Class Field Theory and L Functions: Foundations and Main Results

Description

The book contains the main results of class field theory and Artin L functions, both for number fields and function fields, together with the necessary foundations concerning topological groups, cohomology, and simple algebras. While the first three chapters presuppose only basic algebraic and topological knowledge, the rest of the books assumes knowledge of the basic theory of algebraic numbers and algebraic functions, such as those contained in my previous book, An Invitation to Algebraic Numbers and Algebraic Functions (CRC Press, 2020). The main features of the book are: A detailed study of Pontrjagin’s dualtiy theorem. A thorough presentation of the cohomology of profinite groups. A introduction to simple algebras. An extensive discussion of the various ray class groups, both in the divisor-theoretic and the idelic language. The presentation of local and global class field theory in the algebra-theoretic concept of H. Hasse. The study of holomorphy domains and their relevance for class field theory. Simple classical proofs of the functional equation for L functions both for number fields and function fields. A self-contained presentation of the theorems of representation theory needed for Artin L functions. Application of Artin L functions for arithmetical results. Cover Half Title Title Page Copyright Page Dedication Contents Preface Author Notation 1. Topological groups and infinite Galois theory 1.1. Basics of topological groups 1.2. Topological rings, topological fields and real vector spaces 1.3. Inductive and projective limits 1.4. Cauchy sequences and sequentially completeness 1.5. Profinite groups 1.6. Duality of abelian locally compact topological groups 1 1.7. Duality of abelian locally compact topological groups 2 1.8. Infinite Galois theory 2. Cohomology of groups 2.1. Discrete G-modules 2.2. Cohomology groups 2.3. Direct sums, products and limits of cohomology groups 2.4. The long cohomology sequence 2.5. Restriction, inflation, corestriction and transfer 2.6. Galois cohomology 3. Simple algebras 3.1. Preliminaries on modules and algebras 3.2. Tensor products 3.3. Structure of central simple algebras 3.4. Splitting fields and the Brauer group 3.5. Factor systems and crossed products 3.6. Cyclic algebras 4. Local class field theory 4.1. The Brauer group of a local field 4.2. The local reciprocity law 4.3. Auxiliary results on complete discrete valued fields 4.4. The reciprocity laws of Dwork and Neukirch 4.5. Basics of formal groups 4.6. Lubin-Tate formal groups 4.7. Lubin-Tate extensions 4.8. The reciprocity law of Lubin-Tate 4.9. Abelian extensions of Qp 4.10. Hilbert symbols 5. Global fields: Adeles, ideles and holomorphy domains 5.1. Global fields 5.2. Local direct products 5.3. Adeles and ideles of global fields 5.4. Ideles in field extensions 5.5. S-class groups and holomorphy domains 5.6. Ray class groups 1: Ideal- and divisor-theoretic approach 5.7. Ray class groups 2: Idelic approach 5.8. Ray class characters 6. Global class field theory 6.1. Cohomology of the idele groups 6.2. The global norm residue symbol 6.3. p-extensions in characteristic p 6.4. The global reciprocity law 6.5. Global class fields 6.6. Special class fields and decomposition laws 6.7. Power residues 7. Functional equations and Artin L functions 7.1. Gauss sums and L functions of number fields 7.2. Further analytic tools 7.3. Proof of the functional equation for L functions of number fields 7.4. The functional equation for L functions of function fields 7.5. Representation theory 1: Basic concepts 7.6. Representation theory 2: Class functions and induced characters 7.7. Artin conductors 7.8. Artin L functions 7.9. Prime decomposition and density results Bibliography Subject Index List of Symbols