Algebraic groups and number theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston u.a.
Acad. Press
1994
|
| Series: | Pure and applied mathematics
139 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 614 S. |
| ISBN: | 0125581807 |
Staff View
MARC
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| 100 | 1 | |a Platonov, Vladimir P. |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Algebraičeskie gruppy i teorija čisel |
| 245 | 1 | 0 | |a Algebraic groups and number theory |c Vladimir Platonov ; Andrei Rapinchuk |
| 264 | 1 | |a Boston u.a. |b Acad. Press |c 1994 | |
| 300 | |a XI, 614 S. | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface to the English Edition ix
Preface to the Russian Edition ix
Chapter 1. Algebraic number theory 1
1.1. Algebraic number fields, valuations, and completions .... 1
1.2. Adeles and ideles; strong and weak approximation; the
local global principle 10
1.3. Cohomology 16
1.4. Simple algebras over local fields 27
1.5. Simple algebras over algebraic number fields 37
Chapter 2. Algebraic Groups 47
2.1. Structural properties of algebraic groups 47
2.2. Classification of K iorms using Galois cohomology 67
2.3. The classical groups 78
2.4. Some results from algebraic geometry 96
Chapter 3. Algebraic Groups over Locally Compact Fields 107
3.1. Topology and analytic structure 107
3.2. The Archimedean case 118
3.3. The non Archimedean case 133
3.4. Elements of Bruhat Tits theory 148
3.5. Results needed from measure theory 158
Chapter 4. Arithmetic Groups and Reduction Theory . . 171
4.1. Arithmetic groups 171
4.2. Overview of reduction theory: reduction in GLn(R) . . . 175
4.3. Reduction in arbitrary groups 189
4.4. Group theoretic properties of arithmetic groups 195
4.5. Compactness of Gr/Gz 207
4.6. The finiteness of the volume of Gr/Gz 213
4.7. Concluding remarks on reduction theory 223
4.8. Finite arithmetic groups 229
v
vi Contents
Chapter 5. Adeles 243
5.1. Basic definitions 243
5.2. Reduction theory for Ga relative to Gk 253
5.3. Criteria for the compactness and the finiteness of volume
of GaIGk 260
5.4. Reduction theory for S arithmetic subgroups 266
Chapter 6. Galois cohomology 281
6.1. Statement of the main results 281
6.2. Cohomology of algebraic groups over finite fields 286
6.3. Galois cohomology of algebraic tori 300
6.4. Finiteness theorems for Galois cohomology 316
6.5. Cohomology of semisimple algebraic groups over local fields
and number fields 325
6.6. Galois cohomology and quadratic, Hermitian, and other
forms 342
6.7. Proof of Theorems 6.4 and 6.6: Classical groups 356
6.8. Proof of Theorems 6.4 and 6.6: Exceptional groups . . . 368
Chapter 7. Approximation in Algebraic Groups 399
7.1. Strong and weak approximation in algebraic varieties . . 399
7.2. The Kneser Tits conjecture 405
7.3. Weak approximation in algebraic groups 415
7.4. The strong approximation theorem 427
7.5. Generalization of the strong approximation theorem . . . 433
Chapter 8. Class numbers and class groups of algebraic
groups 439
8.1. Class numbers of algebraic groups
and number of classes in a genus 439
8.2. Class numbers and class groups of semisimple groups of
noncompact type; the realization theorem 450
8.3. Class numbers of algebraic groups of compact type . . . 471
8.4. Estimating the class number for reductive groups .... 484
8.5. The genus problem 494
Chapter 9. Normal subgroup structure of groups of ratio¬
nal points of algebraic groups 509
9.1. Main conjectures and results 509
9.2. Groups of type An 518
Contents vii
9.3. The classical groups 537
9.4. Groups split over a quadratic extension 546
9.5. The congruence subgroup problem (a survey) 553
Appendix A 571
Appendix B. Basic Notation 579
Bibliography 583
Index 609
|
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| id | DE-604.BV008817199 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:25:31Z |
| institution | BVB |
| isbn | 0125581807 |
| language | English |
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| physical | XI, 614 S. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Acad. Press |
| record_format | marc |
| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Platonov, Vladimir P. Verfasser aut Algebraičeskie gruppy i teorija čisel Algebraic groups and number theory Vladimir Platonov ; Andrei Rapinchuk Boston u.a. Acad. Press 1994 XI, 614 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 139 Algebraische Gruppe (DE-588)4001164-1 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 s Zahlentheorie (DE-588)4067277-3 s DE-604 Lineare algebraische Gruppe (DE-588)4295326-1 s DE-188 Rapinčuk, Andrej S. Verfasser aut Pure and applied mathematics 139 (DE-604)BV010177228 139 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005833230&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Platonov, Vladimir P. Rapinčuk, Andrej S. Algebraic groups and number theory Pure and applied mathematics Algebraische Gruppe (DE-588)4001164-1 gnd Zahlentheorie (DE-588)4067277-3 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd |
| subject_GND | (DE-588)4001164-1 (DE-588)4067277-3 (DE-588)4295326-1 |
| title | Algebraic groups and number theory |
| title_alt | Algebraičeskie gruppy i teorija čisel |
| title_auth | Algebraic groups and number theory |
| title_exact_search | Algebraic groups and number theory |
| title_full | Algebraic groups and number theory Vladimir Platonov ; Andrei Rapinchuk |
| title_fullStr | Algebraic groups and number theory Vladimir Platonov ; Andrei Rapinchuk |
| title_full_unstemmed | Algebraic groups and number theory Vladimir Platonov ; Andrei Rapinchuk |
| title_short | Algebraic groups and number theory |
| title_sort | algebraic groups and number theory |
| topic | Algebraische Gruppe (DE-588)4001164-1 gnd Zahlentheorie (DE-588)4067277-3 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd |
| topic_facet | Algebraische Gruppe Zahlentheorie Lineare algebraische Gruppe |
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