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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical 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, from Cambridge University Press. 34 36 39 40 43 45 46 49 50 51 57 59 62 66 69 74 76 77 78 79 80 81 82 83 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 103 104 105 106

Representation theory of Lie groups, M.F. ATIYAH et al Homological group theory, C.T.C. WALL (ed) Affine sets and affine groups, D.G. NORTHCOTT Introduction to H p spaces, PJ. KOOSIS Graphs, codes and designs, PJ. CAMERON & J.H. VAN LINT Recursion theory: its generalisations and applications, F.R. DRAKE & S.S. WAINER (eds) p-adic analysis: a short course on recent work, N. KOBLITZ Finite geometries and designs, P. CAMERON, J.W.P. HIRSCHFELD & D.R. HUGHES (eds) Commutator calculus and groups of homotopy classes, HJ. BAUES Synthetic differential geometry, A.KOCK Techniques of geometric topology, R.A. FENN Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Economics for mathematicians, J.W.S. CASSELS Several complex variables and complex manifolds II, M.J. FIELD Representation theory, I.M. GELFAND et al Symmetric designs: an algebraic approach, E.S. LANDER Spectral theory of linear differential operators and comparison algebras, H.O.CORDES Isolated singular points on complete intersections, E J.N. LOOIJENGA A primer on Riemann surfaces, A.F. BEARDON Probability, statistics and analysis, JJF.C. KINGMAN & G.E.H. REUTER (eds) Introduction to the representation theory of compact and locally compact groups, A. ROBERT Skew fields, P.K. DRAXL Surveys in combinatorics, E.K. LLOYD (ed) Homogeneous structures on Riemannian manifolds, F. TRICERRI & L. VANHECKE Topological topics, I.M. JAMES (ed) Surveys in set theory, A.R.D. MATHIAS (ed) FPF ring theory, C. FAITH & S. PAGE An F-space sampler, N.J. KALTON, N.T. PECK & J.W. ROBERTS Poly topes and symmetry, S.A. ROBERTSON Classgroups of group rings, M.J. TAYLOR Representation of rings over skew fields, A.H. SCHOFIELD Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds) Representations of general linear groups, G.D. JAMES Low-dimensional topology 1982, R.A. FENN (ed) Diophantine equations over function fields, R.C. MASON Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN Localization in Noetherian rings, A.V. JATEGAONKAR Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time techniques for analysts and probabilists, L. EGGHE Groups and geometry, ROGER C. LYNDON Surveys in combinatorics 1985, I. ANDERSON (ed) Elliptic structures on 3-manifolds, C.B. THOMAS A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG Syzygies, E.G. EVANS & P. GRIFFITH

107 108 109 110 111 112 113 114 115 116 117 118 119 121 122 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 158

Compactification of Siegel moduli schemes, C-L. CHAI Some topics in graph theory, H.P. YAP Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) An introduction to surreal numbers, H. GONSHOR Analytical and geometric aspects of hyperbolic space, D.B.A.EPSTEIN (ed) Low-dimensional topology and Kleinian groups, D.B.A. EPSTEIN (ed) Lectures on the asymptotic theory of ideals, D.REES Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG An introduction to independence for analysts, H.G. DALES & W.H. WOODIN Representations of algebras, P.J. WEBB(ed) Homotopy theory, E. REES & J.D.S. JONES (eds) Skew linear groups, M. SHIRVANI & B. WEHRFRITZ Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds) Non-classical continuum mechanics, RJ. KNOPS & A.A. LACEY (eds) Lie groupoids and Lie algebroids in differential geometry, K. MACKENZIE Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZIE Van der Corput's method of exponential sums, S.W. GRAHAM & G. KOLESNIK New directions in dynamical systems, T.J. BEDFORD & J.W. SWIFT (eds) Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W.LIEBECK Model theory and modules, M.PREST Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Whitehead groups of finite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S.PUTCHA Number theory and dynamical systems, M. DODSON & J. VICKERS (eds) Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds) Operator algebras and applications, 2, D. EVANS & M. TAKESAKI (eds) Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds) Geometric aspects of Banach spaces, E.M. PEINADOR and A. RODES (eds) Surveys in combinatorics 1989, J. SIEMONS (ed) The geometry of jet bundles, D.J. SAUNDERS The ergodic theory of discrete groups, PETER J. NICHOLLS Introduction to uniform spaces, I.M. JAMES Homological questions in local algebra, JAN R. STROOKER Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO Continuous and discrete modules, S.H. MOHAMED & B.J. MULLER Helices and vector bundles, A.N. RUDAKOV et al Solitons, nonlinear evolution equations and inverse scattering, M.A. ABLOWITZ & P.A. CLARKSON Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds) Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds) Oligomorphic permutation groups, P. CAMERON L-functions and arithmetic, J. COATES & MJ. TAYLOR (eds) Number theory and cryptography, J. LOXTON (ed) Classification theories of polarized varieties, TAKAO FUJITA Twistors in mathematics and physics, T.N. BAILEY & RJ. BASTON (eds) 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 New York Port Chester Melbourne

Sydney

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 Participants Lectures on automorphic L-functions JAMES ARTHUR and STEPHEN GELBART Gauss sums and local constants for GL(N) COLIN J. BUSHNELL L-functions and Galois modules PH. CASSOU-NOGUES, T. CHINBURG, A. FROHLICH, and M.J. TAYLOR (Notes by D. Burns and N.P. Byott) Motivic p-adic i-functions JOHN COATES The Beilinson conjectures CHRISTOPHER DENINGER and ANTHONY J. SCHOLL Iwasawa theory for motives RALPH GREENBERG Kolyvagin's work for modular elliptic curves BENEDICT GROSS Index theory, potential theory, and the Riemann hypothesis SHAI HARAN Katz p-adic L-functions, congruence modules and deformation of Galois representations H. HIDA and J. TILOUINE Kolyvagin's work on Shafarevich-Tate groups WILLIAM G. MCCALLUM Arithmetic of diagonal quartic surfaces I R.G.E. PINCH and H.P.F. SWINNERTON-DYER On certain Artin L-Series DINAKAR RAMAKRISHNAN

vii viii 1 61 75

141 173 211 235 257

271 295 317 339

The one-variable main conjecture for elliptic curves with complex multiplication KARL RUBIN Remarks on special values of L-functions ANTHONY J. SCHOLL

353 373

PREFACE

This book is the fruit of the London Mathematical Society Symposium on iLfunctions 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 J.V. Armitage, University of Durham J. Arthur, University of Toronto R. Blasubramanian, CIT, Madras P.N. Balister, Trinity College, Cambridge L. Barthel, Weizmann Institute E. Bayer, Universite de Besangon B.J. Birch, University of Oxford D. Blasius, ENS, Paris J. Boxall, Universite de Caen J. Brinkhuis, Erasmus University, Rotterdam D.A. Burgess, University of Nottingham D.J. Burns, University of Cambridge C.J. Bushnell, King's College, London N. Byott, UMIST H. Carayol, Universite Louis Pasteur, Strasbourg P. Cassou-Nogues, Universite de Bordeaux 1 R. Chapman, Merton College, Oxford T. Chinburg, University of Pennsylvania L. Clozel, Universite de Paris-Sud J.H. Coates, University of Cambridge J. Cougnard, Universite de Besancpn J.E. Cremona, University of of Exeter C. Denninger, Universitat Munster F. Diamond, Ohio State University L. Dodd, University of Nottingham H. Dormon, Harvard University B. Erez, Universite de Geneve G. Everest, University of East Anglia D.E. Feather, University of Nottingham M. Flach, St John's College, Cambridge J-M. Fontaine, Universite Paris-Sud A. Frohlich, University of Cambridge Imperial College, London S. Gelbart, Weizmann Institute R. Gillard, Universite de Grenoble 1 C. Goldstein, Universite Paris-Sud D. Grant, University of Cambridge R. Greenberg, University of Washington B.H. Gross, Harvard University S. Haran, University of Jerusalem G. Harder, Universitat Bonn M.C. Harrison, Jesus College, Cambridge D.R. Hayes, Imperial College, London G. Henniart, Universite Paris-Sud H. Hida, University of California, Los Angeles U. Jannsen, Max-Planck Institut, Bonn B. Jordan, Baruch College, CUNY M.A. Kenku, University of Lagos M.A. Kervaire, Universite de Geneve P. Kutzko, University of Iowa R. Ledgard, University of Manchester S. Lichtenbaum, Cornell University

Li Guo, University of Washington S. Ling, University of California, Berkeley W.G. McCallum, University of Arizona M. MacQuillen, Harvard University L.R. McCulloh, University of Illinois, Urbana H. Maennel, Max-Planck Institut, Bonn C. Matthews, University of Cambridge B. Mazur, Harvard University J. Neukirch, Universitat Regensburg J. Oesterle, Universite Paris VI R. Odoni, University of Glasgow D.B. Penman, University of Cambridge R.G.E. Pinch, University of Cambridge A.J. Plater, University of Cambridge R.J. Plymen, University of Manchester D. Ramakrishnan, California Institute of Technology R.A. Rankin, University of Glasgow K.A. Ribet, University of California, Berkeley G. Robert, Max-Planck Institut, Bonn A. Roberston, University of Oxford J.D. Rogawski, University of California, Los Angeles K. Rubin, Ohio State University P. Satge, Universite de Caen N. Schappacher, Max-Planck Institut, Bonn C-G. Schmidt, RUG, Groningen P. Schneider, Universitat Koln A.J. Scholl, University of Durham J-P. Serre, College de France Seymour Kim, Imperial College, London E. de Shalit, University of Jerusalem V. Snaith, McMaster University D. Solomon, UMIST M. Starkings, University of Nottingham A. Srivastav, Universite de Bordeaux 1 H.P.F. Swinnerton-Dyer, University Funding Council M.J. Taylor, UMIST R. Taylor, University of Cambridge J. Tilouine, UCLA E. Todd, King's College, London S. Ullom, University of Illinois, Urbana N.J. Walker, University of Oxford L.C. Washington, University of Maryland U. Weselmann, Universitat Bonn E. Whitley, University of Exeter J.W. Wildeshaus, King's College, Cambridge A. Wiles, Princeton University S.M.J. Wilson, University of Durham K. Wingberg, Universitat Erlangen-Nurnberg C.F. Woodcock, University of Kent R.I. Yager, Macquarie University C.S. Yogananda, CIT, Madras

Lectures on automorphic L-functions JAMES ARTHUR AND STEPHEN GELBART

PREFACE This article follows the format offivelectures that we gave on automorphic Lfunctions. 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 Lfunctions 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 GLn. Much of this 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 representations. 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 lecture §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 counterexamples 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 GLn

Let F be a fixed number field. As usual, Fv denotes the completion of F with respect to a (normalized) valuation v. If v is discrete, ov stands for the ring of integers in Fv, and qv is the order of the corresponding residue class field. We shall write A = f\F for the adele ring of F. In this lecture, G will stand for the general linear group GLn. Then G(f\) is the restricted direct product, over all v, of the groups G(FV) = GLn(Fv). Thus, G(f\) is the topological direct limit of the groups

Gs = n G(F.) • n

Representation theory of Lie groups, M.F. ATIYAH et al Homological group theory, C.T.C. WALL (ed) Affine sets and affine groups, D.G. NORTHCOTT Introduction to H p spaces, PJ. KOOSIS Graphs, codes and designs, PJ. CAMERON & J.H. VAN LINT Recursion theory: its generalisations and applications, F.R. DRAKE & S.S. WAINER (eds) p-adic analysis: a short course on recent work, N. KOBLITZ Finite geometries and designs, P. CAMERON, J.W.P. HIRSCHFELD & D.R. HUGHES (eds) Commutator calculus and groups of homotopy classes, HJ. BAUES Synthetic differential geometry, A.KOCK Techniques of geometric topology, R.A. FENN Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Economics for mathematicians, J.W.S. CASSELS Several complex variables and complex manifolds II, M.J. FIELD Representation theory, I.M. GELFAND et al Symmetric designs: an algebraic approach, E.S. LANDER Spectral theory of linear differential operators and comparison algebras, H.O.CORDES Isolated singular points on complete intersections, E J.N. LOOIJENGA A primer on Riemann surfaces, A.F. BEARDON Probability, statistics and analysis, JJF.C. KINGMAN & G.E.H. REUTER (eds) Introduction to the representation theory of compact and locally compact groups, A. ROBERT Skew fields, P.K. DRAXL Surveys in combinatorics, E.K. LLOYD (ed) Homogeneous structures on Riemannian manifolds, F. TRICERRI & L. VANHECKE Topological topics, I.M. JAMES (ed) Surveys in set theory, A.R.D. MATHIAS (ed) FPF ring theory, C. FAITH & S. PAGE An F-space sampler, N.J. KALTON, N.T. PECK & J.W. ROBERTS Poly topes and symmetry, S.A. ROBERTSON Classgroups of group rings, M.J. TAYLOR Representation of rings over skew fields, A.H. SCHOFIELD Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds) Representations of general linear groups, G.D. JAMES Low-dimensional topology 1982, R.A. FENN (ed) Diophantine equations over function fields, R.C. MASON Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN Localization in Noetherian rings, A.V. JATEGAONKAR Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time techniques for analysts and probabilists, L. EGGHE Groups and geometry, ROGER C. LYNDON Surveys in combinatorics 1985, I. ANDERSON (ed) Elliptic structures on 3-manifolds, C.B. THOMAS A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG Syzygies, E.G. EVANS & P. GRIFFITH

107 108 109 110 111 112 113 114 115 116 117 118 119 121 122 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 158

Compactification of Siegel moduli schemes, C-L. CHAI Some topics in graph theory, H.P. YAP Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) An introduction to surreal numbers, H. GONSHOR Analytical and geometric aspects of hyperbolic space, D.B.A.EPSTEIN (ed) Low-dimensional topology and Kleinian groups, D.B.A. EPSTEIN (ed) Lectures on the asymptotic theory of ideals, D.REES Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG An introduction to independence for analysts, H.G. DALES & W.H. WOODIN Representations of algebras, P.J. WEBB(ed) Homotopy theory, E. REES & J.D.S. JONES (eds) Skew linear groups, M. SHIRVANI & B. WEHRFRITZ Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds) Non-classical continuum mechanics, RJ. KNOPS & A.A. LACEY (eds) Lie groupoids and Lie algebroids in differential geometry, K. MACKENZIE Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZIE Van der Corput's method of exponential sums, S.W. GRAHAM & G. KOLESNIK New directions in dynamical systems, T.J. BEDFORD & J.W. SWIFT (eds) Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W.LIEBECK Model theory and modules, M.PREST Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Whitehead groups of finite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S.PUTCHA Number theory and dynamical systems, M. DODSON & J. VICKERS (eds) Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds) Operator algebras and applications, 2, D. EVANS & M. TAKESAKI (eds) Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds) Geometric aspects of Banach spaces, E.M. PEINADOR and A. RODES (eds) Surveys in combinatorics 1989, J. SIEMONS (ed) The geometry of jet bundles, D.J. SAUNDERS The ergodic theory of discrete groups, PETER J. NICHOLLS Introduction to uniform spaces, I.M. JAMES Homological questions in local algebra, JAN R. STROOKER Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO Continuous and discrete modules, S.H. MOHAMED & B.J. MULLER Helices and vector bundles, A.N. RUDAKOV et al Solitons, nonlinear evolution equations and inverse scattering, M.A. ABLOWITZ & P.A. CLARKSON Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds) Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds) Oligomorphic permutation groups, P. CAMERON L-functions and arithmetic, J. COATES & MJ. TAYLOR (eds) Number theory and cryptography, J. LOXTON (ed) Classification theories of polarized varieties, TAKAO FUJITA Twistors in mathematics and physics, T.N. BAILEY & RJ. BASTON (eds) 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 New York Port Chester Melbourne

Sydney

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 Participants Lectures on automorphic L-functions JAMES ARTHUR and STEPHEN GELBART Gauss sums and local constants for GL(N) COLIN J. BUSHNELL L-functions and Galois modules PH. CASSOU-NOGUES, T. CHINBURG, A. FROHLICH, and M.J. TAYLOR (Notes by D. Burns and N.P. Byott) Motivic p-adic i-functions JOHN COATES The Beilinson conjectures CHRISTOPHER DENINGER and ANTHONY J. SCHOLL Iwasawa theory for motives RALPH GREENBERG Kolyvagin's work for modular elliptic curves BENEDICT GROSS Index theory, potential theory, and the Riemann hypothesis SHAI HARAN Katz p-adic L-functions, congruence modules and deformation of Galois representations H. HIDA and J. TILOUINE Kolyvagin's work on Shafarevich-Tate groups WILLIAM G. MCCALLUM Arithmetic of diagonal quartic surfaces I R.G.E. PINCH and H.P.F. SWINNERTON-DYER On certain Artin L-Series DINAKAR RAMAKRISHNAN

vii viii 1 61 75

141 173 211 235 257

271 295 317 339

The one-variable main conjecture for elliptic curves with complex multiplication KARL RUBIN Remarks on special values of L-functions ANTHONY J. SCHOLL

353 373

PREFACE

This book is the fruit of the London Mathematical Society Symposium on iLfunctions 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 J.V. Armitage, University of Durham J. Arthur, University of Toronto R. Blasubramanian, CIT, Madras P.N. Balister, Trinity College, Cambridge L. Barthel, Weizmann Institute E. Bayer, Universite de Besangon B.J. Birch, University of Oxford D. Blasius, ENS, Paris J. Boxall, Universite de Caen J. Brinkhuis, Erasmus University, Rotterdam D.A. Burgess, University of Nottingham D.J. Burns, University of Cambridge C.J. Bushnell, King's College, London N. Byott, UMIST H. Carayol, Universite Louis Pasteur, Strasbourg P. Cassou-Nogues, Universite de Bordeaux 1 R. Chapman, Merton College, Oxford T. Chinburg, University of Pennsylvania L. Clozel, Universite de Paris-Sud J.H. Coates, University of Cambridge J. Cougnard, Universite de Besancpn J.E. Cremona, University of of Exeter C. Denninger, Universitat Munster F. Diamond, Ohio State University L. Dodd, University of Nottingham H. Dormon, Harvard University B. Erez, Universite de Geneve G. Everest, University of East Anglia D.E. Feather, University of Nottingham M. Flach, St John's College, Cambridge J-M. Fontaine, Universite Paris-Sud A. Frohlich, University of Cambridge Imperial College, London S. Gelbart, Weizmann Institute R. Gillard, Universite de Grenoble 1 C. Goldstein, Universite Paris-Sud D. Grant, University of Cambridge R. Greenberg, University of Washington B.H. Gross, Harvard University S. Haran, University of Jerusalem G. Harder, Universitat Bonn M.C. Harrison, Jesus College, Cambridge D.R. Hayes, Imperial College, London G. Henniart, Universite Paris-Sud H. Hida, University of California, Los Angeles U. Jannsen, Max-Planck Institut, Bonn B. Jordan, Baruch College, CUNY M.A. Kenku, University of Lagos M.A. Kervaire, Universite de Geneve P. Kutzko, University of Iowa R. Ledgard, University of Manchester S. Lichtenbaum, Cornell University

Li Guo, University of Washington S. Ling, University of California, Berkeley W.G. McCallum, University of Arizona M. MacQuillen, Harvard University L.R. McCulloh, University of Illinois, Urbana H. Maennel, Max-Planck Institut, Bonn C. Matthews, University of Cambridge B. Mazur, Harvard University J. Neukirch, Universitat Regensburg J. Oesterle, Universite Paris VI R. Odoni, University of Glasgow D.B. Penman, University of Cambridge R.G.E. Pinch, University of Cambridge A.J. Plater, University of Cambridge R.J. Plymen, University of Manchester D. Ramakrishnan, California Institute of Technology R.A. Rankin, University of Glasgow K.A. Ribet, University of California, Berkeley G. Robert, Max-Planck Institut, Bonn A. Roberston, University of Oxford J.D. Rogawski, University of California, Los Angeles K. Rubin, Ohio State University P. Satge, Universite de Caen N. Schappacher, Max-Planck Institut, Bonn C-G. Schmidt, RUG, Groningen P. Schneider, Universitat Koln A.J. Scholl, University of Durham J-P. Serre, College de France Seymour Kim, Imperial College, London E. de Shalit, University of Jerusalem V. Snaith, McMaster University D. Solomon, UMIST M. Starkings, University of Nottingham A. Srivastav, Universite de Bordeaux 1 H.P.F. Swinnerton-Dyer, University Funding Council M.J. Taylor, UMIST R. Taylor, University of Cambridge J. Tilouine, UCLA E. Todd, King's College, London S. Ullom, University of Illinois, Urbana N.J. Walker, University of Oxford L.C. Washington, University of Maryland U. Weselmann, Universitat Bonn E. Whitley, University of Exeter J.W. Wildeshaus, King's College, Cambridge A. Wiles, Princeton University S.M.J. Wilson, University of Durham K. Wingberg, Universitat Erlangen-Nurnberg C.F. Woodcock, University of Kent R.I. Yager, Macquarie University C.S. Yogananda, CIT, Madras

Lectures on automorphic L-functions JAMES ARTHUR AND STEPHEN GELBART

PREFACE This article follows the format offivelectures that we gave on automorphic Lfunctions. 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 Lfunctions 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 GLn. Much of this 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 representations. 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 lecture §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 counterexamples 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 GLn

Let F be a fixed number field. As usual, Fv denotes the completion of F with respect to a (normalized) valuation v. If v is discrete, ov stands for the ring of integers in Fv, and qv is the order of the corresponding residue class field. We shall write A = f\F for the adele ring of F. In this lecture, G will stand for the general linear group GLn. Then G(f\) is the restricted direct product, over all v, of the groups G(FV) = GLn(Fv). Thus, G(f\) is the topological direct limit of the groups

Gs = n G(F.) • n

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