then for each positive number " one has

Also, when

and

1

Z

0

jf(; P )j2j 1 d  P 2j

j +1=(d j +2)+" :

6 6 d 6 10, one has for each " > 0 the bounds Z 1 jf (; P )j 329 2d d  P 169 2d d+1+" 0

1

Z

0

jf(; P )j 3217 2d d  P 1617 2d

d+" :

Of course, bounds for moments of f(; P ) intermediate between those recorded in the statement of Theorem 1 may be obtained by applying Holder's inequality to interpolate between those above. For comparison, Theorem 2 of Wooley [22] shows that when d > 5 and j is an integer with 1 6 j 6 d 1, one has Z 1 jf (; P )j2j 1 d  P 2j j+ 12 +"; 0 and also provides the estimates Z 1 Z 1 5 2d 5 2d d+1+" 16 8 jf (; P )j d  P and jf(; P )j 169 2d d  P 98 2d d+" : 0 0 Case (iii) of Theorem 1 above plainly provides estimates superior to the latter bounds. On the other hand, case (i) of Theorem 1 is simply a restatement of the rst estimate of [22, Theorem 2]. We note also that when d is greater than or equal to 11, it is possible to apply a trivial variant of Vinogradov's methods in order to obtain conclusions superior to those stemming from Theorem 1 (see [22, x8] for details). Since we are interested primarily in ideas likely to generalise successfully to homogeneous forms in many variables, we discuss Vinogradov's methods no further herein. There are immediate consequences of the estimates recorded in Theorem 1 for the solubility of homogeneous diophantine equations which split as sums of binary forms. We con ne ourselves here to a routine conclusion discussed in detail in [22].

EXPONENTIAL SUMS OVER BINARY FORMS

Theorem 2.

5

3 6 d 6 10, and de ne s0 (d) by 1 ; when d = 3; 4; 2 s0 (d) = 17 d 32 2 ; when 5 6 d 6 10: Let s > s0 (d), and let j 2 Z[x; y ] (1 6 j 6 s) be homogeneous forms of degree d with non-zero discriminants. Let N (B ) = Ns (B ; ) denote the number of solutions Let d be an integer with  d

of the diophantine equation

1(x1 ; y1) +


+ s (xs; ys) = 0; (1.2) subject to jxj j 6 B and jyj j 6 B (1 6 j 6 s). Then provided that the form 1 (x1; y1) +


+ s (xs; ys) is inde nite, one has Ns (B ; ) = C SB 2s d + O (B 2s d Æ ); for some positive number Æ . Here, C denotes the volume of the (2s 1)-dimensional hypersurface determined by theQequation (1:2) contained in the box [ 1; 1]2s . Also,

S denotes the singular series

p vp , where the product is over prime numbers, vp = lim ph(1 2s) Ms (ph ; ); h!1

and Ms (ph ; ) denotes the number of solutions of the congruence

1 (x1; y1) +


+ s(xs; ys)  0 (mod ph ); with 1 6 xj ; yj 6 ph (1 6 j 6 s). We note that the expression C S explicitly describes the \product of local densities", for the problem at hand, previously mentioned in Theorems A, B and C. Given the existence of non-singular real and p-adic solutions of the equation (1.2), the proof of Theorem 2 follows precisely the argument of the proof of [22, Theorem 3], and hence we omit details in the interest of saving space. Following some preliminary reductions in x2, we grapple with basic estimates for the number of integral points on aÆne plane curves in x3. We discuss the main induction in x4, thereby establishing the majority of the estimates recorded in Theorem 1. The closing stages of the induction have a dierent avour, and this we defer to x5, completing the proof of Theorem 1. Throughout this paper, implicit constants occurring in Vinogradov's notation  and  will depend at most on the coeÆcients of the implicit binary forms, a small positive number ", exponents d and k, and quantities occurring as subscripts to the latter notations, unless otherwise indicated. We write f  g when f  g and g  f . When x is a real number, we write [x] for the greatest integer not exceeding x. Also, we use vector notation for brevity. Thus, for example, the stuple (1; : : : ; s) will be abbreviated simply to . In an eort to simplify our exposition, we adopt the convention that whenever " appears in a statement, we are implicitly asserting that the statement holds for each " > 0. Note that the \value" of " may consequently change from statement to statement. The author is grateful to the referee for useful comments.

6

TREVOR D. WOOLEY

2. Preliminary reductions. Let k be an integer with k > 3 and let (x; y) 2 Z[x; y ] be a non-degenerate homogeneous polynomial of degree k . Let P be a large real number, and de ne the exponential sum f () = f(; P ) as in (1.1). We aim initially to transform f () into an associated exponential sum amenable to our dierencing procedure, and the latter goal we achieve by following closely the argument of [22, x2]. When (x; y) 2 Z[x; y], we describe the polynomial as being a condensation of  when the following condition (C ) is satis ed. (C ) We have (u; v) 2 Z[u; v], and the coeÆcients of depend at most on those of . Further, the polynomial (u; v) has the same degree as (x; y), and takes the shape (u; v) = Auk + Buk t vt +

k X j =t+1

Cj uk j v j ;

(2.1)

with AB 6= 0 and 2 6 t 6 k. Lemma 2.1.

There is a condensation

of , and a positive real number X with

X  P , with the property that for every natural number s one has 1

Z

0 where we write

jf(; P )j2sd 

H (; X ) =

X

1

Z

0

jH (; X )j2sd;

X

juj6X jvj6X

e( (u; v )):

(2.2)

This is [22, Lemma 2.3]. The work of [22, x5] takes care of certain special cases that arise in our treatment. We summarise the relevant conclusions of this discussion in the following two lemmata. Lemma 2.2. Suppose that k = 3 or 4 and j is an integer with 1 6 j 6 k, or else that k > 5 and j = 1 or 2. Then for each positive number ", one has Proof.

1

Z

0 Proof.

jf(; P )j2j 1 d  P 2j

j +" :

This estimate is recorded as the rst conclusion of [22, Theorem 2].

EXPONENTIAL SUMS OVER BINARY FORMS

7

Lemma 2.3. Suppose that (u; v) 2 Z[u; v] has the shape (2:1). Suppose also that k > 5, that X is a large real number, and that H (; X ) is de ned as in (2:2). Then for 1 6 j 6 k, and for each positive number ", one has the upper bound 1

Z

0

jH (; X )j2j 1 d  X 2j

j +" ;

provided either that t = k, or else that t = k 1 and Ck = 0. When t = k 1 and Ck 6= 0, meanwhile, then there is a condensation  of with the property that  has the shape k

X (x; y) = A0xk + B 0 xk 2y2 + Cj0 xk j yj ; j =3 with A0 B 0 6= 0, and there is a positive real number Y with Y  X , and satisfy the property that for each natural number s, one has

1

Z

0

jH (; X )j2sd 

Z

0

1

 and Y

jH (; Y )j2sd:

The situations in which t = k, or else t = k 1 and Ck = 0, are dealt with, respectively, in Lemmata 5.2 and 5.3 of [22]. The alternative situation in which t = k 1 and Ck 6= 0, on the other hand, is discussed in the preamble to Lemma 5.3 of [22]. Our deliberations are also greatly simpli ed through a manoeuvre that transforms a polynomial of the shape (2.1) with t = k 2 into a corresponding polynomial in which t = 2 or 3. We begin with an analogue of Lemma 5.3 of [22]. Suppose, temporarily, that (u; v) has the shape (2.1) with t = k 2, so that for some integers a, b, c, d with ab 6= 0, one has (x; y) = axk + bx2 yk 2 + cxyk 1 + dyk : (2.3) Lemma 2.4. Suppose that k > 4, and that (u; v) 2 Z[u; v] has the shape (2:3) with ab 6= 0 and d = 0. De ne the exponential sum H (; X ) as in (2:2). Then for 1 6 j 6 k, and for each positive number ", one has the upper bound Proof.

1

Z

0

jH (; X )j2j 1 d  X 2j

j +" :

(2.4)

Our argument is a variant of the proof of Lemma 5.3 of [22]. We abbreviate H (; X ) simply to H (). Also, when 1 6 j 6 k, we write Proof.

Ij (X ) =

1

Z

0

jH ()j2j 1 d:

(2.5)

8

TREVOR D. WOOLEY

The bound (2.4) is immediate from Lemma 2.2 when j = 1; 2. Suppose then that j is an integer with 2 6 j 6 k 1, and that the inequality (2.4) holds. We seek to show that (2.4) holds with j replaced by j + 1, whence the desired conclusion follows for 1 6 j 6 k by induction. Observe rst that X X k 2 k 2 k 1 jH ()j  X + e((ax + bx y + cxy )) : 16jxj6X jyj6X

De ne the exponential sum hl () = hl (; X ) by hl (; X ) =

X

jyj6X

e((bly k 2 + cy k 1 )):

Then it follows from (2.5) via Holder's inequality that j 1 Ij +1 (X )  X 2 Ij (X ) +

Z 1

2j 1

X

jH ()j jhx (x)j 0 16jxj6X j 1 j 1  X 2 Ij (X ) + X 2 1 N (X );

where

N (X ) =

1

Z

0

jH ()j2j

In the special situation in which Cauchy's inequality, one has X

X

1

d

(2.6)

jhx (x)j2j 1 d:

16jxj6X j = k 1 and c = 0,

(2.7)

we instead note that by

2 e((axk + bx2 y k 2 ))

X

jyj6X 16jxj6X

X

X X

2 e((axk + bx2 y k 2 ))

jyj6X 16jxj6X X X  X3 + X e(b(x21 16jx1 j;jx2 j6X jyj6X x1 6=x2



x22 )y k 2 ) :

Thus, on applying Holder's inequality within (2.5), we now obtain Ik (X )  X 3
2

3

Ik 1 (X ) Z 1 k 3 2 +X jH ()j2k 0 k

2

X

16jx1 j;jx2 j6X x1 6=x2 k 3 k 3  X 3
2 Ik 1 (X ) + X 3
2 2 M (X );

jhx21

x22 ()j

2k 3

d

(2.8)

EXPONENTIAL SUMS OVER BINARY FORMS

where

M (X ) =

1

Z

0

jH ()j2k

By orthogonality, it follows from integral solutions of the equation x

j 2 2X

i=1

bx(yik 2

zik 2 ) + c(yik 1

X

2

jhx21

16jx1 j;jx2 j6X x1 6=x2 (2.7) that N (X )

9

2k 3 d:

x22 ()j

(2.9)

is equal to the number of

j 2
2X k 1 zi ) = ( (ui ; vi ) i=1

(ti ; wi )) ; (2.10)

with 1 6 jxj 6 X , and with each of yi , zi , ui , vi , ti , wi (1 6 i 6 2j 2) bounded in absolute value by X . Let N0 (X ) denote the number of such solutions of (2.10) in which the right hand side of the equation is equal to zero, and let N1 (X ) denote the corresponding number of solutions with the latter expression non-zero. Then one has N (X ) = N0 (X ) + N1 (X ): (2.11) We rst estimate N0(X ). On considering the underlying diophantine equations and recalling (2.5), we have N0 (X )  Ij (X )

X

Z

16jxj6X 0

1

jhx ()j2j 1 d:

But a classical version of Hua's lemma (see Lemma 2.5 of Vaughan [19]) shows that for 2 6 j 6 k 1, one has 1

Z

0

jhx ()j2j 1 d  X 2j

1 j +1+" ;

uniformly in x 6= 0. Thus we deduce that for 2 6 j 6 k 1, one has j 1 N0 (X )  X 2 j +2+" Ij (X ): (2.12) In order to dispose of N1 (X ), we introduce some additional notation. For each integer l, we denote by rj (n; l) the number of representations of the integer n in the form j 2 2X
n=l bl(yik 2 zik 2 ) + c(yik 1 zik 1 ) ; i=1

with jyi j 6 X and jzi j 6 X (1 6 i 6 2j 2). Similarly, for each integer n we write Rj (n) for the number of representations of n in the form n=

j 2 2X

i=1

( (ui ; vi ) (ti ; wi )) ;

10

TREVOR D. WOOLEY

with each of ui , vi , ti , wi (1 6 i 6 2j 2) bounded in absolute value by X . Then on writing = (jbj + jcj)2j , we nd that X X N1 (X ) 6 Rj (n) rj (n; l): ljn jlj6X

16jnj6 X k

On applying an elementary estimate for the divisor function, we therefore deduce from Cauchy's inequality that N1 (X ) 6

X

n2Z

Rj (n)2

1=2 

X

X

2 1=2

rj (n; l)

16jnj6 X k ljn jlj6X X 1=2  X X 1=2 " 2 2 X Rj (n) rj (n; l) : n2Z n2Z16jlj6X

(2.13)

However, on considering the underlying diophantine equations, it is apparent from (2.13) that N1

(X )  X " (I

j +1

 (X ))1=2

X

Z

1

16jlj6X 0

1=2

jhl ()j2j d

:

But the classical version of Hua's lemma (see Lemma 2.5 of [19]) shows that for 1 6 j 6 k 2, one has Z 1 jhl ()j2j d  X 2j j+" ; 0 uniformly in l 6= 0. Moreover, the latter conclusion remains valid for j = k 1 whenever c is non-zero. In either circumstance, we deduce that  N1 (X )  X " (Ij +1 (X ))1=2

X

16jlj6X

1=2 j X 2 j +" :

(2.14)

On combining (2.6), (2.11), (2.12) and (2.14), we nd that for 2 6 j 6 k 2, and also when j = k 1 and c 6= 0, one has  j 1  j j Ij +1 (X )  X 2 + X 2 j +1+" Ij (X ) + X 2 (j +1)=2+" (Ij +1 (X ))1=2 ; whence our inductive hypothesis (2.4) leads to the upper bound j +1 j Ij +1 (X )  X 2 j 1+" + X 2 (j +1)=2+" (Ij +1 (X ))1=2: Thus the estimate (2.4) follows with j +1 in place of j in the current circumstances, and so the conclusion of the lemma has been established in all cases but that in which c = 0 and j = k 1.

EXPONENTIAL SUMS OVER BINARY FORMS

11

We now turn to the nal elusive case wherein c = 0 and j = k 1. By orthogonality, it follows from (2.9) that M (X ) is equal to the number of integral solutions of the equation b(x21 x22 )

k 4 2X

i=1

(yik

2

zik 2 ) =

k 3 2X

i=1

( (ui ; vi) (ti ; wi ));

(2.15)

with 1 6 jx1j; jx2j 6 X and x1 6= x2 , and with each of yi , zi (1 6 i 6 2k 4 ), and ui , vi , ti , wi (1 6 i 6 2k 3 ) bounded in absolute value by X . Let M0 (X ) denote the number of such solutions of (2.15) in which the right hand side of the equation is equal to zero, and let M1(X ) denote the corresponding number of solutions with the latter expression non-zero. Then plainly one has M (X ) = M0 (X ) + M1 (X ): (2.16) We rst estimate M0(X ). On considering the underlying diophantine equations and recalling (2.5), we have M0 (X )  Ik 1 (X )

X

1

Z

16l;m62X 0

jhlm ()j2k 3 d:

But a classical version of Hua's lemma shows that Z 1 jhlm ()j2k 3 d  X 2k 3 k+3+" ; 0 uniformly in lm 6= 0, whence we obtain k 3 M0 (X )  X 2 k+5+" Ik 1 (X ): (2.17) Meanwhile, recycling the notation introduced to treat N1(X ), we see that X X X M1 (X ) 6 Rk 1 (n) T (n; lm); ljn mjn jlj62X jmj62X

16jnj6 X k

where we write T (n; ) for the number of representations of the integer n in the form k 4 2X n = b (yik 2 zik 2 ); i=1 (1 6 i 6 2k 4).

with jyi j 6 X and jzi j 6 X Again applying an elementary estimate for the divisor function, we deduce from Cauchy's inequality that X 1=2   X 2 1=2 X X 2 M1 (X ) 6 Rk 1 (n) T (n; lm) n2Z

 X"

X

n2Z

16jnj6 X k

Rk 1 (n)2

1=2  X

ljn mjn jlj62X jmj62X

X

X

n2Z16jlj62X 16jmj62X

T (n; lm)2

1=2

:

(2.18)

12

TREVOR D. WOOLEY

On considering the underlying diophantine equations, we nd from (2.18) that M1

(X )  X "(I

1=2 k (X ))



X

Z

X

1

16jlj62X 16jmj62X 0

1=2

jhlm ()j2k 2 d

:

Again applying the classical version of Hua's lemma, one has 1

Z

0

jhlm ()j2k 2 d  X 2k

2 k+2+" ;

uniformly in lm 6= 0, whence M1 (X )  X " (Ik (X ))1=2



X

X

X2

k

16jlj62X 16jmj62X

2 k+2+" 1=2 :

(2.19)

On combining (2.8), (2.16), (2.17) and (2.19), we nd that when c = 0, one has Ik (X ) 



 k 3 k 1 k+3+" k 1 3
2 2 X +X Ik 1 (X ) + X 2 k=2+" (Ik (X ))1=2;

whence our inductive hypothesis (2.4) with j = k 1 leads to the upper bound 1 k=2+"

k k Ik (X )  X 2 k+" + X 2

(Ik (X ))1=2: We therefore conclude that (2.4) holds with j = k even when c = 0, and this completes the proof of the lemma. Lemma 2.5. Suppose that k > 4, and that (u; v) 2 Z[u; v] has the shape (2:3) with abd 6= 0. De ne the exponential sum H (; X ) as in (2:2). Then there is a condensation  of with the property that  has the shape (x; y) = A0xk + B 0 xk t yt +

k X j =t+1

Cj0 xk j y j ;

with A0 B 0 6= 0 and 2 6 t 6 3, and there is a positive real number Y with Y and  and Y satisfy the property that for each natural number s, one has

1

Z

0

jH (; X )j2sd 

Z

0

1

jH (; Y )j2sd:

(2.20)  X,

(2.21)

By hypothesis, the coeÆcient d is non-zero, and thus we may make the non-singular change of variable u = kdy + cx, v = x. Write (u; v) = (kdv; u cv); (2.22) Proof.

EXPONENTIAL SUMS OVER BINARY FORMS

13

so that one has (u; v) = (kd)k (x; y). Then it follows from the argument of the proof of Lemma 2.3 of [22] that for some positive real number Y with Y  X , and for every natural number s, one has the upper bound (2.21). The proof of the lemma will therefore be completed on establishing that the polynomial (x; y), de ned in (2.22), has the shape (2.20). In order to establish the latter conclusion, we apply Taylor's theorem to determine whether or not various coeÆcients of (u; v) vanish. Write @z for the dierential operator @=@z. Then the coeÆcient of uk in (u; v) is equal to 1 @ k (kdv; u cv) : k! u (u;v)=(0;0) On writing i;j for @xi @yj (x; y ) ; (x;y)=(0;0) one nds by the chain rule, therefore, that the coeÆcient of uk in (u; v) is equal to 1 0;k = d; (2.23) k! and this is non-zero by hypothesis. Similarly, the coeÆcient of uk 1 v in (u; v) is equal to 1 @ k 1 @v (kdv; u cv) 1 (kd 1;k 1 c 0;k ) = u (k 1)! (u;v)=(0;0) (k 1)! = kdc ckd = 0: (2.24) Next, the coeÆcient of uk 2 v2 in (u; v) is equal to 1 @ k 2 @ 2 (kdv; u cv) 2!(k 2)! u v (u;v)=(0;0) 1 = 2!(k 2)! (kd)2 2;k 2 2kdc 1;k 1 + c2 0;k
= bk2d2 12 k(k 1)dc2 : (2.25) Finally, the coeÆcient of uk 3 v3 in (u; v) is equal to 1 @ k 3@ 3 (kdv; u cv) 3!(k 3)! u v (u;v)=(0;0) = 3!(k 1 3)! (kd)3 3;k 3 3(kd)2c 2;k 2 + 3kdc2 1;k = bk2 (k 2)d2 c + 13 k(k 1)(k 2)dc3:

1


c3 0;k

(2.26)

When c = 0, one nds from (2.25) that the coeÆcient of uk 2 v2 is bk2 d2, and this is non-zero by hypothesis. When c 6= 0, on the other hand, it follows from

14

TREVOR D. WOOLEY

(2.25) and (2.26) that when the coeÆcients of both uk 2 v2 and uk 3 v3 are zero, then necessarily 2kbd = (k 1)c2 and 3kbd = (k 1)c2; whence bd = c = 0, contrary to hypothesis. We therefore conclude from equations (2.23){(2.26) that (x; y) does indeed take the shape (2.20), wherein A0 B 0 6= 0 and t = 2 or 3. This completes the proof of the lemma. We next recall the Weyl dierencing lemma. Let
j denote the j th iterate of the forward dierencing operator, so that for any function of a real variable , one has
1( (); ) = ( + ) (); and when j is a natural number,
j+1 ( (); 1; : : : ; j+1 ) =
1 (
j ( (); 1; : : : ; j ); j+1): We adopt the convention that
0 ( (); ) = (). Lemma 2.6. Let X be a positive real number, and let (x) be an arbitrary arithmetical function. Write

T ( ) =

X

jxj6X

e( (x)):

Then for each natural number j there exist intervals Ii = Ii (h) (1 6 i 6 j ), possibly empty, satisfying

I1 (h1 )  [ X; X ] and Ii (h1 ; : : : ; hi )  Ii 1 (h1 ; : : : ; hi 1 )

(2 6 i 6 j );

with the property that

jT ( )j2j 6 (4X + 1)2j and here we write

Proof.

Tj =

X

x2Ij \Z

j 1

X

jh1 j62X






X

jhj j62X

Tj ;

e(
j ( (x); h1 ; : : : ; hj )):

This trivial variant of Lemma 2.3 of Vaughan [19] is recorded as Lemma 3.2

of [22]. We must also make use of a two dimensional forward dierencing operator
i;j de ned as follows. When (x; y) is a function of the real variables x and y, one de nes
1;0 ( (x; y); ) = (x + ; y) (x; y) and
0;1( (x; y); ) = (x; y + ) (x; y):

EXPONENTIAL SUMS OVER BINARY FORMS

15

When i and j are non-negative integers, one then de nes
i;j ( (x; y); 1; : : : ; i ; 1; : : : ; j ) by taking
0;0( (x; y); ; ) = (x; y), and in general by means of the relations
i+1;j ( (x; y); 1; : : :; i+1 ; 1; : : : ; j ) =
1;0(
i;j ( (x; y); 1; : : : ; i ; 1; : : : ; j ); i+1 ) and
i;j+1 ( (x; y); 1; : : :; i ; 1; : : : ; j+1) =
0;1(
i;j ( (x; y); 1; : : : ; i ; 1; : : : ; j ); j+1): 3. Integral points on aÆne plane curves. Essential to the main body of our argument are estimates for the number of integral points on aÆne plane curves, and in this section we record the estimates required for later use. Our basic tool is the following result of Bombieri and Pila [5]. Lemma 3.1. Let C be the curve de ned by the equation F (x; y) = 0, where F (x; y ) 2 R [x; y ] is an absolutely irreducible polynomial of degree d > 2. Also, let N > exp(d6 ). Then the number of integral points on C , and inside a square [0; N ]  [0; N ], does not exceed N 1=d exp(12(d log N log log N )1=2 ): This is Theorem 5 of Bombieri and Pila [5]. We note that slightly sharper estimates are now available through work of Pila [12], though these new estimates have no impact on the present work. At the request of the referee, we point out that applications of this result of Bombieri and Pila (of a rather dierent avour, involving slicing arguments) may be found in [1], [14], [15] and [16]. An application more akin to that at hand may be examined in x3 of [21] (the argument therein was in fact inspired by the proof of Lemma 3.2 below). We avoid detailed discussion of the absolute irreducibility condition occurring in the above lemma by careful averaging. Here the initial stages of our argument are modelled closely on the method of the proof of [22, Lemma 4.2]. Lemma 3.2. Let X denote a large real number. Suppose that F (x; y) 2 Z[x; y] is a non-degenerate polynomial of degree d > 2, and that X is suÆciently large in Proof.

terms of d. Suppose also that for some xed positive number A, one has that the coeÆcients of F are each bounded in absolute value by X A . Given a polynomial T (x; y ) 2 R [x; y ], denote by rT (n; X ) the number of solutions of the diophantine equation T (x; y ) = n, with (x; y ) 2 [ X; X ]2 \ Z2. Then one of the following two situations must occur, and in each of the bounds which follows, implicit constants

16

TREVOR D. WOOLEY

depend at most on d, " and A, and otherwise are independent of the coeÆcients of F. (i) There exist polynomials G(x; y ) 2 Z[x; y ] and g (t) 2 Q [t] satisfying the following conditions. (a) G is non-degenerate of degree exceeding 1; (b) g has degree exceeding 1; (c) the equation F (x; y ) = g (G(x; y )) is satis ed identically; (d) one has X X n2Z

rF (n; X )2  X 2+1=d+" + X "

n2Z

rG (n; X )2:

(ii) No polynomials G, g exist satisfying the conditions (a), (b), (c), (d) above. Then one has X n2Z

rF (n; X )2  X 2+1=d+" :

Consider an integer n 2 N with rF (n; X ) 6= 0. In view of the hypotheses of the statement of theB lemma, we may suppose that for some xed positive number B , one has jnj 6 X . When i is a non-negative integer, write Zi = fn 2 Z : jnj 6 2i X B g: Also, let N1 denote the set of integers n 2 Z0 for which the polynomial F (x; y) n is absolutely irreducible. Then an application of Lemma 3.1 reveals that for each n 2 N1 , one has rF (n; X ) = O(X 1=d+" ), whence Proof.

X

n2N1

rF (n; X )2  X 1=d+"

X

n2Z0

rF (n; X )  X 2+1=d+" :

Suppose next that n 2= N1 , so that the polynomial F (x; y) product of absolutely irreducible factors, say F (x; y ) n =

l Y j =1

gj (x; y )

m Y k=1

n

(3.1) factors as a

hk (x; y );

where l + m > 2, and where gj (x; y) 2 R [x; y] (1 6 j 6 l), and p hk (x; y ) = uk (x; y ) + vk (x; y ) 1 (1 6 k 6 m); with uk ; vk 2 R [x; y] for each k. Since hk (x; y) is presumed to be absolutely irreducible, we may suppose that uk (x; y) and vk (x; y) have no non-trivial polynomial common divisor over C [x; y]. It therefore follows from Bezout's theorem that the number of solutions of the simultaneous equations uk (x; y) = vk (x; y) = 0 is bounded above by d2 . By considering real and imaginary components, therefore, the number of integral solutions of the equation hk (x; y) = 0 is also bounded above

EXPONENTIAL SUMS OVER BINARY FORMS

17

by d2 . Next, if gj (x; y) is not some constant multiple of a Q -rational polynomial, then since gj (x; y) is necessarily a constant multiple of a polynomial with algebraic coeÆcients, we deduce that the number of integral solutions of the equation gj (x; y ) = 0 is at most d2 . For we may remove the aforementioned constant factor and consider components with respect to some basis for the eld extension containing the coeÆcients of gj (x; y). Then since gj (x; y) is not a constant multiple of a Q -rational polynomial, we nd that the integral zeros of gj (x; y ) = 0 necessarily satisfy at least two linearly independent Q -rational polynomial equations of degree at most d, whence the desired conclusion follows as in the complex case. Let N2 denote the set of integers n 2 Z0 nN1 for which the polynomial F (x; y) n possesses no non-trivial, absolutely irreducible Q -rational polynomial factor. Then the above argument shows that for each n 2 N2, one has rF (n; X ) = O(1), whence X X rF (n; X )2  rF (n; X )  X 2 : (3.2) n2N2

n2Z0

Suppose next that the set Z0 n (N1 [ N2) is non-empty, so that there exists some integer n0 2 Z0 with the property that F (x; y) n0 possesses a non-trivial, absolutely irreducible Q -rational polynomial factor. Since F (x; y) has integer coef cients, it follows that F (x; y) n0 may be written as a product F (x; y ) n0 = 1 (x; y ) : : : m (x; y ); (3.3) with each i (x; y) 2 Z[x; y] irreducible of degree di , say. Moreover, we may suppose without loss of generality that m > 2 and that d1 +


+ dm = d. Furthermore, on writing R(u; ; X ) = R(u1 ; : : : ; um ; 1 ; : : : ; m ; X ) for the number of integer solutions of the system of equations i (x; y ) = ui (1 6 i 6 m); (3.4) with jxj; jyj 6 X , it follows from (3.3) that when Z0 n (N1 [ N2) is non-empty, one has !2 X X X rF (n; X )2 6 R(u; ; X ) : n2Z0 nfn0 g

n2Z1 nf0g u1 :::um =n

Notice here that on the right hand side of the last inequality, we are implicitly applying a shift by n0 to Z0 , and then we note that this shifted set is contained in Z1 . Thus, on combining an application of Cauchy's inequality with an elementary estimate for the divisor function, we obtain X X X rF (n; X )2  rF (n0 ; X )2 + X " R(u; ; X )2 n2Z0

 X2 + X"

X

u

2Z m 1

n2Z1 nf0g u1 :::um =n

R(u;

; X )2:

(3.5)

18

TREVOR D. WOOLEY

Suppose now that m > 2, and that for 1 6 i 6 m the polynomials i (x; y) 2 Z[x; y ] have degree di > 1. Suppose also that these polynomials satisfy the condition that d1 +


+ dm 6 d, that F (x; y) is a polynomial in 1 ; : : : ; m , and that for some j with 1 6 j < d, one has the upper bound X X rF (n; X )2  X 2+" + X " R(u; ; X )2: (3.6) u2Zjm

n2Z0

Note that by (3.3) and (3.5), this condition is already met when  = , wherein we take j = 1. It is possible that the intersection (3.4) is proper for every available choice of u, by which we mean that the intersection over C consists of isolated points only, and in such circumstances an application of Bezout's theorem leads to the bound R(u; ; X ) = O(1) uniformly in u, whence X

u2Zjm

R(u; ; X )2 

X

u2Zjm

R(u; ; X )  X 2 :

If, on the other hand, there exists a choice of u in the summation for which the intersection de ned by (3.4) is improper, say u = u, then the polynomials i ui (1 6 i 6 m) must possess a non-trivial common factor m+1 2 Z[x; y]. Denote by 1 ; : : : ; m 2 Z[x; y ] the quotient polynomials satisfying the equations i (x; y ) ui = m+1 (x; y )i (x; y ) (1 6 i 6 m): (3.7) Then it is apparent that X

m

X R(u; ; X )2  R(ui ; i ; X )2 i=1 u2Zjm

+

X

2(Zj+1 nf0g)m

2Zjm+1+1 vi vm+1 =ui (16i6m)

u

X

v

!2

R(v; ; X ) ;

whence by combining Cauchy's inequality with an elementary divisor function estimate, one obtains X X X R(u; ; X )2  X 2 + X " R(v; ; X )2 u2Zjm

u2(Zj+1 nf0g)m v2(Zj+1 nf0g)m+1 vi vm+1 =ui (16i6m) X 6 X2 + X" R(v; ; X )2: v2Zjm+1+1

(3.8)

Let fi1 ; : : : ; il g denote the subset of f1 ; : : : ; m+1g in which constant polynomials are omitted. Then it is apparent from (3.7) and our initial hypothesis that

EXPONENTIAL SUMS OVER BINARY FORMS

19

F (x; y ) is a polynomial in i1 ; : : : ; il . If the degrees of the latter polynomials are respectively e1 ; : : : ; el , then it is clear from (3.7) also that e1 +


+ el < d1 +


+ dm 6 d:

Also, on combining the hypothesis (3.6) with (3.8), one deduces that X X rF (n; X )2  X 2+" + X " R(w; i1 ; : : : ; il ; X )2: n2Z0

w

2Zjl+1

(3.9)

In view of the above discussion, therefore, we infer from the hypotheses concluding with (3.6) either that X rF (n; X )2  X 2+" ; (3.10) n2Z0

or that (3.9) holds with l = 1, or else that these initial hypotheses again hold, but with j replaced by j + 1, and with the m-tuple  replaced by an m0 -tuple of polynomials with strictly smaller degree in the sense that their sum of degrees is strictly smaller. Since the sum of the degrees of the i must always be at least 1, we conclude that repeated application of this reduction must terminate after at most d steps either with the conclusion (3.10), or else with the conclusion that (3.9) holds with l = 1 and j = d. In the former case we deduce that X rF (n; X )2  X 2+" : (3.11) n2Z

In the latter situation, meanwhile, we may conclude that polynomials G(x; y) 2 Z[x; y ] and g (t) 2 Q [t] exist satisfying the conditions (b), (c) of the statement of Lemma 3.2. If G(x; y) has degree 1, or else is degenerate of degree exceeding 1, moreover, then it follows from conditions (b) and (c) that F (x; y) is itself degenerate, contrary to our earlier hypotheses. Thus condition (a) is also satis ed. Furthermore, our above discussion also yields the bound X X rF (n; X )2  X 2+" + X " rG (n; X )2: (3.12) n2Z

n2Z

On combining the estimates (3.1), (3.2), (3.11) and (3.12), we nd that the conclusion of the lemma follows in all cases. In the later stages of our argument we are reduced to equations quadratic with respect to a subset of the variables. These we handle with the aid of the following elementary estimate. Lemma 3.3. Let a; b; c be integers with abc 26= 0, 2and let S (a; b; c; P ) denote the number of integral solutions of the equation ax + by = c, with jxj 6 P and jy j 6 P . Then for each positive number ", one has S (a; b; c; P )  1 + (jabcjP )". Proof. This well-known estimate can be found in Estermann [8] or Vaughan and Wooley [20, Lemma 3.5]. We now provide the re nement of Lemma 3.2 of such utility in quadratic cases, basing our argument on that occurring in the proof of Lemma 7.1 of [22].

20

TREVOR D. WOOLEY

Lemma 3.4. Let X denote a large real number. Suppose that F (x; y) 2 Z[x; y] is a non-degenerate polynomial of degree d > 2, and suppose also that F (x; y ) has degree precisely 2 in terms of x. Suppose in addition that for no rational numbers  and  is it true that there exists a polynomial f (x; y ) 2 Z[x; y ] for which the equation F (x; y ) = f (x; y )2 +  is satis ed identically. Further, suppose that for some xed positive number A, the coeÆcients of F are each bounded in absolute value by X A . Then in the notation de ned in the statement of Lemma 3:2, one has X

n2Z

rF (n; X )2  X 2+" :

We may rewrite the polynomial F (x; y) in the form F (x; y ) = (y )x2 + (y )x + (y ); (3.13) where (y) is a polynomial in y with integral coeÆcients which is not identically zero, though possibly constant, and (y); (y) 2 Z[y]. Let R1 (X ) denote the number of solutions of the equation F (x1 ; y1 ) = F (x2 ; y2 ); (3.14) with jxi j 6 X , jyi j 6 X (i = 1; 2), in which (yi ) = 0 for i = 1 or 2. De ne the polynomial
(y) by
(y) = (y)2 4(y) (y); (3.15) and let R2(X ) denote the corresponding number of solutions of (3.14) in which (yi ) 6= 0 (i = 1; 2), and one has that
(y ) is identically zero as a polynomial in y . Let R3 (X ) denote the corresponding number of solutions in which (yi ) 6= 0 (i = 1; 2), and
(y) is not identically zero as a polynomial in y, and moreover one has (y2 )
(y1 ) = (y1 )
(y2 ): (3.16) Finally, let R4(X ) denote the corresponding number of solutions with (yi ) 6= 0 (i = 1; 2), and for which the equation (3.16) does not hold. Then plainly, Proof.

4 X 2 rF (n; X ) 6 Ri (X ): i=1 n2Z X

(3.17)

We rst bound R1(X ). Suppose that (yi ) = 0 for i = 1; 2. Since (y ) is 2 not identically zero, it follows that there are at most d permissible choices for y. Since there are trivially O(X 2) possible choices for2x, we nd that the contribution to R1 (X ) from this rst class of solutions is O(X ). Consider next the remaining solutions for which (yi ) = 0 for at most one value of i. By relabelling variables, we

EXPONENTIAL SUMS OVER BINARY FORMS

21

may suppose that (y1) = 0. There are consequently at most d choices permissible for y1. Fix any one such choice, and also x any one of the O(X ) available choices for x1 . Since (y2) is non-zero, it follows from (3.13) and (3.14) that the latter equation is explicit in both x2 and y2, whence a simple counting argument reveals that the number of possible choices for x2 and y2 satisfying (3.14) is at most O(X ). There are thus O(X 2) solutions of this second type, whence R1 (X )  X 2 : (3.18) Consider next the solutions counted by R2(X ). There exist non-trivial polynomials 1 (y); 2(y) 2 Z[y] with the property that (y) = 1 (y)2(y)2, and 1 (y) has no repeated factors over C [y]. Since (y) is a non-trivial polynomial in y, it follows from (3.15) that if
(y) is identically zero as a polynomial in y, then (y) is divisible by the polynomial 1 (y)2(y). Such is immediate when (y) is non-zero, and when (y) is equal to zero one has (y) = 0, and the desired conclusion again follows. But if (y) is divisible by 1 (y)2(y), then the vanishing of
(y) ensures, by (3.15), that (y) is divisible by 1(y). We therefore deduce that for some nonzero integers 1 , 2 , and some polynomial in y with integral coeÆcients, say Æ(y), one has 1 F (x; y ) = 1 (y )(2 2 (y )x + Æ (y ))2 (3.19) identically as a polynomial in x and y. We observe here that since 1 (y) and 2 (y ) are divisors of (y ), it follows that their coeÆcients have absolute values at most O(X A) (see, for example, Granville [9]). One nds in like manner that the coeÆcients of Æ(y), and also 1 and 2 , may be chosen with absolute values at most O(X 2A). Notice also that our hypothesis that F is not a rational multiple of the square of a polynomial ensures that 1 (y) is not a constant polynomial. Let x2 and y2 be any one of the O(X 2) permissible choices counted by R2(X ). Since, by an elementary counting argument, the number of solutions of the equation F (x; y ) = 0 with jxj 6 X and jy j 6 X is O(X ), the total number of solutions x; y counted by R2 (X ) with F (x2 ; y2) = 0 is O(X 2). We may therefore suppose that our aforementioned choice of x2; y2 satis es the condition that F (x2 ; y2) 6= 0, whence 1 F (x2; y2) 6= 0. But it follows from (3.14) and (3.19) that 1 (y1) and 2 2 (y1 )x1 + Æ (y1 ) are both divisors of the xed non-zero integer 2 F (x2 ; y2 ). By elementary estimates for the divisor function, therefore, there are at most O(X ") possible choices for integers d1 and d2 with 1(y1) = d1 and 2 2 (y1)x1 +Æ(y1 ) = d2 . Since 1(y) is not a constant polynomial, the rst of the latter equations shows that there are at most d possible choices for y1 . Given any one xed such choice of y1 , on noting that the non-vanishing of (y1) ensures also that 2(y1 ) 6= 0, one nds that x1 is uniquely determined from the second of these equations. Thus we deduce that R2 (X )  X 2+" : (3.20) Consider next the solutions x; y counted by R3(X ). If, on the one hand, the polynomial equation (3.16) is non-trivial in y1 and y2 , then a simple counting argument shows that there are O(X ) permissible choices for y1 and y2 satisfying

22

TREVOR D. WOOLEY

(3.16). Given any one such choice of y, in view of the presumed non-vanishing of (yi ) (i = 1; 2), it follows from (3.13) that the equation (3.14) is non-trivial in x1 and x2 , whence there are O(X ) permissible choices of x1 and x2 satisfying (3.14). Thus the total number of solutions of this type is O(X 2). If, on the other hand, the polynomial equation (3.16) is trivial in y1 and y2, then it follows that
(y) is a non-zero constant multiple of (y), say
(y) = (y). We may again write 2 (y ) = 1 (y )2 (y ) , with 1 and 2 de ned as in the treatment of R2 (X ). An inspection of (3.15) now reveals that 1 (y )2(y )2 = (y )2 41 (y )2(y )2 (y ); whence (y) is a multiple of 1(y)2(y). Write (y) =  1 1 (y)1(y)2(y), where  is a non-zero integer and 1 (y ) 2 Z[y ]. Note here that, as in the above discussion, one may suppose that the coeÆcients of 1 (y), 1(y) and 2 (y), together with the integer , have absolute values at most O(X 2A). We thus infer that 42 (y) = 1 (y) 1(y)2 2 : On substituting into (3.13), we nd that 42F (x; y) = 1 (y)(22(y)x + 1 (y))2 2 : In particular, our hypothesis that F (x; y) is not a translation of a rational multiple of a square of a polynomial ensures that 1(y) is not a constant polynomial. In this way, it follows that the equation (3.14) takes the shape 1 (y1 )(22 (y1 )x1 + 1 (y1 ))2 = 1 (y2 )(22(y2 )x2 + 1 (y2 ))2 : A comparison between the polynomial 1 (y)(22(y)x + 1 (y))2 and that on the right hand side of (3.19) reveals that we may now apply the argument concluding the treatment of 2+ R2 (X ) above in order to conclude that the number of solutions of this type is O(X "). Thus we have R3 (X )  X 2+" : (3.21) Finally, we discuss the solutions counted by R4 (X ). Let x; y be any solution of (3.14) of the latter type. Then on recalling (3.13), (3.14) and (3.15), we deduce that (y2 )(2(y1 )x1 + (y1 ))2 (y1 )(2(y2 )x2 + (y2 ))2 = (y2)
(y1) (y1)
(y2): (3.22) But in view of our hypotheses relevant to R4(X ), for each of the O(X 2) permissible values of y, one has that the right hand side of (3.22) is a non-zero integer, say

EXPONENTIAL SUMS OVER BINARY FORMS

23

N . Fix any one such choice of y, and note that our hypotheses ensure also that (yi ) 6= 0 (i = 1; 2). But by Lemma 3.3, the number of solutions of the equation (y2 ) 2 (y1 ) 2 = N;

with  and  each bounded in absolute value by a xed power of X , is O(X "). Consequently, the number of possible xi (i = 1; 2) is also O(X "), and thus we conclude that R4 (X )  X 2+" : (3.23) The conclusion of the lemma now follows immediately on collecting together the estimates (3.18), (3.20), (3.21), (3.23) with (3.17). We note that the treatment of K3 (X ; h) in the proof of Lemma 7.1 of [22] contains an oversight in that (y; h) was presumed to be necessarily zero, as a consequence of the argument presented therein. The treatment of R3(X ) above takes care of this oversight, and the diligent reader will nd that there are no substantive diÆculties encountered here. Indeed, one may assume in the above treatment that 2 (y) is identically equal to 1 when applying this argument in the context of the treatment of K3 (X ; h) in the aforementioned work. Before proceeding to the main inductive part of our argument, we require still another estimate of simpler type than those embodied in Lemmata 3.2 and 3.4. Lemma 3.5. Let X denote a large real number. Suppose that F (x; y) 2 Z[x; y] is a non-degenerate polynomial of total degree 2, and suppose also that F (x; y ) has degree precisely 1 in terms of x. Further, suppose that for some xed positive number A, the coeÆcients of F are each bounded in absolute value by X A . Then in the notation de ned in the statement of Lemma 3:2, one has X

n2Z

rF (n; X )2  X 2+" :

We may rewrite the polynomial F (x; y) in the shape F (x; y ) = (y )x + (y ); where (y) is a linear polynomial in y with integral coeÆcients that is not identically zero, and (y) is a quadratic polynomial in y with integral coeÆcients. By 2 considering the putative coeÆcient of x , it is apparent that F (x; y) cannot be the translate of a rational multiple of the square of a polynomial. Consequently, when (y ) has non-vanishing leading coeÆcient we may reverse the roles of x and y , and appeal to Lemma 3.4 in order to establish the conclusion of the lemma. When the leading coeÆcient of (y) is zero, on the other hand, it follows that for some integers a, b, c and d, with a 6= 0, one has F (x; y ) = axy + bx + cy + d; Proof.

24

TREVOR D. WOOLEY

whence But then one has

aF (x; y ) = (ax + c)(ay + b) + ad bc:

X rF (n; X )2 = rG (n; X )2; (3.24) n2Z n2Z where G(x; y) = (ax + c)(ay + b). When n is non-zero and G(x; y) = n, an elementary divisor function estimate shows that there are O(X ") possible choices for ax + c and ay + b, whence also for x and y . When n is zero, on the other hand, one has that ax + c = 0 or else that ay + b = 0, so that the corresponding number of solutions is O(X ). Consequently, X

n2Z

X

rG (n; X )2  rG (0; X )2 + X "

X

n2Znf0g

rG (n; X )  X 2+" ;

and the desired conclusion is again immediate, in view of (3.24). 4. The inductive step. We are now equipped to discuss the main inductive step in the proof of Theorem 1. Consider a non-degenerate binary form (x; y) of the shape (2.1), and de ne the exponential sum H (; X ) as in (2.2). When X is a large real number and s is a positive number, de ne Is (X ) =

1

Z

0

jH (; X )jsd:

Lemma 4.1. Let (x; y) 2 Z[x; y] be a non-degenerate form of degree k, with 3 6 k 6 10, of the shape discussed above. Then one of the following statements is true. (i) For 1 6 j 6 k, and for each positive number ", one has I2j 1 (X )  X 2j

j +" :

(ii) For each positive number s, and for each integer j with for each " > 0 the upper bound

Is+2j (X )  X 2j+1 1 Is (X ) + X 2j+1

1 6 j 6 k 3, one has

12 (j +2 Æ)+" I2s

( (X ))1=2;

where Æ = Æ (j ) is de ned by

Æ (j ) =



1=(k 0;

j ); when 1 6 j < k when j = k 3.

3,

We begin by noting that the conclusion (i) of the lemma is immediate from Lemma 2.2 when k = 3 or 4, and also when k > 5 and j = 1 or 2. When k > 5 and t > k 1, moreover, the conclusion of Lemma 2.3 demonstrates either that Proof.

EXPONENTIAL SUMS OVER BINARY FORMS

25

conclusion (i) holds, or else that conclusion (ii) will follow provided we establish the validity of the latter when is replaced by a condensation  of of the shape (2.1) wherein t = 2. There is therefore no loss of generality in supposing that 2 6 t 6 k 2. Similarly, when k > 5 and t = k 2, the conclusions of Lemmata 2.4 and 2.5 ensure either that conclusion (i) holds, or else that conclusion (ii) will follow provided we establish the validity of the latter when is replaced by a condensation  of of the shape (2.1) wherein t = 2 or t = 3. We therefore deduce that the conclusion of the lemma follows by establishing the inequality recorded in (ii) for those polynomials for which either k = 5 and t = 2 or 3, or else 6 6 k 6 10 and 2 6 t 6 k 3. We suppose henceforth that the latter conditions do indeed hold. We now modify the argument applied in xx6 and 7 of [22], applying a more elaborate dierencing procedure, and considering also moments other than the even ones. Let w be a parameter to be chosen later satisfying the inequalities maxf1; j k + t + 2g 6 w 6 minfj; t 1g: (4.1) The signi cance of these inequalities will become clear in due course. For the moment we remark only that our hypotheses concerning j , t and k ensure that an integral value of w can always be found satisfying (4.1). We rst view the exponential sum H () = H (; X ) as an exponential sum over v , so that on applying Holder's inequality to (2.2), and then making use of Lemma 2.6, we deduce that jH ()j2w

 X 2w 1

X X

2w

e( (u; v ))

juj6X jvj6X X X w+1 w 2 2 X jK (; h; v)j; h2[ 2X;2X ]w v2I (h)

where I = I (h1; : : : ; hw ) is an interval of integers contained in [ X K (; h; v ) = e(
0;w ( (u; v ); h)):

(4.2)

X; X ], and

juj6X

Next applying Lemma 2.6 to the latter exponential sum, we obtain X X jK (; h; v)j2j w  X 2j w j+w 1 e(p(v ; u; g; h)); g2[

g)

2X;2X ]j w u2J (

(4.3)

where J = J (g1; : : : ; gj w ) is an interval of integers contained in [ X; X ], and the polynomial p(v; u; g; h) is de ned by p(v ; u; g; h) =
j w;w ( (u; v ); g; h): (4.4) Wek note for future reference that, on recalling (4.1), and considering the term Bu t v t in (2.1), it is apparent that the polynomial p(v ; u; g; h) is not identically zero.

26

TREVOR D. WOOLEY

On combining (4.2) and (4.3) via Holder's inequality, we conclude that jH ()j2j  X 2j+1

where

G () =

X

m2[

X

2X;2X ]j

j 2 G ();

X

u2J (g) v2I (h)

(4.5)

e(p(v ; u; g; h));

and here, and throughout, we adopt the convention that m = (m1 ; : : : ; mj ); h = (m1 ; : : : ; mw ) and g = (mw+1 ; : : : ; mj ): De ne the exponential sum G1 () by G1 () =

XX

m

u;v

(4.6) (4.7)

e(p(v ; u; g; h));

where the summation is restricted to the values of m; u; v satisfying m 2 [ 2X; 2X ]j ; u 2 J (g); v 2 I (h); (4.8) with p(v ; u; g; h) 6= 0: (4.9) Since p(v; u; g; h) is not identically zero, it follows from an elementary argument that the number of choices of m; u; v satisfying (4.8) and p(v; u; g; h) = 0 is at most O(X j +1 ). Consequently, one has jG () G1 ()j  X j+1: Then in view of (4.5), we obtain Is+2j (X ) =

1

Z

0

jH ()js+2j d  X 2j+1 j 2

 X 2j+1 1

1

Z

0

1

Z

0

G ()jH ()jsd

jH ()jsd + X 2j+1 j 2

Let T denote the mean value

T (X ) =

1

Z

0

Z

0

1

jG1 ()H ()sjd:

jG1 ()j2d:

(4.10)

Then an application of Schwarz's inequality leads us to the estimate Is+2j (X )  X 2j+1 1 Is(X ) + X 2j+1

j 2 (T (X ))1=2(I

2s (X ))1=2:

(4.11)

EXPONENTIAL SUMS OVER BINARY FORMS

27

Next observe that, in view of (4.8) and (4.9), and on considering the diophantine equation underlying (4.10), the mean value T (X ) is bounded above by K(X ), where K(X ) denotes the number of integral solutions of the equation p(x1 ; y1; g1 ; h1) = p(x2 ; y2 ; g2 ; h2 );

(4.12)

with mi 2 [ 2X; 2X ]j , in the sense of (4.7), also with jxi j; jyij 6 X (i = 1; 2), and subject to the conditions p(xi ; yi; gi ; hi) 6= 0 (i = 1; 2). A comparison between (4.11) and the estimate claimed in the statement of the lemma therefore reveals that the desired conclusion follows immediately from the upper bound K(X )  X j+2+Æ+":

(4.13)

We henceforth concentrate our eorts on establishing (4.13). On recalling (4.4), a modicum of computation reveals that p(x; y ; g; h) = m1 : : : mj F (x; y ; m);

where F (x; y ; m) =

k X i=t

(4.14)

Di k i (y ; m) i (x; m);

(4.15)

in which Di is an integer for t 6 i 6 k, and Dt 6= 0, and in which each i (x; m) is a polynomial with integral coeÆcients of degree i w with respect to x, and each k i (y ; m) is a polynomial with integral coeÆcients of degree k i j + w with respect to y. In view of (4.1), one has 2 6 k t j + w 6 k j 1 and t w > 1. Thus F (x; y; m) has degree at least 1 with respect to x, and degree at least 2 and at most k j 1 with respect to y. We note also for future reference that when w = 1 and j = 1, one may take k t (y ; m) = y k t

and

t (x; m) = m

1 ((x + m)t

xt ):

(4.16)

Finally, we observe that the argument surrounding equations (6.17) and (6.18) of [22] easily establishes that when ml 6= 0 (1 6 l 6 j ), then one has that the polynomial F (x; y; m) is non-degenerate with respect to x and y. When m 2 [ 2X; 2X ]j , let (n; m) denote the number of integral solutions of the equation p(x; y; g; h) = n, with jxj; jyj 6 X . Then it follows from (4.14) and (4.12) that K(X ) =

X

X

n2Znf0g jm1 j62X m1 jn






X

jmj j62X mj jn

!2

(n; m) :

28

TREVOR D. WOOLEY

Consequently, on applying Cauchy's inequality in combination with an elementary estimate for the divisor function, one obtains X X K(X )  X " (n; m)2 n2Znf0g m2[ 2X;2X ]j mi 6=0 (16i6j ) X  X j+" max j (n; m)2 m2[ 2X;2X ] mi 6=0 (16i6j ) n2Znf0g = X j+" m2[ max M(X ; m); 2X;2X ]j mi 6=0 (16i6j )

(4.17)

where M(X ; m) denotes the number of solutions of the equation F (x1 ; y1 ; m) = F (x2 ; y2 ; m); (4.18) with jxi j; jyij 6 X (i = 1; 2). We recall at this point that, by hypothesis, one has either k = 5; t = 2 or 3 and j = 1 or 2; or else 6 6 k 6 10; 2 6 t 6 k 3 and 1 6 j 6 k 3: We now divide our argument into a number of cases, our aim being to make a choice of w, satisfying the condition (4.1), for which the estimates of x3 prove eective. (a) (k; t; j ) satis es j = k 3. We take w = t 1. In this situation it is apparent that k t j + w = 2; (4.19) and also that (k j; t w) = 1: (4.20) Consider the shape of the polynomial F (x; y; m) when the conditions (4.19) and (4.20) hold. We isolate the monomial of highest degree with respect to y that has highest degree with respect to x. In view of (4.15) and the associated discussion, this monomial has the shape Dt xt w y k t j +w : (4.21) Suppose, if possible, that there exist polynomials G(x; y) 2 Z[x; y] and g(t) 2 Q [t] satisfying the conditions (a) and (b) of the statement of Lemma 3.2, and also satisfying the condition that F (x; y) = g(G(x; y)). Then the monomial of highest degree with respect to y that has highest degree with respect to x in the hl hm polynomial g(G(x; y)), must necessarily have the shape Cx y , where h is the degree of g(t). But the condition (4.20) implies that (k t j + w; t w) = 1; (4.22)

EXPONENTIAL SUMS OVER BINARY FORMS

29

and so the latter conclusion contradicts (4.21). It follows, in particular, that for no rational numbers  and  is it true that there exists a polynomial f (x; y) 2 Z[x; y ] for which the equation F (x; y ; m) = f (x; y )2 +  is satis ed identically in x and y. But in view of (4.21) and (4.19), the polynomial F (x; y; m) has degree precisely 2 with respect to y. Since, moreover, the coeÆcients of F are each bounded in absolute value by a xed power of X , it follows from Lemma 3.4 that in this case one has M(X ; m)  X 2+": (4.23) On recalling (4.17), we nd that (4.13) holds with Æ = 0, and thus the proof of the lemma in the case j = k 3 is complete. (b) (k; t; j ) satis es t 1 6 j < k 3. We take w = t 1, and nd that (4.20), and hence also (4.22), remain true. In view of the discussion in case (a) above, it follows that there can exist no polynomials g and G that satisfy the hypotheses (a), (b), (c) of Lemma 3.2(i). We therefore deduce from Lemma 3.2(ii) that in this case one has M(X ; m)  X 2+1=(k j)+": (4.24) Recalling (4.17) again, we now nd that (4.13) holds with Æ = 1=(k j ), and hence the proof of the lemma follows in the case currently under consideration. (c) (k; t; j ) = (5; 3; 1). We take w = 1, and nd that (4.19) holds. Then it follows from (4.21) that in the polynomial F (x; y; m), the monomial of highest degree with respect to y, that has highest degree with respect to x, has the shape Cx2 y2. It is possible that Lemma 3.4 succeeds in supplying the bound (4.23). If such is not the case, then there exists a polynomial f (x; y) 2 Z[x; y], and rational numbers  and , for which F (x; y; m) = f (x; y)2 + . Moreover, it is apparent that  must be non-zero, and our previous discussion ensures that f (x; y) must be non-degenerate of total degree 2, with degree precisely one in terms of y. Thus we deduce that X M(X ; m) 6 rf (n; X )2; n2Z

whence by Lemma 3.5 one again obtains the conclusion (4.23). Recalling (4.17), we now nd that (4.13) holds with Æ = 0, and thus the proof of the lemma again follows. By combining the conclusions of cases (a) and (b) above, one nds that when k = 5 and j = 1 or 2, our hypothesis that t = 2 or 3 leaves only the situation in which (k; t; j ) = (5; 3; 1) to consider. But the latter case is resolved in case (c) above, and so henceforth we may suppose that 6 6 k 6 10. In the latter circumstances, cases (a) and (b) also dispose of all cases in which t = 2, and also all cases wherein j > t 1. Thus we may suppose henceforth that 6 6 k 6 10; 3 6 t 6 k 3 and 1 6 j 6 t 2:

30

TREVOR D. WOOLEY

We treat the remaining allowable cases by hand. (d) (k; t; j ) = (6; 3; 1), (7; 4; 2). We take w = j , and nd that (4.20) holds, since (5; 2) = 1. The argument of part (b) therefore yields the bound (4.24), and hence also (4.13). (e) (k; t; j ) = (8; 3; 1), (8; 4; 1), (8; 5; 1). We take w = 1, and nd that (4.20) holds, since (7; t 1) = 1 for t = 3; 4; 5. The argument of part (b) therefore yields the bound (4.24), and hence also (4.13). (f) (k; t; j ) = (8; 5; 3). We take w = 3, and nd that (4.20) holds, since (5; 2) = 1. The argument of part (b) therefore yields the bound (4.24), and hence also (4.13). (g) (k; t; j ) = (9; 4; 1), (9; 6; 1). We take w = 1, and nd that (4.20) holds, since (8; t 1) = 1 for t = 4; 6. The argument of part (b) therefore yields the bound (4.24), and hence also (4.13). (h) (k; t; j ) = (9; 4; 2), (9; 5; 2), (9; 6; 2). We take w = 2, and nd that (4.20) holds, since (7; t 2) = 1 for t = 4; 5; 6. The argument of part (b) therefore yields the bound (4.24), and hence also (4.13). (i) (k; t; j ) = (10; 3; 1), (10; 5; 1), (10; 6; 1). We take w = 1, and nd that (4.20) holds, since (9; t 1) = 1 for t = 3; 5; 6. The argument of part (b) therefore yields the bound (4.24), and hence also (4.13). (j) (k; t; j ) = (10; 4; 2), (10; 6; 2). We take w = 1, and nd that (4.20) holds, since (8; t 1) = 1 for t = 4; 6. The argument of part (b) therefore yields the bound (4.24), and hence also (4.13). (k) (k; t; j ) = (10; 5; 2), (10; 7; 2). We take w = 2, and nd that (4.20) holds, since (8; t 2) = 1 for t = 5; 7. The argument of part (b) therefore yields the bound (4.24), and hence also (4.13). (l) (k; t; j ) = (10; 5; 3), (10; 6; 3), (10; 7; 3). We take w = 3, and nd that (4.20) holds, since (7; t 3) = 1 for t = 5; 6; 7. The argument of part (b) therefore yields the bound (4.24), and hence also (4.13). (m) (k; t; j ) = (9; 6; 4). We take w = 3, and nd that (4.19) and (4.20) both hold, since (5; 3) = 1. The argument of part (a) now establishes the bound (4.23), and hence also (4.13). (n) (k; t; j ) = (10; 7; 5). We take w = 4, and nd that (4.19) and (4.20) both hold, since (5; 3) = 1. The argument of part (a) now establishes the bound (4.23), and hence also (4.13). (o) (k; t; j ) = (7; 4; 1). We take w = 1, and nd from (4.21) that in the polynomial F (x; y ; m), the monomial of highest degree with respect to y , that has highest degree with respect to x, has the shape Cx3 y3. It is possible that Lemma 3.2 succeeds in supplying the bound (4.24). If such is not the case, then there exist polynomials G(x; y) 2 Z[x; y] and g(t) 2 Q [t] satisfying the conditions (a), (b), (c) of Lemma 3.2. It is evident, moreover, that in such circumstances the degree of g must be 3, and the total degree of G(x; y) must be 2, and also the degree of G(x; y ) with respect to y must be 1. In the latter circumstances, it follows from Lemma 3.5 that one has the estimate X rG (n; X )2  X 2+" ; (4.25) n2Z

EXPONENTIAL SUMS OVER BINARY FORMS

whence by Lemma 3.2(i), M(X ; m)  X 2+1=(k

j )+" + X "

 X 2+1=(k j)+":

X

n2Z

31

rG (n; X )2

(4.26)

Then in any case, one has the upper bound (4.24), and hence (4.13). (p) (k; t; j ) = (8; 5; 2). We take w = 2, and nd from (4.21) that in the polynomial F (x; y ; m), the monomial of highest degree with respect to y , that has highest degree with respect to x, has the shape Cx3 y3. The desired bound (4.24), and hence (4.13), now follows by the argument of case (o). (q) (k; t; j ) = (9; 5; 3). We take w = 2, and nd from (4.21) that we may again apply the argument of case (p). (r) (k; t; j ) = (9; 6; 3). We take w = 3, and nd from (4.21) that we may again apply the argument of case (p). (s) (k; t; j ) = (10; 6; 4). We take w = 3, and nd from (4.21) that we may again apply the argument of case (p). (t) (k; t; j ) = (10; 7; 4). We take w = 4, and nd from (4.21) that we may again apply the argument of case (p). (u) (k; t; j ) = (7; 3; 1). We take w = 1, and nd from (4.21) that in the polynomial F (x; y ; m), the monomial of highest degree with respect to y , that has highest degree with respect to x, has the shape Cx2 y4. It follows that if there exist polynomials G(x; y) 2 Z[x; y] and g(t) 2 Q [t] satisfying the conditions (a), (b), (c) of Lemma 3.2, then g must have degree 2. Moreover, in the polynomial G(x; y ), the monomial of highest degree with respect to y , that has highest degree with respect to x, must have the shape C 0 xy2. In such circumstances, one may apply the argument of case (a) above to obtain the bound (4.25). Then the estimate (4.24) follows in all circumstances from Lemma 3.2, and this suÆces to establish (4.13). (v) (k; t; j ) = (8; 4; 2). We take w = 2, and nd from (4.21) that in the polynomial F (x; y ; m), the monomial of highest degree with respect to y , that has highest degree with respect to x, has the shape Cx2 y4. The desired bound (4.24), and hence (4.13), now follows by the argument of case (u). (w) (k; t; j ) = (9; 5; 1). We take w = 1, and nd from (4.21) that in the polynomial F (x; y ; m), the monomial of highest degree with respect to y , that has highest degree with respect to x, has the shape Cx4 y4. It follows that if there exist polynomials G(x; y) 2 Z[x; y] and g(t) 2 Q [t] satisfying the conditions (a), (b), (c) of Lemma 3.2, then g must have degree either 2 or 4. When the degree of g is 4, the polynomial G(x; y) must have total degree 2, and the degree of G(x; y) with respect to y must be 1. In these circumstances, Lemma 3.5 establishes the estimate (4.25). When the degree of g is 2, meanwhile, we may suppose without loss of generality that there are no rational numbers  and  for which a polynomial f (x; y) 2 Z[x; y] exists satisfying G(x; y) = f (x; y)2 +  (if such a polynomial f were to exist, then we would be in the situation already considered wherein the degree of g was presumed to be 4). But then, in the polynomial

32

TREVOR D. WOOLEY

G(x; y ),

the monomial of highest degree with respect to y, that has highest degree with respect to x, has the shape C 0 x2y2. The hypotheses of Lemma 3.4 are therefore satis ed with F replaced by G, and the upper bound (4.25) again follows. We may therefore conclude from Lemma 3.2(i) that in either case, one has the estimate (4.26). If no such polynomials G and g exist, on the other hand, then the estimate (4.26) is immediate from Lemma 3.2(ii). (x) (k; t; j ) = (10; 4; 1). We take w = 1, and nd from (4.21) that in the polynomial F (x; y ; m), the monomial of highest degree with respect to y , that has highest degree with respect to x, has the shape Cx3 y6. It follows that if there exist polynomials G(x; y) 2 Z[x; y] and g(t) 2 Q [t] satisfying the conditions (a), (b), (c) of Lemma 3.2, then g must have degree 3. Moreover, in the polynomial G(x; y ), the monomial of highest degree with respect to y , that has highest degree with respect to x, has the shape C 0 xy2 . Thus we may proceed as in case (u) to obtain the desired estimate (4.13). (y) (k; t; j ) = (9; 3; 1). We take w = 1, and nd from (4.21) that in the polynomial F (x; y ; m), the monomial of highest degree with respect to y , that has highest degree with respect to x, has the shape Cx2 y6. It follows that if there exist polynomials G(x; y) 2 Z[x; y] and g(t) 2 Q [t] satisfying the conditions (a), (b), (c) of Lemma 3.2, then g must have degree 2, and G(x; y) must have the shape G(x; y ) = xy 3 + y 3 + H (x; y ); (4.27) with H (x; y) of degree at most 2 with respect to y. In view of (4.16), one nds that with a suitable non-zero constant K , one has that F (x; y ; m) = Ky 6 (3x2 + 3xm + m2 ) + I (x; y ; m); (4.28) where I (x; y; m) is a polynomial of degree at most 5 with respect to y. Since we may suppose that g has degree 2, it follows from (4.27) and (4.28) that there is a non-zero number a with a(xy 3 + y 3 )2 = Ky 6(3x2 + 3xm + m2 ): On equating coeÆcients of powers of x, we nd that a2 = 3K; 2a = 3Km; a 2 = Km2 ; whence 9K 2 m2 = 4(a2)(a 2) = 12K 2m2 : This yields a contradiction whenever m 6= 0, as we may suppose. In this way we nd that no such polynomials G, g exist, and hence Lemma 3.2(ii) establishes that the estimate (4.26) holds. (z) (k; t; j ) = (10; 7; 1). We take w = 1, and nd from (4.21) that in the polynomial F (x; y ; m), the monomial of highest degree with respect to y , that has highest degree with respect to x, has the shape Cx6 y3. It follows that if there exist

EXPONENTIAL SUMS OVER BINARY FORMS

33

polynomials G(x; y) 2 Z[x; y] and g(t) 2 Q [t] satisfying the conditions (a), (b), (c) of Lemma 3.2, then g must have degree 3, and G(x; y) must have the shape G(x; y ) = yx2 + yx + y + H (x);

(4.29)

with H (x) a polynomial independent of y. In view of (4.16), one nds that with a suitable non-zero constant K , one has that F (x; y ; m) =Ky 3 (7x6 + 21x5 m + 35x4 m2 + 35x3 m3 + 21x2 m4 + 7xm5 + m6 ) + I (x; y; m); (4.30)

where I (x; y; m) is a polynomial of degree at most 2 with respect to y. Since we may suppose that g has degree 3, it follows from (4.29) and (4.30) that there is a non-zero number a with a(x2 y + xy + y )3 = Ky3(7x6 + 21x5m + 35x4m2 + 35x3 m3 + 21x2m4 + 7xm5 + m6 ):

On equating coeÆcients of powers of x, we nd that a3 = 7K;

3a2 = 21Km;

a(32 + 3 2 ) = 35Km2 ; a 3 = Km6 :

Thus we deduce that 276a2 Km6 = a3 (32 )3 = (35Km2 3a 2 )3 ; whence 3375 (Km)6 = (35Km2(a3 ) 3(a2 )2 )3 = (245K 2m2 147K 2m2 )3 = (98K 2m2 )3: Since 33 75 6= 983, we obtain a contradiction whenever m 6= 0, as we may suppose. In this way, we nd that no such polynomials G, g exist, and hence Lemma 3.2(ii) establishes that the estimate (4.26) holds. On collecting together the conclusions of cases (a){(z), we nd that the estimate (4.13) holds in all circumstances under consideration. The conclusion of the lemma now follows immediately from (4.11). 5. The completion of the proof of Theorem 1. We are now prepared to complete the proof of Theorem 1. We begin with an induction based on the use of Lemma 4.1 for the small moments.

34

TREVOR D. WOOLEY

Lemma 5.1. Let (x; y) 2 Z[x; y] be a non-degenerate form of degree k, with 5 6 k 6 10, of the shape discussed in the opening paragraph of x4. Then for each j with 3 6 j 6 k 2, and for each positive number ", one has 1

Z

0

jH (; X )j2j 1 d  X 2j

j +1=(k j +2)+" :

The conclusion of the lemma is either immediate from part (i) of Lemma 4.1, or else we may apply part (ii) of that lemma. By part (i) of Theorem 1, which we have already established in Lemma 2.2, one has Proof.

1

Z

0

Thus we have

jH (; X )j2d  X 2+": I2 (X )  X 2+":

Suppose that, in fact, one has the estimate I2j 1 (X )  X 2j j+1=(k j+2)+" ; (5.1) for 2 6 j 6 J , where J is an integer with 2 6 J 6 k 3. We apply part (ii) of Lemma 4.1 with j = J 1 and s = 2J 1 in order to obtain Is+2J 1 (X )  X 2J 1 Is (X ) + X 2J 21 (J +1 Æ)+" (I2s(X ))1=2; with Æ = 1=(k J + 1). On employing the inductive hypothesis (5.1), we obtain I2J (X )  X 2J +1 J 1+1=(k J +2)+" + X 2J 12 (J +1 Æ)+" (I2J (X ))1=2; whence I2J (X )  X 2J +1 J 1+1=(k J +1)+": This establishes the inductive hypothesis for j = J + 1, and thus the conclusion of the lemma follows by induction. Lemma 5.2. With the hypotheses of the statement of Lemma 5:1, one has 1

Z

0

jH (; X )j 329 2k d  X 169 2k

k+1+" :

As in the proof of the previous lemma, the desired conclusion is either immediate from part (i) of Lemma 4.1, or else we may apply part (ii) of that lemma. By the conclusion of Lemma 5.1 with j = k 2, one has Proof.

I2k 3 (X ) =

1

Z

0

jH (; X )j2k 3 d  X 2k

2 k+9=4+" :

(5.2)

EXPONENTIAL SUMS OVER BINARY FORMS

35

On applying part (ii) of Lemma 4.1 with s = 2k 3 and j = k 3, one obtains I2k 2 (X )  X 2k 2 1 I2k 3 (X ) + X 2k 2 (k 1)=2+" (I2k 2 (X ))1=2; whence by (5.2), I2k 2 (X )  X 2k 1 k+5=4+" : (5.3) An application of Holder's inequality establishes the upper bound Z 1 I 325 2k (X ) = jH (; X )j 325 2k d 0

6

1=4 Z 1 3=4 k 2 2 jH (; X )j d jH (; X )j2k 3 d : 0 0

Z

1

Thus, by (5.2) and (5.3) we deduce that (5.4) I 325 2k (X )  X 165 2k k+2+" : A second application of part (ii) of Lemma 4.1, now with s = 325 2k and j = k 3, gives the estimate  1=2 I 329 2k (X )  X 2k 2 1 I 325 2k (X ) + X 2k 2 (k 1)=2+" I 165 2k (X ) : But a trivial estimate for H (; X ) demonstrates that Z 1 k 4 2 jH (; X )j 329 2k d = X 2k 4 I 329 2k (X ): I 165 2k (X )  X 0 In view of (5.4), therefore, we obtain 1=2 9 2k k+1+" 9 2k (k 1)=2+"  16 32 I 329 2k (X )  X +X I 329 2k (X ) ; and the conclusion of the lemma follows immediately. The large moments are estimated via the Hardy-Littlewood method by means of a treatment contained, in all essentials, within the proof of Lemma 7.4 of [22]. We include an account of the proof for the sake of completeness. We rst require a major arc estimate stemming from our version of Weyl's inequality. Lemma 5.3. Suppose that (x; y) 2 Z[x; y] is a non-degenerate form of degree d > 3, and let  2 R . (i) Suppose that there exist r 2 Z and q 2 N with (r; q ) = 1 and j r=q j 6 q 2 . Then for each " > 0, one has X X
22 d e((x; y ))  X 2+" q 1 + X 1 + qX d :

16x6X 16y6X (ii) Whenever r 2 Z and q 2 N satisfy 1 6 q 6 X and jq rj 6 X 1 d , one has X X 2 d e((x; y ))  X 2+" (q + X d jq rj) 2 : 16x6X 16y6X

The rst conclusion is immediate from Theorem 1 of [22], and the second conclusion is Lemma 7.3 of [22]. Proof.

36

TREVOR D. WOOLEY

Lemma 5.4.

With the hypotheses of the statement of Lemma 5:1, one has

1

Z

0

jH (; X )j 3217 2k d  X 1716 2k

k+" :

For the sake of concision, we abbreviate H (; X ) to H (). When r 2 Z and q 2 N , write M(q; r) =  2 [0; 1) : jq rj 6 X 1 k : Take M to be the union of the intervals M(q; r) with 0 6 r 6 q 6 X and (r; q) = 1. Note that the intervals occurring in the latter union are disjoint. Also, write m = [0; 1) n M. Since Lemma 5.3(i) yields the estimate sup jH ()j  X 2 22 k +"; 2m Proof.

and Lemma 5.2 establishes that 1

Z

0

we deduce that Z

m

jH ()j 329 2k d  X 169 2k

j 3217 2k d 

jH ()

k+1+" ;

2k 2 Z



sup jH ()j 2m

 X 1617 2k k+" :

1 0

jH ()j 329 2k d

(5.5) On making use of Lemma 5.3(ii) and the de nition of M, on the other hand, we obtain Z

M

q Z X k k +" X 17 17 2 2 j 32 d  X 16 a=1 j j6(qX k 1) 1 16q6X (a;q )=1 q X k k+" X 17 2 16 X q 2 a=1 16q6X (a;q )=1 k 17  X 16 2 k+2":

jH ()

(q + X k qj j)

2 d

(5.6) Consequently, on combining the estimates (5.5) and (5.6), we arrive at the upper bound 1

Z

0

jH ()

j 3217 2k d

=

Z

j 3217 2k d

jH ()

M17

 X 16 2k

k+2" ;

Z

+ jH ()j 3217 2k d m

EXPONENTIAL SUMS OVER BINARY FORMS

37

and so the conclusion of the lemma follows immediately. On recalling the conclusion of Lemma 2.1, and noting that all of the moments occurring in the statement of Theorem 1(iii) are even, one nds that the upper bounds provided in Theorem 1(iii) are immediate from Lemmata 5.1, 5.2 and 5.4. The same is true also for the rst three bounds recorded in Theorem 1(ii), but in this case, for the second two estimates, one combines Lemmata 5.1, 5.2 and 5.4 via Holder's inequality in the respective shapes 1

Z

0

jf(; P )j8d  

and

1

Z

0

jf(; P )j10d  

1

Z

0

jH (; X )j8d

Z

1 0

1

Z

0

1=5 Z

1

jH (; X )j4d

0

4=5

jH (; X )j9d

;

jH (; X )j10d

Z

1 0

7=8 Z

jH (; X )j9d

1 0

1=8

jH (; X )j17d

:

The nal estimate of Theorem 1(ii), on the other hand, may be established along the lines of the proof of Lemma 5.4, now working from the 10th moment 1

Z

0

jf(; P )j10d  P 127=8+";

together with the minor arc bound sup jf (; P )j  P 15=8+"; 2m which is immediate from Lemma 5.3.

References

1. M. A. Bennett, N. P. Dummigan and T. D. Wooley, The representation of integers by binary additive forms, Compositio Math. 111 (1998), 15{33. 2. B. J. Birch, Forms in many variables, Proc. Roy. Soc. Ser. A 265 (1961), 245{263. 3. B. J. Birch and H. Davenport, Note on Weyl's inequality, Acta Arith. 7 (1961/62), 273{277. 4. K. D. Boklan, The asymptotic formula in Waring's problem, Mathematika 41 (1994), 329{ 347. 5. E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), 337{357. 6. J. Brudern and T. D. Wooley, The addition of binary cubic forms, R. Soc. Lond. Philos. Trans. Ser. A. Math. Phys. Eng. Sci. 356 (1998), 701{737. 7. S. Chowla and H. Davenport, On Weyl's inequality and Waring's problem for cubes, Acta Arith. 6 (1961/62), 505{521.

38

TREVOR D. WOOLEY

8. T. Estermann, Einige Satze uber quadratfrei Zahlen, Math. Ann. 105 (1931), 653{662. 9. A. Granville, Bounding the coeÆcients of a divisor of a given polynomial, Monatsh. Math. 109 (1990), 271{277. 10. D. R. Heath-Brown, Weyl's inequality, Hua's inequality, and Waring's problem, J. London Math. Soc. (2) 38 (1988), 216{230. 11. L.-K. Hua, On Waring's problem, Quart. J. Math. Oxford 9 (1938), 199{202. 12. J. Pila, Density of integer points on plane algebraic curves, Internat. Math. Res. Notices (1996), 903{912. 13. W. M. Schmidt, The density of integer points on homogeneous varieties, Acta Math. 154 (1985), 243{296. 14. C. M. Skinner and T. D. Wooley, Sums of two kth powers, J. Reine Angew. Math. 462 (1995), 57{68. 15. C. M. Skinner and T. D. Wooley, On the paucity of non-diagonal solutions in certain diagonal diophantine systems, Quart. J. Math. Oxford (2) 48 (1997), 255{277. 16. W. Y. Tsui and T. D. Wooley, The paucity problem for simultaneous quadratic and biquadratic equations, Math. Proc. Cambridge Philos. Soc. 126 (1999), 209{221. 17. R. C. Vaughan, On Waring's problem for cubes, J. Reine Angew. Math. 365 (1986), 122{178. 18. R. C. Vaughan, On Waring's problem for smaller exponents, Proc. London Math. Soc. (3) 52 (1986), 445{463. 19. R. C. Vaughan, The Hardy-Littlewood Method, second edition, Cambridge University Press, 1997. 20. R. C. Vaughan and T. D. Wooley, Further improvements in Waring's problem, Acta Math. 174 (1995), 147{240. 21. R. C. Vaughan and T. D. Wooley, Further improvements in Waring's problem, IV: higher powers, Acta Arith. 94 (2000), 203{285. 22. T. D. Wooley, On Weyl's inequality, Hua's lemma, and exponential sums over binary forms, Duke Math. J. 100 (1999), 373{423. Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, U.S.A.

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