Algebraic number fields: (L-functions and Galois properties) : proceedings of a symposium

712 Pages·1977·79.012 MB·English

by  organised by the London Mathematical Society with the support of the Science Research Council and the Royal Society , edited by A. Fröhlich.

Algebraic Number Fields
(L-functions and Galois properties)
Proceedings of a Symposium organised by the London Mathematical
Society with the support of the Science Research Council and the
Royal Society
Edited by
A. FROHLICH
King's College, University of London
1977
ACADEMIC PRESS
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Copyright ©1977 by
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Library of Congress Catalog Card Number: 76-016966
ISBN: 0-12-268960-7
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List of Contributors
C.J. Bushnell,
Department of Mathematics,
King’s College London,
Strand,
London WC2R 2LS
W. Casselman,
Department of Mathematics,
University of British Columbia,
2075 Westbrook Place,
Vancouver, B.C.,
Canada.
J. Coates,
Department of Pure Mathematics and Mathematical Statistics,
University of Cambridge,
16 Mill Lane,
Cambridge.
J. Cougnard,
Math emat i que s ,
University de Besanqon,
Route de Gray - La Bouloie,
25030 Besancon Cedex,
France.
A. Frohlich,
Department of Mathematics,
King’s College London,
Strand,
London WC2R 2LS
H. Koch,
ADW der DDR,
ZI fur Mathematik und Mechanik,
DDR 108 Berlin,
Mohrenstr. 39.
V
vi
LIST OF CONTRIBUTORS
J.C. Lagarias,
Bell Laboratories ,
Murray Hill, N.J. 0797^,
U.S.A.
J. Martinet,
Department de Mathematiques,
Universite de Bordeaux,
351 Cours de la Liberation,
33^+05 Talence,
France.
J. Masley,
Department of Mathematics,
University of Illinois at Chicago Circle,
Chicago, Ill. 60680,
U.S.A.
L.R. McCulloh,
Department of Mathematics,
University of Illinois at Urbana,
Urbana, Ill. 61801,
U.S.A.
0. Neumann,
DAW-Inst.-komplex Mathematik,
IRM,
DDR 1199 Berlin-Adlershof,
Rudower Chaussee 5«
A.M. Odlyzko,
Bell Laboratories ,
Murray Hill, N.J. 0797^,
U.S.A.
R. Odoni,
Department of Mathematics,
University of Exeter,
North Park Road,
Exeter EXU i+QE
LIST OF CONTRIBUTORS vii
J-P. Serre,
College de France,
Paris,France.
H. Stark,
Department of Mathematics,
MIT,
Cambridge, Mass. 02139,
U.S.A.
J. Tate,
Department of Mathematics,
Harvard University,
1 Oxford Street,
Cambridge, Mass. 02138,
U.S.A.
M.J. Taylor,
Department of Mathematics,
King’s College London,
Strand,
London WC2R 2LS
A.A. Terras,
Department of Mathematics,
University of California at San Diego,
P.O. Box 109,
La Jolla, Calif. 92037,
U.S.A.
S.V. Ullom,
Department of Mathematics,
University of Illinois at Urbana,
Urbana, Ill. 61801,
U.S.A.
H.W. van der Waal 1,
Mathematisch Institut,
Katholieke Universiteit,
Toernooiveld,
Nijmegen,
Holland.
Preface
This volume is the outcome of a symposium on L-functions
and Galois properties of algebraic number fields, held from
2 to 12 September 19755 in the University of Durham. It was
organised by the London Mathematical Society, with the
generous financial support of the Science Research Council,
aided further by a grant from the Royal Society. The smooth
running of the conference was made possible by the helpful
attitude of the authorities of Durham University and the hard
work of the symposium secretary, Dr. S.M.J. Wilson.
Almost all the lectures given at the symposium are
recorded here. In many cases the presentation has been
expanded and new relevant material added. My gratitude is
due to the lecturers for making publication of this volume
possible by their willing cooperation, as well as for their
original contribution to the success of the meeting itself.
I also wish to express my thanks to Mrs. J. Bunn, who
typed the whole volume ready for publication, to Mrs. E.
Smith, who edited the manuscripts, to Dr. J.C. Bushnell
for help on all fronts and to Academic Press London for
continued cooperation.
A. Frohlich
IX
Contents
Page
List of Contributors v
Preface 1X
J. Martinet, Character theory and Artin
L-functions. 1
J.T. Tate (prepared in collaboration with
C.J. Bushnell and M. Taylor),
Local constants. 89
A. Frohlich, Galois module structure. 133
J-P. Serre (prepared in collaboration with
C.J. Bushnell), Modular forms of
weight one and Galois representations. 193
J. Coates, p-adic L-functions and Iwasawa’s
theory. 269
H.M. Stark, Class fields for real quadratic
fields and L-series at 1. 355
A.M. Odlyzko, On conductors and discriminants. 377
J.C. Lagarias and A.M. Odlyzko, Effective
versions of the Chebotarev density
theorem. U09
J. Masley, Odlyzko bounds and class number
problems. ^65
A. Terras, A relation between SK(s) and
^(s-1) for algebraic number field K. ^75
xi
R. Odoni, Some global norm density results
obtained from an extended Chebotarev
density theorem. ^85
S.V. Ullom, A survey of class groups of
integral group rings. ^97
J. Martinet, H . 525
o
J. Cougnard, Un contre-exemple a une
conjecture de J. Martinet. 539
L.R. McCulloh, A Stickelberger condition
on Galois module structure for Kummer
extensions of prime degree. 561
A. Frohlich, Stickelberger without Gauss
sums. 589
H. Koch, Fields of class two and Galois
cohomology. 609
0. Neumann, On p-closed number fields and
an analogue of Riemann’s existence
theorem. 625
R.W. van der Waall, Holomorphy of
quotients of zeta-functions. 6^9
W. Casselman, GL . 663
Character theory and Artin L-functions
J. Martinet
I. NON ABELIAN L-FUNCTIONS
The aim of this chapter is to describe the theory of
Artin's non abelian L-functions, taking for granted the
theory of abelian L-functions. This chapter owes much to a
talk by Serre (Fonctions L non abeliennes, Seminaire de
Theorie des Nombres, Bordeaux, 10 avril 1973).
§1 . Frobenius
Two papers of Frobenius, both dating back to 1896, play a
key role in the theory we are going to describe. The first
one is devoted to what is now called the "Frobenius sub­
stitution". Let E/K be a finite normal extension of number
fields with Galois group G, and let p be a finite prime of
K. Assume E/K is unramified at p. For every prime P of E
lying above p, there is a unique element Op e G (the
1