L-Functions and Arithmetic

400 Pages·1991·7.86 MB·English

by  J. Coates

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES
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Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England
The books in the series listed below are available from booksellers, or, in case of difficulty,
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34 Representation theory of Lie groups, M.F. ATIYAH et al
36 Homological group theory, C.T.C. WALL (ed)
39 Affine sets and affine groups, D.G. NORTHCOTT
40 Introduction to H spaces, PJ. KOOSIS
p
43 Graphs, codes and designs, PJ. CAMERON & J.H. VAN LINT
45 Recursion theory: its generalisations and applications, F.R. DRAKE & S.S. WAINER (eds)
46 p-adic analysis: a short course on recent work, N. KOBLITZ
49 Finite geometries and designs, P. CAMERON, J.W.P. HIRSCHFELD & D.R. HUGHES (eds)
50 Commutator calculus and groups of homotopy classes, HJ. BAUES
51 Synthetic differential geometry, A.KOCK
57 Techniques of geometric topology, R.A. FENN
59 Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI
62 Economics for mathematicians, J.W.S. CASSELS
66 Several complex variables and complex manifolds II, M.J. FIELD
69 Representation theory, I.M. GELFAND et al
74 Symmetric designs: an algebraic approach, E.S. LANDER
76 Spectral theory of linear differential operators and comparison algebras, H.O.CORDES
77 Isolated singular points on complete intersections, E J.N. LOOIJENGA
78 A primer on Riemann surfaces, A.F. BEARDON
79 Probability, statistics and analysis, JJF.C. KINGMAN & G.E.H. REUTER (eds)
80 Introduction to the representation theory of compact and locally compact groups, A. ROBERT
81 Skew fields, P.K. DRAXL
82 Surveys in combinatorics, E.K. LLOYD (ed)
83 Homogeneous structures on Riemannian manifolds, F. TRICERRI & L. VANHECKE
86 Topological topics, I.M. JAMES (ed)
87 Surveys in set theory, A.R.D. MATHIAS (ed)
88 FPF ring theory, C. FAITH & S. PAGE
89 An F-space sampler, N.J. KALTON, N.T. PECK & J.W. ROBERTS
90 Poly topes and symmetry, S.A. ROBERTSON
91 Classgroups of group rings, M.J. TAYLOR
92 Representation of rings over skew fields, A.H. SCHOFIELD
93 Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds)
94 Representations of general linear groups, G.D. JAMES
95 Low-dimensional topology 1982, R.A. FENN (ed)
96 Diophantine equations over function fields, R.C. MASON
97 Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN
98 Localization in Noetherian rings, A.V. JATEGAONKAR
99 Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE
100 Stopping time techniques for analysts and probabilists, L. EGGHE
101 Groups and geometry, ROGER C. LYNDON
103 Surveys in combinatorics 1985, I. ANDERSON (ed)
104 Elliptic structures on 3-manifolds, C.B. THOMAS
105 A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG
106 Syzygies, E.G. EVANS & P. GRIFFITH
107 Compactification of Siegel moduli schemes, C-L. CHAI
108 Some topics in graph theory, H.P. YAP
109 Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds)
110 An introduction to surreal numbers, H. GONSHOR
111 Analytical and geometric aspects of hyperbolic space, D.B.A.EPSTEIN (ed)
112 Low-dimensional topology and Kleinian groups, D.B.A. EPSTEIN (ed)
113 Lectures on the asymptotic theory of ideals, D.REES
114 Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG
115 An introduction to independence for analysts, H.G. DALES & W.H. WOODIN
116 Representations of algebras, P.J. WEBB(ed)
117 Homotopy theory, E. REES & J.D.S. JONES (eds)
118 Skew linear groups, M. SHIRVANI & B. WEHRFRITZ
119 Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL
121 Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds)
122 Non-classical continuum mechanics, RJ. KNOPS & A.A. LACEY (eds)
124 Lie groupoids and Lie algebroids in differential geometry, K. MACKENZIE
125 Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZIE
126 Van der Corput's method of exponential sums, S.W. GRAHAM & G. KOLESNIK
127 New directions in dynamical systems, T.J. BEDFORD & J.W. SWIFT (eds)
128 Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU
129 The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W.LIEBECK
130 Model theory and modules, M.PREST
131 Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds)
132 Whitehead groups of finite groups, ROBERT OLIVER
133 Linear algebraic monoids, MOHAN S.PUTCHA
134 Number theory and dynamical systems, M. DODSON & J. VICKERS (eds)
135 Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds)
136 Operator algebras and applications, 2, D. EVANS & M. TAKESAKI (eds)
137 Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds)
138 Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds)
139 Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds)
140 Geometric aspects of Banach spaces, E.M. PEINADOR and A. RODES (eds)
141 Surveys in combinatorics 1989, J. SIEMONS (ed)
142 The geometry of jet bundles, D.J. SAUNDERS
143 The ergodic theory of discrete groups, PETER J. NICHOLLS
144 Introduction to uniform spaces, I.M. JAMES
145 Homological questions in local algebra, JAN R. STROOKER
146 Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO
147 Continuous and discrete modules, S.H. MOHAMED & B.J. MULLER
148 Helices and vector bundles, A.N. RUDAKOV et al
149 Solitons, nonlinear evolution equations and inverse scattering, M.A. ABLOWITZ &
P.A. CLARKSON
150 Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds)
151 Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds)
152 Oligomorphic permutation groups, P. CAMERON
153 L-functions and arithmetic, J. COATES & MJ. TAYLOR (eds)
154 Number theory and cryptography, J. LOXTON (ed)
155 Classification theories of polarized varieties, TAKAO FUJITA
156 Twistors in mathematics and physics, T.N. BAILEY & RJ. BASTON (eds)
158 Geometry of Banach spaces, P.F.X. MULLER & W. SCHACHERMAYER (eds)
London Mathematical Society Lecture Note Series. 153
L-functions and Arithmetic
Edited by
J. Coates
University of Cambridge
and
M J. Taylor
UMIST
The right of the
University of Cambridge
to print and sell
was granted by
Henry VIII in 1534.
The University has printed
and published continuously
since 1584.
CAMBRIDGE UNIVERSITY PRESS
Cambridge
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Published by the Press Syndicate of the University of Cambridge
The Pitt Building, Trumpington Street, Cambridge CB2 1RP
40 West 20th Street, New York, NY 10011, USA
10, Stamford Road, Oakleigh, Melbourne 3166, Australia
© Cambridge University Press 1991
First published 1991
Library of Congress cataloguing in publication data available
British Library cataloguing in publication data available
ISBN 0 52138619 5
Transferred to digital printing 2004
CONTENTS
Preface vii
Participants viii
Lectures on automorphic L-functions 1
JAMES ARTHUR and STEPHEN GELBART
Gauss sums and local constants for GL(N) 61
COLIN J. BUSHNELL
L-functions and Galois modules 75
PH. CASSOU-NOGUES, T. CHINBURG, A. FROHLICH,
and M.J. TAYLOR (Notes by D. Burns and N.P. Byott)
Motivic p-adic i-functions 141
JOHN COATES
The Beilinson conjectures 173
CHRISTOPHER DENINGER and ANTHONY J. SCHOLL
Iwasawa theory for motives 211
RALPH GREENBERG
Kolyvagin's work for modular elliptic curves 235
BENEDICT GROSS
Index theory, potential theory, and the Riemann hypothesis 257
SHAI HARAN
Katz p-adic L-functions, congruence modules and
deformation of Galois representations 271
H. HIDA and J. TILOUINE
Kolyvagin's work on Shafarevich-Tate groups 295
WILLIAM G. MCCALLUM
Arithmetic of diagonal quartic surfaces I 317
R.G.E. PINCH and H.P.F. SWINNERTON-DYER
On certain Artin L-Series 339
DINAKAR RAMAKRISHNAN
The one-variable main conjecture for elliptic curves with
complex multiplication 353
KARL RUBIN
Remarks on special values of L-functions 373
ANTHONY J. SCHOLL
PREFACE
This book is the fruit of the London Mathematical Society Symposium on iL-
functions and Arithmetic', which was held at the University of Durham from
June 30 until July 11, 1989. The Symposium attempted to bring together
the many and diverse aspects of number theory and arithmetical algebraic
geometry which are currently being studied under the general theme of the
connections between //-functions and arithmetic. In particular, there were
series of lectures on each of the following topics:
(i) descent theory on elliptic curves;
(ii) automorphic i-functions;
(Hi) Beilinson's conjectures;
(iv) p-adic cohomology and the Bloch-Kato conjectures on the Tamagawa
numbers of motives;
(v) Iwasawa theory of motives;
(vi) /-adic representations attached to automorphic forms;
(vii) L-functions and Galois module structure of rings of integers and groups
of units.
While the editors were not able to persuade all the lecturers to write up
the material of their lectures,the present volume represents a major portion
of the proceedings of the Symposium. The editors wish to express their
warm appreciation to all the lecturers. The Symposium was sponsored by
the London Mathematical Society, and benefited from the generous financial
support of the Science and Engineering Research Council. We are grateful to
Tony Scholl and the Department of Mathematics of the University of Durham
for their help in organising the meeting; we particularly wish to thank Grey
College for its kind hospitality.
J. Coates
M.J. Taylor
PARTICIPANTS
A. Agboola, Columbia University Li Guo, University of Washington
J.V. Armitage, University of Durham S. Ling, University of California, Berkeley
J. Arthur, University of Toronto W.G. McCallum, University of Arizona
R. Blasubramanian, CIT, Madras M. MacQuillen, Harvard University
P.N. Balister, Trinity College, Cambridge L.R. McCulloh, University of Illinois, Urbana
L. Barthel, Weizmann Institute H. Maennel, Max-Planck Institut, Bonn
E. Bayer, Universite de Besangon C. Matthews, University of Cambridge
B.J. Birch, University of Oxford B. Mazur, Harvard University
D. Blasius, ENS, Paris J. Neukirch, Universitat Regensburg
J. Boxall, Universite de Caen J. Oesterle, Universite Paris VI
J. Brinkhuis, Erasmus University, Rotterdam R. Odoni, University of Glasgow
D.A. Burgess, University of Nottingham D.B. Penman, University of Cambridge
D.J. Burns, University of Cambridge R.G.E. Pinch, University of Cambridge
C.J. Bushnell, King's College, London A.J. Plater, University of Cambridge
N. Byott, UMIST R.J. Plymen, University of Manchester
H. Carayol, Universite Louis Pasteur, Strasbourg D. Ramakrishnan, California Institute
P. Cassou-Nogues, Universite de Bordeaux 1 of Technology
R. Chapman, Merton College, Oxford R.A. Rankin, University of Glasgow
T. Chinburg, University of Pennsylvania K.A. Ribet, University of California, Berkeley
L. Clozel, Universite de Paris-Sud G. Robert, Max-Planck Institut, Bonn
J.H. Coates, University of Cambridge A. Roberston, University of Oxford
J. Cougnard, Universite de Besancpn J.D. Rogawski, University of California,
J.E. Cremona, University of of Exeter Los Angeles
C. Denninger, Universitat Munster K. Rubin, Ohio State University
F. Diamond, Ohio State University P. Satge, Universite de Caen
L. Dodd, University of Nottingham N. Schappacher, Max-Planck Institut, Bonn
H. Dormon, Harvard University C-G. Schmidt, RUG, Groningen
B. Erez, Universite de Geneve P. Schneider, Universitat Koln
G. Everest, University of East Anglia A.J. Scholl, University of Durham
D.E. Feather, University of Nottingham J-P. Serre, College de France
M. Flach, St John's College, Cambridge Seymour Kim, Imperial College, London
J-M. Fontaine, Universite Paris-Sud E. de Shalit, University of Jerusalem
A. Frohlich, University of Cambridge V. Snaith, McMaster University
Imperial College, London D. Solomon, UMIST
S. Gelbart, Weizmann Institute M. Starkings, University of Nottingham
R. Gillard, Universite de Grenoble 1 A. Srivastav, Universite de Bordeaux 1
C. Goldstein, Universite Paris-Sud H.P.F. Swinnerton-Dyer, University Funding
D. Grant, University of Cambridge Council
R. Greenberg, University of Washington M.J. Taylor, UMIST
B.H. Gross, Harvard University R. Taylor, University of Cambridge
S. Haran, University of Jerusalem J. Tilouine, UCLA
G. Harder, Universitat Bonn E. Todd, King's College, London
M.C. Harrison, Jesus College, Cambridge S. Ullom, University of Illinois, Urbana
D.R. Hayes, Imperial College, London N.J. Walker, University of Oxford
G. Henniart, Universite Paris-Sud L.C. Washington, University of Maryland
H. Hida, University of California, U. Weselmann, Universitat Bonn
Los Angeles E. Whitley, University of Exeter
U. Jannsen, Max-Planck Institut, Bonn J.W. Wildeshaus, King's College, Cambridge
B. Jordan, Baruch College, CUNY A. Wiles, Princeton University
M.A. Kenku, University of Lagos S.M.J. Wilson, University of Durham
M.A. Kervaire, Universite de Geneve K. Wingberg, Universitat Erlangen-Nurnberg
P. Kutzko, University of Iowa C.F. Woodcock, University of Kent
R. Ledgard, University of Manchester R.I. Yager, Macquarie University
S. Lichtenbaum, Cornell University C.S. Yogananda, CIT, Madras
Lectures on automorphic L-functions
JAMES ARTHUR AND STEPHEN GELBART
PREFACE
This article follows the format of fivel ectures that we gave on automorphic L-
functions. The lectures were intended to be a brief introduction for number
theorists to some of the main ideas in the subject. Three of the lectures
concerned the general properties of automorphic L-functions, with particular
reference to questions of spectral decomposition. We have grouped these
together as Part I. While many of the expected properties of automorphic L-
functions remain conjectural, a significant number have now been established.
The remaining two lectures were focused on the techniques which have been
used to establish such properties. These lectures form Part II of the article.
The first lecture (§1.1) is on the standard L-functions for GL . Much of this
n
material is familiar and can be used to motivate what follows. In §1.2 we
discuss general automorphic L-functions, and various questions that center
around the fundamental principle of functoriality. The third lecture (§1.3) is
devoted to the spectral decomposition of L2(G(F) \ G(f\)). Here we describe
a conjectural classification of the spectrum in terms of tempered represen-
tations. This amounts to a quantitative explanation for the failure of the
general analogue of Ramanujan's conjecture.
There are three principal techniques that we discuss in Part II. The lec-
ture §11.1 is concerned with the trace formula approach and the method of
zeta-integrals; it gives only a skeletal treatment of the subject. The lecture
§11.2, on the other hand, gives a much more detailed account of the theory of
theta-series liftings, including a discussion of counterexamples to the general
analogue of Ramanujan's conjecture. We have not tried to relate the coun-
terexamples given by theta-series liftings with the conjectural classification
of §1.3. It would be interesting to do so.
These lectures are really too brief to be considered a survey of the subject.
There are other introductory articles (references [A.I], [G], [B.I] for Part I)
2 Arthur & Gelbart - Lectures on automorphic L-functions: Part I
in which the reader can find further information. More detailed discussion is
given in various parts of the Corvallis Proceedings and in many of the other
references we have cited.
PARTI
1 STANDARD L-FUNCTIONS FOR GL
n
Let F be a fixed number field. As usual, F denotes the completion of F with
v
respect to a (normalized) valuation v. If v is discrete, o stands for the ring
v
of integers in F , and q is the order of the corresponding residue class field.
v v
We shall write A = f\ for the adele ring of F.
F
In this lecture, G will stand for the general linear group GL . Then G(f\)
n
is the restricted direct product, over all v, of the groups G(F ) = GL (F ).
V n v
Thus, G(f\) is the topological direct limit of the groups
Gs = n G(F.) • n <?(•.),
ves vgs
in which S ranges over all finite sets of valuations of F containing the set SQQ
of Archimedean valuations.
One is interested in the set H(G(f\)) of equivalence classes of irreducible,
admissible representations of G(A). (Recall that a representation of G(f\) is
admissible if its restriction to the maximal compact subgroup
K= n U(n,C)x n O(n,R)x JJ GL (o )
n v
v complex v real v discrete
contains each irreducible representation of K with only finite multiplicity.)
Similarly, one has the set U(G(F )) of equivalence classes of irreducible ad-
V
missible representations of G(F ). It is known [F] that any TT £ II(G?(A)) can
V
be decomposed into a restricted tensor product
(gK, TT eIL(G(F)),
V v
V
of irreducible, admissible representations of the local groups.
The unramified principal series is a particularly simple subset of U(G(F ))
V
to describe. Suppose that the valuation v is discrete. One has the Borel
subgroup
/&, ••• *
0