Table Of ContentTheorie des nombres
Number Theory
Theorie des nombres
Number Theory
Comptes Rendus de la Conference internationale de
Theorie des nombres tenue ä l'Universite Laval,
5-18 juillet 1987
Proceedings of the International Number Theory
Conference held at Universite Laval, July 5 -18, 1987
Editeurs/Editors
Jean-Marie De Köninck
Claude Levesque
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DE
G
Walter de Gruyter · Berlin · New York 1989
Editors
Jean-Marie De Köninck
Claude Levesque
Departement de mathematiques
et de statistique
Universite Laval
Quebec, Canada G1K 7P4
1980 Mathematics Subject Classification (1985 Revision): 11-06; llAxx, llBxx, llDxx, llExx, llFxx,
llJxx, llKxx, llLxx, llMxx, llNxx, llPxx, llRxx, llSxx, llYxx, 14Cxx, 14Kxx.
Library of Congress Cataloging-in-Publication Data
International Number Theory Conference (1987: Universite Laval)
Theorie des nombres : comptes rendus de la Conference
internationale de theorie des nombres tenue ä l'Universite Laval en
1987 / editeurs, Jean-Marie De Köninck, Claude Levesque = Number
Theory : proceedings of the International Number Theory Conference
held at Universite Laval in 1987 / editors, Jean-Marie De Köninck,
Claude Levesque
p. cm.
English and French.
Held July 5 -18, 1987, Quebec.
ISBN 0-89925-510-8 (U.S. : alk. paper)
1. Numbers, Theory of — Congresses. I. De Köninck, J.M., 1948-
II. Levesque, Claude, 1946- . III. Title. IV. Title: Number theory.
QA241.I59 1987 89-1181
512 '.7~dcl9 CIP
Deutsche Bibliothek Cataloging-in-Publication Data
Theorie des nombres : comptes rendus de la Conference
internationale de Theorie des nombres tenue ä l'Universite
Laval, 5-18 Juillet, 1987 = Number theory / ed. Jean-Marie De
Köninck ; Claude Levesque. - Berlin ; New York : de Gruyter, 1989
ISBN 3-11-011791-6
NE: International Number Theory
Conference <1987, Quebec>; Universite Laval <Quebec>; PT
® Printed on acid-free paper.
© Copyright 1989 by Walter de Gruyter & Co.. Berlin. All rights reserved, including those of translation
into foreign languages. No part of this book may be reproduced in any form - by photoprint, microfilm
or any other means nor transmitted nor translated into a machine language without written permission
from the publisher. Printing: Bosch-Druck, Landshut. Binding: Dieter Mikolai, Berlin.
Cover design: Thomas Bonnie. Printed in Germany.
Introduction
Du 5 au 18 juillet 1987, a From July 5 to 18, 1987,
l'Universite Laval, se tenait dans at Universite Laval, an interna-
le cadre enchanteur de la ville de tional number theory conference
Quebec, une conference interna- took place in the lovely city of
tionale portant sur tous les do- Quebec; all aspects of number the-
maines de la theorie des nombres: ory were covered: elementary, al-
theorie elementaire, alg^brique et gebraic and analytic number the-
analytique des nombres; theo- ory; constructive (computational)
rie constructive (calculatoire) des number theory; elliptic curves and
nombres; formes modulaires, cour- modular forms; abelian varieties,
bes elliptiques; varietes abeliennes; arithmetical geometry.
geometrie arithmetique.
Pres de 300 personnes provenant Close to 300 mathematicians
d'une vingtaine de pays se sont ins- coming from about 20 countries
crites, et on trouvera dans la liste registered, and the list of partici-
des participants les noms de ceux pants shows the names of all who
qui sont efFectivement venus (les effectively came (the names of Uni-
noms des etudiants de l'Universite versite Laval students not studying
Laval n'etudiant pas en theorie des in number theory being excluded).
nombres etant exclus).
Le leitmotiv des organisateurs The leitmotiv of the confer-
de la conference etait: plus on a du ence organizers was the following:
plaisir, plus les echanges mathema- the merrier, the more fruitful the
tiques sont fructueux. Tous furent mathematical exchange is. Every-
unanimes ä dire qu'ils ont eu beau- body agreed: everyone had fun.
coup de plaisir.
II y eut 155 exposes presentes 155 lectures were delivered by at
par au moins 144 mathemati- least 144 mathematicians. Three
ciens(nes), dans un cadre perma- parallel sessions were run excepted
nent de trois sessions paralleles, for the following so-called general
sauf pour les conferences suivantes audience lectures:
declarees d'interet general:
vi INTRODUCTION
(1) Dr. Karl RUBIN, The arithmetic of elliptic curves with complex
multiplication. Parts I, II, III:
"In these lectures we will describe in detail descent on elliptic curves
with complex multiplication, to relate arithmetic properties of such a curve
with the value of its L-function at s = 1. Our main goal will be to prove the
following theorem (the first assertion of which is due to Coates and Wiles):
Theorem. Let Ε be an elliptic curve defined over an imaginary quadratic
field K, with complex multiplication by K. If L(Ejj, 1) φ 0, then E(K) is
finite and the p-part of the Tate-Shafarevich group of Ε is trivial for almost
all primes ρ of K.
This theorem will allow us to give examples of finite Tate-Shafarevich
groups."
(2) Dr Jean-Marc DESHOUILLERS, Coefficients de Fourier de formes
modulaires et applications a la theorie analytique des nombres.
Partie I: Coefficients de Fourier de formes modulaires.
Partie II: Applications a la theorie analytique des nombres.
Partie III: Valeurs propres du Laplacien pour le sous-groupe To(Q).
(3) Dr. Carl POMERANCE, Algorithmic aspects of the fundamental
theorem of arithmetic.
Part I: Primality testing.
Part II: Factoring.
Part III: Applications of elliptic curves.
(4) Dr. Peter SCHNEIDER, An introduction to Beilinson conjectures.
Parts I, II.
(5) Dr Michel WALDSCHMIDT, Recent developments on transcen·
dance theory.
(6) Dr. Aleksandar IVIC, Some recent results on the Riemann zeta
function.
II y eut meme une conference Even a popular lecture was
d'interet populaire prevue pour le scheduled for lay persons, publi-
public, annoncee et commentee par cized and commented by news me-
les medias de la presse par lee, ecrite dia: radio, newspapers and televi-
et televisee: sion:
(7) Dr Paulo RIBENBOIM, Les nombres: des amis qui nous donnent
des problemes.
INTRODUCTION vii
Mentionnons en terminant que It is worth mentioning that all
tous les articles soumis pour pu- papers submitted for publication in
blication dans ces Comptes Rendus these Proceedings were refereed
furent arbitr£s par des experts: ne by experts: only papers meeting
furent retenus que ceux qui satisfai- high standards were accepted.
saient a des criteres severes.
Les editeurs, The editors,
Jean-Marie De Köninck
Claude Levesque
Octobre 1988 October 1988
Remerciements Acknowledgements
Nous voulons d'abord re- Our first thanks go to the Cana-
mercier la Societe Mathematique dian Mathematical Society (CMS)
du Canada (SMC) pour avoir for sponsoring this Conference ω
parraine notre conference comme the CMS 1987 annual summer sem-
etant le seminaire d'ete annuel 1987 inar; in particular we thank Dr.
de la SMC; en particulier, nous re- James G. Timourian and Dr. Stan-
mercions Dr James G. Timourian ley O. Kochman.
et Dr Stanley O. Kochman.
Ce fut un plaisir de nous associer We were fortunate and pleased
avec la celebre maison d'edition to deal with the famous publisher
Walter de Gruyter & Co., et Walter de Gruyter & Co., and
nous exprimons notre reconnais- we express our gratitude to the
sance a toute l'equipe de Walter de Walter de Gruyter team.
Gruyter.
Cette conference de theorie des This Number Theory Con-
nombres fut rendue possible grace ference was made possible with
a, des subventions provenant du grants coming from Natural Sci-
Conseil de recherches en sci- ences and Engineering Re-
ences naturelles et en genie search Council of Canada, from
du Canada, du Ministere des the Minist&re des communi-
communications du Quebec, cations of the government of
du Fonds pour la formation Quebec, from Quebec FCAR
de chercheurs et l'aide k programme, and from our main
la recherche du Quebec et supporter 1ST, a computer ser-
de notre commanditaire principal vices company with Mr Andr£
1ST, societe de services informa- Thompson acting as its enthou-
siastic president. Several copo-
tiques dont le dynamique president
rations joined 1ST to support
est monsieur Andr£ Thomp-
the conference: Amdhal; Xerox;
son. Se sont associes ä IST
Laurent Gagnon; Raymond,
les entreprises: Amdhal; Xerox;
Chabot, Martin, Pari, c.a.;
Laurent Gagnon; Raymond,
Price Water House; Fiducie,
Chabot, Martin, Pari, c.a.;
Pret et Revenu; Air Canada;
Price Water House; Fiducie,
En Route; Marcel Jobin.
Pret et Revenu; Air Canada;
En Route; Marcel Jobin.
χ
REMERCIEMENTS / ACKNOWLEDGEMENTS
Nous avons beneficie de tout le We benefitted from all the
support possible de l'Universit^ possible support of University
Laval, l'hote de cette conference, Laval, host of the conference, and
et en particulier du Departement in particular of Departement de
de mathematiques et de statis- Mathematiques et de statistique;
tique; nous remercions son di- we thank its head, Dr. Robert
recteur, Dr Robert Cote. Cote.
Nous avons regu de precieux We received valuable advices
conseils des mathematiciens sui- from the following mathematicians:
vants: Dr Ram Murty, Dr Kumar Dr. Ram Murty, Dr. Kumar
Murty, Dr Michel Waldschmidt et Murty, Dr. Michel Waldschmidt
Dr Richard Mollin. Nous leur and Dr. Richard Mollin. We
sommes reconnaissants. Nous re- are grateful to them. We also
mercions aussi Dr Ernst Kani, Dr thank Dr. Ernst Kani, Dr. Hugh
Hugh Williams et Dr Victor Snaith Williams and Dr. Victor Snaith for
pour leur aide. their help.
Nous voulons vivement re- We heartfully thank all au-
mercier tous les auteurs des ar- thors of submitted papers, as well
ticles soumis, de meme que tous as all the referees for their de-
les arbitres pour leurs rapports tailed reports frequently assorted
detailles et souvent pleins de sug- with many suggestions.
gestions.
Tous les articles furent redacty- All the papers were typewritten
lographies avec T^jjX, nous remer- with the TßX facilities. We thank
cions tous ceux qui les ont tapes. all those who typed them.
Enfin, nous ne saurions pas suf- Finally, but definitely not least,
fisamment insister pour remercier we are much grateful to Dr.
Dr Ärmel Mercier (Chicoutimi) Armel Mercier (Chicoutimi) for
pour les nombreuses heures qu'il the many hours he spent to format
a passees pour rendre possible the whole book with I^TJJX.
1'edition finale de ces Comptes
Rendus avec JATgX.
Les editeurs, The editors,
J.M. De Köninck
C. Levesque
