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
http://dx.doi.org/10.1090/pspum/009
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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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, 318,338,345
addle ring, 114 adelized
group, 115 variety, 114
adjoint group, 38,42 representation, 5
admissible pair, 72 PEL-structure, 319
affine algebraic group, 4,5 or extended Dynkin diagram, 34 Weyl group, 35,68,74
algebraic matrix group, 3 torus, 7 transformation group, 5
algebras L with a positive involution, 319
anisotropic over K, 7 K-torus over, 7 kernels, 39 reductive groups, 13 reductive kernel, 39 semisimple kernel
antihermitian form, p-, 317 arithmetic groups, 20,99,104 automorphic
factor, 387 form, 200,201,204,326,391
automorphism group, 38 automorphy
canonical factor, 202,388,391,392 factor, 201-203
J3iV-pair, 71 Borel subgroup, 11 boundary components, 287 bounded
subgroup, 63 symmetric domain, 319,320
canonical automorphy factor, 202,388,391,392 numbering, 286
Cayley transforms, natural compacti-fications, 285
center, 38 character, 6 Chevalley groups, 143,148 class
field, 322 numbers, 188 of o*~lattices, 323
classical groups, 66 cocycle of r, 49 cohomologous, innerly, 49 cohomology
exact sequences in noncommu-tative, 153
Galois, 164,190,192 noncommutative, 151,164
commensurability subgroup of r, 20 compactification, 281,292,284 complex multiplication, 320,322 condition (Hi), 352 conjecture
Ramanujan -Peterson, 264 and spherical functions, 258
conjugacy, Frobenius classes, 267 connected, simply
algebraic group, 154 covering, 42 group, 68 universal covering, 38
cusp-forms, 222,226,230,233,331 spectral decomposition, 225
cylindrical sets, 296
d-cohomology, 398 groups, 394
of vector valued forms, 388 decomposition
Jordan, 6,26 spectral cusp-forms. 225
421
422 INDEX
deformation of Cr-structure, 407 desingularization, problem of, 301-303 dimension, fields of, 154 discrete
series, 262 subgroups, 26S, 272
divisor, basic, 314 duality, Tate-Nakayama, 127 Dynkin diagram, 33-35, 37,41,46,53
Eisenstein -Poincare series, 284,285,291 series, 146,206,208, 209,214,215, 221,235, 236,247,251, 252
-Siegel series, 140 element, i?-regular, 414 exact sequences, 173
in noncommutative cohomology, 153 Euler product, 327
fibre system of abelian varieties, 323 variety, 326,327
field of dimension, 154 moduli, 282,321
Fock representation, 381,363 Fourier-Jacobi series, 288, 296 Frobenius
conjugacy classes, 267 reciprocity law, 372
fundamental set, 21
r-automorphic right spherical function, 261
Galois cohomology, 164,190,192 generalized Tits system, 71, 74, 79
harmonic theory, 392 Hasse
principle for semisimple simply connected groups, 155
zeta-function, 312 Hecke
algebra, 79 ring, 78, 79, 326 operators, 326,327
Hermitian mapping
quasi-, 288 (V-),297
modular group, 283 Hilbert
modular forms, 358-360
group, 358 -Siegel modular group, 282 variety, 358
Hodge group of an abelian variety, 348 of a complex torus, 347
homogeneous, principal space of a linear algebraic group, 149
index, 39 inner and outer form, 39 innerly cohomologous, 49 involution
of an algebra, 313 opposition, 36
isogenous, (strictly) over k or k-isogenous, 34,42
isogeny, 34,313
Jordan decomposition, 6,26
k anistropic torus over, 7 or ^-isogenous, strictly over, 34,42 -primary, 353 -rank, 13 representation strongly rational over, 18
roots, 13 split torus, 7 Weyl group relative to, 13
L-functions, 333,334 Laplace transform, 212 lattice, 400 Lefschetz imbedding theorem, 307,308 Lie algebra of an algebraic group, 5 linear algebraic groups, 3 locally trivial, 389
map connecting, 171 G, 406,408
maximal family of polarized abelian varieties, 315
modular form, 308 function, 212 group, Hilbert-Siegel, 282 imbeddings, 358-360
moduli variety, 324
for PEL-structures, 321 modulus, supersingular, 265
INDEX
moronism of algebraic groups, 5
Nakayama-Tate duality, 127 natural compactification, 284, 285
and Cayley transforms, 285 Navier-Stokes equations, 16,45,47,49 nilpotent, 9
solvable algebraic group, 8 noncommutative
cohomology, 151,164 exact sequences in cohomology, 153
normal PEL-type, 321,322
1-cocycle of r, 49 opposition involution, 36 orbits, distinguished, 39 orthogonal group, special, 107
parabolic standard £-subgroups, 14 subgroup, 9,38,41
partial Cayley transformations, 286 desingularizations theorem, 302,303
PEL -structure, 347,318
admissible, 319 moduli for, 321
-type, 318 normal, 321,322
Peterson-Ramanujan conjecture, 264 Poincare
-Eisenstein series, 284,285,291 series, 204,205
polarization, 306,314 polarized
abelian variety, 314 normally, 282
positive, 313 involution, 316
principal homogeneous space of a linear algebraic group
matrices, 306 series, 211,261
quaternion algebra totally definite, 319 totally indefinite, 319
p-antihermitian form, 317 p-compactification, Satake, 413,415 radical (of an algebraic group), 9
Ramanujan, conjecture and spherical functions, 258
-Peterson conjecture, 264 rank
of an algebraic group, 10 M3
rational boundary components, 284,289,290 points, 184 representation, 6
reciprocity law, Frobenius, 372 reductive
algebraic group, 9 anistropic
groups, 13 kernel, 39
regular element, 30 relative Weyl group, 40 representation
adjoint, 5 induced, 370 infinitesimal, 370 rational, 6 special, 275 strongly rational over kt 18
Riemann conditions, 298,306 form, 313
rigid at °°, 407 rigidity, 418 ring of Thetanullwerthe, 309 root
M 3 system, 11,12
s-semiautomorphism, 176 Satake p-compactification, 413,415 schemes, semisimple group, 99 Schrodinger representation, 361 semiautomorphism, s-, 176 semisimple
algebraic group, 9 anistropic kernel, 39 element, 414,416 group schemes, 99
sequences, exact, 173 Siegel
domain, 22-24,296 -Eisenstein series, 140 formula, 133,136,141,143 -Hilbert modular group, 282 modular form, 358-360 sets, 297 variety, 358
424
simple, 313 almost, 33
singularity theorem, 302,303 solvable
nilpotent algebraic group, 8 split group, 9
spherical r-automorphic right function, 261 zonal function, 258
split groups, 66 fc-torus, 7 solvable group, 9
structure theorem, 272 supersingular, 266,268
modulus, 265 supplementary series, 261 Sylow subgroup, 76 symmetric domain, 339
bounded, 319,320 symplectic
pair, 339 representations, 358
Tamagawa number, 124,127,128,133 Tate, Nakayama-, duality, 127 tensor, twisted product, 79
^-function, 311 Thetanullwerthe, 308
ring of, 309 Tits system, 68,69,72
(BN-pair), 71 generalized, 71,74, 79 saturated, 72
torus, 68 algebraic, 7 anistropic over A, 7 split*-, 7
trace formula, 329 transform, Laplace, 212
INDEX
transformation group, algebraic, 5 truncated
Siegel sets, 290 domains, 297
twisted tensor product, 79 highly, 194
twisting, 151
U -function, 291 unbounded realizations, 285 unimodular group, 123 unipotent, 122
algebraic group, 8 radical (of an algebraic group), 9
Unitary, special group, 107
variety, adelized, 114 vertices, 34
weight group, 36 Weil's conjecture, 123,331 Weyl chamber, 12,14,203,249
chamber, 12,14,203,249 commutations relations, 366 group, 11,12,14,15,16,33,35,40, 68,71,73,194,235,414
affine, 35,68, 74 extended, 194 generalized, 71 relative, 40 relative to k, 13,416
Zariski dense, 11,26 density, 104,105
Zeta function, 329,331 Hasse, 312
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.