Volume 139. Algebraic Groups and Number Theory

(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. P