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

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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 fivemajo r 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, 173 isogenous, 34,42 in noncommutative cohomology, 153 isogeny, 34,313 Euler product, 327 Jordan decomposition, 6,26 fibre system of abelian varieties, 323 k variety, 326,327 anistropic torus over, 7 field of or ^-isogenous, strictly over, 34,42 dimension, 154 -primary, 353 moduli, 282,321 -rank, 13 Fock representation, 381,363 representation strongly rational Fourier-Jacobi series, 288, 296 over, 18 Frobenius roots, 13 conjugacy classes, 267 split torus, 7 reciprocity law, 372 Weyl group relative to, 13 fundamental set, 21 L-functions, 333,334 r-automorphic right spherical Laplace transform, 212 function, 261 lattice, 400 Galois cohomology, 164,190,192 Lefschetz imbedding theorem, 307,308 generalized Tits system, 71, 74, 79 Lie algebra of an algebraic group, 5 linear algebraic groups, 3 harmonic theory, 392 locally trivial, 389 Hasse principle for semisimple simply map connected groups, 155 connecting, 171 zeta-function, 312 G, 406,408 Hecke maximal family of polarized abelian algebra, 79 varieties, 315 ring, 78, 79, 326 modular operators, 326,327 form, 308 Hermitian function, 212 mapping group, Hilbert-Siegel, 282 quasi-, 288 imbeddings, 358-360 (V-),297 moduli modular group, 283 variety, 324 Hilbert for PEL-structures, 321 modular forms, 358-360 modulus, supersingular, 265 INDEX moronism of algebraic groups, 5 Ramanujan, conjecture and spherical func• Nakayama-Tate duality, 127 tions, 258 natural compactification, 284, 285 -Peterson conjecture, 264 and Cayley transforms, 285 rank Navier-Stokes equations, 16,45,47,49 of an algebraic group, 10 nilpotent, 9 M3 solvable algebraic group, 8 rational noncommutative boundary components, 284,289,290 cohomology, 151,164 points, 184 exact sequences in cohomology, 153 representation, 6 normal PEL-type, 321,322 reciprocity law, Frobenius, 372 reductive algebraic group, 9 1-cocycle of r, 49 anistropic opposition involution, 36 groups, 13 orbits, distinguished, 39 kernel, 39 orthogonal group, special, 107 regular element, 30 relative Weyl group, 40 parabolic representation standard £-subgroups, 14 adjoint, 5 subgroup, 9,38,41 induced, 370 partial infinitesimal, 370 Cayley transformations, 286 rational, 6 desingularizations theorem, 302,303 special, 275 PEL strongly rational over kt 18 -structure, 347,318 Riemann admissible, 319 conditions, 298,306 moduli for, 321 form, 313 -type, 318 rigid at °°, 407 normal, 321,322 rigidity, 418 Peterson-Ramanujan conjecture, 264 ring of Thetanullwerthe, 309 Poincare root -Eisenstein series, 284,285,291 M3 series, 204,205 system, 11,12 polarization, 306,314 polarized s-semiautomorphism, 176 abelian variety, 314 Satake p-compactification, 413,415 normally, 282 schemes, semisimple group, 99 positive, 313 Schrodinger representation, 361 involution, 316 semiautomorphism, s-, 176 principal semisimple homogeneous space of a linear algebraic group, 9 algebraic group anistropic kernel, 39 matrices, 306 element, 414,416 series, 211,261 group schemes, 99 sequences, exact, 173 quaternion algebra Siegel totally definite, 319 domain, 22-24,296 totally indefinite, 319 -Eisenstein series, 140 formula, 133,136,141,143 -Hilbert modular group, 282 p-antihermitian form, 317 modular form, 358-360 p-compactification, Satake, 413,415 sets, 297 radical (of an algebraic group), 9 variety, 358 424 INDEX simple, 313 transformation group, algebraic, 5 almost, 33 truncated singularity theorem, 302,303 Siegel sets, 290 solvable domains, 297 nilpotent algebraic group, 8 twisted split group, 9 tensor product, 79 spherical highly, 194 r-automorphic right function, 261 twisting, 151 zonal function, 258 split U -function, 291 groups, 66 unbounded realizations, 285 fc-torus, 7 unimodular group, 123 solvable group, 9 unipotent, 122 structure theorem, 272 algebraic group, 8 supersingular, 266,268 radical (of an algebraic group), 9 modulus, 265 Unitary, special group, 107 supplementary series, 261 Sylow subgroup, 76 variety, adelized, 114 symmetric domain, 339 vertices, 34 bounded, 319,320 symplectic weight group, 36 pair, 339 Weil's conjecture, 123,331 representations, 358 Weyl chamber, 12,14,203,249 chamber, 12,14,203,249 Tamagawa number, 124,127,128,133 commutations relations, 366 Tate, Nakayama-, duality, 127 group, 11,12,14,15,16,33,35,40, tensor, twisted product, 79 68,71,73,194,235,414 ^-function, 311 affine, 35,68, 74 Thetanullwerthe, 308 extended, 194 ring of, 309 generalized, 71 Tits system, 68,69,72 relative, 40 (BN-pair), 71 relative to k, 13,416 generalized, 71,74, 79 saturated, 72 torus, 68 Zariski algebraic, 7 dense, 11,26 anistropic over A, 7 density, 104,105 split*-, 7 Zeta function, 329,331 trace formula, 329 Hasse, 312 transform, Laplace, 212 Zonal spherical function. 258 Authors

This is a list of the authors of the various papers in the book, together with their permanent addresses.

Dr. Nelo D. Allan, Department of Mathematics, The University of Chicago, Chicago, Illinois, and Department of Mathematics, DePaul University, Chicago, Illinois. Professor Walter L. Baily, Jr., Department of Mathematics, University of Chicago, Chicago, Illinois. Professor Armand Borel, School of Mathematics, The Institute for Advanced Study, Princeton, New Jersey. Professor Francois Bruhat, 80 Boulevard Pasteur, Paris XV°, France. Professor Pierre Cartier, Department of Mathematics, University of Strasbourg, Strasbourg, France. Professor Howard Garland, School of Mathematics, The Institute for Advanced Study, Princeton, New Jersey. Professor R. Godement, Institut Henri Poincare, Rue Pierre Curie, Paris, Ve, France. William F. Hammond, 5007 Falls Road Terrace, Baltimore, Maryland 21210. Professor Jun-ichi Igusa, 911 Breezewick Road, Towson, Maryland. Professor Yasutaka Ihara, School of Mathematics, The Institute for Advanced Study, Princeton, New Jersey. Professor Nagayoshi Iwahori, Department of Mathematics, University of California, Berkeley, California Martin L. Kneser, Merkelstrasse 39, 34 Gottingen, Germany. Professor Bertram Kostant, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Professor Michio Kuga, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo, Japan. Dr. Robert P. Langlands, Fine Hall, Princeton University, Princeton, New Jersey. Professor J. G. M. Mars, Mathematique Institut, Boothstraat 17, Utrecht, The Netherlands. Professor Hideya Matsumoto, Institut Henri Poincare, 11, Rue Pierre Curie, Paris 5e, France. Professor George D. Mostow, Department of Mathematics, Yale University, New Haven, Connecticut. Professor David Mumford, Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138. Professor Shingo Murakami, Department of Mathematics, Osaka University, Osaka, Japan. Professor Takashi Ono, Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104. Professor Ichiro Satake, Department of Mathematics, University of Chicago, Chicago, Illinois 60637. 425 426 AUTHORS

Professor Goro Shimura, Department of Mathematics, Princeton University, Princeton, New Jersey. Professor T. A. Springer, Mathematisch Institut, Boothstraat 17, Utrecht, The Netherlands. Professor Tsuneo Tamagawa, Department of Mathematics, Yale University, New Haven, Connecticut. Professor Jacques L. Tits, Mathematisches Institut der UniversitAt, Wegelerstrasse 10, 53 Bonn, Germany.