Algebraic Groups and Number Theory (eBook)

Front Cover 1
Algebraic Groups and Number Theory 4
Copyright Page 5
Contents 6
Preface to the English Edition 10
Preface to the Russian Edition 10
Chapter 1. Algebraic number theory 14
1.1. Algebraic number fields, valuations, and completions 14
1.2. Adeles and ideles strong and weak approximation
1.3. Cohomology 29
1.4. Simple algebras over local fields 40
1.5. Simple algebras over algebraic number fields 50
Chapter 2. Algebraic Groups 60
2.1. Structural properties of algebraic groups 60
2.2. Classification of K-forms using Galois cohomology 80
2.3. The classical groups 91
2.4. Some results from algebraic geometry 109
Chapter 3. Algebraic Groups over Locally Compact Fields 120
3.1. Topology and analytic structure 120
3.2. The Archimedean case 131
3.3. The non-Archimedean case 146
3.4. Elements of Bruhat-Tits theory 161
3.5. Results needed from measure theory 171
Chapter 4. Arithmetic Groups and Reduction Theory 184
4.1. Arithmetic groups 184
4.2. Overview of reduction theory: reduction in GLn (R) 188
4.3. Reduction in arbitrary groups 202
4.4. Grouptheoretic properties of arithmetic groups 208
4.5. Compactness of GR/GZ 220
4.6. The finiteness of the volume of GR/GZ 226
4.7. Concluding remarks on reduction theory 236
4.8. Finite arithmetic groups 242
Chapter 5. Adeles 256
5.1. Basic definitions 256
5.2. Reduction theory for GA relative to GK 266
5.3. Criteria for the compactness and the finiteness of volume of GA/GK 273
5.4. Reduction theory for S-arithmetic subgroups 279
Chapter 6. Galois cohomology 294
6.1. Statement of the main results 294
6.2. Cohomology of algebraic groups over finite fields 299
6.3. Galois cohomology of algebraic tori 313
6.4. Finiteness theorems for Galois cohomology 329
6.5. Cohomology of semisimple algebraic groups over local fields and number fields 338
6.6. Galois cohomology and quadratic, Hermitian, and other forms 355
6.7. Proof of Theorems 6.4 and 6.6: Classical groups 369
6.8. Proof of Theorems 6.4 and 6.6: Exceptional groups 381
Chapter 7. Approximation in Algebraic Groups 412
7.1. Strong and weak approximation in algebraic varieties 412
7.2. The Kneser-Tits conjecture 418
7.3. Weak approximation in algebraic groups 428
7.4. The strong approximation theorem 440
7.5. Generalization of the strong approximation theorem 446
Chapter 8. Class numbers and class groups of algebraic groups 452
8.1. Class numbers of algebraic groups and number of classes in a genus 452
8.2. Class numbers and class groups of semisimple groups of noncompact type the realization theorem
8.3. Class numbers of algebraic groups of compact type 484
8.4. Estimating the class number for reductive groups 497
8.5. The genus problem 507
Chapter 9. Normal subgroup structure of groups of rational points of algebraic groups 522
9.1. Main conjectures and results 522
9.2. Groups of type An 531
9.3. The classical groups 550
9.4. Groups split over a quadratic extension 559
9.5. The congruence subgroup problem (a survey) 566
Appendix A 584
Appendix B. Basic Notation 592
Bibliography 596
Index 622