Automorphic Forms, Representations, and L-functions

http://dx.doi.org/10.1090/pspum/033.1

Proceedings of Symposia in

PURE MATHEMATICS

Volume 33 , Part 1

Automorphic Forms, Representations, and L-functions Symposium in Pure Mathematics held at Oregon State University July 11-August 5, 1977 Corvallis, Oregon

A. Borel W. Casselman

ft American Mathematical Society Providence, Rhode Island

^VDED

PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT OREGON STATE UNIVERSITY CORVALLIS, OREGON

JULY 11-AUGUST 5, 1977

Prepared by the American Mathematical Society with partial support from National Science Foundation Grant MCS 76-24539

Library of Congress Cataloging-in-Publication Data

Symposium in Pure Mathematics, Oregon State University, 1977. Automorphic forms, representations, and L-functions. (Proceedings of symposia in pure mathematics; v. 33) Includes bibliographical references and index. 1. Automorphic forms — Congresses. 2. Lie groups — Congresses. 3. Representations of

groups — Congresses. 4. L-functions — Congresses. I. Borel, Armand. II. Casselman, W., 1941- III. American Mathematical Society. IV. Title. V. Series. QA33LS937 1977 512'.7 78-21184 ISBN 0-8218-1435-4 (v.l) ISBN 0-8218-1437-0 (v.2)

ISBN 0-8218-1474-5 (set)

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CONTENTS

Foreword ix

Part 1

I. Reductive groups. Representations

Reductive groups 3 By T. A. SPRINGER

Reductive groups over local fields 29 By J. TITS

Representations of reductive Lie groups 71 By NOLAN R. WALLACH

Representations of GL2(R) and GL2(C) 87 By A. W. KNAPP

Normalizing factors, tempered representations, and L-groups 93 By A. W. KNAPP and GREGG ZUCKERMAN

Orbital integrals for GL2(R) 107 By D. SHELSTAD

Representations of p-adic groups: A survey I11 By P. CARTIER

Cuspidal unramified series for central simple algebras over local fields 157 By PAUL GÉRARDIN

Some remarks on the supercuspidal representations of P-adic semisimple groups 171

By G. LUSZTIG

II. Automorphic forms and representations

Decomposition of representations into tensor products 179 By D. FLATH

Classical and adelic automorphic forms. An introduction 185 By 1. PIATETSKI-SHAPIRO

Automorphic forms and automorphic representations 189 By A. BOREL and H. JACQUET

On the notion of an automorphic representation. A supplement to the preceding paper 203

By R. P. LANGLANDS

Multiplicity one theorems 209 By I. PIATETSKI-SHAPIRO

Forms of GL(2) from the analytic point of view 213 By STEPHEN GELBART and HERVÉ JACQUET

Eisenstein series and the trace formula 253 By JAMES ARTHUR

0-series and invariant theory 275 By R. HOWE

v

Vi CONTENTS

Examples of dual reductive pairs 287 By STEPHEN GELBART

On a relation between SL2 cusp forms and automorphic forms on orthogonal groups 297

By S. RALLIS

A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups 315

By R. HOWE and I.I. PIATETSKI-SHAPIRO

Part 2

III. Automorphic representations and L-functions

Number theoretic background 3 By J. TATE

Automorphic L-functions 27 By A. BOREL

Principal L-functions of the linear group 63 By HERVÉ JACQUET

Automorphic L-functions for the symplectic group GSp4 87 By MARK E. NOVODVORSKY

On liftings of holomorphic cusp forms 97 By TAKURO SHINTANI

Orbital integrals and base change 111 By R. KOTTWITZ

The solution of a base change problem for GL(2) (following Langlands, Saito, Shintani) 115

By P. GÉRARDIN and J.-P. LABESSE

Report on the local Langlands conjecture for GL2 135 By J. TUNNELL

IV. Arithmetical algebraic geometry and automorphic L-functions

The Hasse-Weil ^-function of some moduli varieties of dimension greater than one 141

By W. CASSELMAN

Points on Shimura varieties mod P 165 By J. S. MILNE

Combinatorics and Shimura varieties mod p (based on lectures by Langlands) 185

By R. KOTTWITZ

Notes on L-indistinguishability (based on a lecture by R. P. Langlands) . . . . 193 By D. SHELSTAD

Automorphic representations, Shimura varieties, and motives. Ein Märchen 205

By R. P. LANGLANDS

CONTENTS Vii

Variétés de Shimura: Interpretation modulaire, et techniques de construction de modeles canoniques 247

By PIERRE DELIGNE

Congruence relations and Shimura curves 291 By YASUTAKA IHARA

Valeurs de fonctions L et périodes d'intégrales 313 By P. DELIGNE

with an appendix: Algebraicity of some products of values of the I' function 343

By N. KOBLITZ and A. OGUS

An introduction to Drinfeld's "Shtuka" 347 By D. A. KAZHDAN

Automorphic forms on GL2 over function fields (after V. G. Drinfeld) 357 By G. HARDER and D. A. KAZHDAN

Index

Foreword

The twenty-fifth AMS Summer Research Institute was devoted to automorphic forms, representations and L-functions. It was held at Oregon State University, Corvallis, from July 11 to August 5, 1977, and was financed by a grant from the National Science Foundation. The Organizing Committee consisted of A. Borel, W. Casselman (cochairmen:), P. Deligne, H. Jacquet, R. P. Langlands, and J. Tate. The papers in this volume consist of the Notes of the Institute, mostly in revised form, and of a few papers written later.

A main goal of the Institute was the discussion of the L-functions attached to automorphic forms on, or automorphic representations of, reductive groups, the local and global problems pertaining to them, and of their relations with the L-functions of algebraic number theory and algebraic geometry, such as Artin L-functions and Hasse-Weil zeta functions. This broad topic, which goes back to E. Hecke, C. L. Siegel and others, has undergone in the last few years and is undergo­ing even now a considerable development, in part through the systematic use of infinite dimensional representations, in the framework of adelic groups. This devel­opment draws on techniques from several areas, some of rather difficult access. Therefore, besides seminars and lectures on recent and current work and open problems, the Institute also featured lectures (and even series of lectures) of a more introductory character, including background material on reductive groups, their representations, number theory, as well as an extensive treatment of some relatively simple cases.

The papers in this volume are divided into four main sections, reflecting to some extent the nature of the prerequisites. I is devoted to the structure of reductive groups and infinite dimensional representations of reductive groups over local fields. Five of the papers supply some basic background material, while the others are concerned with recent developments. II is concerned with automorphic forms and automorphic representations, with emphasis on the analytic theory. The first four papers discuss some basic facts and definitions pertaining to those, and the passage from one to the other. Two papers are devoted to Eisenstein series and the trace formula, first for GL2 and there in more general cases. In fact, the trace formula and orbital integrals turned out to be recurrent themes for the whole Institute and are featured in several papers in the other sections as well. The main theme of the last four papers is the restriction of the oscillator representation of the metaplectic group to dual reductive pairs of subgroups, first in general and then in more special cases.

Ill begins with the background material on number theory, chiefly on Weil groups and their L-functions. It then turns to the L-functions attached to automor­phic representations, various ways to construct them, their (conjectured or proven) properties and local and global problems pertaining to them. The remaining papers are mostly devoted to the base change problem for GL2 and its applications to the proof of holomorphy of certain nonabelian Artin series.

Finally, IV relates automorphic representations and arithmetical algebraic geometry. Over function fields, it gives an introduction to the work of Drinfeld for

ix

X FOREWORD

GL2, which constructs systems of /-adic representations whose L-series is a given automorphic L-function. Over number fields, it is mainly concerned with problems on Shimura varieties: canonical models, the point of their reductions modulo prime ideals, and Hasse-Weil zeta functions.

This Institute emphasized representations so that, at least formally, the primary object of concern was an automorphic representation rather than an automorphic form. However, there is no substantial difference between the two, and this should not hide the fact that the theory is a direct outgrowth of the classical theory of automorphic forms. In order to give a comprehensive treatment of our subject matter and yet not produce too heavy a schedule, it was decided to omit a number of topics on automorphic forms which do not fit well at present into the chosen framework. For example, the Institute was planned to have little overlap with the Conference on Modular Functions of One Variable held in Bonn (1976). The reader is referred to the Proceedings of the latter (Springer Lecture Notes 601, 627) and to those of its predecessor (Springer Lecture Notes 320, 350, 476) for some of those topics and a more classical point of view. Also, some topics of considerable interest in themselves such as reductive groups, their infinite dimensional representations, or moduli varieties, were discussed chiefly in function of the needs of the main themes of the Institute.

These Proceedings appear in two parts, the first one contains sections I and II, and the second one sections III and IV.

A. BOREL

W. CASSELMAN