Séminaire de Théorie Des Nombres, Paris 1988-89 3764334932, 9783764334932

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Seminaire de | Theorie des Nombres, Paris 1988—1989 Edited by Catherine Goldstein

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WITHDRAWN

WRIGHT STATE UNIVERSITY LIBRARIES

Progress in Mathematics Volume 91

Series Editors J. Oesterlé A. Weinstein

Séminaire de Théorie des Nombres, Paris 1988-1989 Edited by Catherine Goldstein

1990

Birkhauser Boston : Basel « Berlin

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ty =,

1970 ” ‘ Catherine Goldstein Département de Mathématiques Université de Paris-Sud Centre d’Orsay F-91405 Orsay Cedex France

Printed on acid-free paper.

© Birkhauser Boston, 1990 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use, or the internal or personal use of specific clients, is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. 3493-2/90 $0.00 + .20 ISBN 0-8176-3493-2 ISBN 3-7643-3493-2 Camera-ready copy prepared by the editor using TeX. Printed and bound by Edwards Brothers, Inc., Ann Arbor, Michigan. Printed in the U.S.A.

Oty tf Ws) Gh) 74Il

Ce volume est dédié a la mémoire de Georges Poitou, Cofondateur de ce Séminaire.

Digitized by the Internet Archive in 2022 with funding from Kahle/Austin Foundation

https://archive.org/details/seminairedetheorO000semi_j2c0

Les textes qui suivent sont pour la plupart des versions écrites de conférences données

pendant l'année

1988-89

au Séminaire

de Théorie des Nombres

de

Paris. Ce séminaire est organisé par la S.D.I.6180 du C.N.R.S. qui regroupe des arithméticiens de plusieurs universités et est dotée d’un conseil scientifique et

éditorial. Ont été aussi adjoints certains textes dont la mise à la disposition d’un large public nous a paru intéressante. Les papiers proposés ici exposent soit des

résultats nouveaux, soit des synthèses originales de questions récentes; ils ont

en particulier tous fait l’objet d’un rapport. Ce recueil doit bien sur beaucoup à tous les participants du séminaire et à ceux qui ont accepté d’en réviser les textes. Il doit surtout à Monique Le Bronnec

qui s’est chargée du secrétariat et de la mise au point définitive du manuscrit; son efficacité et sa très agréable collaboration ont été cruciales dans l'élaboration de ce livre. Nous remercions toutes deux chaleureusement Raymond Séroul qui nous a apporté un précieux concours

lors de notre initiation aux arcanes de

TeX.

Georges Poitou, cofondateur et pilier actif du séminaire de théorie des nombres, est décédé le 14 décembre 1989.

Nous dédions ce volume à sa mémoire.

Pour le Conseil éditorial et scientifique C. GOLDSTEIN

Contributors

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205 221

Contents

Sur les réseaux unimodulaires pour eCog Re C. BACHOC

IDR PRCRPENINICS OVLE ere PRE eee R. BALASUBRAMANIAN and M. RAM MURTY

RS

CES

ome

Oe

Le

A subexponential algorithm for the determination of class groups and Realm DE aise Dralc number ficlds TM norme ec adencarantye austen J. BUCHMANN Surfaces rationnelles fibrées en coniques de degré 4 ..............................

J.-L. COLLIOT-THÉLÈNE De Rham cohomology for Drinfeld modules ....................................... E.-U. GEKELER Le produit de Petersson et de Rankin p-adique ..................................... H. HIDA Cocher couiecion foe CUI CUTVES. normes dote de4s M. HINDRY and J.H. SILVERMAN Formes linéatres. d'fntécrales elliptiques N. HIRATA-KOHNO

7e

rte 4 POEM

Computing the Hodge group of an abelian variety ................................. V. KUMAR MURTY Développement de la loi de groupe sur une cubique .............................. N. SCHAPPACHER

x

CONTENTS 8

The structure of the minus class groups of abelian number fields............... R. SCHOOF On the fibration method for proving the Hasse principle and weak APPIOXIMAUON scree veeeve wewaete, RP cao Em

ae ee

185

205

A.N. SKOROBOGATOV SLE WIN DrODISIMECMETAOSIC G. TENENBAUM

PATATE

PEER

221

An effective version of Hilbert’s irreducibility theorem........................... T. EKEDAHL

241

iste. des: conferenclers Masseuse ashe eee

251

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Séminaire de Théorie des Nombres

Paris 1988-89

Sur les réseaux unimodulaires pour Trace (X?)

Christine BACHOC

L'exposé du 16 janvier 1989 a consisté en un compte rendu de résultats obtenus dans un travail commun avec Boas Erez ([B.E]) sur la forme Trace (X ch Dans cet article, nous décrivons succinctement ces résultats, puis en démon-

trons une généralisation.

1. — INTRODUCTION

Si K est un corps de nombres, tout idéal J est un réseau (i.e., un Z-module

libre muni d’une forme quadratique) pour la forme quadratique Tracex /Q(27), qui sera notée Trig.

Ainsi l'anneau

des entiers

Ox est un réseau, dont le dual est Dj’, la

codifférente, et dont le discriminant est le discriminant du corps.

Le dual d’un idéal J quelconque est I Tipe ; en particulier,

J est unimodu-

laire pour Tri g si et seulement si J vérifie la relation : J? = Dj’. Il existe donc au plus un idéal unimodulaire

pour Trx/g; une

condition

suffisante pour son existence est que K soit galoisien sur Q et de degré impair. En effet, il suffit de regarder la formule donnant la valuation de la différente

en un idéal premier p de K, en fonction de la suite (G;(p));>0 des groupes de ramification en p :

vp(Dx) = D (|Gi(p)| — 1). i>0

On se place désormais sous cette hypothèse.

Soit donc G le groupe de Galois de K/Q,

|G| impair, et

K unimodulaire pour Trxg : la relation A7 = D

Ax l'unique idéal de

montre que AK est un

2

C. BACHOC t

Z[G]-module. La forme Trx/q étant invariante par G, on dit que (Ax, Trx/@) est un Z[G]-réseau. Les Z[G]-réseaux sont classés par les Z[G]-isométries c’est-à-dire les isomorphismes de Z[G]-modules qui sont des isométries des formes quadratiques. Les résultats obtenus dans l'étude de la classe de Z[G]-isométrie du réseau unimodulaire (Ax,'Trx/@) sont les suivants : —

Cherchant à comparer (Ax, Trx/q) à (Z[G], qo) où go est la forme

standard : qo()) ag) = >} aj, B. Erez a démontré : THEOREME.

—

Si K/Q

est

une

extension

abélienne

de

degré

impair

(Ax, Trx/q) est Z[G]-isométrique à (Z[G], qo) si et seulement si K/Q est peu ramifiée, c’est-à-dire : un nombre premier p quelconque est soit modéré, soit son

indice de ramification est égal à p. —

Nous nous sommes

intéressés au cas très sauvage suivant

: G est

cyclique d'ordre p”, avec p premier impair et n > 2, et p est totalement ramifié

dans X. Alors, AK est toujours isométrique à un Z[G]-réseau, ne dépendant que de p et de n, dont un représentant est décrit au paragraphe 2 (cf. [B.E.]);

le corollaire 2, dont l’assertion 1 a été démontrée en collaboration avec B. Erez, ne figure pas dans [B.E]. —

Les résultats précédents s’étendent au cas où :

(H) : K/Q est une extension abélienne, de degré impair, dans laquelle tout nombre

premier

p est soit modéré,

soit a pour

degré de ramification

une

puissance de p. C’est l'objet du troisième paragraphe, dans lequel nous donnons la démons-

tration du fait que le réseau Ax est une somme orthogonale de produits tensoriels des réseaux précédents (§3, thm. 2).

De plus, un résultat plus précis sur la structure de Z[G]-module de Ax montre que, sous l’hypothése (H), Ax est libre sur son ordre associé.

Enfin, un court paragraphe 4 est consacré 4 quelques exemples.

SUR LES RESEAUX

UNIMODULAIRES

3

2. — G est cyclique d'ordre p", p premier impair, n > 2, et p est totalement

ramifié dans X

Notations :

— K; est l'unique sous-corps de K tel que [K; : Q] = p' — Gi est une racine primitive p'-ème de l'unité —

m est l'ordre maximal de l'algèbre Q[G]; si (ei)o hh € ([[ B) NH =(1) Donc

hah

69,9) = DC FA, fh) = (ff). i ken

Donc (@ Ar. ,b) x (@ Ag; ® qE; )-

On obtient encore, pour la structure de Z[G]-module de Ax :

10

C. BACHOC %

COROLLAIRE 3.3. — Si l'extension K/Q vérifie les hypothèses (H), alors Ax est libre sur son ordre associé.

En effet, avec les notations du paragraphe 2, on peut prendre Ay,, = Di,

qui est un ordre de l'algèbre Q[J,,]. Alors Ax est, d’après le théorème

2, Z[G]-isomorphe

au

Z[G]-module

Z(G](® Dr,. ), qui est aussi un ordre. C’est donc l'ordre associé à AK, sur lequel 1

Ax est donc libre.

4, — Exemples

Explicitons, sur quelques exemples, les réseaux obtenus par le théorème 2 :

Notons R(p”) le réseau décrit dans le théorème 1 :

R(p") = Z L B;(p) L --- L Bo(p) ou Z? 1 B3(p) L --- L B,(p) suivant la parité de n.

Exemple 1 :

|G| = pip2::-p, et e(pi) = 1 ou pi. Alors G est cyclique, l'extension est peu

ramifiée, et d’après [E], Ax ~ ZP1P2"Pr, Exemple 2 :

|G| = p” et e(p) = p° avec 2 < 5 2, et Ax ~ (R(p"))!.

SUR LES RESEAUX

UNIMODULAIRES

11

Exemple 4 : |G| = p*q? avec (p,q) = 1, e(p) = p? et e(q) = q?. Alors Ax ~R(p*)@ R(q?) ~ (zi B,(p)) ® (zae B,( q))

Ax

©Z1 B(p)1 Ba(q) 1 (B2(p) ® B,(q)).

Manuscrit recu le 18 septembre 1989

12

C. BACHOC .)

BIBLIOGRAPHIE

[B.E.] C. Bachoc et B. Erez. — Forme trace et ramification sauvage, a paraitre dans

Proceeding of the London Mathematical Society. [C.S.]

J.-H. Conway

et NJ.A.

Sloane.

—

Sphere

packings,

Lattices

and

Groups,

Springer-Verlag, New York, 1988. [E] B. Erez. — Structure galoisienne et forme Trace dans les corps de nombres, Thése,

Genève, 1987. [F] A. Frôhlich.

—

Invariants

for modules

over

commutative

separable

orders,

Quart. J. Math. 16, (1965), 193-232. [S] J.-P. Serre. — Corps locaux, Hermann, Paris, 3éme édition, 1968.

[W] L. Washington. — Introduction to cyclotomic fields, GTM 83, Springer, New York, 1982:

>

C. Bachoc

Département de Mathématique Université de Bordeaux I 351, Cours de la Libération

33045 TALENCE

Séminaire de Théorie des Nombres Paris 1988-89

Elliptic pseudoprimes, II R. BALASUBRAMANIAN

and M. RAM MURTY

1. — Introduction Given an integer n, the problem of determining whether n is prime or not, in the most efficient manner, is fundamental in mathematics. It is still an open

problem whether this can be done in polynomial time. The interested reader may

consult the work of Adleman and Huang [7] or Goldwasser and Kilian [8] for the current status of the problem.

Fermat’s criterion, namely that if p is prime, then 2?~' = 1(modp), can be used to test

a number n for primality. However, it still may happen that n is

composite and

(1)

2"-1 = 1(modn).

For example, for n = 561, (1) is satisfied. Such numbers are called pseudoprimes

(to the base 2). If P;:(x) denotes the number of such composite numbers n satisfying (1), then Erdôs [2] showed and

that for some positive constants c;

¢2,

© logx < P,(x) < rexp (—c2 Vlog x loglog«) À Pomerance [6] improved these bounds to

os (log) « Pale) 51)

where

L(x) = exp (log x log log log x/ log log x).

14

R. BALASUBRAMANIAN

and M. RAM

MURTY

8

Analogous results have been obtained by Erdés, Kiss and Sarkozy [9] for Lucas pseudoprimes.

Recently, elliptic curves have been introduced into the theory of primality

testing. Gordon [3], for example, introduced the analogue of the Fermat test (1) above and defined the concept of an elliptic pseudoprime. For instance, let E be an elliptic curve over Q with complex multiplication by the Gaussian ring of

integers, Z[i]. If E has good reduction at p and p = 3(mod4), then E(F,) has size p + 1. If

(2)

ais a rational point of infinite order on E, then

(p+1l)ja=0

in

E(F,).

This is an elliptic analogue of (1) and so can be used to formulate a pseudoprimality test for composite numbers n = 3(mod4). More precisely, let Y, denote the n-th division polynomial. The equation (2) can be rephrased as plYp+1(a).

We therefore say that a composite number

n is an elliptic pseudoprime

for

E if

n|%n+41(@). More generally, let E be an elliptic curve over Q with complex multiplication

by Q(./—d). Denote by A the discriminant of E. Let us fix a rational point P

of infinite order in E(Q). (If E has no such points, then we have no test.) A composite number n satisfying

renan ME

is called an elliptic pseudoprime for E to the base P if %nti(P) = 0(modn).

Gordon

[3] has shown that there are infinitely many elliptic pseudoprimes. Assuming a generalised Riemann hypothesis, he proved [3] that the number of elliptic pseudoprimes up to z is

oe log log= log” x

ELLIPTIC PSEUDOPRIMES,

II

15

I, Miyamoto and M. Ram Murty [5] showed unconditionally that the number of

elliptic pseudoprimes up to z is


3r/R(x) because otherwise, the penultimate sum above would be T

É R(z) log x Now L = p and n = sp together with n + 1 = 0(mode,) imply

sp +1= s(a, — 1) +1 = 0(mode,). This means that

s(a —1)+1>e,

> VrR(x).

Since |a,| < 2,/x, we find s > R(x)/2. Therefore, n = sp > 32/2 > x. Therefore, with our choice of parameters, class (4) is empty.

Thus far, our analysis is similar to [5] except for our choice of parameters. In class (5), we would like to count the composite numbers n < x such that n = sp

where p splits in K and e, satisfies

Vz/R(z) < ey < VrR(x).

22

R. BALASUBRAMANIAN

and M. RAM

MURTY

.

The number of such n does not exceed the number of 4-tuples

(j,r, 8, k) where

retds*=p,

Vr/R(x) y, Indag. Math. 13, (1951), 50-60. [2] P. Erdés. — On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen

4, (1956), 201-206. [3] D.M.Gordon. — Pseudoprimes on elliptic curves, Math. Comp. 52 (115), (1989), 231-245.

[4] R. Gupta and M.Ram Murty. — Primitive points on elliptic curves, Compositio Mathematica 58, (1986), 13-44.

[5] I. Miyamoto and M. Ram Murty. —

Elliptic pseudoprimes,

Math.

Comp. 52

(187), (1989), 415-430. [6] C. Pomerance. — On the distribution of pseudoprimes, Math. Comp. 37, (1981),

587-593. [7] L.M. Adleman and M.A. Huang. — Recognizing primes in random polynomial time, Proc. CRYPTO

86, (1986), 0-0.

[8] S. Goldwasser and J. Kilian. — Almost all primes can be quickly certified, Proc. 18th STOC, Berkeley , (1986), 316-329.

[9] P. Erdôs, P. Kiss, and A. Sarkézy. — A lower bound for the counting function of Lucas pseudoprimes, Math. Comp., (1988).

ELLIPTIC PSEUDOPRIMES,

II

25

[10] J. Silverman. — Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Springer-Verlag, New York, 1986.

R. BALASUBRAMANIAN Institute for Mathematical Sciences Madras, India

M. RAM MURTY

McGill University Montréal, Canada

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Séminaire de Théorie des Nombres Paris 1988-89

A subexponential algorithm for the determination of class groups and regulators of algebraic number fields Johannes BUCHMANN

Abstract : A new

probabilistic

algorithm

for the determination

of class groups

and regulators of

an algebraic number field F is presented. Heuristic evidence is given which shows that the expected running time of the algorithm is exp( og Dloglog D)°+2() where D is the absolute

discriminant of F, where c € Ryo is an absolute constant, and where the o(1)-function depends on the degree of F.

1. Introduction Computing the class group and the regulator of an algebraic number field F are two major tasks of algorithmic algebraic number theory. In the last decade, several regulator and class group algorithms have been suggested (e.g. [16], [17], [18], [3]). In [2] the problem of the computational complexity of those algorithms

was adressed for the first time. This question was then studied in [2] in great detail. The theoretical results and the computational experience show that com-

puting class groups and regulators is a very difficult problem. More precisely, it turns out that even under the assumption of the generalized Riemann hypothe-

sis (GRH) the fastest known deterministic algorithm is of complexity D1/°+°) where D is the absolute discriminant of F. This running time is exponential in the length of the input data since this length is polynomially bounded in log D. Computationally hard problems in number

theory are a very good source

for secure cryptosystems as is well known for the factoring problem. Therefore, in [7], [4], [8] and [9] the use of quadratic fields in cryptology was suggested

and discussed. Stimulated by this new application, Hafner and McCurley [11] invented a new probabilistic class group algorithm for imaginary quadratic fields

28

J. BUCHMANN L

and they proved under the assumption of the GRH that the expected running

time of this new method is L(D)V2+°{1) where L(D) = exp Vlog Dloglog D. In this paper we will generalize the idea of Hafner and McCurley to arbitrary number fields. If we use an additional assumption about the number of “smooth

reduced ideals”, then we obtain the computational complexity L(D)!7+°0) where the o(1)-function depends on the degree of the field F.

2. The idea The algorithm of Hafner and McCurley makes use of the fact that under the

GRH the class group CI of an imaginary quadratic field F is generated by the

prime ideals of norm below 6(log D)?. By means of a probabilistic method the algorithm finds a basis for the lattice L C Z* of relations for a generating system

for Cl containing k elements. Since Z*/L =~ Cl, the knowledge of this basis implies the knowledge of the structure and the basis of C1. The fact that a basis for the full lattice

L has been found can be verified by means of the analytic

class number formula by which an approximation h* to the class number h of

F with

Re-Gh gh can be determined in time polynomial in log D (A linearly independent system

of k lattice vectors is a basis of L if its determinant is less than h*). For arbitrary number fields

F the prime ideals of F whose norms do not

exceed 12(log D)? will also generate Cl (see [6]) (assuming again the GRH). As we will show below, it is still possible to generate random relations among a small

generating system for Cl. It is, however, not known how to verify in polynomial time that the proper class number has been found because in the general case the analytic class number formula only yields a polynomial time method

for

calculating a number h* with hR (log D) “à. In order to estimate vol(2Q* N V;), we choose an orthonomal basis b,,..

FS

of V;. Let w € 20+ NV. Then we can write d; = (w1,... »Wk(c)+r) = ya Ab;. CN:

Here we have |w;| < 3D (for sufficiently large D and 1 < i < k(c) +r). This inequality implies

Ai] = |(@,b,)]< [lol S$ 3DY/k(c) +r and therefore

vol (20+ NV;) < (sD Vie) + r). This completes the proof.

We define \(D) = VD(log D)"-!.

©

ALGORITHM

LEMMA 3.4. —

If dj =

FOR CLASS GROUPS

AND REGULATORS

37

k(c) + r, then there is a lattice basis of L; in

(0, c5A(D)I |}, Proof : By a theorem of Siegel we have det L; = h-R-I;

< c5A(D)I;.

Hence, there is a basis of L; whose integer entries are bounded by cs \(D)I;. Since the determinant of the sublattice of Log Oj; embedded in L; is bounded by RI;, it follows that there is a basis of that sublattice contained in [0, cgA(D)I;]". This also implies that each vector in the basis of L; can be modified such that

its logarithm vector part belongs to [0,c;\(D)J,;]" and this concludes the proof. LEMMA 3.5. — If d; = k(c) + r, then we have

Ny (D —cs(D)*( D(log D) * — e5(D))" /(AR). Proof: Let F be the fundamental parallelepiped of the basis of L. from Lemma

3.4. Then for every e € ON L, the intersection of the box

(0, D — cs\(D)]" x [0, D(log D) * — cs(D)]" is not empty and this implies the assertion. COROLLARY 3.1. —

pis = pjn(D) > ..:2E;2E;-12>:::2

fixe

d’un

# 9, et soit F,

2-sous-groupe

de

Sylow

Comme H est un groupe nilpotent, en une chaine d'extensions de degré 2 : Ey) =F. Supposons T(E;) # (. En appli-

quant le lemme-clé au niveau du corps E;, on voit que Z contient une droite

définie sur E;, et donc Zz,_, contient deux droites définies sur une extension au plus quadratique de E;_, et globalement rationnelles sur E;_,. Comme par ail-

leurs Zp,_, a des points rationnels lisses dans tous les complétés de E;_; puis-

qu'il

en

remarque

est 13.2.1)

ainsi

de

de [5] sur

Z

sur

k,

le

les intersections

théorème complètes,

13.2

(et

la

géométriquement

J.-L. COLLIOT-THELENE

52

possède intégres et non coniques, de deux quadriques, assure que Z C PT est Zz;_, Ainsi des points rationnels (lisses) sur le corps de nombres E;-1. E;_,-unirationnelle ([5], Prop. 2.3), donc Z (E;-1) est Zariski-dense dans Zg,_, (cela résulte aussi de la validité de l'approximation faible pour ZB aa 40l

Thm. 3.11), et de l'équivalence k-birationnelle de T à Pj, x Pj x Z, on déduit que T(E;-;) est non vide. Une récurrence descendante évidente montre alors que Z posséde des points

rationnels (lisses) sur Ey = F. Comme l'extension F/k a un degré impair, un théoréme de Amer [1] et Brumer [2] assure alors que la variété Z posséde un singulier un _ point possède Z Si k. sur rationnel point k-rationnel,

la projection depuis ce point dans

P? induit un

isomorphisme

k-birationnel de Z avec une quadrique ([5], Prop. 2.1), quadrique qui possède des points lisses dans tous les complétés de k, et est donc k-birationnelle à

un espace projectif. Ainsi, Z possède un point k-rationnel lisse, et l'on conclut

comme ci-dessus que les points k-rationnels sont Zariski-denses dans Z. Ceci

implique T(k) # @ et donc T,(k) # 0, ce qui établit le principe de Hasse pour 7;,. Une nouvelle application du lemme assure que l'intersection de deux

quadriques Z contient une droite rationnelle sur k. On peut alors utiliser [5],

Prop. 2.2, et conclure que Z est k-birationnelle à un espace projectif, donc aussi

T et T,. La k-variété lisse T, étant k-rationnelle, elle satisfait l’approximation faible. Remarques :

(1) Une fois connue l'existence sur Z d’une droite définie sur F, on peut aussi conclure en utilisant le théorème d’Amer [1], qui est une généralisation du théorème de Brumer, pour en déduire directement l'existence d’une droite définie sur k. Ceci évite la seconde utilisation du lemme-clé.

(2) Lorsque la surface fibrée en coniques X est aussi une surface de Del Pezzo de degré 4, la k-rationalité des torseurs universels possédant un k-point est le principal résultat de [6]. Le résultat de [6] vaut sur un corps de caractéristique zéro arbitraire. La présente démonstration redonne ce résultat dans cette généralité :

Soit T, un torseur universel avec T,(k) 4 0, et soit T le torseur de type À

associé. Alors T(k) est non vide, et le théorème 2.6.4 de [5] assure que T est

SURFACES

RATIONNELLES

FIBREES EN CONIQUES

53

k-birationnelle à un produit C x Pi x Z, avec C une conique lisse. Par ailleurs,

l'hypothèse T(k) 4 @ assure comme ci-dessus que T(k) est Zariski-dense dans T. On en déduit C(k) 4 0, et donc C ~ P}, et le lemme-clé, valable sur un corps de caractéristique zéro quelconque, assure alors que Z contient une droite

k-rationnelle : Z est donc k-birationnelle 4 un espace projectif, et il en est de méme de T, puis de T,. (3) I serait souhaitable de donner une preuve plus intrinsèque du lemme-

clé et de son analogue chez Salberger [13].

Manuscrit recu le 7 septembre 1989

J-L. COLLIOT-THELENE

54

BIBLIOGRAPHIE

[1] M. Amer. —

Quadratische Formen

tiber Funktionenkorpen,

Dissertation, Mainz,

1976.

[2] A. Brumer. — Remarques sur les couples de formes quadratiques,

C.R. Acad.

Sc. Paris, série A, 286, (1978), 679-681. [3] J.-L. Colliot-Théléne et J.-J. Sansuc. — La descente sur les surfaces rationnelles

fibrées en coniques, C.R. Acad. Sc. Paris, Série I, 303, (1986), 303-306. [4] J.-L. Colliot-Théléne et J.-J. Sansuc. — La descente sur les variétés rationnelles

II, Duke Math. J. 54, (1987), 375-492.

[5] J.-L. Colliot-Thélène, J.-J. Sansuc and Sir Peter Swinnerton-Dyer. — Intersec-

tions of two quadrics and Châtelet surfaces, J. für die reine und ang. Math. I, Bd. 373, (1987), 37-107; Il, Bd. 374, (1987), 72-168. [6] J.-L. Colliot-Théléne and A.N. Skorobogatov. — R-equivalence on conic bundles

of degree 4, Duke Math. J. 54, (1987), 671-677. [7] V.A. Iskovskih. — Propriétés birationnelles des surfaces de degré 4 dans P%,

Mat. Sbornik (N.S.) 88, (1972) = Math. USSR-Sbornik 17, (1972), 575-577. [8] V.A. Iskovskih. — Modéles minimaux des surfaces rationnelles sur des corps

arbitraires, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 19-43, (engl. transl. : Math.

USSR Izv. 14, (1980), 17-39).

[9] B.E. Kunyavskii, A.N. Skorobogatov, M.A. Tsfasman. — Del Pezzo surfaces of degree 4, Mémoires de la Société Mathématique de France, n° 37, Supplément au Bulletin de la S.M.F. 117, (1989), Fascicule 2.

[10] Yu. I. Manin. — Formes cubiques : algébre, géométrie, arithmétique, Nauka, Moscou 1972. Trad. anglaise : Cubic Forms, algebra, geometry, arithmetic : Second edition, North-Holland 1986.

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RATIONNELLES

FIBREES EN CONIQUES

55

[11] P. Salberger. — Sur l’'arithmétique de certaines surfaces de Del Pezzo, C.R. Acad. Sc. Paris, Série I, 303, (1986), 273-276.

[12] P. Salberger. —

On the arithmetic of conic bundle surfaces, in Séminaire de

théorie des nombres de Paris 1985-86, éd. C. Goldstein, Progr. Math. Birkhauser 71, (1987), 175-197.

[13] P. Salberger. — Lettre a A.N. Skorobogatov, 27/3/87. [14] P. Salberger. —

Zero-cycles on rational surfaces over number

fields, Invent.

Math. 91, (1988), 505-524. [15] P. Salberger. — Some new Hasse principles for conic bundle surfaces, in Sémi-

naire de théorie des nombres de Paris 1987-88, éd. C. Goldstein, Progr. Math. Birkhauser 81, (1989), 283-305.

J.-L. Colliot-Thélène C.N.R.S. Mathématiques, Batiment 425

Université de Paris-Sud F-91405 Orsay Cedex

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Séminaire de Théorie des Nombres

Paris 1988-89

De Rham cohomology for Drinfeld modules Ernst-Ulrich GEKELER

Introduction The aim of this paper is to present the newly developed theory of "de Rham

cohomology”

for Drinfeld modules

(due to G. Anderson,

P. Deligne, and the

author, with some remarkable contributions by J. Yu), and to state and prove

some fundamental theorems in this theory. Roughly speaking, a Drinfeld module is the global function field analogue of the motive associated with an elliptic curve (or, in a weaker sense, of an abelian

variety), in the sense that (a) torsion

points of certain

fixed Drinfeld

modules

may be used

to

construct algebraic field extensions of global function fields with prescribed ramification

properties

(e.g., abelian

class field theory of function

fields is

subsumed in the theory of Drinfeld modules of rank one), and

(b) the cohomology of modular schemes for Drinfeld modules provides automorphic representations of certain adele-valued groups over global function fields. In some cases, this leads to (abelian and non-abelian) reciprocity laws.

(See the introduction of [7] for a more detailed discussion as well as for references.)

As with motives over global number fields, where one disposes of different

cohomology theories (H}= Betti,

Hf = {-adic, Hj p= de Rham cohomology)

and their comparison isomorphisms, one has to find the "correct” definitions in the Drinfeld module setting, for which one may try to prove at least those properties that can be guessed by analogy. Note that for an abelian variety A,

58

E.-U. GEKELER

of B, é, or all the cohomology modules H(A), where ”?” stands for either one merely will we ®, DR, are determined by H}(A). In fact, for a Drinfeld module

define modules H3(®) that are similar to H7(A), but no higher Hi(@),7 >1.In

particular, H*($) will not be defined as the cohomology of a cochain complex. But apart from this, H7(@) will enjoy almost all the properties one would expect from a first cohomology module of an abelian variety. As far as the analogues of Betti and ¢-adic cohomologies are concerned, the definitions and basic properties are either obvious or treated elsewhere

(see,

e.g. [1], [5], [9]). The situation for Hj,p, however, is different, in that even the definition is highly non-obvious. This is why we first discuss H D r Of an elliptic

curve, in order to motivate the definition of H5,A(®), due to Anderson Deligne, which is given in section 3. Our main results on Hj, are:

and

(i) For © fixed, H7,R(®) is a contravariant functor (on affine schemes, say) that commutes with arbitrary base changes, i.e., if ® is defined over a ring B and B — B' is an extension, the canonical map

B! @p Hip(®) — Hpp(® x B!) is an isomorphism (Theorem 4.5).

(ii) Let ® be defined over the function field analogue C of the complex numbers

C. Then there is a canonical "de Rham

isomorphism”

DR

:

Hi p(®) — H3(4) induced by cycle integration (Theorem 6.6). (ii) Let ® be defined over a field L that is algebraically closed and complete with respect to a non-archimedean valuation. There exists a short exact

sequence of "vanishing cycles” that relates H},,(®) with the Tate parametriza-

tion data of ®, i.e., with the de Rham module H*,,(%) of the stable reduction @ of @ (Theorem 7.12).

(iv) Over the fields C and L of (ii) and (iii), it is natural to introduce

“analytic” versions H5%° of Hj,p. In fact, we obtain GAGA-like isomorphisms between the algebraic and the analytic de Rham modules. In the “complex” case, this is a simple consequence of the de Rham isomorphism (see Remark 6.7), but over L, additional arguments ar needed (Theorem 7.14).

A further interesting topic around

H br (some

kind of ”Kodaira-Spencer

isomorphism” relating H7, p(®) with the tangent space of some modular scheme

DE RHAM

COHOMOLOGY

59

at the point determined by ®, and with modular forms) will be treated in a different paper [10].

The proof of (i) is given in section 4. Combined with the description of Hyp over a field, it implies that for any Drinfeld module & of rank r over a scheme

S, H5r(®) defines a locally free sheaf of rank r on S. Since (ii) is proved elsewhere

[8], we limit ourselves to a brief discussion (see section 6) of that

result, presenting only those details that are also needed for the proofs of (iii)

and (iv), which are given in section 7. In both cases, we use non-archimedean analysis over the base field C or L, respectively, and investigate convergent

power series that satisfy a certain kind of functional equations. The author hopes that the material presented here will enable further pro-

gress in the theory of motives of Drinfeld type over function fields. In particular,

(iii) seeems to be a necessary pre-requisite for describing the behavior of Drinfeld modules at the boundary of modular schemes. It is a pleasure for me to thank the IHES at Bures-sur-Yvette for its great hospitality. Also,

I am

indebted

to Deutsche

Forschungsgemeinschaft

for its

support through a Heisenberg grant.

1 Notations. — Throughout the paper, X denotes a function field in one variable over the

finite field F, with g elements, and ”oo” a place of K fixed once for all. Further, A is the subring of elements of K regular away from oo, K. the completion of

K at oo, and |? |=|? |. the associated absolute value, normalized in the usual way. The corresponding degree function deg : K* — Z satisfies g*°8 * =| a |; its

value group is 6Z, where 6 is the degree of oo over Fy. (1.1) Let L be a field with a structure 7 : A — L as an A-algebra. A Drinfeld module of rank r € N over L (or over S = Spec L) is a commutative

group scheme G/S together with an action ® : A—

Ends(G)

ar,

of A on G (i.e., an S-scheme in A-modules) such that the following hold : (o) G is isomorphic with the additive group scheme G,;

60

E.-U. GEKELER

with the (i) The induced action of A on the Lie algebra Lie(G) of G coincides one given by 7;

of (ii) For each non-zero element a of A, the group scheme ker(®,) is finite degree |a |" over S. Let a: G — G, be an isomorphism of group schemes. Then ple). A — > Ends(Ga)

aao®,0oa!

defines a rank r Drinfeld module Frobenius endomorphism

non-commutative rule rr =

structure on G,. Let

x ++ x. In Endz(G,)

=

r : Gz —

Ends(G,),

G, be the

7 generates a

polynomial ring L{r} over L, subject to the commutation

zx1r for constants x € L. We always write coefficients to the left

of monomials in 7, and let deg, be the degree of a polynomial in r. Since A

contains F,, for any a € A

HV

ar

(ai € Lay #0)

i L (thus for all such isomorphisms), form

of?) has the

Br = A(T) + Sy art, 1 fi(xi) = 1. In other words, we have to A @ A) — A@ À is a perfect duality. This

follows from [4], VII, section 1, 6, Prop. 10 and section 3, 2, Prop. 1. Now suppose

(b). In this case, B @ A is Dedekind and Ig the maximal ideal ker(B @ A — B). We may assume

N

=

B @ A, and the assertion follows as above since 18 is

invertible.

Let now again ® = (G,®) be a Drinfeld module over the affine A-scheme

S = Spec B. First note that M(®) = Hom((G, ®)/S, G,/S) is a covariant functor in B (or a contravariant functor in S). The same holds for N(@), D(®), D;(®), D,i(®), Hpp(®), in view of the description given above. In case the necessity arises, we write M(®) = M(®,B) etc...

68

E-U. GEKELER 4.4 Lemma. — The canonical map M(&) —

D;($) is injective.

Proof :

The kernel consists of those m € M(@) that satisfy y(a)m = mo

Qa, all

a € À. Locally choosing isomorphisms of G with G,, thus reducing to the case (3.6), and comparing degrees, one sees this forces m to be trivial.

In particular, M(@) + D,(®) and N(®) => D.i(®). 4.5 THEOREM. — The functors M(®), N(&), D(®), Di(®), Dsi(®), Hpp(@) on A-algebras commute with arbitrary base changes B — 1329 Proof:

(a) M(®). If G= G, then M($,B) = B{r}, hence M(9, B’) = B'{r} = B'@8 B{r} = B' ®p M(4,B). The general case follows by standard glueing techniques, restricting to open affine subschemes

S; of S = Spec B where G

becomes isomorphic with G,.

(b) N(@). The statement follows from (a). (c) D(®). We have »

D(®,B) = Der(A, N(®, B)) = Homgega(Ip, hence

N(®, B)),

B' @p D(4, B) = B' ®g Homgea(Ip, N(4, B)) © Hompea(Ip, B'@ N($,B)) = Homgea(Ip, N(, B’))

= Homg’g (Ip, N(&,B')) (OR)

where only (*) needs explanation. First note that on B @ A-modules, ”B’@p,?” is the same as ”(B' ® A)®8@4?”. But J is an invertible A @ A-module, thus projective of rank 1, and the same

A) @a@a I. Thus taking

holds for the B @ A-module

R= B@ A, R' = B' QA,

I B=

(B@

P= Tz, (*) results from

R' @r Homp(P,?) = Homr(P,R'@r?), which is valid at least for finitely generated projective R-modules P. Note that

the statement was trivial for B'/B flat.

DE RHAM

COHOMOLOGY

69

(d) D;($) and (e) D,;(®). In view of Lemma 4.4, the base change property follows from (a) and (b).

(f) HS R(®). Here, the result follows from (c) and (e). 4.6 PROPOSITION. — Assume the Drinfeld module ® of rank r is defined over

S = Spec B, where B is a field. Then N(®, B) (and thus D(®, B)) is a projective B ® A-module of rank r. Proof :

We may assume degree d > 0, the Let Ao = F, [7] be is finite. If N has

that G = G,, hence N = N(®,B) finite set {r'|1 < i < rd} generates a subring generated by one element torsion as a B @ A-module, it has

= B{r}r. Ifa € A has N as a B ® A-module. T and such that A/Ao so as a module under

B @ Ap = BIT]. So let b = 5 b;T' € BIT] and n € N satisfy D= bn = J bino br. Comparing degrees in 7, this forces b = 0 or n = 0. Therefore, N is torsionfree,

hence projective since B @ A is Dedekind. If A = Ap = F, [7], we easily see that

{r']1 0 (or even > 0) by assumption.

(5.2) We define 7 to be reduced (strictly reduced) if deg,na (deg, na < r. deg a). For n reduced, we let

< r.dega

def 1 =r. deg a — deg, nq be the defect of 7. (All of this is independent of the choice of a.) Further, we

let D,(®), D.,(®) be the sub-B-modules of D(®) of reduced, strictly reduced

DE RHAM

COHOMOLOGY

71

derivations, respectively. By the above, each class in H},p(®) contains a unique reduced representative, and D(®) has the canonical B-decomposition

(5.3)

D(&) = Dyr(B) © By” @ Dyi().

Note that the defect defines a filtration on D,(®)

—>

HypR(®), and that

D,,(®) = {n € D,(®)|def » > 1} is a complement of D;(®)/D,;(®) in H4,p(®). (5.4) Next, we allow ® = (G,%) to be defined over an arbitrary A-scheme

S. Locally choosing isomorphisms a : Lie(G) +

Lie(G,), we arrive at the

situation of (5.1)-(5.3), so we may define reducedness and strict reducedness of local sections of the sheaf D(®) (this does not depend on the choice of a).

We obtain subsheafs (actually, direct summands) D,($) and D,,(®), whose sections are the reduced or strictly reduced sections of D(®), respectively.

Clearly, D.(®) — Hpp(®). (5.5) An isomorphism a : Lie(G) —

Lie(G,) defines a section 7(°? of

D,(®). Let G = Lie(G) be the line bundle whose underlying additive group is G. Viewing a as a nowhere vanishing section of the dual bundle G% (see (1.5)),

the map a +—> 7°) defines an injection of G’ into D,(#). (As usual, we do not distinguish between “vector bundles” and “locally free sheaves”.) From the local

considerations, it follows that

(5.6)

D($) = D,,(%) OH, oD,(®),

where H, is the image of G, i.e., the subsheaf generated by the sections 7(°). In particular, G* + H, is contained as a direct summand in D,(®) — H}p(®). In [10], we will describe the complementary sheaf D,.(@) —> H, + Hpp(®) through some sort of Kodaira-Spencer isomorphism. If for example r equals 2,

the line bundle H,, will be isomorphic with G. 6 The de Rham isomorphism. — Here we discuss the case where the Drinfeld module

® has an analytical

"Weierstrass parametrization”.

(6.1) Let C be the completion of the algebraic closure K.. with respect to the natural extension, also denoted by ”|?|”, of the absolute value on Koo.

72

E.-U. GEKELER

Recall that by Krasner’s lemma, C itself of rank r in C (shortly : an r-lattice) is a A-submodule A of C of rank r. Discrete” each bounded set in C is finite. For such

is algebraically closed. An A-lattice discrete, finitely generated projective means that the intersection of A with A, let

eq(z) = ZIlogaea(1 — 2/4) be the exponential (or lattice) function associated with A. It is not difficult to verify that the product converges, uniformly on bounded sets, thereby defining

an entire, surjective, F,-linear function ea : C — C. The zeroes of e, are the points of A C C, all of multiplicity one. Furthermore, for À € A, Cal ee ea(z + À), ie., ea is A-periodic. An r-lattice defines in a natural way a Drinfeld module of rank r over C :

Let

0 # a € A, and consider the following commutative

determines the element

diagram that

4 =a+--- of C{r}:

LG |:

fr

[+

Ge

By the snake lemma, ker(@*) © A/aA, thus deg, 64 = r.dega. Further, a +—» &À is a ring homomorphism,

and defines a Drinfeld module structure

on G,/C. Up to isomorphism, each Drinfeld module over C arises this way, and the category of rank r Drinfeld modules over C is equivalent with the category (morphisms = multiplications) of r-lattices in C. All the properties collected here are due to Drinfeld; proofs may be found in [6], [7], or [5].

(6.2) From now on, we fix an r-lattice A with exponential function e = €A

and Drinfeld module

6 = 4. As is motivated from the comparison with the

Weierstrass parametrization

of complex elliptic curves, we define for an A-

module X

H,(®,X)=AQ,X H*(®, X) = Hom,(A, X), which we view as a Betti homology resp. cohomology module of ®.

DE RHAM

COHOMOLOGY

73

(6.3) Next, we define a bilinear "cycle integration map”

Hs r(®) x À — C

HA

|7 BY

or, equivalently, a de Rham map

DR:

Hpr(®,C) —

Homa(A, C) = H*(&,C).

Let C{{r}} be the C-algebra of (non-commutative) formal power series in 7 and Cent{{r}} the subalgebra of those power series f = 5 a;t’ that converge

everywhere, i.e., that define an entire, F,-linear function f(z) = > a;z% on C. For example, e = ea € Cent{{7}}. The proof of the next lemma is given in [8]. 6.4 Lemma. — For n € D(®) there exists a unique F, € Ceni{{7}}7 such that for a € A, we have the functional equation

(+)

Fn(az) — aF;,(z) = na(e(z))(For any given non-constant a, (*) has a unique formal solution F,. Estimates

on the coefficients show F, is entire, and the derivation rule forces F,, = F, to be independent of a.)

Entire F,-linear functions on C satisfying a system of functional equations like (*) are similar to quasi-periodic functions for a complex lattice.

(6.5) Since e vanishes on A, we read off from (*) that the restriction x” of

F, to A is A-linear. Hence we have a map 7 +> x” from D(®) to H*(®,C). It factors through H5n(®,C) = D($)/D,;($) since for n = n°" strictly inner, F,(z) = —n(e(z)), Le., F,la = 0. We let DR : Hpp(®,C) — H*(®,C) be the induced de Rham map. Note that Hj, is defined in purely algebraical terms

(i.e., without reference to A), whereas H* is given by A, without reference to the Drinfeld module structure. 6.6 THEOREM. — DR is always an isomorphism. We will not give a complete proof (which may be found in [8]), but merely

outline the ideas. Injectivity of DR requires an estimate of the coefficients of

74

E.-U. GEKELER

is done in Thm. power series with a system (*) of functional equations, which already prove would this 3.1, loc. cit. Knowing the dimension of Hjp by (4.7), tion of the theorem. But surjectivity of DR may be shown by direct construc a pre-image of y : A —>

C, which will be useful in later applications. More

precisely, we construct a function F : C — C that satisfies

(a) Fla = x: (b) F is entire and F,-linear, i.e., lies in Cent{{7 }}7; (c) For a € A, there exists n, € C{r} such that (*) holds (then a +> tha will define an element of D(®)). For a natural number h, let F,,, be defined by Le)

= ye ayy". AEA

Using polynomial approximations, it is fairly easy to show that F), , converges,

uniformly on bounded sets, defines an entire and F,-linear function on C, and

restricts to x on A, i.e., (a) and (b). It is somewhat more difficult to show (c). In sections 4 and 5 of loc. cit., it is shown that (c) holds for F,,,, more precisely, the associated derivation 7 satisfies deg, 7 < r.dega +h — 1. In particular, the derivation that corresponds to F) , is reduced. Further, analyzing the coefficients of F},,, one may describe the effect under

DR

of the defect

filtration on H7, p (loc. cit., section 6). 6.7 REMARK

: We may introduce an analytical version of de Rham

mology of ® = 6, Namely, replace the C @ A-module M(®) M°"(®)

= C{r} by

= Cent{{r}} (or, to make it coordinate-free, by Hom*"(G,C)

of F,-linear morphisms

coho-

= set

of the additive group underlying the 1-dimensional

C-vector space G to C that are given by entire functions, with its natural structure as a C’ @ A-module), N(®) by N°"(@) = Cem{{r}}r, D(®) by

De"(®) = Der(A, N(®)), DA(&) by D3(%) = {nn € N(S)}, and App(®)

by H5% (®) = D2"(&)/D2"(S). The proof of Thm. 6.6 ([8], Thm. 3.1)

shows that the natural map c :

Hj}, —+ H}%" is injective. Furthermore, Lemma

6.4 also holds for 7 € D*"(®), which means that DR factors through c. Thus by dimension

reasons,

GAGA-type result.

c is an isomorphism,

a fact that may be viewed

as a

DE RHAM

COHOMOLOGY

75

7 Tate objects. — In this section, we let + : A —>

L be an A-field, complete with respect to

a rank one valuation v with valuation ring B and residue class field L(v). We require that y take values in B. In contrast with our general notation, ”|?|” will denote an absolute value associated with v. Furthermore (although this is not necessary, see Remark 7.15), we suppose L to be algebraically closed. The

reduction map from B to L(v) and everything derived from it will be denoted by

abar "7". (7.1) Given a Drinfeld module @ : A —>

L{r} of rank r € N over L, we

can find c € L* such that the isomorphic module ©’ defined by ®/, = c@, oc”! takes values in B{r}, and such that ®' : A —+ L(v){r} is different from the

homomorphism + : A —+ L(v). In that case, © is again a Drinfeld module of some rank r; < r. Its isomorphism class depends only upon that of ® (not on the choice of c); it is called the stable reduction of ®. If r; = r, we say that ® has good reduction. Let ® have good reduction of rank rj, and let A be a lattice in ®, ie, a

discrete projective A-submodule of (L, ®) of some finite rank r > 0. As in (6.1),

we form the exponential function e,, and define elements YA € L{r} by the commutativity of the diagram with exact rows

I.

EN |.

ee

ee

a

LE a |»

Sh

|«

sor else, 110,

Then (see [6], 7.2) :

(7.2) (i) a+ W4 is a Drinfeld module Y = WA of rank r = r1 + r2 over L. (ii) Ÿ has stable reduction of rank r1.

76

E.-U. GEKELER

of (iti) (6, A) +> W defines a bijection from the set of isomorphism classes

pairs (®, A), where @ has good reduction of rank ry and A is a lattice of rank r in ®, with the set of isomorphism classes of Drinfeld modules of rank r and stable reduction of rank rj.

We call Ÿ the Tate object determined by the Tate data (®, A). In what follows, we assume that Ÿ of rank r has coefficients in B and stable reduction Ÿ of rank

r. Then © has coefficients in B, too, we have Ÿ = ®, and EEE

B{{r}} nN Tent

(hats

where Leni{{T}} consists of those power series that induce entire functions on L. We want to relate the de Rham modules of Y and of ©.

7.3 Lemma. — Let n € Lent{{T}} satisfy a functional equation

(1)

n(La(z)) — 7(a)n(z) = F(2),

where f € L{r} and a € A is non-constant. Then actually n € L{r}. Proof

let

Si nr

and

Ÿ,

=

i>0

DA air”,

et

"rane

SU RLer

1] of © (see (5.3)) is always mapped by j to the canonical class [n:?)] of &. This is because for a € A,

no)o€ = (y(a)—Ya)oe = y(a)e—e0%, = 7(a)—%_ +7(a)(e—1)—(e—1) 0B = nfo? + nf? with n =

e—1 € N*"(®). We ignore what happens to def j([n]) if

def n > 0. Let now

7* be the derivation associated with F,,, where

homomorphism

y is some

À-

from A to L. From the proof of Thm. 7.7, we know that 7*

is reduced. To get the filtration induced on HomA(A, L), we had to determine deg, 7X. Having a closer look on the proof, one sees the equivalence of the following statements : (a) n* is strictly reduced; (b) ax

= Lent{{r}}T:

(c) S(1,x) = 0, since

SG,x)= SS xA)/A OAAEA

is minus the Taylor coefficient of z in Fo for r; =

r2 =

filtration on

,(z). Simple examples show that even

1, the filtration given by 7.12 does not agree with the defect

Hj, p(¥).

Manuscrit reçu le 25 août 1989

84

E.-U. GEKELER

BIBLIOGRAPHY

[1] G. Anderson. — t-motives, Duke Math. J. 53, 1986, 457-502.

[2] G. Anderson. — Talks given at the Seminar on Drinfeld modules, IAS, Princeton, 1987.

[3] N. Bourbaki. — Algébre, Hermann, Paris, 1970. [4] N. Bourbaki. — Algébre commutative, Hermann, Paris, 1965.

[5] P. Deligne, D. Husemolle. — Survey of Drinfeld modules,

Contemp.

Math. 67,

1987, 25-91.

[6] V.G. Drinfeld. — Elliptic modules (Russian), Math. Sbornik 94, (1974), 594-627: English translation : Math. USSR-Sbornik

23, (1976), 561-592.

[7] E.-U. Gekeler. — Drinfeld modular curves, Lecture Notes in Mathematics

1231,

Springer-Verlag. Berlin-Heidelberg-New York, 1986.

[8] E.-U. Gekeler. — On the de Rham isomorphism for Drinfeld modules, J. Reine angew. Math. 401, 1989, 188-208. [9] E.-U. Gekeler. — On finite Drinfeld modules, to appear in J. Algebra.

[10] E.-U. Gekeler. — De Rham cohomology and the Gauss-Manin connection for Drinfeld modules, to appear in the Proceedings volume of the Conference on P-adic Analysis Trento (1989) (Lecture Notes in Mathematics).

[11] N. Katz. — P-adic properties of modular schemes and modular forms, Lecture Notes in Mathematics 350, Springer-Verlag, Berlin-Heidelberg-New York, 1973.

DE RHAM

[12] B. Mazur,

W. Messing.

—

Universal

COHOMOLOGY

extensions

and one

85

dimensional

crystal-

line cohomology, Lecture Notes in Mathematics 370, Springer-Verlag, BerlinHeidelberg-New York, 1974.

Ernst-Ulrich Gekeler Institut des Hautes Etudes Scientifiques

35, route de Chartres 91440 Bures-sur-Yvette

re

7

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|

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Prue

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Line

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Giada

NC

Lutpye

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eng

ye

ance) 7

t

(e

0*

SUT ra" Là

.

te

a) =

pee)

rm

bee

fr.

|

© sn

:~

cx

;

ts anand

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| L

o@d Ber =

js

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“a DNA

Séminaire de Théorie des Nombres

Paris 1988-89

Le produit de Petersson et de Rankin p-adique

Haruzo HIDA'

0. Soit p > 5 un nombre premier et soit A = O[[X]] où O désigne l’anneau des entiers p-adiques d'une extension finie K/Q,. On fixe un générateur topologique

u du groupe multiplicatif 1 + pZ,. Pour chaque couple (F,G) de formes modulaires A-adiques ordinaires (voir ci-dessous pour la définition), on a construit dans [H1] un produit de Rankin p-adique D,, qui interpole p-adiquement les

valeurs spéciales du produit de Rankin D(s, f,g) pour deux spécialisations f et

g de F et G, et qui est le quotient d’une série de trois variables ®(X,Y,Z) par une série d’une variable H(X ), avec la propriété d’interpolation pour tout triplet

critique d’entiers (k,£,m) : _ G(u* — 1,ut— 1,u" — 1) a Dm, fer geo) D,(u* —1,u* —1,u" ~—1)=

H(u* — 1)

Gate)

‘

ou “x” est une constante canonique (voir Théorème 2 ci-dessous), et f;, (resp.

ge) est la spécialisation de F (resp. G) au poids k (resp. @), et gejw~™ est la torsion de g, par une puissance

w ”*

du caractère de Teichmüller w. L'utilité

de cette série est déjà évidente d’après les travaux de Perrin-Riou et Tilouine (par exemple, [P-R] et [T]) dans lesquels l'existence de cette série joue un rôle

important. D'autre part, la construction de D, donnée dans [H1] n'est pas si facile à comprendre parce que l’idée simple est en fait cachée par la complexité

inéluctable dans le traitement du cas général des formes modulaires J-adiques

pour toute extension finie J de A. Dans cette petite note, en se restreignant au cas des formes modulaires A-adiques avec

“Neben-typus” primitif et en se

concentrant sur seulement deux variables relatives 4 F et G, on se propose de

construire D, assez simplement.

H* HIDA

88

g € S¢(S L2(Z)) avec 1. On commence par une forme modulaire parabolique son développement de Fourier : co

(q = exp(2riz)).

oe + a(n,g)q” n=0

Considérons d’autre part la série d’Eisenstein :

EA(2z)=

DE

(mz+n) * (K > 2).

(m,n)E(Z?—(0))/{+1} Si g est une forme parabolique de poids £, la série gE, est une forme parabolique

de poids k = « + €. Notons que S;(SL2(Z)) a une base B = {f} consistant en des formes propres pour tout opérateur de Hecke. On veut regarder le coefficient

c( f)dans la décomposition :

gEx = D c(f)f. f

Sous le produit de Petersson (f,h) = Sst.ay\u Foy" *dedy, la base B est orthogonale parce que (f|T(n),h) = (f,h|T(n)) pour tout opérateur de Hecke. Donc, par la formule de Shimura [Sh1, §2], ona

Cf) = ee a ee Se Gyo he) (f,f)

où D(k— 1, f,9) = (2s +2—k- 0) D, an, flan, gn". Pour étudier le nombre c(f), on peut changer le produit scalaire (,) en gardant la propriété (f|T(n),h) = (f,h|T(n)). Donc on veut définir un nouveau produit bilinéaire (algébrique) sur l'espace des formes modulaires. On va faire une telle construction. un peu plus généralement en incluant le cas de Neben-Typus non-trivial. Soit N un entier premier à p et soit 4 un caractère de Dirichlet modulo Np. On suppose que le conducteur de w est divisible par N. On note

Mx(To(Np),#) l'espace des formes modulaires holomorphes avec caractères 1). Alors, pour chaque sous-anneau A de € contenant les valeurs de 4, on définit

M(A) = Mi(To(Np), #; A) comme le sous-espace des formes modulaires f avec coefficients a(n,f) dans A. On note S(A) = Sx(To(Np), 7%; À) le sous-espace

PRODUIT

DE RANKIN p-ADIQUE

89

des formes paraboliques dans M4(To(Np),#; A). L'effet de l'opérateur de Hecke

T(n) sur M;(To(Np),~; A) est donné sur les coefficients de Fourier par

a(m, fIT(n)=

D d**y(a)a(—.f). 0*. Soit } un caractère modulo Np avec #(—1) = 1. Une série formelle de deux variables F = F(q) = 51% A(n; F)(T)q” € A[[q]] est dite une forme A-adique avec caractère si F(u* — 1) € M;(%w-*; K) pour tout k assez grand. Alors

PRODUIT

DE RANKIN p-ADIQUE

97

on peut définir l'action de l'opérateur de Hecke T(n) sur une forme A-adique F par la formule

A(m,F|T(n))=

V2

d'yx(d)a (= F). “a2?

0 2 et l'ensemble {Px}k>2 est Zariski-dense dans Spec(A), on voit que A(#; A) @, L (L désignant le corps des fractions de A) est semi-simple. On peut donc définir une forme bilinéaire de la même façon que dans le cas déjà traité :

Ga : SL) x S(b;L) — (S(H:L) = S(H;A)Sa L). Soit F une forme propre dans S°"4(4); A) pour tout opérateur de Hecke et g € Say; QMO). On considère e(g + E(p~*p)) et

(F) = D,(T;F,

F,e(g x

g) = nn

E(y7

a

a

= (F,e(g * E(p"")))a

EL

PRODUIT

DE RANKIN

p-ADIQUE

99

parce que (F, F), = 1. Notons que

e(f*E(p'b)) =c(F)F +X

pour

X € (LF);

alors g * E(y~*p)(u* — 1) = e(F)(u* —1)F(u* —1) + X(u* —1) +h, où h est tel que e(h) = 0 par construction. Donc on a

THEOREME 1. — Sig € S;(y~; QN O) est une forme propre ordinaire, il existe un unique élément D,(T; F,g) € L tel que, pour tout k > @ > 2,

D,(u* — 1; F; 9) = (—N)**S( fx) Ep(fe, 9) Dk — OT(k — 1, fe 9) (ri) (ar) Gp we (FE, FR) où fr = F(u* —1) et E, et S( fx) sont les facteurs d’Euler supplémentaires en p définis au paragraphe 2. Maintenant,

on va essayer de faire varier g. Soit G une forme propre dans

$°r4(5: As), où As = O[[S]] pour la variable S. On continue de considérer E(g”-1#) € M°'4(y;Ar) pour Ar = OJ[T]]. Formons le produit GE(y71y)

dans O([T, S]][[g]] et posons

Ge E(p'v)(T, S) = GE(p""P)((1+ T)(1 +S)" - 1,5).

Ona G+ E(p-'y)(T, uf —1) = Gu’ —1)* E(y-1¥)(T) € S(4; O[[T]]) pour tout £2.

js

Si on a une série formelle H(T,S)(g)

=

>> A(n,H)(T,S)q”

telle que

n=1

H(T,ut — 1) € S(#; O[[T]]) pour tout £ > 2, alors H(T,S) — H(T,u? — 1) est divisible par S; = tisfait la même H\i(T, S) =

=

H(T,S) — H(T,u? —1) ee

Si

condition pour tout 4 > 3, et donc on peut définir H,

Hi(T, u? =

So

S — (u? — 1) et My

1)

Sa-

=

pour S, = S — (u* — 1). En répétant cette démarche,

on voit

H= > H,(T,o"** —1)S,e;-- 05, n=0

W. HIDA

100

A, (Thee D où S; = S—(u't! — 1) et Anyi = Hu(LS) n+1

ea

(HT, sine

de H(T, S)). Le développement de H ci-dessus est convergent sous la topologie

l'idéal maximal de O[[T, S]], et on voit

H € S(v; Ol[T]]) So OMT, SI]On peut donc appliquer la projection e a H. En particulier, on a

e(G x E(p714)) € S°™(d; O[[T]) Soyry OUT, SI]On

peut

étendre

le

produit

scalaire

(,),

linéairement

sur

S°"d(p; O[T]]) Sorry CLT, SI] et de nouveau, on note (,)a ce produit scalaire à valeurs dans le corps des fractions de O[[T, S]]. Maintenant, on définit, dans

le corps des fractions de O[[T, S]],

D,(T, 5; F, G) = (F,e(G + E(g "#)))aOna alors

THEOREME 2. — Il existe une unique fonction D,(T,S; F,G) € Frac(O|[T, S]]) telle que, pour tout k > L > 2,

D,(u* ul

G)

1 Ff — €)P(k — 1)D(k —1, fi, = (—N)*-*5(f,) AON OrT'(k e reenr Pour simplifier, on suppose que N = 1 et on suppose aussi que w = ¢. Alors

(TE(éd))(0) = (1- 1)log(u). Done e(G+(TE(id))\(T, T) = (1- 1)log(u)G(T). Donc, notant que (F, F), = 1, on a une formule de résidu. THÉORÈME 3. — Si N = 1, alors

LCD)

EN

D SO)

re — (a= =)log(u)(F, Ga.

C'est un analogue p-adique de la formule de résidu complexe :

Ress D(s, f,g) = T(K) 12" 1(4r)(f,9)si f,g € Sa(SLa(Z)).

PRODUIT

Remarque

: On

peut

définir

DE RANKIN

le

module

p-ADIQUE

Cy

de

101

congruence

de

F

par

Serd(N; A)/{FA @ (FL+ 1 S°r4(N; A))}, qui est un A-module de type fini et de torsion. Ce module C est isomorphe à celui défini dans [H1, II, §4], et par

définition, il existe une série H € O[[T]] telle que

C = A/HA. Alors, il est fa-

cile de vérifier que HD, € O[[T, S]]. Alors que la fonction L p-adique de trois variables est obtenue dans [H1, II] dans un cadre très général, on ne sait pas si le résultat d’holomorphie comme ci-dessus est vrai pour celle qui interpole la

fonction L primitive complexe en général sans l'hypothèse (4b). Le théorème 2

donne donc le seul cas connu sous l'hypothèse (4b) pour le problème de l'holomorphie. Si on peut calculer en général les facteurs d’Euler supplémentaires

en Np

sans utiliser l'équation fonctionnelle, on peut probablement démontrer

cette holomorphie en général par la voie employée dans [H3].

Manuscrit reçu le 12 juin 1990 * p. 87 : Supporté partiellement par NSF.

102

W. HIDA

BIBLIOGRAPHIE

[H1] H. Hida. — A p-adic measure attached to the zeta functions associated with two

elliptic modular forms I, II, I, Invent. Math. 79, (1985), 159-195; II: Ann. Inst. Fourier, Grenoble, 38, 3 (1988), 1-83.

[H2] H. Hida. —

Iwasawa

modules attached

to congruences

of cusp forms,

Ann.

Scient. Ec. Norm. Sup. 4ème série 19, (1986), 231-273.

[H3] H. Hida. — p-adic L-functions for base change lifts of GL2 to GL3, “Variétés de Shimura, Représentations galoisiennes et fonctions L automorphes”, Proc. d’une conférence a Ann Arbor, Michigan 1988.

[J] H. Jacquet. — Automorphic forms on GL (2), II, Lecture Notes in Math. Springer, 278, 1972.

[M] T. Miyake. — Modular forms, Springer 1989. [P-R] B. Perrin-Riou. — Dérivée de fonctions L p-adiques et points de Heegner, Invent.

Math. 89, (1987), 456-510. [Sh1] G. Shimura. — The special values of the zeta functions associated with cusp forms, Comm.

Pure Appl. Math. 29, (1976), 783-804.

[T] J. Tilouine. — Sur la conjecture principale anticyclotomique, Duke Math. J. 59, (4

(1989), 629-673.

[W] A. Wiles. — On ordinary \-adic representations associated to modular forms, Invent. Math. 94, (1988), 529-573.

H. Hida Department of Mathematics

University of California, Los Angeles Los Angeles, Ca. 90024

U.S.A.

Séminaire de Théorie des Nombres

Paris 1988-89

On Lehmer’s conjecture for elliptic curves Marc HINDRY and Joseph H. SILVERMAN*

Introduction

Let h : Q — [0, 00) be the (absolute, logarithmic) height function. The classical Lehmer conjecture says that there is an absolute constant c > 0 such that

h(a) > ee

~ [@(a) : Q]

oral

€ Q* except roots of unity.

There is a natural analogue of this conjecture for other group varieties, such as elliptic curves. Thus let Æ be an elliptic curve defined over a number field k, and

let À : E(k) — [0,00) be the canonical height on E. (See [11], especially VIII §9, for basic facts about elliptic curves and the canonical height.) Then the elliptic

Lehmer conjecture says that there should exist a constant c = c(E/k) > 0 so that

(1)

h(P) >

Cc

[k(P) : k]

for all non-torsion P € E(k).

The first non-trivial results in this direction were obtained by Anderson and Masser [1], who proved the lower bound

=

"2 GPF og PY

(To ease notation, we set d(P) = [k(P) : k].) This was improved by Masser [9] to

G)

“

Cc

OZ AP ogd(P)

and J.H. SILVERMAN

M. HINDRY

104

and in the special case that E has complex multiplication, Laurent [8] extended Dobrowolski’s method [3] to prove

P

c

[(loglogd(P)\°

amy (Fea )MP)2

©

The proofs of (2), (3) and (4) use techniques from transcendence theory. Recently Zhang [12] has given a new proof of (3) based on a Fourier series averaging technique whose origins go back to the work of Blanksby-Montgomery [2] on the classical Lehmer problem and which has been used in the context of elliptic curves (although not for the Lehmer problem) by Elkies [5], [7] and the authors [4]. In this note we will use Zhang’s ideas together with a more careful analysis

of the relevant Fourier series to improve the exponent in (3) to d(P)* under the assumption that the j-invariant of E is non-integral. (We have not been able to

improve on Masser’s and Zhang’s estimate (3) in the case that j(£) is integral.) This improvement will be an easy consequence of the following bound for the

number of points of small height on an elliptic curve.

THEOREM

0.1. —

Let k/Q be a number field, and

let E/k be an elliptic

curve with non-integral j-invariant. There is an effectively computable constant

c= c(E/k) > 0 such that for any finite extension K/k of degree at most d,

(5)

# {a€ E(K) : R(Q)
Pe k(P;) St1 fret A(P; — Px) 4

1

i

SD ee

(parallelogram law)

me

2 Fda (Pi =a: JÉK

vEMKx

We now break the sum over v into several pieces. For the archimedean places we use Proposition 1.1, for the non-archimedean places other than vp we use

Proposition 1.2, and for vp we use Proposition 1.3 as applied above in (13). (Actually, for the non-archimedean places v not dividing w we will only use the

negative part of Proposition 1.2. Also note that if vt(j—~!) = 0, then E has good reduction at v, so !,(P) > 0 for all points P.) This yields

4N(N + 1) max À (Q)

>=)

Ny (=) YO me

+9

n IlogiV+) D Be+(N vEMR

veMe

>

NIN +1)

ra

12

Vy

vlw,vvo

re nl F1 ea

ml) re are

ueST

112

_

log(N + 1) &

v|w,vfAv0

Using (11) to estimate v, in terms of n,, we finally obtain the estimate (14)

5

1

Hes h(Q) = 7

CeNy, + CT

pie v|w,v#vo

1

=

tig! Ga)

ee de Dav +/,-1)

ny

a

log(N + 1)

lee

aman

M. HINDRY

112

arid J.H. SILVERMAN

positive The following elementary lemma provides the crucial estimate for the

terms in the right-hand side of (14). LEMMA 3.1. — Let a, > 0 be fixed. There is a constant k = k(a,B) > 0 so that the following holds : Let D,n,n:1,...,n, be integers satisfying n+n+-.+n.

=D

andquadn

> max

ee

Then

Proof : Applying the arithmetic-harmonic inequality (i.e. the arithmetic mean is greater than the harmonic mean,) we find ii

1

an + B

—

iF2

> on +

B——— =

ere

sr

LE

r2

Dir

PE

n

Next, the condition n > n; implies that D < (r +1)n, so

an + B

2 i = > ant D Æ

Bs n

n

:

Now a little elementary calculus can be used to check that s

D

1

quel ey ala 0 DURE © Resuming the proof of Theorem 0.1, we apply Lemma

3.1 to the right-hand

side of (14). Note that 37,4, no = [X : k] = d/[k : Q], and (10) ensures that n,, is the largest of the n,’s. Hence we find

à Gen h(Q)

1 > Se

log(N + 1) =

Se

Non

So choosing #2 and (so by (12)) N sufficiently large gives the result

#D> ey d*/3) ogd il

—

j

er

la

a9

This last implication is an alternative way of stating the conclusion of Theorem 0.1.

©

ON LEHMER’S

CONJECTURE

FOR ELLIPTIC CURVES

113

Proof : (of Corollary 0.2.) Let P € E(k) be a point of infinite order, let K = k(P), and let d = d(P) = [K : Q]. Suppose that >

15)

RP)

oy

(P)

1

< ———;,

4c3d? log? d

where c = c(E/k) is the constant from Theorem 0.1. We derive a contradiction.

From (15) we see that

‘ : 1 h(nP) = n°h(P) < spr

frall 1, Lo un nombre réel > 0, a;; (1 1,on a

Max |Dj 0 --- 0 Dief(z)| |t|=T

1, f1,---,

fn des fonctions analytiques

dans un ouvert U de €”. Il existe un entier rationnel c,; > 0 ayant la propriété suivante : soient K, un corps de nombres de degré fini et L,---, Dn, ti,---,tn des entiers positifs ou nuls avec |t| = t1 + ---+ tn > 0. On suppose que les

dérivations or (1 0 tel que pour |t| < T, Ossie Smonait log| Df F(su)| < —c22c2U.

Démonstration de la proposition 3.14 : D'abord on déduit de (3.12) et de la proposition 3.13,

log| Dg F(sw)| < 03 @/?U Pour le cas de non-torsion, on fixe t; avect;

pour

(t,s)€&.

< T (1 T le long de W et que DF (su) eA: SiLo < T alors on a pour L = Max

(Iy,:::,La):

log |DÉF(su)| d

> —c50D {log H +Lolog B+Tlog(T+L)+Lolog $+) ~LS? log

a

i=1 Démonstration du lemme 3.17 : On se ramène au cas où les degrés de P par rapport a Xo, Xi,---,Xq sont exactement Lo, Lj,---,Lq. Si su; € 0; ona

A(i(sui)) = 0 et sinon, le lemme 4.8 de [R] donne h(#:;(su;)) < c31S?log (1

V;

C, namely

Vec=]|[v. Let us write

L = L(A)(C) and H = H(A)(C). Each V, is both an L and H

module. In fact, V, is a simple

H module and we have that

Vo, = Vo, as H — modules

&

Vo, © Vo, as

L — modules.

Indeed, it follows from the definitions that

Endy V = Endy V = End(A) @ Q. Since K is a maximal commutative semisimple subalgebra, it follows that Endy

x Vi

End;

x Veen

COMPUTING

THE HODGE

GROUP

153

The assertion follows from this.

Step 2. Write

K = K, x --- x K,. This gives a corresponding decomposition up

to isogeny Aw

A, X::+XÀ,

with each A; a power of a simple abelian variety, say A; ~ B; () Then, B; is of type IV since each K; is a CM field and m is odd. (For the reader’s convenience,

we quickly recall the different types. Let B be a simple abelian variety,

D =

End(B) @ Q, and F the center of D. Then there are four possibilities : TypelI

: D = F isatotally real field.

Type Il : F is totally real, and D @R = M,(R)F). Type III : F is totally real, and D @R = HF). Type IV

: D is a division algebra over the CM field F.

In the above M2(R) denotes the algebra of 2 x 2 matrices over R and H denotes the Hamilton quaternions.) Step 3. Let us set

S = H,, and R = L,,. For each homomorphism o : kK®C —

€, let us denote by S, and R, the projection of S and R (respectively) to GL(V,

).

By definition, we have that S, C R,. Let us denote by p complex conjugation in

C. By the explicit calculation of L, we know that

v=) where V, denotes the dual space, and that

i, =S8UV,).

Since dime V, = m is odd, either m = 1 or m > 3. In the first case, it would follow from what we have said above that S$, = À, =

1. From now on, let us

therefore assume that m > 3. Serre [12] shows that in this case,

So oad Si.¢ 5,50 Sr

with each S;, a simple group of type A. Moreover, there is a corresponding decomposition

Vo = Vo

8° @Viw

V. KUMAR

154

MURTY

with each V;, being a fundamental representation of S;,,.

Step 4. Let us assume for a moment that H(A) = D(A). Then,

H*(A, QUA) = H(A) = D(A) = H*(A, QE. Therefore, for any sequence {i(c)}._ of positive integers, we have

(craie fe (@ AX Ve) Under our weaker assumption that all Hodge classes on A of codimension < 3

are generated by divisors, the above is still valid provided the sum 5° 7(c) is less than 6. Using V,, © V, for any o, then for any 1 s, alors (r,s) = r{s,our{s=rdc+ts

le signe étant choisi tel que

3 r is. Par conséquent, l’ensemble de tous les points qui se déduisent de 1

par la méthode des tangentes et sécantes est {k € Z| k > 1,3/ k}. Il calcule (sur une forme normalisée de la cubique) que les coordonnées de k sont des

fonctions rationnelles de degré k? des coordonnées de 1.1% 48 Sylvester affiche, dans un autre contexte, une certaine prédilection pour “... an intuitional proof ... without any recourse to concepts drawn from reticulated arrangements, as in the applications of geometry to arithmetic made by Dirichlet and Eisenstein.” [Sylvester 1879/80,

344]

11 Dans [Sylvester 1879/80, 353ff] le mot “group” désigne un ensemble de points fermé par 8 B rapport à la méthode des tangentes et sécantes. 12 Le lecteur trouvera un exposé très clair de cette théorie dans le traité de Salmon : [Salmon 1879, §§158-161, pp. 136-140]. La notion clé est la suivante. “If two systems of points [nous

dirions aujourd’hui : deux diviseurs] ... together [1.e., leur somme]

make up the complete

intersection with the cubic of a curve of any order, one of these [i.e., chacun des deux] is said

to be the residual of the other.” [Salmon 1879, 136] 13 Cette constatation est accompagnée, [Sylvester 1879/80, 356], par cette note en bas de page :

N. SCHAPPACHER

170

de Sylvester. Un point d’inflexion I étant fixé sur la résulte de J et de courbe, Sylvester appelle opposé d’un point p le point qui Deuxième

Observation

l'ensemble formé p par la méthode des tangentes et sécantes : p’ = (I,p). Alors opposés est fermé des points considérés dans la premiere observation et de ses

sous la méthode des tangentes et sécantes.

Sylvester pose (1',3i — 1) = 3: '* et vérifie explicitement cas par cas que {I} U{k| k > 1} U {k'| k > 1} est fermé sous l'opération (, ). Il se convainc

aussi de ce que le degré de l'expression rationnelle des coordonnées des points

k et k' est k?, pour tout k. Traduisons ce que trouve Sylvester en termes de la loi de groupe sur la courbe relative à l'élément neutre J. L'opération k +> k’ est aussi le passage au négatif dans le groupe. Si P est le point de départ 1, alors on trouve pour tout UE

(31) =3iP,

(3i+1)=(3i+1)P,

(3i+2) = (3i+2)P.

Les notations adoptées par Sylvester ont donc, pour un oeil moderne, tendance a voiler la présence du groupe, isomorphe a (un quotient de) Z, engendré par P.

“The proof here supplied is sufficiently exact to dispel any reasonable doubt as to the truth a the law; but an exact proof which does not assume but demonstrates the non-existence of atent common measures .... will be furnished under Title 5 — one of the most surprising feats

of demonstration which it has ever fallen to the author’s lot to accomplish.” — I] revient sur ce ge Pages plus loin où il avoue avoir démontré seulement que le degré de k est borné

tk ee ae

Le

: “... before I come to an end of the discussion I trust to be able to establish

- :tan rigour that the order is actually equal to the square ...” —

phrase à laquelle

rajoutée la note en bas de page : “This anticipation (for it was only such when these words were written) will be found fully realised under Title 5.” 14 + a a Corriger les fautes d’impression [Sylvester 1979/80, 361, ligne 12].

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3. La tradition analytique Dans ce numéro je répéte une observation de Scriba sur les théorémes d’Euler relatifs aux

intégrales elliptiques et sur une

petite note apparemment

peu

connue de Jacobi. Commençons avec Euler. Je cite un passage de [Scriba 1984,

25f] relatifà l'équation y? = f(x) = ax° + bz? + cx 4d: “Beim Studium elliptischer Integrale hatte Euler bemerkt, daf, falls man

für einen Punkt A(z, y) ... definiert

n(4= |ooraeY =fooer

ae

WV F(E)

es zu zwei entsprechend definierten Punkten A und B immer einen dritten

C auf [der Kurve] gibt, daf

II(4) + 0(B) = 1(C), wobei

sich

die Koordinaten

von

C rational

aus

denen

von

A und

B

ausdrücken lassen [Euler 1912/13]. Zu diesem Additionstheorem tritt das

Eulersche Multiplikationstheorem

(D) = n.I(C), d.h. es gilt auch hier, daf sich für ganzzahliges n die Koordinaten von D rational aus denen von À ausdrücken

lassen. Sind also einer oder zwei

rationale Punkte bekannt, kann man [mit diesen Sätzen] weitere finden.”

Mais

cette façon analytique

de construire

partir d'un ou deux points donnés

de nouveaux

n'était jamais

points rationnels

a

appliquée aux problèmes

diophantiens par Euler. Ceci étonnait Jacobi tant qu'il rédigea un petit article de propagande pour l‘usage de la théorie des intégrales elliptiques et abéliennes

dans l'analyse diophantienne’ : [Jacobi 1835]. Il généralise les deux théorèmes

d’Euler en un troisième concernant des Z-combinaisons linéaires arbitraires,

I(x) = mll(z1) +... + mall(zn), où zx et

,/f(z) s'expriment encore rationnellement en 2;, \/ Fer);

N. SCHAPPACHER

172

de Jacobi. Sinon, J'ignore s’il y a un seul travail qui reprend la suggestion inutile et toute est nde faudrait-il conclure que, en mathématiques, la propaga ?!° Implicisuggestion doit étre menée jusqu’au bout par celui qui la propose dévelopdu partie grande tement le message de Jacobi se transmet dans une

pement de la théorie des intégrales abéliennes au siécle dernier —voir passim le rapport tout à fait impressionnant [Brill, Noether 1892/93]. En particulier on peut mentionner Clebsch — voir [Clebsch 1863], [Klein 1926, 298ff]. Tout ce développement forme l'arrière-plan du travail de Poincaré avec lequel nous entrons dans notre siècle.

4. Poincaré

Il ne s’agit pas ici de rendre justice à toutes les idées du long article de Poincaré sur l’arithmétique des courbes algébriques, [Poincaré 1901]. Le lecteur intéressé consultera surtout les notes des éditeurs des œuvres de Poincaré.

Je me bornerai

essentiellement au traitement de la méthode des tangentes et sécantes. Toutefois il faut souligner que cette méthode est discutée par Poincaré dans un cadre nouveau : celui de l'étude des points rationnels de courbes algébriques dans la perspective de l’invariance birationnelle.

à

“Les propriétés arithmétiques de certaines expressions et, en particulier, celles des formes quadratiques binaires, se rattachent de la façon la plus

étroite à la transformation de ces formes par des substitutions linéaires à coefficients entiers. Je n'ai pas à insister ici sur le parti qui a été tiré

de l'étude de ces substitutions et qui est assez connu de tous ceux qui s'intéressent à l'Arithmétique.

On peut supposer que l'étude de groupes de transformations analogues est appelée à rendre de grands services à l’Arithmétique. C’est ce qui m’engage à publier les considérations suivantes, bien qu'elles constituent plutôt

un programme d'études qu'une véritable théorie. Je me suis demandé si beaucoup de problèmes d'Analyse indéterminée ne peuvent pas être rattachés les uns aux autres par un lien systématique, grâce à une classification nouvelle des polynomes homogènes d'ordre supé-

rieur de trois variables, analogue à certains égards à la classification des u est-ce que les temps ont changé?

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formes quadratiques. Cette classification aurait pour base le groupe des transformations bira-

tionnelles, à coefficients rationnels, que peut subir une courbe algébrique.” [Poincaré 1901, Introduction, 483f]

Le genre’® étant un invariant birationnel, Poincaré se lance dans son vaste programme

en commençant

par les “courbes unicursales”,

1.e., les courbes de

genre 0. Sur ce point il a été devancé par Hilbert et Hurwitz [Hilbert, Hurwitz

1890] qui classifient complétement les ensembles possibles de points rationnels

des courbes de genre 0 (méme avec singularités). Dans [Hurwitz 1917, 446, note 1], Hurwitz dit que Poincaré avait retrouvé indépendamment une partie de leurs résultats. Ainsi Poincaré établit le fait [Poincaré 1901, 488] qu’une courbe de genre O est toujours birationnellement équivalente à une droite ou une conique.

Ceci remonte

a Noether [Noether 1884] et est précisé dans [Hilbert, Hurwitz

1890]. La méthode des tangentes et sécantes fait son apparition, bien sûr, quand Poincaré passe aux courbes de genre 1. “On voit avec

quelle facilité se traite le cas des courbes

unicursales.

Passons maintenant aux courbes de genre 1 et d’abord aux plus simples d’entre elles, je veux dire aux cubiques. Etudions d’abord la distribution des points rationnels sur ces courbes. J’observe que la connaissance de deux points rationnels sur une cu-

bique rationnelle suffit pour en faire connaitre un troisiéme. En effet, la droite qui joint deux points rationnels donnés va couper la cubique en un troisième point qui, étant unique, est encore rationnel. De méme,

si nous connaissons un point rationnel, nous pouvons en

déduire un second; la tangente à la cubique en un point rationnel est une droite rationnelle qui coupe la cubique en un autre point rationnel.”

[Poincaré 1901, 490] Pour voir ce que donne cette méthode il exprime les opérations en termes

des arguments elliptiques des points rationnels. Ainsi il constate qu’en partant 16 Cette notion est, bien sûr, un des fruits les plus importants du développement des mathématiques au dix-neuvième siècle. Faute d’en pouvoir dire plus ici, je renvoie sommairement à (Brill,

Noether 1892/93] et a [Scholz 1980].

N. SCHAPPACHER

174

e des tangentes et du point qui correspond au nombre complexe a, la méthod nombres de la t aux sécantes engendre précisément les points correspondan : forme (3k + 1).a, k € Z. Et en partant de plusieurs points il trouve “Plus généralement, si les points d’arguments elliptiques a,

1,

A2,

«++,

Ag

sont rationnels, il en est de méme de tous les points dont les arguments elliptiques sont compris dans la formule

(1)

a + 3na + pi(ay — a) + p2(a2 — a) +... + Pg (Gq — à),

où n et les p sont entiers.” [Poincaré 1901, 492]

Nulle part, Poincaré n’essaie de combler les lacunes dans

cette série : Il

ne se demande pas si toutes les Z-combinaisons linéaires des arguments des points de départ appartiennent a des points rationnels, et par quelle opération géométrique on peut les atteindre. Ceci malgré le fait qu'il sait très bien qu™“une cubique qui a un point rationnel est toujours équivalente à une cubique qui a

un point d’inflexion rationnel.” [Poincaré 1901, 538] Poincaré n’atteint donc pas du tout le niveau de l'analyse de la méthode des tangentes et sécantes que nous avons vu dans Sylvester —

et pourtant ses

notations sont beaucoup plus suggestives, étant guidées par la paramétrisation

analytique, que celles de Sylvester. En particulier, Poincaré ne découvre pas la loi

de groupe géométrique sur les points rationnels d’une cubique non-singulière. Il ne dispose dans sa description ni d’un élément neutre ni de l'inverse, et l'opération

binaire qu'il étudie n'est pas associative.

Comment

s’est donc développée la légende selon laquelle Poincaré aurait

“montré [Poincaré 1901] qu’...on peut introduire, dans l'ensemble des points rationnels de la courbe, l'opération d'addition, de manière que cet ensemble

soit muni d'une structure de groupe abélien” [Bachmakova 1966, 294]? Même Scriba qui est beaucoup plus soigneux!” semble pris dans le piège de ne pas pouvoir nier une découverte de plus à un esprit aussi fécond que Poincaré. Ainsi il remarque bien que Poincaré associe le point d’argument — = Né

: A 5 A la fin de sa discussion de Fermat il souligne, “da8 auch Fermat — wie Diophantos — nur algebraische Substitutionen und Transformationen ausführte. Wie stark er sich dabei vielleicht

von geometrischen Bildern leiten lieB, wird nicht offenkundig.” [Scriba 1984, 24]

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—(a + @) aux points d'arguments a, 8. Mais il fait comme si ceci n'avait pas

d'importance.l$ Cette erreur, qu’on pardonnerait facilement à un mathématicien Soucieux de trouver des sources d'inspiration, semble essentielle dans le travail d'un historien professionnel des mathématiques.

Pour finir notre discussion

de l'article de Poincaré,

regardons le passage

célèbre où il définit sa notion du rang d'une cubique, qui suit le dernier passage

cité de [Poincaré 1901] : “On peut se proposer de choisir les arguments (2)

AS

@1,

G2,

.:.,

Ads

de telle façon que la formule (1) comprenne tous les points rationnels de la

cubique. Les ¢+1

points rationnels qui ont les arguments (2) forment alors

ce que nous appelerons un système de points rationnels fondamentaux. Il est clair que l’on peut choisir d’une infinité de manières le système des points rationnels fondamentaux.

On doit tout d’abord, dans ce choix,

s’arranger de telle façon que le nombre g + 1 des points fondamentaux soit

aussi petit que possible. Cette valeur minimum de ce nombre q + 1 est ce que j'appelerai le rang de la cubique; c’est évidemment un élément très important de la classification des cubiques rationnelles.” [Poincaré 1901,

492f] Dans la mesure où Poincaré n’envisage point la situation où aucun nombre fini de points fondamentaux ne suffit à engendrer tous les points rationnels on

peut voir dans ce texte une façon de conjecturer le théorème de la base finie des points rationnels, démontré dans [Mordell 1922].!° Il est aussi possible que, étant intéressé par une analyse plutôt constructive des points rationnels, il ait

exclu tout de suite le cas d’un rang infini.?° Comme

Poincaré ne dispose ni d’un élément neutre ni de l'inverse dans sa

structure, sa notion de rang n’est pas bien calibrée de notre point de vue. En

fait, le rang n’est pas un invariant birationnel pour la raison triviale que, de 18 “Da Poincaré ... das Integral .. mit entgegengesetztem Vorzeichen versah, ist y durch —+ zu ersetzen.” [Scriba 1984, 36] 19 Cf. la formulation du jeune A. Weil [Weil [1929], 47] : “Il y a quelques années, Mordell a démontré un théorème remarquable, qui avait été entrevu déjà par Poincaré ....” 20 C’est E. Brieskorn qui me proposait cette interprétation dans une discussion.

N. SCHAPPACHER

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avoir un point rationnel deux cubiques équivalentes, une, mais pas l'autre, peut

par Beppo d'inflexion.2! Cet inconvénient allait être bientôt remarqué et corrigé nous passons Levi dans [Levi 1906/08, 758f] — travail impressionnant auquel

maintenant.

5. Beppo Levi (et Hurwitz)

Levi se place, avec [Levi 1906/08], consciemment dans la tradition de Sylvester et Poincaré, bien que ses informations sur Sylvester ne soient apparemment

basées que sur les indications indirectes des Nouvelles Annales; il ne cite pas le mémoire américain [Sylvester 1879/80]. Plus généralement et plus géométriquement que ses prédecesseurs, il com-

mence par la définition [Levi 1906/08, 752f] selon laquelle

“un punto razionale o un gruppo [on dirait aujourd’hui : ensemble] razionale di punti di una cubica a coefficienti razionali è dedotto razionalmente

dai punti razionali A, A2,..., A, quando è univocamente determinato mediante intersezioni della cubica con curve a coefficienti razionali, completamente definiti dai punti A,, A2,..., A, e dai valori (razionali) di un certo

gruppo di coefficienti che vi compaiono come parametri; per assegnare questi parametri costringeremo

generalmente

la curva a passare

per al-

trettanti punti razionali fissati arbitrarimente sul piano.” Mais il montre aussitôt, en faisant intervenir les paramètres analytiques, que

cette notion a prior: plus générale de la déduction rationnelle de points sur

une cubique, se ramène à l'application itérée de la méthode des tangentes et sécantes. En corrigeant la notion de rang comme nous l'avons vu, Levi s'approche de

ce que nous appelons aujourd’hui le rang (libre) du groupe abélien des points rationnels.?? Toutefois, il ignore les problèmes de la torsion : un groupe d'ordre 6, par exemple,

admet comme

ensemble

de générateurs indépendants

—

“tal:

che nessuno di essi possa dedursi razionalmente da altri” [Levi 1906/08, 758] — soit un élément d'ordre 6, soit deux, d'ordres 2 et 3 respectivement

: Le rang

TT

21

; : : P Il n’est donc pas étonnant que Poincaré ne démontre pas l’invariance birationnelle du rang, comme remarque Scriba [Scriba 1984, 36]. 22 [Levi: RU “ 1906/08, 758f, en particulier la premiere formule p. 759]. Noter pourtant qu’il considére toutes les courbes dans une classe d’équivalence birationnelle à la fois.

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de Levi n'est pas bien défini en général. Cette bavure peut surprendre?* dans la mesure où ce que nous appelons la torsion dans le groupe des points rationnels, est le sujet principal de l'article [Levi 1906/08]. Mais il ne faut pas oublier que Levi est surtout un géométre, dans la tradition italienne, et il voit ce que nous traitons prosaiquement comme les éléments d'ordre fini d'un groupe, comme des

configurations finies de points sur une cubique** qui sont fermées par rapport aux tangentes et sécantes.”° C'est avec

ce langage

géométrique

que

Levi se lance

dans

l'exploration

systématique de ce que nous appelons les sous-groupes de torsion qu’on peut rencontrer dans les groupes des points rationnels de courbes elliptiques sur Q. Nous savons, bien sûr, qu'il n'y a que très peu de possibilités [Mazur 1978, Thm 2] : les groupes cycliques d'ordre 1,2, ..., 10,12, et les groupes isomorphes à

Z/2Z x Z/2vZ, pour v = 1,...,4. Il est impressionnant de voir Levi se rapprocher de cette liste complète par des analyses détaillées des configurations finies sur les cubiques rationnelles.

Plus précisément, Levi discute essentiellement les configurations engendrées par un point d'ordre t dans le groupe des points rationnels. (Ses normalisations

sont pourtant différentes; il part systématiquement d’un point de paramétre elliptique de la forme =, ou w est une période : voir [Levi 1906/08, 101], ainsi que [Levi 1906/08, 676-680].) Ceci revient plus ou moins à traiter l'existence d’un sous-groupe cyclique d’ordre ¢ du groupe des points rationnels.

Commencant par le cas t = 2”, il trouve que de telles configurations existent

sur certaines cubiques rationnelles, pour v = 1, 2,3. Mais qu'il n'y en a pas pour v = 4, Levi démontre l’impossibilité du cas t = 16 — qui implique l'impossibilité = Généralement, le travail de Levi me semble assez fiable. Voir les quelques remarques làdessus au cours de ma discussion de Levi. — En particulier, je n’ai trouvé nulle part, ni dans

[Levi 1906/08], ni dans [Levi 1909], la faute, concernant la forme normale d’une cubique ayant un point rationnel, dont Cassels accuse Levi [Cassels 1986, 33] : Chaque fois que Levi utilise l’équation dont Cassels remarque qu’elle n’est pas valable en général, il impose soigneusement une condition supplémentaire. La formulation malicieuse de Cassels, “Levi ... appears to claim …” donne d’ailleurs impression trompeuse que les travaux de Levi sont rédigés de façon embrouillée. — Toutefois, je n’ai pas vérifié tous les détails des démonstrations de Levi cités plus loin.

24 Configurazioni arborescenti et Configurazioni poligonali (semplici ou misti) [Levi 1906 /08, 101ff]. Quand il aborde le cas que nous décrivons comme Z/2Z x Z/3Z, il prend, bien sûr, un

seul générateur comme base : [Levi 1906/08, 420]; cf. [Levi 1906/08, 681]. 5 Sylvester avait déjà souligné à plusieurs endroits l’éventualité que la méthode des tangentes et sécantes revienne sur elle même après un nombre fini d'opérations, en partant de certains

points rationnels : [Sylvester 1879/80 : 314 (*), 340 (*), 351, 354].

N. SCHAPPACHER

178

d’ordre 16 — en déduisant d’une courbe elliptique sur Q avec un point rationnel s admettant une d’abord une équation d’une courbe paramétrisant les cubique issons comme reconna nous que ce telle configuration. Autrement dit, il déduit

un modèle de la courbe modulaire X1(16) :

[y? =] A =(c? +1)(c— 1)(c? — c? — 3c — 1) [Levi 1906/08, 113], cf. [Washington 1990, dernier §]. Ensuite il montre par une descente infinie [Levi 1906/08, 113-115] qu'il n’y a pas de point rationnel sur cette “courbe modulaire” correspondant à une cubique non dégénérée admettant

une configuration de points avec t = 16.7° Pour les petites valeurs impaires de t, Levi trouve des équations paramétrisant les familles (“fascio”) des cubiques ayant un point rationnel d'ordre t.

Mais dans le cas t = 11, il n’arrive pas à conclure [Levi 1906/08, 417ff]. Plus précisément, il écrit l'équation de la cubique paramétrisant les cubiques ayant un point rationnel d'ordre 11, :.e., il écrit un modèle de la courbe que nous ap-

pelons aujourd'hui X:(11), [Levi 1906/08, no (7), p. 418]; et il repère là-dessus les cinq points rationnels que nous reconnaissons comme

étant les pointes de

cette courbe modulaire.”" Mais il n'arrive pas à démontrer que ce sont les seuls points rationnels. Finalement, en traitant les ‘configurations polygonales mixtes’ — qui corres-

pondent aux sous-groupes de la forme Z/2”Z xZ/t'Z —, Levi démontre l’impossibilité sur Q du cas v = 1, t' = 7 et conjecture la même chose pour v = 1,t! > 7. Si v = 2, il montre que t' = 3 est possible, mais t’ = 5 ne l'est pas. Il se résigne

à ne pas aborder le cas v = 3 : “questa recerca presenta notevoli difficolta arithmetiche e noi l’abbandiamo per ora” [Levi 1906/08, 434].

Vu les résultats considérables de Levi, l’article de Hurwitz [Hurwitz 1917], qui

aborde essentiellement les mêmes problèmes, mérite à peine d’être mentionné. 267R Schoof a remarqué dans cet argument des lacunes que L. Washington m’a obligeamment communiquées : Levi a tendance à oublier des possibilités de signes variés dans les identités quadratiques. Ainsi, on pourrait aussi avoir —q et —r? au lieu de q, r? dans [Levi 1906/08 114, l.— 10], pourtant ceci ne ferait qu’échanger les rôles de p, q dans les équations qui suivent.

Nos De même, les omissions analogues gues [[Levi 1906/08, /08, 113 , no (8)] 8 et [Levii 1906/08, 115, 1.5] sont

27= Je dois is à R. aR. Schoof Sc ‘oof| l’observation i

que l’équati ’équati on de Levi donne en fait un modèle de Xj(11).

(C’est même ce qu’on appelle aujourd’hui un modèle minimal sur Z.) — Levi ignore, bien sûr, le contexte analytique des courbes modulaires qu’il étudie. Mais il repère les “pointes” comme étant les points rationnels correspondant à des cubiques dégénérées.

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Hurwitz explique [Hurwitz 1917, 446, note 2] qu'il a eu connaissance des notes de Levi seulement

aprés la rédaction de son article. Contrairement a Levi il

ne démontre aucun théorème excluant l'existence de cubiques ayant certaines

structures finies de points rationnels. Et ses résultats et démonstrations ne surpassent en rien ceux de Levi — exception faite peut-être de l'écriture lucide de la forme générale des “vollständigen Gruppen”

(finis) de points rationnels

[Hurwitz 1917, 451], et la facilité avec laquelle Hurwitz travaille sur des corps de nombres, par exemple [Hurwitz 1917, 458ff]. Toutefois, c'est l'exposé très clair de Hurwitz qui servit comme point d’appui

au travail de M.I. Logsdon, [Logsdon 1925] — “une mathématicienne de Chica-

go”? dont le passage à Rome fit connaître à André Weil le mémoire de Mordell [Mordell 1922]. 6. De Mordell a Weil

Le théorème de finitude démontré dans [Mordell 1922] s’exprime très bien sans

référence à la notion de groupe abélien : “I shall now prove that if an ... equation

.. [of genus one has] an infinite number of solutions, then the method of infinite descent applies, that is to say, all the solutions can be expressed rationally in

terms of a finite number by means of the classic method,” 1.e., par la méthode des tangentes et sécantes. [Mordell 1922, 108] Mordell ne montre pas clairement si oui ou non il dispose de la notion de groupe abélien dans ce contexte. D’autre part, la présentation de la démonstration par André Weil dans le petit ‘digest’ [Weil [1929]] est parfaitement moderne. Il

nous semble donc que la notion du groupe abélien a été définitivement introduite

dans

la théorie des points rationnels sur les cubiques

(courbes elliptiques)

seulement vers le milieu des années 1920. Weil était peut-étre le premier auteur

qui en faisait un usage systématique. Nous ne discutons pas en détail l’article fondamental mais assez particulier

de Mordell. Pour une analyse intéressante de l’arrangement de la démonstration de Mordell, voir [Cassels 1986]. De méme nous passons sous silence la these de Weil et les travaux ultérieurs sur les courbes de genre supérieur, resp. leurs variétés jacobiennes.

La démonstration du théorème de Mordell (ainsi que de sa généralisation aux

28 Voir [Weil [1917c], [1928]*, 524].

N. SCHAPPACHER

180

naturellement variétés abéliennes sur les corps de nombres par Weil) se coupe

Dans en deux parties. D’abord on établit le ‘théorème de Mordell(-Weil) faible’.

la formulation actuelle, celui-ci dit que le groupe quotient C(Q)/2.C(Q) — plus généralement C(Q)/n.C(Q), pour tout n > 1 — est un groupe fini. C’est un énoncé un peu lourd a exprimer en termes de la seule méthode des tangentes et sécantes. Ensuite il faut mesurer le comportement de la ‘taille’ des

points rationnels quand on leur applique la méthode de la tangente — plus généralement, la multiplication par n. C’est aujourd’hui le début de la ‘théorie

des hauteurs’.?° Concluons ce rappert — sans doute préliminaire — sur la méthode des tangentes et sécantes par une réflexion générale. Le théorème de Mordell marque, par sa généralité, un point tournant dans l’histoire de ce domaine des mathématiques : c’est un énoncé fondamental qui vaut pour toutes les courbes elliptiques sur Q. Traditionnellement,

l’analyse diophantienne se pratiquait par bribes :

équation par équation, descente par descente. C’est Poincaré qui a énonçé le

programme pour sortir de cette industrie désceuvrée, et entamé l'étude systématique de l'arithmétique des courbes algébriques. Ce n'était pas la technique de

démonstration qui manquait aux prédécesseurs de Mordell, c'était l'esprit théorique qui faisait défaut dans une branche mathématique qui allait à la dérive, avant son renouement avec le développement de la géométrie algébrique.

Le monde avec lenteur, arrive à la vertu.

Fr. Ancillon Consid. sur la philos. de l’hist., Paris 1796

Manuscrit reçu le 20 décembre 1989 corrigé le 16 mai 1990

——_—_————

29

: : ‘ P Nous reconnaissons aujourd’hui des débuts de la notion de hauteur dans les démonstrations par descente infinie : cf. notre remarque (note 4 plus haut) sur la façon dont Fermat mesure la

taille des triangles rectangles dans sa démonstration mentionnée au 81. Le mot “Hohe” apparaît

dans ce contexte dans [Hurwitz 1917, 458].

DEVELOPPEMENT

DE LA LOI DE GROUPE

SUR UNE CUBIQUE

181

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sequel to A Treatise on Conic Sections; troisieme édition (cité d’après la reproduction photographique par G.E. Stechert, New York 1934). N. Schappacher [1989], Neuere Forschungsergebnisse in der Arithmetik ellipti-

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und XVII

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Leipzig (Teubner).

Norbert Schappacher

Max-Planck-Institut für Mathematik Gottfried-Claren-Str. 26

D-5300 Bonn 3

Séminaire de Théorie des Nombres

Paris 1988-89

The structure of the minus class groups of abelian number fields

René SCHOOF*

Abstract. abelian

It is shown

number

always determine

that sometimes one can read off the structure of the minus class groups of

fields from certain Stickelberger elements;

the question is raised whether one can

the structure of these class groups from Stickelberger elements.

Some numerical

and theoretical evidence for an affirmative answer is presented.

1. — Introduction Ideal class groups of cyclotomic or abelian number fields have been a subject of study for a long time [5,14]. The problem naturally falls apart in two. The class groups of real abelian number field are still relatively poorly understood. But about the other parts, the minus parts of the class groups of imaginary

abelian number fields, much more is known. For an imaginary abelian number

field K the minus class group Cé;, of K is defined to be C£x/im(C{x+) where KT is the maximal real subfield of K. The analytic class number formula gives an expression for the cardinality of C’¢;, in terms of certain generalized Bernoulli numbers which are defined in terms of the Galois group Gal(Q”” /Q). This formula is quite practical and can be used to compute the cardinalities of minus

class groups of abelian fields of small conductor, see [7,12,16]. More precise results were recently obtained by B. Mazur and A. Wiles [9]. They took the

action of the Galois group Gal(Q/Q) into account. They obtained formulas for the cardinalities of certain eigenspaces for this action. Apart from their results there does not appear to be a general way to describe the structure of the minus class groups as abelian groups in similar terms. For instance, in the case of a

complex quadratic field X, there does not seem to be a way to tell, in terms of

186

R. SCHOOF

structure of generalized Bernoulli numbers or Stickelberger elements what the Céx as an abelian group is.

In this paper we will discuss a way that might possibly lead to a procedure to describe the structure of minus class groups of abelian number fields in terms

of Stickelberger elements. In the case of a complex quadratic field and an odd prime p this leads to a necessary and sufficient criterion for the p-part of the class group to be a cyclic group [8]. This criterion is elementary and solely in terms of

Gal(Q”” /Q). While for quadratic fields there are definitely more practical ways to compute the structure of the class group, it seems that for abelian fields of high degree our method is quite practical.

Section 2 contains a preliminary discussion of Fitting ideals. In section 3 we introduce our “Stickelberger ideal” and we pose the question whether it is equal to a certain “Fitting ideal”. An affirmative answer to this question would

imply that one can completely describe the structure of the odd parts of the minus class groups of abelian number fields in terms of Stickelberger elements. One might even hope that this, in combination with an effective Cebotarev Density Theorem, leads to an efficient way to determine this structure. Finally, in section 4 we present some numerical examples indicating that the answer to our question is affirmative. Another indication in this direction is the result

mentioned above, joint with Hendrik Lenstra, on quadratic fields. The details of the proof will be published elsewhere.

I would like to thank Serge Lang for stimulating discussions concerning this work and the Department of Mathematics of the University of California at Berkeley, where part of this research was done, for its hospitality.

After this paper was written I became aware of Kolyvagin’s results [4,11] on p-class groups of Q((,). He gives a new proof of the theorem of Mazur and Wiles and he gives in addition a description of the Galois module structure of these groups in terms of certain higher “Stickelberger elements”. His results can easily

be generalized to abelian fields F for which p does not divide [F : Q]. His paper does, however, not seem to contain an explicit answer to questions (3.2) and (3.2)’.

THE STRUCTURE

OF THE MINUS CLASS GROUPS

187

2. — Fitting ideals For the definition and the basic properties of the R-Fitting ideals of finitely

generated R-modules A we refer to the books by Lang and Northcott [5,10] and the appendix to the paper by Mazur and Wiles [9]. We let R denote a discrete valuation ring with uniformizing element x. Any

finitely generated R-module is a finite product of copies of R and modules of

the form R/(x"). The R-Fitting ideal Fitr(A) of an R-module A measures the “size” of A : if A admits R as a direct summand then Fitr(A) = 0 and when

A= 4, R/(rx"i) then Fitr(A) = (r") where m = Se n;. The R-Fitting ideal does, in general, not reveal the entire R-structure of the module. One has, for

instance, that Fit p(R/(x?)) = Fitr(R/(x) x R/(x)) = (r?). We can, however, recover the R-isomorphism class of a finitely generated R-module A from the

Fitting ideal of A with respect to a larger ring as follows. We let A = R[[T]] denote the ring of power series with coefficients in R. We turn every R-module into a

A-module by letting T act as zero on A. PROPOSITION 2.1. — Let À and n1 < n2 < n3 < ... denote non-negative integers.

For the R-module

A=R°

À

d

ny

© @ R/(r")

one has that

Fita(A) = {9 aT'eA:a=0for0 1 we define I.{n) to be the ideal generated in A/((1+7)?" —1) by the images of the Stickelberger elements ¢, ¢(T’) for all primes £ = 1 (mod p")

and all possible identifications of the rings R,[G/G?"] and A/((1+ 7)?" — 1): I)0 = (4,,e(T) : 2 = 1 (mod p")). Finally we define the A-ideal I(x) :

I(x) = (E(x) + (A+ TP = 1)A). n>1

As we already remarked the “Stickelberger ideal” I() is contained in the “Fitting

ideal” Fit, C(x).

Numerical experiments suggest that, given C£(x), the only

restriction on the R,{G]-structure of the modules M(x) is the fact that the norm map M(x) — Cé(x) is surjective and that the G-cohomology of M(x) is as described in Lemma(4.1)

below. One would therefore, apart from these

restrictions, expect “random” behavior of the R,[G]-isomorphism classes of M(x) and of the Stickelberger elements. So one is tempted to ask the following question :

Question 3.2. — Is I(x) = Fit, C(x) for characters x which are not powers of the Teichmuller character? When y is a power of w the answer to the question would be negative because

I(x) € (T) while Fit,(C(x)) ¢ (T). In this case we have a modified question : Question 3.2. — When: # 1 is I(w*) = Fit, C£(w') N(T)? An affirmative answer to Questions(3.2) and (3.2) would enable one to recover

the structure of the class groups Cf(x) from certain Stickelberger elements. For x not equal to a power of the Teichmüller character this is clear from Prop.(2.1).

For powers of w one can recover the isomorphism class of the p-group Cl(w")

from the ideal Fit, C@(w*) (WT) and the class number #Cé(w"). We recall that Cl(w) = 0. We have some results that suggest that perhaps the answer to the questions is always "yes”. In the next section we will present some numerical examples

and some theoretical evidence.

192

R.'SCHOOF

de

4, — Examples

We recall that p denotes an odd prime, x an odd p-adic character of Gal(Q/Q) and £ a prime which we suppose to be 1 (mod p). We let F denote the fixed field of ker(y) and K the maximal p-power degree extension of F inside F((4). The x-part of the p-part of the class group of K is denoted by M (x). It is isomorphic to the y-part of the p-part of the class group of F((,) as we observed earlier. We let G = Gal(K/F) and A = Gal(F/Q). Clearly the Galois group of K over Q is isomorphic to the direct product G x A. One can view x as a character of A. The

Z,(A]-algebra Z, [im x] will be denoted by R, and the powerseries ring R,|[T]] will be denoted by A. For the basic facts on cohomology of groups and of class field theory that we will use see [2]. We first prove a useful lemma.

LEMMA 4.1. — When x is not the Teichmtller character w then the Tate coho-

mology groups of the G-module M(x) are isomorphic to R,/(p") or O according

as x(£) = 1 or not. (Here p” denotes the order of G.) Proof : Because A and G have coprime order the action of A commutes with

G-cohomology. More precisely :

H1(G, A) = H9(G,A%) H1(G,A)(x) ¥ H4(G, A(x))

for every Z[G x A] — module A, for every Z,[G x A] — module A.

We compute the cohomology of M(x) by taking x-parts of the cohomology groups of the G x A-modules that occur in the exact sequence 0—

Of

—

UK

—

Cx

—

Cle

— 0.

Here 0% denotes the unit group of the ring of K -integers, Cx denotes the group of idéle classes of K and Ux denotes the subgroup of idéles that have trivial valuation at the finite primes of K.

By global class field theory we have that H1(G,Cx) q € Z and therefore, since x £ 1 that

(1)

H(G,Cx)(vy)=0

foralqez.

= HAT)

for all

THE

STRUCTURE

OF THE

MINUS

CLASS

GROUPS

193

Since yx is odd and not equal to w we have that

(2)

HG, O%)(x)=0

forall q EZ.

We compute the cohomology of the idèle unit group Ux by means of local class field theory. It is known [2] that

H4(G,Ux) = @ HG», O%,) vl£

where the sum runs over the primes v of F over £ and G, denotes the decomposition group of any such v in G. We have, of course, that G, = G for every v. By w we denote a prime of K over v and by O%, the ring of integers of the completion K

of K at w. The Galois group of the local field K,, over Q, is just G, x Ay

where A, denotes the decomposition group of a prime v in F over £. The group

A: C A acts trivially on the cohomology groups H 1(G,, Z) and therefore, by lo-

cal class field theory, it acts trivially on the groups H4(G,, K*) = H!~?(G»,Z) as well. It follows from the long cohomology sequence of

0—

OP —

KU, —z—

0

and the fact that ged(#G,,#Ay,) = 1 that A, acts trivially on the cohomology

groups H1(G,, O:,). Each of the groups H1(G,, O%) is cyclic of order #G,

= p” and A/A¢

permutes the sumands in the sum above. We conclude that for all q € Z

H(G,UK) =Z/p"Z[A/A,] Since x : À —

Q, is injective we find

(3)

H1(G,UxK)(x) = {Eu

rs

The lemma

n

as A-modules.

Ww)

jae

Li

; =e 4 #1.

now follows from (1), (2), (3) and the long cohomology sequences

associated to the four term exact sequence above.

194

R.'SCHOOF

COROLLARY 4.2. — If x #w and x(£) # 1 then there exists an exact sequence

0—

VV

—+ M(x) —

0

where V is R,[G]-free of rank d = the minimal number of R,-generators of C(x). Proof : Since

x is odd and not equal to w, it follows from the proof of

Lemma(4.1) that H!(G,Oÿ)(x) = 0. This implies that the canonical map M(x) is injective and it follows easily that the module — Cl(x) + —

M(x)/(

1 )M(x) is isomorphic to = Cé(x). (Here 7 denotes a genera-

tor of the cyclic group G.) Therefore there is, by Nakayama’s lemma, an exact

sequence 0—

A—

V —

M(x)

—

0

where V is R,[G]-free of rank d. By Lemma(4.1) we have H4(G, M(x)) = 0. Therefore since G is a p-group and M(y) is finite we have that M(x) is a cohomologically trivial G-module. Therefore A C V is cohomologically trivial and

one can show that À is a projective R,[G]-module in a way similar to the proof

of Théorème 8 of Chapitre IX of [13]. Since R,[G] is a local ring one concludes that A is free and since M(x) is finite it is free of rank d. This proves (4.2). The following theorem can often be used to prove that Cé(x) is cyclic over lies THEOREM 4.3. — If there is a prime { = 1 (mod p") for which the Stickelberger

element ¢,,¢(T’) has Weierstrass degree 1 then CU(x) is cyclic over R, and

I) (x) = FitaCe(x) (mod (1+7)?"—1) I (w') = FityClw')()(T) (mod (1+ TP"

when x fu’, 1) fori #1.

Proof: In the proof we will write h for p raised to the length of the R,-module

C(x). In other words we have that #CC(X) = #R,/(h). By Weierstrass’ Preparation Theorem we have that

Sticky (T) =(T—a)-unit

in A/((1+T)" —1).

We have that, upto a unit,

Gas)! we(Z/efz)*

(R CE) e

ze(Z/f2)*

NC=) (X(0) =Dh.

THE STRUCTURE

OF THE MINUS CLASS GROUPS

195

If x(€) = 1 this implies that a = 0. By Stickelberger’s Theorem [16] we see

that T kills M(x) i.e. the module M( ) is G-invariant. Therefore the zero-th Tate cohomology group is M(x)/p" M(x). By Lemma(4. 1) this group is cyclic over Ry.

This implies that the class group Cé(x) is also cyclic over R, and therefore its

A/((1+T)?" — 1)-Fitting ideal is equal to (T,h). Since the constant terms of all Stickelberger elements are either 0 or a unit times h, we clearly have that

I) (yx) € Fit,C(x) (mod (1+ 7)" — 1). Since T € I‘")(x) we are now done when x is a power of w. In all other cases there exists a prime {' = 1 (mod p”)

for which x(¢’) # 1. By (4) the Stickelberger element ¢, ¢(T) has constant term equal to a unit times À which shows that h € Stick“ (x) as required. If x(£) # 1 we have by Lemma(4.1) that M(x) is a cohomologically trivial G-module. In the notation of Corollary(4.2) we let f = det(o). The Fitting ideal Fitr,(c(M(x)) is generated by f. Since M(x)/TM(x) = Cé(x) we see that f(0) = unit x h. By Stickelberger’s theorem M(x) is annihilated by ¢, ¢(T) = unit*(T—a). Since R,[G]/(T—a) is cyclic, the ideals between (T—a) and R, are linearly ordered. The smallest ideal strictly larger than (T — a) is (T—a, ap"~').

The ,[G]-annihilator is one of the ideals between (T — a) and R,. If it were not equal to (T — a) then (T — a,ap"~*) C Anng, jq(M(x)). It follows easily that (1 el — 1 kills M(y) and hence that the subgroup H = GP" acts trivially on M(x). This implies that the H-cohomology groups of M(x) are non-trivial, contradicting the cohomological triviality of M(x).

We conclude that Anng (G(M(x)) = (T — «) and hence that of T — a. Since, upto units, we have that f(0) = ¢,,¢(0) = h = that f = (T — a) * unit. It follows that both M(x) and Cé(x) modules. As in the previous case we have that Fitr (qjCé(x) =

f is a multiple a we conclude are cyclic R,(T, h) and the

obvious inclusion I‘”)(y) C FitaC{{(x) (mod (1+T7)?" — 1). To prove the other inclusion we choose another isomorphism

R(G] = RyITI/(A+ TY" — 1) by replacing T+1 by (T+1)2. We see that (7+1)?—1—a = T? +2T—a € I\)(x). It follows at once that T and a = unit + A are in I(x) as required.

The following Corollary says that if the Stickelberger ideal Z(x) is very large, the answers to questions (3.2) and (3.2)’ are affirmative.

196

R. SCHOOF

rass degree 1 CoRoLLARY 4.4. — If there is a power series in I(x) of Weierst then

;

I(x) = FitaC&(x)

when x Fw",

fori #1.

I(w') = FitaC€(w')( \(T)

Proof : If there is a power series in ¢,,¢ of Weierstrass degree 1 then there

must be a prime £ = 1 (mod p) for which ¢,,¢(T) has Weierstrass degree 1. It follows from the previous Theorem that Cé(x) is cyclic over R, and hence that Fit,Cé(x) = (T,h) in the notation of the proof of Theorem{4.3). It is obvious that I(x) € Fit, C(x) and the other inclucion follows by arguments similar to the ones employed in the proof of Theorem(4.3). This proves (4.4). THEOREM 4.5. — If y is quadratic, not equal to the character of conductor 3,

and Cé(y) is a non-trivial cyclic group then

I(x) = Fit, Ce(x).

Proof : It is easy to see that y cannot be a power of the Teichmüller character.

In [8] it is shown that foreach n > 1 there exist primes € = 1 (mod p”) for which ¢y,e(T) has its linear coefficient not divisible by p. As in the proof of Theorem

(4.3) one concludes that T € I") (x) for every n > 1. It follows that I() contains T. The Theorem therefore follows from the previous Corollary.

In [8] it is shown that the set of primes

£ =

1 (mod p”) for which the

linear coefficient of ¢, (T°) is not divisible by p has positive Cebotarev density. Moreover, the two subsets of primes for which in addition x(£) = 1 or x(£) #1 respectively each have positive density.

Remark 4.6 : If x is not a power of the Teichmüller character and Cé(y) = 0 then the answer to questions (3.2) is affirmative. This follows easily from the fact

that there is a prime ¢ congruent to 1 (mod p) for which y(é) 4 1. By (1) the Stickelberger element ¢, ,(7’) has constant term a unit. We conclude that both the Fitting and the Stickelberger ideal are the unit ideal. I do not have such a complete result in the case where x is a power of the Teichmiiller character. See example(4.6) for some numerical results.

THE STRUCTURE

OF THE MINUS CLASS GROUPS

197

We proceed by presenting some numerical evidence for a positive answer to

Questions (3.2) and (3.2). The calculations involved are quite straightforward and not too lengthy.

EXAMPLE 4.7. — p-parts of the class groups of Q(¢,).

It is known that all w'-eigenspaces of the p-parts of the class groups of Q(¢p) are cyclic whenever p
ind(z)Bi(x) # 0 (mod p) ? z=1

(Here B,(t) denotes

the k-th Bernoulli polynomial and ind, the index with

respect to some primitive root mod £.)

By Theorem(4.3) an affirmative answer to this question would imply that the

first Stickelberger ideal I“) (w') is equal to Fit, Cl(w') (\(T) modulo (1 + T)? —1 for 1 # 1 (mod p — 1). It would, independently of the truth of Vandiver's conjecture, also imply that the w'-eigenspaces of the p-part of the class group of

Q(¢,) are cyclic groups. Our criterion is similar to Washington's [16,Prop 8.19] but it appears to be independent For a few primes p and characters w' we checked the above. In all cases

considered

there appeared

to exist a prime { =

1 (mod p) for which

the

linear coefficient of the Stickelberger element ¢,,i ¢(T) is not zero mod p for all : simultaneously. We list the first few odd primes p and the corresponding

smallest such @. Often, but not always, @ is just the smallest prime congruent

to 1 (mod p).

198

R.'SCHOOF

P 83 89 97 101 103 107 109 113 127 131

£ P 83 41 947 43 283 47 107 53 709 59 1709 61 269 67 569 71 439 73 70217

l Pp ie 5 29 7 Oe tl 79 13 178157 191 19 139 23 59 29 ST SRE 37 149

£ 167 1069 1553 809 1031 857 2399 1583 509 263

EXAMPLE 4.8. — Various fields of prime conductor. In [7] D.H. Lehmer and J. Masley computed the minus class numbers of the

cyclotomic fields Q((;) for the primes f < 509. In many cases one can determine the structure of these class groups as abelian groups by exploiting the action of

Gal(Q(¢;)/@). In some cases the action of this Galois group does not help very much. For instance, when x is an odd character of degree d and p is a prime

congruent to 1 (mod d) dividing #C4(x) more than once it is not immediately clear how to decide whether C£(x) is cyclic over R, or not. In these cases we were always able to find an auxiliary prime ¢ for which the Weierstrass degree

of #,,e(T) is equal to 1. We conclude from Theorem(4.3) that in all these cases the group C(x) is cyclic over Ry.

Below we list the results in a small table. In the column "g +> y(g)” we list a primitive root g mod f and the value of the character y(g) mod p.

is

pl

degy gt x(g)

£

1390147 RAGE 2-9 25 lees Pe RE

283 83

443

272

26

2H

7

461 491

5? 11

4 1078

2-3 26

là a

491

11?

100

327

23

a

THE STRUCTURE

OF THE MINUS CLASS GROUPS

199

Mutatis mutandis everything we said above also holds in the case f = 443. In this case, however, “cyclic” means cyclic over the ring Z3[¢13] which is of degree

3 over Z3. The character y is given by a = x(g) which mod 3 is determined by

a + a? —a@+1=0. The class group involved is cyclic over this ring, but as an abelian group it is isomorphic to Z/9Z x Z/9Z x Z/9Z. EXAMPLE 4.9. — p-class groups of quadratic fields.

Sofar we encountered only cyclic class groups in our examples. There are

heuristics on the statistical behavior of the structure of class groups of number

fields that suggest that non-cyclic groups C£(x) are rare [3]. Probably any character x for which the minimal number of R,-generators of C¢(y) is merely moderately large will have a large conductor.

The only examples we present are quadratic characters x. In this case C(x) is for each odd prime p just the p-part of the class group of the quadratic field

Sr: Thanks to the efforts of D. Shanks and others many examples of class groups of complex quadratic fields and small primes p are known for which the prank is somewhat large. We computed for some quadratic characters x and some

odd primes p several Stickelberger elements ¢, ¢(T). In the table for a fixed prime é a few Stickelberger elements are listed, each made with a different primitive root mod @. For computational convenience we chose in all our examples the conductor

f to be prime. We computed

the Fitting ideals from the structure

of the p-class groups which we took from Buell’s tables [1]. We found that the

Stickelberger elements generate the A-Fitting ideal modulo ((1 + T)?” — 1) for certain small powers p”. When the exact power of p dividing ¢ — 1 is p”, the

Stickelberger element was computed modulo ((1 +7)?" —1,p"T).

R. SCHOOF

134059

classgroup

£

3x9

163

9x9

by,e(T)

487

34

811

34

1297

34

1459

35

1621

34

1783

34

163

34

487

34

811

34

27 + 15T + 49T? +... 27 + 48T + 61T? +... 27 + 24T + 70T? +... 45T + 201T? + 120T% + 47T4 + 144T + 75T? + 99T? + 167T4 +... TOT PISTE ALT PAT 4 oe 27 + 42T + 58T°? +... 27 + 21T + 70T? +... 27-LOITH2STI LS 27 + 157 4 2477 4 62T? 4 27 +.78T +2777 +287? +... 2+ 39T 4.787? + 50T* 40: 27 + 489T + 12877 +... 27 + 609T + 62T? +... 27 + 669T + 39577 +... 127. 2177 197 EE 6T + 2477 + 41T$ +... 37-6672 AST ole BNE YatBMS Se GOT 25570. BTM SY gi ey 36T + 43T? + 477 4+... IST + 6722 HIT te OT) 45577 sae 814 45T + 94T? 4+...

81 + 144T + 109T? + 81 + 72T + 70T? +... SAT + 35T +... 27T + 5AT? + 557? + SAT ATT a.

i

classgroup

e

351751

9 x 27

163 … st

497

pF

3¢

S11

3

1297

3‘

oy,e(T)

243 + 457 + 5777 + 567? + 243 + 63T + 39T? + 28T$ + 243 + 72T+ 12T? + 117$ +... 243 + 90T + 99T? + 57T% + 20T4 +...

243 243 63T T2T 36T 243

+ 45T + 1357? + 13273 + 233T4 + + 144T + 180T? + 487° + 2374 +... + 12T? + 80T4 +... + 6677 + 217° +.56T? +... + 487? + 3673 + 1474 +... + IT? + 28T$ +...

243 + 63T? + 537% +... 243 + 36T? + 467% +

Bazi GOR

3 KE

PocTe!

1458

3°

603T + 608T? + 666T + 350T? +... 3337 + 551171...

CAS.

«34

567 + OT + 277? + 517% + 60T* + 87° +1

Gg

WK

81103

12451

5x5

1297

34

1459

3°

162)

3°

1783

3‘

251

53

567 + 18T + 367? + 30T° + 60T* + 30T + 567 + 63T + 547? + 427? + GT4 + 12TS + 517? + 1957? 1... 195T? + 232T? +... 231T? + 357% +... 36T + 367? + 38T? +... 18T + 45T? + 167% +... OT + OT? + 11T$ +... 567 + 2777 + 9TS + 52T4 + 567 + 27T? + 72T* +1074 + 567 + 2777 + 637? + 474+... 567 + 171T + 2527? + 399T? + 866T4 +.. 567 + 450T + 315T? + 348TŸ + 317T* +.. 567 + 225T + 387T? + 336T° + 1774 +... 63T + 457? + 667 + 6374 + 52T° + 72T + 547? + 157° + 72T4 + 77T* +... 36T + 457? + 577? + 4574 + 10T° +... 63T + 42T? + 70T? +... 72T + 33T? + 2677 +... 36T + 60T? + 7T? +... 25 + 90T + 1217? +... 25 + 45T + 19T? +

R. SCHOOF

202

y 63499

classgroup

pl

oy,e(T)

aan

ie

14T + 47T? + 32T? +... TT + 107? + 4877 +...

272231

11583

1016083

13 x 13

727 677

11?

«117 4+ 87? +...

152

66T + 46T2 +... 13T + 15177 -£:.. OIT-PTISTIEERE

Manuscrit reçu le 28 septembre 1989

* p. 185 supported by the Netherlands Organization of Scientific Research.

THE STRUCTURE

OF THE MINUS CLASS GROUPS

203

BIBLIOGRAPHY

[1] D.A. Buell. — Class groups of quadratic fields, Math. Comp. 30, (1976), 610-

623. [2] J.W.S. Cassels and A. Frôhlich. — London,

Algebraic number

theory, Academic Press,

1967.

[3] H. Cohen and J. Martinet. — Class groups of number fields : Numerical heuristics, Math.

Comp. 48, (1987), 123-137.

[4] V.A. Kolyvagin. — Euler systems, To appear.

[5] S. Lang. — Algebra, 2nd edition, Addison-Wesley, Menlo Park, 1984. [6] S. Lang. — Cyclotomic fields, Graduate Texts in Math. 59, Springer-Verlag, New York 1978.

[7] D.H. Lehmer and J. Masley. — Table of the cyclotomic class numbers h*(p) and their factors for 200 < p < 521, Math. Comp. 32, (1978), 577-582, microfiche supplement.

[8] H.W. Lenstra and R.J. Schoof. — Class groups of imaginary quadratic number fields, in preparation.

[9] B. Mazur and A. Wiles. — Class fields of abelian extensions of Q, Invent. Math.

76, (1984), 179-330.

[10] J. Northcott. — Finite free resolutions, Cambridge Tracts in Math. 71, Cambridge University Press, Cambridge 1976.

[11] K. Rubin.

—

|

Kolyvagin's system of Gauss

sums,

To appear in Arithmetic

algebraic geometry Texel Birkatiser 1990. [12] G. Schrutka

von Rechtenstamm.

—

Tabelle der (Relativ)-Klassenzahlen

der

Kreiskôrper, deren ¢-Funktion des Wurzelexponenten (Grad) nicht grésser als

256 ist, Abh. Deutschen Akad. Wiss. Berlin, KI. Math. Phys. 2, (1964), 1-64. [13] J.-P. Serre. —

Corps Locaux, Hermann, Paris, 1968.

204

R. SCHOOF

[14] J.W. Tanner and S. Wagstaff. — New congruences for Bernoulli numbers, Math. Comp. 48, (1987), 341-350.

[15] S. Wagstaff. — The irregular primes to 125,000, Math. Comp. 32, (1978), 583-

591. [16] L.C. Washington. — Introduction

to cyclotomic fields, Graduate texts in Math.

83, Springer-Verlag, New York, 1982.

R. SCHOOF Mathematisch Instituut

Rijksuniversiteit Utrecht 3508 TA Utrecht The Netherlands

Séminaire de Théorie des Nombres

Paris 1988-89

On the fibration method for proving the Hasse principle and weak approximation A.N. SKOROBOGATOV

1. — Introduction

Let 4 be a class of varieties over a number

Xsmooth(kv)

field k. If for any

# Ÿ for all places v of k implies Xsmootn(k)

X € ¥

# 0, we say that

the varieties of 1 satisfy the smooth Hasse principle. In this case we call the

varieties of 4 HP-trivial. Let Xsmootn(k) # 0. Then X is said to satisfy the weak

approximation property, if for any finite set S of places Xsmootn(k) is dense in the product I,esXsmooth(ky). We call the varieties of 4 WA-trivial if they are HP-trivial and satisfy weak approximation.

The fibration method for proving the Hasse principle and weak approximation is described in the introduction to [7]. Various instances of this technique occur in [7], [5]. In §2 of this paper we present a theorem which subsumes many of these previous results. Although essentially known to the experts, it has not yet appeared in such a generality. Some applications of the theorem are studied in § 3. In §4 we apply the theorem to weak approximation on complete intersections

of three quadrics in P#, n > 11, as soon as they have a smooth k-rational point. In the Appendix written jointly with B.E. Kunyavskii we obtain a cohomological criterion for weak approximation on linear algebraic groups based on similar

ideas.

A.N. SKGROBOGATOV

206

2. — Main theorem

if for some For the purpose of this paper we call a subset U Cc k”" Hilbertian A}, and À in étale covering p : R — Z, where Z is non-empty and Zariski-open is a geometrically irreducible k-variety, U is the set of k-points x of Z for which with p7!(x) is connected in the scheme-theoretic sense. If p is a Galois covering

the Galois group G, this says that U is the set of k-points x of Z for which the residue algebra of the semilocal ring of R in p (x) is a Galois extension of k with the group G. Hilbert’s irreducibility theorem states that U is not empty. Its effective version implies that U satisfies both weak and strong approximation

[9].

Let & be an algebraic closure of k, X = X x, k, A" = AT. THEOREM

1. — Let

k. Let f : X —+

X be a geometrically integral variety over a number field

Y be a projective morphism with geometrically integral fibres,

f(X) = Y. Assume that Y is open in A7, and dim(A; \ Y) < r — 2. Further assume that the following geometric condition holds over k:

(Sect) If n C A” is the embedding of the generic point of A’

C

A’, then

X,,smooth(n) # 0, for some non-empty Zariski-open set of the variety of affine lines in A’.

Then ifthe fibres of f over a Hilbertian subset are H P-trivial (resp. WA-trivial,

X is also H P-trivial (resp. WA-trivial. Proof : Step 1 is the reduction to the case r = 1. This is done by a trick of

P. Salberger (cf. [5], Thm. 3.1). Let p : R —>

Z, Z C Aj, be an étale covering

defining our Hilbertian set U, with R geometrically integral. Let A be the variety

of affine lines in Aj, and let A’ C A be the subvariety of lines contained in Y. Since dim(Aj, \ Y) < r — 2, A’ is a dense open subset of A. It follows from the statement known as "the first Bertini theorem” (cf. [1], III.7) that for some nonempty open subset A; C A and ¢ € U, the inverse image p~1(@) is geometrically integral. The same argument applied to f yields another non-empty open set Uz € A’. Let U = WNU NU C A’, where W is an open set defined over k,

such that W x, k satisfies the condition (Sect). Let us fix M € Y(k) such that some line passing through M belongs to U/. Since X y is geometrically integral, for almost all non-archimedean places v of k a reduction of X M atu hasa smooth point by the Lang-Weil theorem. By Hensel's lemma it lifts to a smooth

ON THE FIBRATION

METHOD

207

ky-point on X y. Let S be a finite set of places of k containing all archimedean

places and all the non-archimedean ones for which X M smooth(kv) = 0 (if we are proving weak approximation we further assume that S contains the initially given places). Now we would

like to obtain a smooth

approximate M, € Xsmooth(kv),

k-point on X (resp. to

v € S, by a smooth k-point on X). To this end

let us approximate f(M,), v € S, by a k-point N in At, which is as close to each f(M,) as we wish, and is such that MN

€ UW. (Due to our choice of M,

lines of / through M form a non-empty Zariski-open subset of the space of lines passing through M. It remains to apply the weak approximation theorem.) By

the implicit function theorem for k, there exists an open neighbourhood Y, of M, in v-adic topology, such that the restriction of f to V, is homeomorphic to

a direct product of a neighbourhood of M,

in its fibre and a neighbourhood of

f(M,). Thus the fibre Xy for N € f(¥,), v € S, has smooth k,-points for v € S.

These points are smooth on f~!(M/N). Since p-! (MN ) is geometrically integral

too we can apply the theorem for r = 1,Y = MN the Hilbertian set

= Ai, X = f-!(MN), and

UN MN.

From now on let r =

1. Step 2 is also well-known

(cf. [7], Thm.

3.10). It

consists of a reduction of the theorem to the following statement : (*) there exists a finite set of places S containing all the archimedean ones, such

that ifS C S' andt € k = Aj(k) is an S'-integer (ie. integer away from 5"), then the fibre X, has smooth k,-points for allv ¢ S'. Suppose (*) is true. By condition (Sect) f has a k-section s over an open

subset V C Aj, such that s(t) belongs to the smooth locus of X; for t € V. This section is defined over a finite extension L/k. By Chebotarev’s density theorem there exists an infinity of places w, such that L ®, k, contains k, as a direct

summand. Therefore we obtain a smooth k,,-point in each k-fibre over V. Now

we would like to approximate smooth k,-points M, on X, v € S', by a smooth k-point. Choose w as above not in S’. Let us approximate f(M,), v € S’, by

N € V(k) NU

with a S’ U {w}-integral coordinate. This is possible due to the

fact that the strong approximation theorem holds for Hilbertian sets ([9], Thm. 1.3). Reasoning in the same way as in step 1 we see that Xy has smooth k,-

points for v € S’ by the implicit function theorem. X y also contains a smooth k,-point, and smooth k,-points for v ¢ S’ U {w} by (*). Since Xj is H P-trivial (resp. W A-trivial) we are done.

208

A.N. SKOROBOGATOV flat over Al. Step 3 is devoted to the proof of (*) above. First note, that X is

be Indeed, X is geometrically integral, and f is dominant ([10], III.9.7). Let Ox the ring of integers of k. X extends to a projective scheme X over Aj,, where V is a non-empty open subscheme of Spec(O;). Since X is flat over Al, by the open nature of flatness ([EGA 4], 11.1.1), ¥ is flat over an open neighbourhood of Al

in A4. In other words, + is flat over A}, for some non-empty open subscheme hi CV. Let Y C Al, be the closed subscheme, such that fibres of f:X — A, outside Y are geometrically integral. Since Y does not intersect the generic fibre

A1 of Al,, we see that V is contained in “vertical” divisors of Aj,. Therefore shrinking V, further to V, C Vi we can assume that all fibres of f : 4 —

Aj,

are geometrically integral.

Let Z C X be a closed reduced subscheme, such that f : X \Z —

Aj isa

smooth morphism ([10], III.10.5. Here we use the assumption char.k = 0.) Let Z = Z, U Z2 be the decomposition of Z into irreducible components, such that

each component of Z; dominates A}, while Z2 is contained in fibres of f. By ([10], III.9.7) Z; is flat over Al. In the same way as above, Z; extends to a flat closed subscheme

Z; C 4 over AY, V3 C Y2. On the other hand, Z: extends

to a closed subscheme Z2 C +. Clearly for some open subscheme

V4 C V3 the

restriction of f to ¥ \ (Z1 U Z2) is smooth over Aj,,. (Indeed, if an irreducible component of the union of singular loci of fibres of 1 is not contained in vertical divisors, then it intersects with the generic fibre X. This intersection is contained in Z, thus the component in question belongs to the closure of Z in 1’, i.e. to

Z1 U Z2,) Now let us use the fact that the Hilbert polynomial is constant in a flat projective family ([10], III.9.9). In particular, the degree and the dimension are

constant. The Lang-Weil theorem [13] states the existence of a constant C(n, r, d) depending on n, r and d, such that for any geometrically integral closed variety X of degree d and dimension r in P” over a finite field F, we have

IX(Fq) — q"| < C(n,r, dr". In particular, X has an F,-point as soon as q > C(n,r,d)*. It follows that for

some open subscheme V; C V4 and any closed point p € AY..

© Xai V t(p)

has a t(p)-point, where r(p) is the residue field at p. Moreover, reasoning in

ON THE FIBRATION

METHOD

209

the same way we see that # XaL, r(p) has a r(p)-point outside Z; XaL, r(p), for some

Vg open in V; (here we use the constancy of the Hilbert polynomial

and the Lang-Weil theorem for each irreducible component of Z;). Shrinking Ve

further to V; we can assume that for any p € Aj,, V XA1, t(p) has a r(p)-point outside Z2 Xa1 : t(p). In view of the previous discussion we conclude that for

any p € Ay, # XAL. t(p) has a smooth r(p}-point. Summing up, we have proved that there exists a finite set S of places of k

(containing all the archimedean ones), such that if t is integral outside S, then

X;, extends to a projective scheme over Spec(O x5) with smooth points in all the closed fibres. By Hensel’s lemma these lift to smooth k,-points on X;, for all

v ¢ S. We can replace S by any S' 2 S. Thus the condition (*) is satisfied, and the theorem is proved.

©

Remarks :

1. Let k(t) be the function field of a generic line in AT. By Tsen’s theorem K(t)

is a C:-field, hence the condition (Sect) is satisfied if the fibres of f over a non-empty Zariski-open set are rational curves, smooth quadrics in P”, n > 2, smooth cubic hypersurfaces in P", n > 3, smooth intersections of m quadrics

in P”, n > 2m, etc.

2. The condition (Sect) is satisfied if the fibres of f over

a non-empty Zariski-

open set are smooth proper rational surfaces ([4], Prop. 2).

3. The condtion (Sect) is satisfied when X;,,) is birationally equivalent to a principal homogeneous G. Indeed,

space under a connected linear algebraic k(t)-group

by a theorem

of Springer, any principal homogeneous

space is

trivial since k(t) is a Cj-field of characteristic 0 (cf. [16], III. 82). According to

(SGA 3], XIV.6.10) G is unirational, hence k(z)-points of G are dense in G, thus X52), emooth(*(=)) ca 0.

3. — Miscellaneous applications

Let G be a finite group,

and

U2 (G, N) POY Ker[H?(G,N) >

N be a G-module

of finite type. Define

H?(< g >,N)], where < g > is the cy-

clic subgroup generated by g.

Let K be a field and T be a K-torus. Let K be an algebraic closure of K. Recall that the minimal splitting field L of T is the invariant field of the kernel

A.N. SKOROBOGATOV

210

of T xx K (19), of the natural action of Gal(K /K) on the module of characters 3.36). Let T be the Gal(L/K)-module of characters of T. Define

2(P) = U2, (Gal(L/K),T). When K is a number field, the vanishing of I? (T) implies the W A-triviality of principal homogeneous spaces under T ([19], 6.38). Let G be a connected semisimple K-group with fundamental group B. Let

L be the invariant field of the kernel of the natural action of Gal(K/K) on B.

Define

Il? (B) = I? (Gal(L/K), B).

When K is a number field, the vanishing of III?,(B) implies the W A-triviality of principal homogeneous spaces under G ([15], 5.1, 5.5, [3]). THEOREM 2. — Let X be a geometrically integral variety over a number field k. Let f : X —

Y bea

projective morphism with geometrically integral fibres,

f(X) = Y. Assume that Y is an open subvariety of Aj, dim(A; \ Y) < r —2. Assume that the generic fibre of f is birationally equivalent to a variety of one of the following types :

à

a) a Del Pezzo surface of degree at least 5;

b) a principal homogeneous space under a torus T with IL2(T)

= 0,

where T is the module of characters of T ; c) a principal homogeneous

space under a connected semisimple group

G with I? (B) = 0, where B is the fundamental group of G (e.g. G is simply connected or adjoint).

Then X is W A-trivial (in particular, H P-trivial). Proof :

a) It is enough to consider k-minimal Del Pezzo surfaces, that is those of degree 5, 6, 8 or 9 (cf. [11]). Del Pezzo K-surfaces of degree 5 are K-rational

[18], and there is nothing to prove in this case. Del Pezzo surfaces of degree d are sometimes birational with principal homogeneous spaces under 2-dimensional tori : this is the case for d = 9 (well-known), d = 6 [14], and d = 8 (forms of P!xP!, see e.g. [17]). According to the classification of [11] it remains to consider

rational ruled surfaces. These are known to be K-rational. Now 2-dimensional

ON THE FIBRATION

METHOD

211

K-tori are K-rational ([19], 4.74). Since III2,(T) = 0 for a K-rational torus T (cf. [6], 9.5, [19], 4.57), a) is a particular case of b).

b) Let k(n) be the field of rational functions in r variables, and let L be the minimal splitting field of the k(n)-torus T. There exist an open dense subset

V c Aj and a V-torus 7 such that T is the generic fibre of T and X is V-birational with a principal homogeneous

space over V under 7. Let us

consider the Hilbertian set U C V(k) consisting of points x € V(k) for which the k-torus 7, has the splitting field k, with the Galois group Gal(k,/k) isomorphic to Gal(L/k(n)). The Gal(k,/k)-module T,, c EU, is naturally identified with

the Gal(L/k(n))-module T, thus III2(7,) = 0. By ({19], 6.38) k-fibres of f over U are WA-trivial (in particular,

H P-trivial). Now use Theorem 1 and Remark 3

to complete the proof. c) The

proof here is the same

as in b) with ([15], 5.1) used

in place of

([19], 6.38). The W A-triviality of principal homogeneous spaces under simply connected

groups

is a theorem

of M. Kneser and G. Harder

(cf. [12]). The

analogous result for adjoint groups without a factor of type Es is proved in [15], 5.4. The case of factors of type Eg is treated in [3] (cf. [15], 5.5).

©

4. — Weak approximation on intersections of three quadrics

Let k be an arbitrary field, char.k # 2. Let

X = Q1NQ2NQ3

C Pi be

an intersection of three quadrics. We shall always assume that X satisfies the following properties : a) X is geometrically integral and pure of codimension 3;

b) Xsmooth(k) # 0:

c) n > 8, and rk(\yQi + \2Q2 + A3Qs) > 7 for any (Ay, 2, À3)€ & \(0,0,0);

d) the polynomial det(r1Q1 + r2Q2 + 303) is square-free of degree n + 1. Let us denote by II = P? the space of quadrics spanned by Qi, Q2, Q3. Choose a smooth k-point 0 on X. The points of II are in 1 — 1 correspondence with hyperplanes

passing through the tangent space TX

: to each quadric

Q:, t € II, one associates its tangent space at 0. Let II be the dual plane to II. Its points are in 1 — 1 correspondence with lines in I], i.e. with pencils of

hyperplanes passing through T,X. To each line u C II we associate the common locus of corresponding hyperplanes : L, F0 ToQ1. Clearly this yields a 1 — 1

A.N. SKOROBOGATOV

212

passing through correspondence between points of Il’ and codimension 2 spaces

ToX. Our goal is to show that under assumptions a)-d) X is birationally equivalent to a fibration into intersections of two quadrics satisfying the conditions of Theorem 1. This is inspired by ([7], 3.2).

Choose a hyperplance H C P?, 0 ¢ H. Let p: II’ x H — II’ be the first projection. Let Y © II’ x H be the closed subset such that Y, = YN peal: u € II’, is the intersection of two quadrics in L, NH

= P”"-3, which is the

projection of the cone LM ash Q: = à (Q:N ToQ+) from 0 to H. Fort € u let us denote by R;,, the base of the cone LA N Q:, so that y, pris Riu. It is easy to

see (cf. [7], 1.16) that rk(Reu) = rk(Lu N Qt) = rk(Q:) — 4. By the assumption c) it is at least 3.

LEMMA 1. — For any u € Il, Y, C L, N H is pure of codimension 2. Proof : If for some u, Y, C L, N H were of codimension 1 or 0, then the R;,,’s

for all t € u would have a common factor. Since rk(R;,.) > 3, this is not so.

o

Let S C II be the curve defined by det(r1Q1 + r2Q2 + z3Q3) = 0. By the

assumption d) S is reduced. LEMMA 2. —

If a projective line u C

à II is not contained in S, then Y, is

geometrically integral.

Proof : For such u there exists t € u such that Q; has maximal rank n + 1 (by the assumption d)). Thus rk(R;,) > n — 3 > 5. On the other hand, for any t, rk(Rt1) > 3. If follows from ([7], 1.11) that Y, is geometrically integral. © Now we introduce the desired fibration into intersections of two quadrics.

Let Comp(S)

C II' be the set of lines in II which are components of S. Let

Z = Y Np "(Il \ Comp(S)). Since the fibres ofp : Z —

II \ Comp(S) have

equal dimensions (by Lemma 1) and are geometrically integral (by Lemma 2) we conclude that Z itself is geometrically integral.

PROPOSITION 1. — X is birationally equivalent to Z.

Proof: Let X; = X\(X NT X), and let 7 be the morphism X, — Il! sending x to u, if L, is the span of x and TX. We claim that there is a commutati ve

ON THE FIBRATION

METHOD

213

diagram : LS

nA

Indeed, x~'(u) is an open subset of the intersection of the cone L,N A, Q: and

a quadric

Q; for some

t’ ¢ u. Since 0 is smooth

on X, L, N Q» is

smooth in 0. Any ruling of the cone L,/N A, Q:, which is not in the tangent

plane to L, N Qe, intersects Q, in 0 and another point. Thus a projection of Qe N(LaN 2 Q:) from 0 to the base of the cone ZL,N A, Q; is an isomorphism away from T5Q4 N Ly = To X. This identifies X, with an open dense subset of

Ls

©

LEMMA 3. — If a projective line u intersects S in deg(S) = n + 1 distinct points (and by the assumption d) such lines form an open dense subset of II'), then Zu = ZNY, is not a cone. Proof : Let us choose a coordinate system, in which 0 is given by zp =

z; = 0 for:

1,

4 0, and L, is given by rz; = x2 = 0. Let Q; and Q2 be quadrics

passing through X, such that T)Q, is z1 = 0, and T)Q2 is r2 = 0. Then Z, is given by zp = ty = z2 = Q; = Q2 = 0. Suppose that Z, is a cone. By a change

of variables we can assume that its vertex is given by z3 = 1, x; = 0 for? # 3. The matrix of the quadratic form Q, + vQz is as follows :

Bey

Poet

L

*

VAR 0

uz

ood *

V3

*

*

0

0

0

vz;

Up

*

dv

Are Us

O

uz

aka

A

ELU

where u3, v3, the coefficients of A and the asterisks are linear forms in y and v. A straightforward computation yields :

det(uQi + vQ2) = (uv3 — vus) det(A).

A.N. SKOROBOGATOV

214

contradiction However by our choice of u this polynomial must be separable. This © proves that Z, is not a cone. Now we are in a position to prove the main result of this section.

THEOREM 3. — Let k be a number field. Assume that either Dir =elion

ii) n > 9, and smooth models of geometrically integral intersections of two quadrics (pure of codimension 2 and non-conical in PF, m > 6, satisfy the Hasse

principle (this is ([7], 16, Conjecture 1)). Then intersections of three quadrics in P? satisfying the assumptions a)-d) (e.g. smooth intersections of three quadrics

with a k-poini) are W A-trivial. Remark that this theorem is an analog of ([7], 3.11).

Proof : In order to apply Theorem

1 to p : Z —

that geometrically integral non-conical of two quadrics in Pf are

II' \ Comp(S) we notice

(pure of codimension

2) intersections

H P-trivial for n > 8 ([7], 11.1), and satisfy weak

approximation for n > 6 ([7], 3.11). By virtue of Lemma 3 such fibres are located

over a dense open subset of II’. It remains to verify the condition (Sect). Let

n © II' be the embedding of the generic point of a line £ C II’. Since k(n) is a C\-field, the intersection of two quadrics Z, has a k(n)-point as soon as dim(Z,) = n — 5 > 2. If this point is smooth we are done. Assume it is not. Let t be a point in II corresponding to £. If t¢ S,;,, then some line u, t € u CII,

intersects S in n +1 points. Thus Z, is non-conical for some u € £ (by Lemma 3), hence so is Z,. According to ([7], 2.1) Z, containing a k(7)-rational non-conical singular point is birational with a geometrically integral quadric. By the C:-

property of k(n) this quadric has a smooth k(n)-rational point, hence (Sect) is satisfied.

©

Remarks : 1. For n > 11 (assuming rk(Q;) > 9 for any t) we can do without 1. Instead we may use a theorem of Dem'yanov and Birch-LewisMurphy ([8], [2]), which states that two quadratic forms in at least 9 variables

Theorem

over a p-adic field have a common non-trivial zero. The proof imitates ([7], 3.9),

using ([7], 10.3).

ON THE FIBRATION

METHOD

215

2. As was pointed out by J.-L. Colliot-Théléne, our proof of Theorem 3 does not

require the full strength of Theorem

1. In particular, it seems likely that the

conditional result in ii) can be improved to n > 7 using somewhat more elaborated technique and a stronger conjecture, namely that the Manin obstruction to the Hasse principle and weak approximation is the only one for smooth in-

tersections of two quadrics in P# (cf. [7], 15). It seems

possible that the process could in principle be applied to obtain

weak approximation

for n-dimensional intersections of m quadrics, were the

Hasse principle and weak approximation known for a Hilbertian dense family of

(n — m + 1)-dimensional intersections of m — 1 quadrics. The

author

is deeply grateful to J.-L. Colliot-Théléne

for many

valuable

remarks, helpful discussions and encouragement, and to B.E. Kunyavskii for his interest in this paper.

APPENDIX

A criterion for weak approximation on linear algebraic groups

by B.E. Kunyavskii and A.N. Skorobogatov

Let G be a connected linear algebraic k-group, Ryu C G be its unipotent radical, and ¢ : G —>

G, be the natural projection onto the corresponding

reductive group G, = G/R,. Let Go C G, be the Zariski-open subset of regular semisimple elements. Let T be a maximal k-torus in G, and N be the normalizer of ¢(T) in G,. Maximal tori in G, are conjugate, that is they are parametrized by

the variety X = G,/N. Each element of Gp belongs to a unique maximal torus, thus we have a natural morphism f : Go — À whose fibres are open dense subsets of maximal tori of G,. Since k-tori are unirational ([SGA 3], XIII.3.4),

A.N. SKOROBOGATOV

216

of X. Note k-points are dense on each k-fibre of f. Let 7 be the generic point ([SGA 3], XIV.6.1). that by the Chevalley-Grothendieck theorem X is k-rational

Let T be the k(n)-torus corresponding to the generic fibre of f, L be its splitting field, and T its module of characters. We shall refer to T as "the generic torus” of G.

THEOREM.

—

Let G be a connected

linear group over a number field k. If

2 (T) = 0, where T is the generic torus of G, then G satisfies weak approximation.

Proof : Let S be a finite set of places of k, and M, € G(k,), v € S, bea set of local points which we would like to approximate by a k-point. Since G is

k-isomorphic to the product of k-varieties R, x G,, and R, is an affine space, we are reduced to the case G = G,. Now G is open and dense in G, hence we can assume that M, € Go(k,). By T, let us denote the maximal k-torus in G

corresponding to x € X(k). Let U be the Hilbertian set of points x € X(k) for which the splitting field k, of T, has the Galois group isomorphic to the splitting

group Gal(Z/k(n)) of the generic torus T. (In the language of [20] such maximal tori have no affect.) By virtue of Hilbert’s irreducibility theorem of T. Ekedahl [9]

and k-rationality of X, U is dense in the product of X(k,), v € S. By the implicit function theorem for k,, locally f has a structure of a direct product. Thus if we choose z € U very close to f(M,), T, contains a k,-point close to M,. Since the

Gal(k, /k)-module T, is naturally isomorphic to the Gal(L/k(7))-module T, we have IH) = 0. This implies weak approximation for T, ([19], 6.38). © CoROLLARY

(Kneser-Harder-Sansuc).

—

The following types of semisimple

groups over number fields satisfy weak approximation :

a) simply connected groups :

b) adjoint groups; c) absolutely almost simple groups. This was first proved in ([12], 7.1, [15], 5.4, cf. references in that paper).

Proof : The vanishing of IL (T) of the generic torus T for adjoint groups of classical type and simply connected groups of type A,, is proved in ([21], cf. [20],

ON THE FIBRATION

METHOD

217

Thm. 3). A uniform proof for each of the three cases is obtained independently

by A.A. Klyachko and M.A. Borovoi (unpublished).

©

Manuscrit reçu le ler septembre 1989

A.N. SKOROBOGATOV

218

REFERENCES

IV, [EGA4] A. Grothendieck and J. Dieudonné. — Elements de géométrie algébrique, Etude locale des schémas et des morphismes de schémas, Publ. Math. IHES, 20

(1964), 24 (1965), 28 (1966), 32 (1967). [SGA3] A. Grothendieck

and

M.

Demazure.

—

Séminaire

de géométrie

algébrique,

Schémas en groupes, I, II, III, Lecture Notes in Math., Springer-Verlag, Heidelberg, 151, 152, 153, (1970). [1] M. Baldassarri. — Algebraic varieties, Springer-Verlag, Berlin, (1956). [2] B.J. Birch, D.J. Lewis and T.G. Murphy. —

Simultaneous

quadratic forms,

American J. Math. 84, (1962), 110-115.

[3] V.I. Chernousov. — On the Hasse principle for groups of type Es (In Russian),

Doklady Akad. Nauk SSSR 306, (1989), 1059-1063. [4] J.-L. Colliot-Thélène.

—

Arithmétique

des variétés rationnelles

birationnels, Proc. Int. Congr. Math., Berkeley Ca. USA

et problèmes

1, (1986), 641-653.

[5] J.-L. Colliot-Théléne and P. Salberger. — Arithmetic on some

singular cubic

hypersurfaces, Proc. London Math. Soc. (3) 58, (1989), 519-549. [6] J.-L. Colliot-Théléne and J.-J. Sansuc. — Principal homogeneous spaces under

flasque tori, applications, J. Algebra 106, (1987), 148-205. [7] J.-L. Colliot-Théléne, J.-J. Sansuc and Sir Peter Swinnerton-Dyer. — Intersections of two quadrics and Châtelet surfaces, J. reine angew. Math., I, 373 (1987)

37-107, II, 374 (1987) 72-168. [8] V.B. Dem’yanov. — Pairs of quadratic forms over a complete field with discrete norm with a finite residue field, (in Russian), Izv. Akad. Nauk SSSR 20, (1956), 307-324.

[9] T. Ekedahl. — An effective version of Hilbert’s irreducibility theorem, Preprint, 1987, to appear in this seminar.

ON THE FIBRATION

METHOD

219

[10] R. Hartshorne. — Algebraic geometry, Springer-Verlag, Heidelberg, (1977). [11] V.A. Iskovskih. — Minimal models of rational surfaces over arbitrary fields, Izv. Akad.

Nauk SSSR Ser. Mat.

43, (1979),

19-43,

(Engl. transl.

: USSR-Izv

14,

(1980), 17-39). [12] M. Kneser. — Starke Approximation in algebraischen Gruppen. I, J. reine angew.

Math. 218, (1965), 190-203. [13] S. Lang and A. Weil. — Number of points of varieties in finite fields, Amer. J.

Math. 76, (1954), 819-827.

[14] Yu.I. Manin. — Cubic forms, North-Holland, 2™ ed., (1986). [15] J.-J. Sansuc.

—

Groupe de Brauer et arithmétique

linéaires sur un corps de nombres,

[16] J.-P. Serre. —

des groupes algébriques

J. reine angew. Math. 327, (1981), 12-80.

Cohomologie Galoisienne, Lecture Notes in Math. 5, Springer-

Verlag, Heidelberg, (1964). [17] A.N. Skorobogatov. — Arithmetic on certain quadric bundles of relative dimension 2, I, (in preparation). [18] H.P.F. Swinnerton-Dyer. — Rational points on del Pezzo surfaces of degree 5,

Proc. 5" Nordic Summer School in Math., Woltersand Noordhoff, Groningen , (1972), 287-290.

[19] V.E. Voskresenskii. — Algebraic tori, (in Russian), Nauka, Moscow, 1977. [20] V.E. Voskresenskii.

—

Maximal

tori without affect in semisimple algebraic

groups, (in Russian), Mat. Zametki 44 :3, (1988), 309-318.

[21] V.E. Voskresenskii and B.E. Kunyavskii. —

On maximal tori in semisimple

algebraic groups, (in Russian), Preprint deposited in VINITY, N° 1269-84, 1984.

A.N. SKOROBOGATOV Institute for Problems of Information Transmission U.S.S.R. Academy of Sciences 19 Ermolovoy, Moscow 101447

U.S.S.R.

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Séminaire de Théorie des Nombres Paris 1988-89

Sur un problème d’Erdés et Alladi Gérald Tenenbaum

1. Introduction Désignons par P(n) le plus grand facteur premier d’un entier générique n,

avec la convention P(1) = 1. L'ensemble SES

Pin) Sy}

fait l’objet d’une abondante littérature et a suscité ces dernières années un important regain d'intérêt. Nous renvoyons le lecteur à [2,3,5-7,9, 10] pour une liste

plus complète de références bibliographiques et nous nous contentons de men-

tionner

ici

deux

résultats

concernant

la

fonction

de

compte

U(z,y) := |S(z,y)|. Dans tout l’article nous employons systématiquement la notation

=

]

abe log y

(fie

2),

(a) La formule asymptotique de Hildebrand [6]

(1.1)

Y(z,y) = 2p(u){1 +

log(u + SE

OCF

valable uniformément, pour chaque € > 0, dans le domaine

(He)

r>3, exp{(log,z)$**}

1 du problème différentiel aux différences

p{u)=0 up'(u) +

(—o

2)

CE H(u)

=

essere

yy

(u > 0).

Dans tout l’article cg, c,,--- désignent des constantes absolues positives.

THEOREME

1. — Soit € > 0. On a uniformément pour x > y > 2

(1.7)

M(2,u) € Ve

H(u)° Bly + awa} Ye=H

Lorsque u n’est pas trop grand, on peut approcher M(z,y) par une fonction

à variations régulières. Désignons par w(u) la fonction de Buchstab, solution continue pour u > différences

1 et dérivable pour u > 2 du problème différentiel aux

w(u)

uw(u) (uw(u))’

=0

(—o0o < u < 1)

=1 =w(u—1)

(1 0, 0 < 6 < $7”, et chaque entier k>0,ona w I) (y

(1.12)

A(z,y)= “ye A

H(u)-ô

+ Oz A Wasser)

uniformément sous les conditions

(Ge)

r>3, exp{(logz)#+*} 0. On a uniformément dans le domaine (G:)

(1.15) Nous

M(z,y) = A(z,y) + O(U(2,y)Le(y) *). terminons

cette introduction

logiques. Deux techniques

par quelques

commentaires

méthodo-

sont essentiellement disponibles pour aborder un

problème comme celui de l’évaluation de Ÿ(x,y) ou M(z, y) : la méthode élémentaire d’équation fonctionnelle et la méthode d'intégration complexe.

La première comporte elle-même deux variantes, selon la nature de l'équation fonctionnelle utilisée. On peut avoir recours à l’identité de Buchstab — cf. [1, 2] — ou procéder, comme le fit Wirsing [13], en introduisant le coefficient pondéral

log n et en égalant les expressions obtenues par sommation d’Abel d’une part, et

par insertion de la relation de convolution log n = $74), A(d) d'autre part. Cette voie a été inaugurée par Hildebrand dans [6] et a permis l'extension du domaine

de validité (1.5) de la majoration initiale de Alladi. La technique analytique est subordonnée à l'emploi de la méthode du col et de ses avatars —

cf. [11]. Elle a permis l'obtention des résultats (1.2) et (1.6).

Elle est encore à la base des démonstrations des théorèmes

travail —

encore que, comme

1 et 3 du présent

on le verra, l’abscisse d'intégration ne soit pas

ici choisie exactement optimale. Nous obtenons ainsi un traitement unifié de la

somme M(z,y). Les détails sont semblables à ceux de [3] (théorèmes 4 et 5), où il s'agissait d'étudier la répartition des entiers de S(x,y) dans les progressions arithmétiques. En fait, l'étude de M(z, y) est analogue à celle de

Yo x(n) n€S(z,y)

où x est un charactére de Dirichlet. Le zéro de ¢(s)~1 en s = 1 joue le rôle de

l'éventuel zéro de Siegel de L(s, x), mais certaines complications supplémen-

taires sont dues au fait que, contrairement à la série L, la fonction ¢(s)~! n'est pas holomorphe dans la bande critique 0 < a < 1.

L'exposé oral tenu à ce Séminaire avait pour but de présenter les principaux résultats de l’article [3] écrit en collaboration avec Fouvry. A l'heure de rédiger un compte rendu, nous avons préféré éviter les redondances et illustrer la méthode en développant une application nouvelle.

SUR UN PROBLEME D’ERDOS ET ALLADI

227

2. Lemmes

Rappelons la notation ¢(s,y) = [[, 2 T8. Il suit

Ro € VU(z,y)T~* € V(x, y)E(z,y). Si u > (logy)®@-*), nous avons recours au procédé général, décrit dans la démonstration

du théoréme

1 de [11], et qui repose sur le calcul de la

transformée de Fourier de

(t)

ba soit

te

O(T) := /

—co

1 sat ee li acy27 Uee hae LVT

rl

eT w(t)dt = a max(1 — pp):

230

G. TENENBAUM *

On a

me

e chee (D'un = xehyo [Lee +inner

> POS

aie

\C(a + ir, y)|dr il

+17,

< 2° (a) {> ne cots d’après

le Lemma

8 de

[9]. Compte

tenu

(2.9),

de

cette

majoration

est

< Y(x,y)E(zx,y).

Il nous reste à montrer que, lorsque T # L, c’est-à-dire u > (log y}sU—e), la partie R; de l'intégrale de (2.8) correspondant

au domaine

d'intégration

L < |r| < T peut être englobée par le terme d'erreur de (2.6). Cela découle aisément du lemme

1, qui fournit, pour ces valeurs de 7, »

(s,y) 1 & C(a,yje2" K C(a,y)E(z, y). Il suffit alors de reporter cette estimation dans l’intégrale et d’appliquer (2.9).

Le lemme suivant permet de relier, pour o assez proche de 1 et |r| < L.(y), la

quantité ¢(s,y)~1 à la valeur en (s — 1) log y de la transformée de Laplace de la fonction de Buchstab. Rappelons (cf. par exemple [3], éq. (6.7)) que si l’on pose

@()ic= i e**w(t)dt 0

alors ona

(2.10) avec

1+0(s) = e7)

(o> 0),

SUR UN PROBLEME D’ERDOS ET ALLADI

231

On peut montrer (cf. [3], lemme 6.1) que si s n'est pas réel négatif ou nul, on a

(2.11)

(a) +7 +logs = ~I(-8)

- | 0

2 wel as v

Il découle donc de (2.10) et (2.11) que

(2.12)

1+8(s) = s~' exp{—y — I(—s)}.

Cela définit un prolongement analytique de @(s) en une fonction méromorphe

sur C ayant pour seule singularité un pôle simple en s = 0. On peut encore remarquer, mais nous n’en aurons pas besoin, que le calcul classique de p(s) permet de réécrire (2.12) sous la forme

sp(s){1+@(s)} = 1. LEMME 3. — Soit € > 0. On a uniformément poury > 2, o > 1—(log y),

et

Ir < Le(y), (2.13)

CS, y)" = C(s)"*{1 + B((s — 1) log y)}(1 + O(L.(y)~*)).

Remarques : Le terme ¢(s)~! a bien un sens car les hypothèses impliquent que s est situé dans la région sans zéro de Vinogradov. Lorsque

s = o < 1 le

terme entre accolades de (2.13) doit être interprété en faisant appel à (2.12), le

pôle en s = 1 étant compensé par le zéro de ¢(s)7?. Démonstration : D’après la proposition 1 de [10], ona

¢(s,y) = ¢(s)(s — 1) log yexp{y + I((1 — s)logy))}(1 + O(Le(y)~*)) uniformément pour 1 — (log “1ns 5 1 par une

en restant dans le domaine

V:={s:0 >1-c7(log(3+ [r|)) 7}.

SUR UN PROBLEME D’ERDOS ET ALLADI

235

Désignons par a +21, les points d'intersection de la frontière de Y et de la droite a = a, et considérons le chemin Z, symétrique par rapport à l'axe réel, dont la

partie supérieure est constituée du segment vertical [a + iL,a + iT] et de la partie r > Tp de la frontière de V. On déduit de ce qui précède que l’on a

L

(3.7)

= B(log x) —- — x

ffB(s)x “ds

et la majoration (3.6) implique que la derniére intégrale est

€ rLy,(z)* € V(x,v)Le(y) dans (G.). Par (3.4), on a B(logx) = A(x,y). La démonstration du théorème 3 est ainsi complétée.

4. Preuve du Théorème 2. Nous

pouvons

supposer

sans

perte de généralité que y est assez

grand.

Remarquons d’abord que la représentation intégrale (3.3) implique

(4.1)

aj= "Sf m(erje =

(—1)

ps

todo

eS erty,

Fe

(§ = 0,1,-->). DE

LA

Bien entendu, as = 0. Nous appliquons ensuite le lemme 6 et intégrons l'identité

(2.12) sur R relativement à la mesure M(y°)y ‘dv. Compte tenu de (4.1), nous obtenons A(z,y)

er) cs +

Si + So

avec

+00

Ci

j=0

j'

emi ie (v + mu) M(y")y "dv,

1 (log y),

Ly (y*~*) « (logy) “2.

La majoration précédente pour 5; est donc encore valable. Pour estimer S2, nous faisons appel au lemme 5. On a < aoe (log ca La

(4.4)

< aon

+00 Î

w (k+1) ut |

[> ]

Hiei

1

1 dz

Lye ST

Jo) (uu— ta (ut) dt.

< eflo (UD) Le }ela ia plu —t)H(u—t)~*Ls = ,(y*)a1 “tt où cg est une constante absolue telle que 6 < cg < 272.

On a pour 0 tué

SUR UN PROBLEME

D’ERDOS ET ALLADI

237

Il suit que l'intégrale de (4.4) est

< p(u)H(u)"® Î exp{—t(3cg + 3logu — u?°)}dt 0

< p(u)H(u)S. Cela implique bien l'estimation

H(u)

S2 < SU) et achève la démonstration du Théorème 2.

5. Preuve du Théorème

1.

Nous distinguons deux cas, selon la taille de u. Observons d’abord qu’en

appliquant le théorème 2 avec k = 0, on obtient, sous l'hypothèse (G 1e) et pour

u> à,

H(u) SOU

rene TE

Lorsque 1 < u < 3, on peut écrire A

Fa L(y”) (eu (log y)3— 3°, nous faisons appel au lemme 2. Le terme d'erreur de (2.6) est clairement acceptable, et nous estimons le terme principal grâce au

lemme 1. Par (2.4), ona a+iL

ds

/ aC et < 2%(a,y)H(a)-* logL € U(z,y){H(u)

Ley) + ¥0"}

ou nous avons utilisé (2.9). Cela achéve la démonstration.

texte recu le 28 juillet 1989

238

G. TENENBAUM

BIBLIOGRAPHIE

[1] K. Alladi. — Asymptotic estimates for sums involving the Moebius function II, Trans. Amer. Math. Soc. 272, (1982), 87-105.

[2] N.G. De Bruijn. — On the number of positive integers < x and free of prime factors > y, Nederl. Akad.

Wetensch. Proc. A 54, (1951), 50-60.

[3] E. Fouvry et G. Tenenbaum. — Entiers sans grand facteur premier en progressions arithmétiques, prépublication.

[4] A. Hildebrand. — On a problem of Erdôs and Alladi, Monat. Math. 97, (1984),

119-124. [5] A. Hildebrand. — Integers free of large prime factors and the Riemann Hypothesis, Mathematika 31, (1984), 258-271.

[6] A. Hildebrand. —

On the number of positive integers < x and free of prime

factors > y, J. Number

Theory 22, (1986), 289-307.

[7] A. Hildebrand. — On the number of prime factors of integers without large prime divisors, J. Number

[8] A. Hildebrand.

—

Theory 25, (1987), 81-106.

The asymptotic

behaviour of the solutions

of a class of

differential equations, J. London Math. Soc., a paraitre. [9] A. Hildebrand and G. Tenenbaum. — On integers free of large prime factors, Trans. Amer. Math. Soc. 296, (1986), 265-290.

[10] E. Saias. — Sur le nombre des entiers sans grand facteur premier, J.

Number

Theory 32, (1989), 78-99. [11] G. Tenenbaum.

—

La méthode

du col en théorie

analytique

des nombres,

Séminaire de Théorie des nombres, Paris 1986-87, Prog. Math. Birkhauser, 75, (1989), 411-441. [12] E.C. Titchmarsh.

—

The

theory of the Riemann

revised by D.R. Heath-Brown, Oxford, (1986).

zeta function,

2nd edition,

SUR UN PROBLEME

[13] E. Wirsing. —

D’ERDOS

ET ALLADI

Das asymptotische Verhalten von Summen

239

über multiplikative

Funktionen II, Acta Math. Acad. Scient. Hungar. 18, (1967), 411-467.

G. TENENBAUM Université de Nancy I

BP 239 54506 VANDŒUVRE Cedex

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Séminaire de Théorie des Nombres

Paris 1988-89

An effective version of Hilbert’s irreducibility theorem Torsten EKEDAHL

In this note we will give a proof of Hilbert’s irreducibility theorem which differs from the standard the technique

otherwise.

one. There are mainly two reasons for doing this. Firstly,

used enables one to obtain refinements

not easily obtainable

For instance, it is seen that the parameter variety does not have

to be rational; it suffices that the set of rational points has a certain weak approximation property. Secondly, not only will we give an effective proof, but it will be seen that the proof is very efficient. We will demonstrate the last property

by showing that the smooth cubic surface

X := {1lw* + 56w?z + 332° + 2827y + 11y° + 14y2z + 28wz? + 11z? = 0} considered as a surface over the rational numbers,

has the largest possible

Galois action on Pic(X@). This hence gives an explicit example of a Galois

extension of Q with Galois group the Weyl group W(Es). I would like to thank J.-L. Colliot-Théléne for pointing out that [Fr] already

contains the basic idea of proof, as well as for picking out parts of a primitive version of this paper needing clarification.

1. — If G is a finite group and S is a set of conjugacy classes of elements and

subgroups of G we will say that S generates G if any choice of elements in all members of S generates G. We will need the following well known result (which goes back to Jordan).

LEMMA 1.1. — Let G be a finite group. Then the set of all conjugacy classes of

elements of G generates it.

242

T. EKEDAHL

Proof : This is equivalent to saying that no proper subgroup H meets all

conjugacy classes of G or, equivalently, that the conjugates of H do not cover G

which is immediately seen by counting ({G : H](|H| — 1) +1 < |G}). Remark

: In practice a finite group

is generated

by a small

© number

of

conjugacy classes. We will see an example of this later on.

By a number ring, we mean the localization of the ring of integers in a number field at finitely many primes. LEMMA

1.2. — Let x : X —

SpecR be a morphism of finite type, where R is

a number ring, and let p : Y — X be an étale Galois cover with Galois group G and suppose that tp has a geometrically irreducible generic fiber. Let

C be a

conjugacy class ofG and let c:=| C |/ | G |.Foreachp € max(R) let t(p) be the number of x € X(k(p)) for which the Frobenius element F, belongs to C. Then

t(p)/|X(&(p))| — c = O(|k(p)|~'/”). Proof : We may ignore a finite number of p’s and so we may assume, after replacing Spec À by an open subset, that all the fibers of xp are geometrically connected. We may also assume

that there is a prime @ which is invertible in

Ox. Let E be a finite extension of Q, whicH is a splitting field for G. For each irreducible G-representation x over E we can twist it by p to get an étale E-sheaf

V, on X. By the Lefschetz trace formula ([Del :Thm 3.2]), for p € maz(R),

D,

zEX(k(p))

xFe)=

D

0