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http://dx.doi.org/10.1090/pspum/009 PROCEEDING S O F SYMPOSI A IN PUR E MATHEMATIC S Volum e 9 Algebrai c Group s an d Discontinuou s Subgroup s Arman d Bore l an d Georg e D . Mostow , Editor s AMERICA N MATHEMATICA L SOCIET Y PROVIDENCE , RHOD E ISLAN D PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY HELD AT THE UNIVERSITY OF COLORADO BOULDER, COLORADO JULY 5 - AUGUST 6, 1965 Prepared by the American Mathematical Society under National Science Foundation Grant GP-3983 and Office of Naval Research Grant Nonr(G)00055-65 International Standard Serial Number 0082-0717 International Standard Book Number 0-8218-1409-5 Library of Congress Catalog Number 66-18581 COPYING AND REPRINTING. Individual readers of this publication, and nonprofit li• braries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publi• cation (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright © 1966 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. @ 109 8 7 6 54 95 94 93 92 91 Foreword This book is an outgrowth of the twelfth Summer Mathematical Institute of the American Mathematical Society, which was devoted to Algebraic Groups and Discontinuous Subgroups. The Institute was held at the University of Colorado in Boulder from July 5 to August 6, 1965, and was financed by the National Science Foundation and the Office of Naval Research. The present volume consists of the Institute lecture notes, in part slightly revised, and of a few papers written somewhat later. From the beginning, it was understood that a comprehensive exposition of the arithmetic aspects of algebraic groups should be a central aim of the Institute. In order to survey effectively the topics chosen for discussion, some important parts of the theory of Lie groups and algebraic groups had to be omitted, and the program was concentrated around five major themes: linear algebraic groups and arithmetic groups, adeles and arithmetic properties of algebraic groups, auto- morphic functions and spectral decomposition of L2-spaees of coset spaces, holomorphic automorphic functions on bounded symmetric domains and moduli problems, vector valued cohomology and deformation of discrete subgroups. The lectures fulfilled diverse needs, and accordingly the papers in this book are intended to serve various purposes: to supply background material, to present the current status of a topic, to describe some basic methods, to give an exposition of more or less known material for which there is no convenient reference, and to present new results. It is hoped that this collection of papers will facilitate access to the subject and foster further progress. A. Borel G. D. Mostow in Contents I. Algebraic Groups, Arithmetic Groups Linear Algebraic Groups 3 BY ARMAND BOREL Reduction Theory for Arithmetic Groups 20 BY ARMAND BOREL Rationality Properties of Linear Algebraic Groups 26 BY ARMAND BOREL AND T. A. SPRINGER Classification of Algebraic Semisimple Groups 33 BY J. TITS p-adic Groups 63 BY FRANCOIS BRUHAT Generalized Tits System (Bruhat Decomposition) on p-Adic Semisimple Groups 71 BY NAGAYOSHI IWAHORI On Rational Points on Projective Varieties Defined Over a Complete Valuation Field 84 BY TSUNEO TAMAGAWA Groups Over Z 90 BY BERTRAM KOSTANT Subgroups of Finite Index in Certain Arithmetic Groups 99 BY H. MATSUMOTO The Problem of the Maximality of Arithmetic Groups 104 BY NELO D. ALLAN II. Arithmetic Properties of Algebraic Groups. Adele Groups Ad&es 113 BY TSUNEO TAMAGAWA On Tamagawa Numbers 122 BY TAKASHI ONO The Siegel Formula for Orthogonal Groups. I 133 BY J. G. M. MARS The Siegel Formula for Orthogonal Groups. II 138 BY J. G. M. MARS The Volume of the Fundamental Domain for Some Arithmetical Subgroups of Chevalley Groups 143 BY R. P. LANGLANDS V VI CONTENTS Galois Cohomology of Linear Algebraic Groups 149 BY T. A. SPRINGER Hasse Principle for Hl of Simply Connected Groups 159 BY MARTIN KNESER Nonabelian H2 in Galois Cohomology 164 BY T. A. SPRINGER Inseperable Galois Cohomology 183 BY PIERRE CARTIER Strong Approximation 187 BY MARTIN KNESER III. Automorphic Functions and Decomposition of L2(G/r) Introduction to Automorphic Forms 199 BY ARMAND BOREL The Decomposition of L2(G/T) for T = SL(2, Z) 211 BY R. GODEMENT The Spectral Decomposition of Cusp-Forms 225 BY R. GODEMENT Eisenstein Series 235 BY R. P. LANGLANDS Dimension of Spaces of Automorphic Forms 253 BY R. P. LANGLANDS Spherical Functions and Ramanujan Conjecture 258 BY ICHIRO SATAKE Algebraic Curves Mod p and Arithmetic Groups 265 BY YASUTAKA IHARA Discrete Subgroups of PL(2, kp) 272 BY YASUTAKA IHARA IV. Bounded Symmetric Domains, Holomorphic Automorphic Forms, Moduli On Compactifications of Orbit Spaces of Arithmetic Discontinuous Groups Acting on Bounded Symmetric Domains 281 BY WALTER L. BAILY, JR. Fourier-Jacobi Series 296 BY WALTER L. BAILY, JR. On the Desingularization of Satake Compactifications 301 BY JUN-ICHI IGUSA Classical Theory of ^-functions 306 BY WALTER L. BAILY, JR. Moduli of Abelian Varieties and Number Theory 312 BY GORO SHIMURA CONTENTS Vii Hecke's Polynomial as a Generalized Congruence Artin L-function 333 BY MICHIO KUGA Fiber Varieties over a Symmetric Space Whose Fibers Are Abelian Varieties 338 BY MICHIO KUGA Families of Abelian Varieties 347 BY DAVID MUMFORD Symplectic Representations of Algebraic Groups 352 BY ICHIRO SATAKE The Modular Groups of Hilbert and Siegel 358 BY WILLIAM F. HAMMOND Quantum Mechanical Commutation Relations and Theta Functions 361 BY PIERRE CARTIER V. Quotients of Symmetric Spaces. Deformations Cohomologies of Vector-values Forms on Compact, Locally Symmetric Riemann Manifolds 387 BY SHINGO MURAKAMI On Deformations of Lattices in Lie Groups 400 BY HOWARD GARLAND On Deformations of Discrete Groups in the Noncompact Case 405 BY HOWARD GARLAND On the Conjugacy of Subgroups of Semisimple Groups 413 BY G. D. MOSTOW Index 421 Authors 425 Index abelian variety, 312,313,315,316, canonical 318,338,345 automorphy factor, 202,388,391,392 addle ring, 114 numbering, 286 adelized Cayley transforms, natural compacti- group, 115 fications, 285 variety, 114 center, 38 adjoint character, 6 group, 38,42 Chevalley groups, 143,148 representation, 5 class admissible field, 322 pair, 72 numbers, 188 PEL-structure, 319 of o*~lattices, 323 affine classical groups, 66 algebraic group, 4,5 cocycle of r, 49 or extended Dynkin diagram, 34 cohomologous, innerly, 49 Weyl group, 35,68,74 cohomology algebraic exact sequences in noncommu- matrix group, 3 tative, 153 torus, 7 Galois, 164,190,192 transformation group, 5 noncommutative, 151,164 algebras L with a positive involu• commensurability subgroup of r, 20 tion, 319 compactification, 281,292,284 anisotropic complex multiplication, 320,322 over K, 7 condition (Hi), 352 K-torus over, 7 conjecture kernels, 39 Ramanujan reductive groups, 13 -Peterson, 264 reductive kernel, 39 and spherical functions, 258 semisimple kernel conjugacy, Frobenius classes, 267 antihermitian form, p-, 317 connected, simply arithmetic groups, 20,99,104 algebraic group, 154 automorphic covering, 42 factor, 387 group, 68 form, 200,201,204,326,391 universal covering, 38 automorphism group, 38 cusp-forms, 222,226,230,233,331 automorphy spectral decomposition, 225 canonical factor, 202,388,391,392 cylindrical factor, 201-203 sets, 296 J3iV-pair, 71 d-cohomology, 398 Borel subgroup, 11 groups, 394 boundary components, 287 of vector valued forms, 388 bounded decomposition subgroup, 63 Jordan, 6,26 symmetric domain, 319,320 spectral cusp-forms. 225 421 422 INDEX deformation of Cr-structure, 407 group, 358 desingularization, problem of, 301-303 -Siegel modular group, 282 dimension, fields of, 154 variety, 358 discrete Hodge group series, 262 of an abelian variety, 348 subgroups, 26S, 272 of a complex torus, 347 divisor, basic, 314 homogeneous, principal space of a duality, Tate-Nakayama, 127 linear algebraic group, 149 Dynkin diagram, 33-35, 37,41,46,53 index, 39 Eisenstein inner and outer form, 39 -Poincare series, 284,285,291 innerly cohomologous, 49 series, 146,206,208, 209,214,215, involution 221,235, 236,247,251, 252 of an algebra, 313 -Siegel series, 140 opposition, 36 element, i?-regular, 414 isogenous, (strictly) over k or k- exact sequences,
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