The Congruence Subgroup problem: An elementary approach aimed at applications

315 Pages·2003·26.65 MB·English

by  B. Sury (auth.)

24
TEXTS AND READINGS
IN MATHEMATICS
The Congruence
Subgroup Problem
An elementary approach
almed at applicatlons
Texts and Readings in Mathematics
Advisory Editor
C. S. Seshadri, Chennai Mathematical lnst., Chennai.
Managing Editor
Rajendra Bhatia, Indian StatisticalInst., New Delhi.
Editors
V. S. Borkar, Tata lnst. of Fundamental Research, Mumbai.
Probai Chaudhuri, Indian Statisticalinst., Kolkata.
R. L. Karandikar, Indian Statisticalinst., New Delhi.
M. Ram Murty, Queen's University, Kingston.
C. Musili, University of Hyderabad, Hyderabad.
V. S. Sunder, lnst. of Mathematical Sciences, Chennai.
M. Vanninathan, TlFR Centre, Bangalore.
T. N. Venkataramana, Tata lnst. of Fundamental Research, Mumbai.
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The Congruence
Subgroup problem
An elementary approach
aimed at applications
B. Sury
Indian Statistical Institute
Bangalore
HINDUSTAN
fl::[g1IQ0l o
U L!::!J UB OOK AGENCY
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Produced from camera ready copy supplied by the Author.
ISBN 978-81-85931-38-8 ISBN 978-93-86279-19-4 (eBook)
DOI 10.1007/978-93-86279-19-4
TO
VIVUR
FOR PROVUCINg
T1lE :FIRST SOCUTION oN PAPER
Contents
Preface xi
1 A review of background material 1
1.1 Congruenees. . . 1
1.2 A topology on 'll. ......... 2
1.3 Roots of unity. . . . . . . . . . . 3
1.4 Free produets and invariant faetors 4
1.5 Deeompositions for matrix groups 10
1.6 Group eohomology and eentral extensions 14
1.7 Profinite groups. . . . . . . . . . . 25
1.8 Completions of topological groups 31
1.9 Golod-Shafarevieh type theorems . 35
1.10 Congruenee subgroups . . . . . . . 38
1.11 Group theory vis-a-vis number theory 39
1.12 Algebraie number theory: reeolleetions 43
1.13 Theorems of Dirichlet & Chebotarev . 49
1.14 Adeles, ideles and strang approximation 52
1.15 Moore's loeal uniqueness theorem . . . . 58
1.16 Moore's uniqueness of global reciproeity laws 60
2 Solvable groups 63
2.1 The additive group 63
2.2 The multiplicative group . 64
2.3 Chevalley's theorem 65
2.4 Wehrfritz's theorem ... 69
viii CONTENTS
2.5 Upper triangular group .. 72
3 SL2 - The negative solutions 75
3.1 CSP for SL2(Os) - naive formulation. 76
3.2 SL2 over semilocal rings - a positive case 77
3.3 Structure of SL ('7l,) . . . . . . . . . . . . 79
2
3.4 Congruence subgroup problem for SL ('7l,) . 85
2
3.5 Level versus index - a criterion . . 87
3.6 Remarks on the CSP for SL ('7l,) 89
2
3.7 The CSP - modern formulation . 91
3.8 CSP - Some necessary conditions 94
3.9 Structure of SL2(0~(V-D)) ... 96
3.10 Fundamental sets for SL2( Od) 97
3.11 Grunewald-Schwermer's theorem 100
3.12 Failure of CSP for SL2(0~(V-D)) 108
3.13 Another proof that CSP fails for SL2(0) 110
3.14 Normal subgroups of infinite index 112
4 SLn(Os) - Positive cases of CSP 113
4.1 The Steinberg groups. . . . . . 115
4.2 A presentatioll for SLn('7l,/k'7l,) 124
4.3 Presentation for SLn('7l,), n ~ 3 130
4.4 Normal subgroups of SLn('7l,), n ~ 3 135
4.5 CSP - the modus operandi. . . . . . 137
4.6 Centrality of C(S) from Steinberg relations 140
4.7 CSP for SL ('7l,s) . . . . . . . . . . 147
2 Os .....
4.8 Centrality for infinite 159
4.9 C(S) versus the metaplectic kernel 163
4.10 Calvin Moore's theory . . . . . . . 166
4.11 Schur multiplier of SL (K) for infinite K 171
2
4.12 Topological central extensions of SLn(K) 179
4.13 Metaplectic kernel of SLn . . . . . . • . . 186
CONTENTS ix
5 Applications of the CSP 194
5.1 An application to Hecke operators 195
5.2 A congruence criterion in SL ('D.) . 202
2
5.3 CSP and generators for arithmetic groups 204
5.4 A criterion for linearity 212
5.5 CSP and super-rigidity ... 214
5.6 Phantom finite groups ... 217
5.7 Subgroup growth and CSP 223
5.8 Congruence subgroups and Mersenne primes. 226
5.9 Bounded generation Vs CSP 230
5.10 Character variety . . . . 235
5.11 Adelic profinite groups . 236
5.12 Probabilistic methods 240
6 CSP in general algebraic groups 244
6.1 Arithmetic groups ... 245
6.2 Reduction theory ...... 253
6.3 Arithmeticity of lattices .. 254
6.4 Group-theoretic properties . 254
6.5 Deformations and rigidity 255
6.6 Structure of algebraic groups 256
6.7 CSP - Status and methods . 263
Appendix 273
Bibliography 286
Index 297
Preface
"In truth, it is not knowledge, but learning, not possessing, but
production, not being there, but travelling there, which provides
the greatest pleasure. "
Letter from C.F.Gauss to W.Bolyai on Sept.2, 1808.
This important subject can be approached at many different
levels of mathematical maturity. On the first glance, it seems to
require an enormous amount of prior knowledge of the basics of
subjects like the theory of linear algebraic groups and arithmetic
subgroups on the one hand, and of abelian class field theory on
the other.
A student of number theory, say, who is keen to learn about
the congruence subgroup problem, could possibly be overawed on
being confronted with algebraic geometry which she might view
as a technical block. A popular opinion is that aprerequisite to
our subject matter is a reasonably good grounding in the theory
of algebraic groups.
This lIlonograph seeks to convey the view that at least so me
aspects of this subject cau be dealt with profitably without the
knowledge of algebraic group theory. We attempt to do this through
judiciously chosen examples and through a development of some of
their relations with other brauches. In this book, we discuss some
specific cases of the congruence subgroup problem which are still
generic iu asense. Moreover, more than the solution ofthe congru
ence subgroup problem itself, it is, arguably, the so-many different
aspects of the problem in terms of its relations with other subjects