UID:
almahu_9949697508302882
Format:
1 online resource (629 p.)
ISBN:
1-281-76882-0
,
9786611768829
,
0-08-087459-2
Series Statement:
Pure and applied mathematics ; v. 139
Uniform Title:
Algebraicheskie gruppy i teorii͡a chisel.
Content:
This milestone work on the arithmetic theory of linear algebraic groups is now available in English for the first time. Algebraic Groups and Number Theory provides the first systematic exposition in mathematical literature of the junction of group theory, algebraic geometry, and number theory. The exposition of the topic is built on a synthesis of methods from algebraic geometry, number theory, analysis, and topology, and the result is a systematic overview ofalmost all of the major results of the arithmetic theory of algebraic groups obtained to date.
Note:
Description based upon print version of record.
,
Front Cover; Algebraic Groups and Number Theory; Copyright Page; Contents; Preface to the English Edition; Preface to the Russian Edition; Chapter 1. Algebraic number theory; 1.1. Algebraic number fields, valuations, and completions; 1.2. Adeles and ideles; strong and weak approximation; the local-global principle; 1.3. Cohomology; 1.4. Simple algebras over local fields; 1.5. Simple algebras over algebraic number fields; Chapter 2. Algebraic Groups; 2.1. Structural properties of algebraic groups; 2.2. Classification of K-forms using Galois cohomology; 2.3. The classical groups
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2.4. Some results from algebraic geometryChapter 3. Algebraic Groups over Locally Compact Fields; 3.1. Topology and analytic structure; 3.2. The Archimedean case; 3.3. The non-Archimedean case; 3.4. Elements of Bruhat-Tits theory; 3.5. Results needed from measure theory; Chapter 4. Arithmetic Groups and Reduction Theory; 4.1. Arithmetic groups; 4.2. Overview of reduction theory: reduction in GLn (R); 4.3. Reduction in arbitrary groups; 4.4. Grouptheoretic properties of arithmetic groups; 4.5. Compactness of GR/GZ; 4.6. The finiteness of the volume of GR/GZ
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4.7. Concluding remarks on reduction theory4.8. Finite arithmetic groups; Chapter 5. Adeles; 5.1. Basic definitions; 5.2. Reduction theory for GA relative to GK; 5.3. Criteria for the compactness and the finiteness of volume of GA/GK; 5.4. Reduction theory for S-arithmetic subgroups; Chapter 6. Galois cohomology; 6.1. Statement of the main results; 6.2. Cohomology of algebraic groups over finite fields; 6.3. Galois cohomology of algebraic tori; 6.4. Finiteness theorems for Galois cohomology; 6.5. Cohomology of semisimple algebraic groups over local fields and number fields
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6.6. Galois cohomology and quadratic, Hermitian, and other forms6.7. Proof of Theorems 6.4 and 6.6: Classical groups; 6.8. Proof of Theorems 6.4 and 6.6: Exceptional groups; Chapter 7. Approximation in Algebraic Groups; 7.1. Strong and weak approximation in algebraic varieties; 7.2. The Kneser-Tits conjecture; 7.3. Weak approximation in algebraic groups; 7.4. The strong approximation theorem; 7.5. Generalization of the strong approximation theorem; Chapter 8. Class numbers and class groups of algebraic groups; 8.1. Class numbers of algebraic groups and number of classes in a genus
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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; 8.4. Estimating the class number for reductive groups; 8.5. The genus problem; Chapter 9. Normal subgroup structure of groups of rational points of algebraic groups; 9.1. Main conjectures and results; 9.2. Groups of type An; 9.3. The classical groups; 9.4. Groups split over a quadratic extension; 9.5. The congruence subgroup problem (a survey); Appendix A; Appendix B. Basic Notation; Bibliography; Index
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English
Additional Edition:
ISBN 0-12-558180-7
Language:
English
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