Number Theory Related to Fermat’s Last Theorem: Proceedings of the conference sponsored by the Vaughn Foundation

366 Pages·1982·10.284 MB·English

by  P. Ribenboim (auth.), Neal Koblitz (eds.)

Progress in Mathematics
Vol. 26
Edited by
J. Coates and
S. Helgason
Springer Science+Business Media, LLC
Number Theory
Related
to Fermat's
Last Theorem
Proceedings of the conference
sponsored by the
Vaughn Foundation
Neal Koblitz, editor
Springer Science+Business Media, LLC 1982
Editor:
Nea 1 Kob 1 itz
Department of Mathematics
University of Washinoton
GN-50
Seattle, WA 98195
Library of Conoress Catalooino in Publication Data
Main entry under title:
Number theory related to Fermat's last theorem.
( Prooress in mathemati es ; v. 26)
1. Numbers, Theor.v of--Addresses, essays, 1e ctures.
2. Fermat's last theorem--Addresses, essays, lectures.
I. Koblitz, Neal, 1948- . li. Series: Prooress in
mathematics (Cambridne, ~1ass.) ; v. 26.
QA24l.N876 1982 512'.7 82-19736
ISBN 978-0-8176-3104-8
CIP-Kurztitelaufnahme der Deutschen Bib1iothek
Number theory re1ated to Fermat's last theorem/
Neal ed.
~oblitz,
(Proqress in mathematics ; Vol. 26)
ISBN 978-0-8176-3104-8 ISBN 978-1-4899-6699-5 (eBook)
DOI 10.1007/978-1-4899-6699-5
NE: Koblitz, Neal (Hrsq. ); GT
All riqhts 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, photocopyinn,
recording or otherwise, without prior permission of the copyriqht
owner.
©Springer Science+Business Media New York 1982
Originally published by Birkhäuser Boston,lnc. in 1982
Softcoverreprint of the hardcover 1st edition 1982
ISBN 978-0-8176-3104-8
TABLE OF CONTENTS
Preface
THE WORK OF KUMMER ON FERMAT'S LAST THEOREM
P. Ribenboim • 1
EISENSTEIN, AND HIGHER RECIPROCITY LAWS
KUMI~ER,
H. Edwards • • • • . • • 31
GEOMETRY OF FERMAT VARIETIES
T. Sh ioda .. 45
THE DEGENERATE FIBRE OF THE FERMAT CURVE
W. McCallum •...•• 57
SOME REMARKS ON WEIERSTRASS POINTS
D. Rohrlieh . • • . • . • • 71
DIOPHANTINE PROPERTIES OF THE PERIODS OF THE FERMAT CURVE
79
M. Waldschmidt •..•.••.
THE FERMAT EQUATION AND TRANSCENDENCE THEORY
A. Baker. • • • • 89
AN ARCHIMEDEAN ANALOGUE Of THE ROOT NUMBER
G. Anderson 97
PERIOD INTEGRALS OF EISENSTEIN COHOMOLOGY CLASSES AND
SPECIAL VALUES OF SOME L-FUNCTIONS
G. Harder . . . . • . . . . • . . • • . 103
CUSPIDAL AUTOMORPHIC REPRESENTATIONS ASSOCIATED TO
PARABOL I C SUBGROUPS AND CONJECTURE
RA~IANUJ.A.N
I. Piatetski-Shapiro. . • . . . . • . . . . 143
ON AUTOMORPHIC FUNCTIONS Of HALF-INTEGRAL WEIGHT WITH
APPLICATIONS TO ELLIPTIC CURVES
D. Goldfeld et al. • . • • . • • • • . . 153
ON THE SATO-TATE CONJECTURE
V. Murty •. 195
VALUES OF L-FUNCTIONS OF JACOBI-SUM HECKE CHARACTERS
OF ABELIAN FIELDS
S. Lichtenbaum ••.•• 207
ON THE CONJECTURE OF BIRCH AND SWINNERTON-DYER FOR
ELLIPTIC CURVES WITH MULTIPLICATION
COI~PLEX
B. Gross. • • . . . . • . • . • • . 219
MORDELL-WEIL GROUPS OF ELLIPTIC CURVES OVER
CYCLOTOMIC FIELDS
K. Rubin, A. Wiles ..••.••• 237
REMARKS ON EQUATIONS RELATED TO FERMAT'S LAST THEOREM
S. Chowla, M. Cowles. • • • . . • . • . . 255
THE CUBIC THUE EOUATION
J. Si1verman 263
IWASAWA THEORY FOR SEVERAL PRIMES AND A CONNECTION TO
WIEFERICH'S CRITERION
E. Fr i edman • . . . . 269
IWASAWA'S THEORY AND p-ADIC L-FUNCTIONS FOR IMAGINARY
QUADRATIC FIELDS
R. Greenberg . . . . . . . . 275
REGULATORS, IWASAWA MODULES, AND TroE MAIN CONJECTURE
FOR p=2
L. Federer . . . . . . 287
ZEROES OF p-ADIC L-FUNCTIONS, II
S. Wagstaff .... 297
SOllE AR I THIIET I C Pr..OBLEIIS ON AUTOIIORPH I St\S OF ALGEBRA I C
VARIETIES
C. R. t-latthews 309
ELLIPTIC UNITS IN FUNCTION FIELDS
D. Hayes ... . . . . . . . . 321
PR I~ lALl TY TEST I NG \II TH ART IN SYIIBOLS
H.\~. Lenstra, Jr. 341
THE COATES-WILES THEOREM REVISITED
H.l1. Stark. . . . . 34S
PREFACE
Some years ago, Mr. James Vaughn suggested to me that it might be
worthwhile to have a meeting on current mathematical work relating to
Fermat's last theorem. I was del ighted with the idea and hoped that
1631 was the year Fermat really made his conjecture so that we could
celebrate its 350th anniversary. But, alas, Dickson's History puts the
year at 1637-without any real justification.
ln any event, the problern has stood for approximately 350 years,
and as anyone can see by a hasty perusal of this volume, it touches al 1
modern branches of mathematics. Dur aim has been to stress several
areas of number theory in which recent work associated to Fermat's last
theorem has been most promising; not necessarily for a solution to the
theorem, but to the development of mathematics. The themes of the
conference centered araund "geometry of Fermat varieties," "lwasawa
theory of cyclotomic fields," "Grössencharakter and special values of
L-functions," and "history." Much inspiration has come from the simpler
theory of el 1 iptic curves where many deep problems concerning rational
points have recently been solved. lt is hoped that these methods may
yield insights to problems related to rational points on the Fermat
curves.
would like to thank the other organizers of this conference:
Harold Edwards, t~icholas Katz, Neal Kobl itz, Barry Mazur, and Andrew
Wiles, without whose combined efforts we could not have held this
meeting. Particular thanks are due to Neal Kobl itz for the coordinating
and editing of the volume.
Also, special thanks go to the staff at Endicott House for having
made our stay in Dedham so enjoyable, and to Renate D'Arcangelo for her
outstanding typing of the manuscripts. Klaus Peters of Birkhäuser
Boston was especially helpful in making the meeting so successful.
Finally, we express our deepest appreciation to the Vaughn Foundation
for its most generaus support, not only to this particular conference,
but to pure mathematics i8 general.
Dorian Goldfeld
May 1982
vii
CONFERENCE PARTICIPANTS (U.S.A.)
G. ANDERSON W. JOHNSON
N. ANKENY B. JORDAN
M. ARTIN N. KATZ
J. BADDOURA N. KOBLITZ
D. BROWNAWELL D. KUBERT
S. CHOWLA R. KUHN
B. CIPRA s. LICHTENBAUM
R. COLEMAN J. LUBIN
G. CORNELL B. MAZUR
M. COWLES W. McCALLUM
D. CREWE D. MEUSER
B. DWORK P. MONSKY
H. EDWARDS D. MUMFORD
L. J. FEDERER K. MURTY
E. FRIEDMAN A. OGUS
S. P. GALOVICH D. ROHRUCH
J. M. GANDHI t M. ROSEN
D. GOLDFELD K. RUBIN
J. GOLDMAN R. RUMLEY
R. GREENBERG K. SCHMIDT
B. GROSS A. SELBERG
R. GUPTA J. SILVERMAN
M. HAKOSALO H. STARK
S. HARAN G. STEVENS
M. HARRIS J. TATE
D. HAYES J. VAUGHN
J. HOFFSTEIN P. VOJTA
P. HRILJAC S. WAGSTAFF
K. IWASAWA L. WASHINGTON
N. JOCHNOWITZ D. B. ZAGIER
ix
CONFERENCE PARTICIPANTS (FOREIGN)
A. BAKER (ENGLAND)
B. BIRCH (ENGLAND)
J. COATES (FRANCE)
J. M. FONTAINE (FRANCE)
G. HARDER (WEST GERMANY)
M. A. KENKU (NIGERIA)
H. v/. LENSTRA (NETHERLANDS)
c. MATTHEWS (ENGLAND)
T. ODA (JAPAN)
B. PERRIN-RIOU (FRANCE)
I. PIATETSKI-SHAPIRO (ISRAEL)
P. RIBENBOIM (CANADA)
A. J. SCHOLL (ENGLAND)
T. SHIODA (JAPAN)
c. SOULE (FRANCE)
M. WALDSCHMIDT (FRANCE)
X