# Algebraic Groups and Number Theory

**By**

- Vladimir Platonov
- Andrei Rapinchuk
- Rachel Rowen

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.

**Audience**

Research mathematicians; graduate students; and libraries.

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Published: October 1993

Imprint: Academic Press

ISBN: 978-0-12-558180-6

## Contents

- (Chapter Heading): Algebraic Number Theory. Algebraic Groups. Algebraic Groups over Locally Compact Fields. Arithmetic Groups and Reduction Theory. Adeles. Galois Cohomology. Approximation in Algebraic Groups. Class Numbers andClass Groups of Algebraic Groups. Normal Structure of Groups of Rational Points of Algebraic Groups. Appendix A. Appendix B: Basic Notation.
**Algebraic Number Theory:**Algebraic Number Fields, Valuations, and Completions. Adeles and Ideles; Strong and Weak Approximation; The Local-Global Principle. Cohomology. Simple Algebras over Local Fields. Simple Algebras over Algebraic Number Fields.**Algebraic Groups:**Structural Properties of Algebraic Groups. Classification of K-Forms Using Galois Cohomology. The Classical Groups. Some Results from Algebraic Geometry.**Algebraic Groups over Locally Compact Fields:**Topology and Analytic Structure. The Archimedean Case. The Non-Archimedean Case. Elements of Bruhat-Tits Theory. Results Needed from Measure Theory.**Arithmetic Groups and Reduction Theory:**Arithmetic Groups. Overview of Reduction Theory: Reduction in GLn(R).Reduction in Arbitrary Groups. Group-Theoretic Properties of Arithmetic Groups. Compactness of GR/GZ. The Finiteness of the Volume of GR/GZ. Concluding Remarks on Reduction Theory. Finite Arithmetic Groups.**Adeles:**Basic Definitions. Reduction Theory for GA Relative to GK. Criteria for the Compactness and the Finiteness of Volume of GA/GK. Reduction Theory for S-Arithmetic Subgroups.**Galois Cohomology:**Statement of the Main Results. Cohomology of Algebraic Groups over Finite Fields. Galois Cohomology of Algebraic Tori. Finiteness Theorems for Galios Cohomology. Cohomology of Semisimple Algebraic Groups over Local Fields and Number Fields. Galois Cohomology and Quadratic, Hermitian, and Other Forms. Proof of Theorems 6.4 and 6.6: Classical Groups. Proof of Theorems 6.4 and 6.6: Exceptional Groups.**Approximation in Algebraic Groups:**Strong and Weak Approximation in Algebraic Varieties. The Kneser-Tits Conjecture. Weak Approximation in Algebraic Groups. The Strong Approximation Theorem. Generalization of the Strong Approximation Theorem.**Class Numbers and Class Groups of Algebraic Groups:**Class Numbers of Algebraic Groups and Number of Classes in a Genus. Class Numbers and Class Groups of Semisimple Groups of Noncompact Type; The Realization Theorem. Class Numbers of Algebraic Groups of Compact Type. Estimating the Class Number for Reductive Groups. The Genus Problem.**Normal Subgroup Structure of Groups of Rational Points of Algebraic Groups:**Main Conjecture and Results. Groups of Type An. The Classical Groups. Groups Split over a Quadratic Extension. The Congruence Subgroup Problem (A Survey). Appendices: Basic Notation. Bibliography. Index.