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El. knyga: Class Field Theory and L Functions: Foundations and Main Results

(University of Graz, Austria)
  • Formatas: 584 pages
  • Išleidimo metai: 13-Mar-2022
  • Leidėjas: CRC Press
  • Kalba: eng
  • ISBN-13: 9780429014734
Kitos knygos pagal šią temą:
Class Field Theory and L Functions: Foundations and Main ResultsDidesnis paveikslėlis
  • Formatas: 584 pages
  • Išleidimo metai: 13-Mar-2022
  • Leidėjas: CRC Press
  • Kalba: eng
  • ISBN-13: 9780429014734
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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 Pontrjagins 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.
Preface ix
Author xi
Notation xiii
1 Topological groups and infinite Galois theory
1(86)
1.1 Basics of topological groups
1(13)
1.2 Topological rings, topological fields and real vector spaces
14(4)
1.3 Inductive and projective limits
18(15)
1.4 Cauchy sequences and sequentially completeness
33(7)
1.5 Profinite groups
40(10)
1.6 Duality of abelian locally compact topological groups 1
50(11)
1.7 Duality of abelian locally compact topological groups 2
61(13)
1.8 Infinite Galois theory
74(13)
2 Cohomology of groups
87(68)
2.1 Discrete G-modules
87(14)
2.2 Cohomology groups
101(8)
2.3 Direct sums, products and limits of cohomology groups
109(7)
2.4 The long cohomology sequence
116(12)
2.5 Restriction, inflation, corestriction and transfer
128(17)
2.6 Galois cohomology
145(10)
3 Simple algebras
155(46)
3.1 Preliminaries on modules and algebras
155(6)
3.2 Tensor products
161(6)
3.3 Structure of central simple algebras
167(7)
3.4 Splitting fields and the Brauer group
174(5)
3.5 Factor systems and crossed products
179(14)
3.6 Cyclic algebras
193(8)
4 Local class field theory
201(90)
4.1 The Brauer group of a local field
202(7)
4.2 The local reciprocity law
209(12)
4.3 Auxiliary results on complete discrete valued fields
221(11)
4.4 The reciprocity laws of Dwork and Neukirch
232(8)
4.5 Basics of formal groups
240(7)
4.6 Lubin-Tate formal groups
247(10)
4.7 Lubin-Tate extensions
257(9)
4.8 The reciprocity law of Lubin-Tate
266(13)
4.9 Abelian extensions of Qp
279(2)
4.10 Hilbert symbols
281(10)
5 Global fields: Adeles, ideles and holomorphy domains
291(78)
5.1 Global fields
291(6)
5.2 Local direct products
297(4)
5.3 Adeles and ideles of global fields
301(17)
5.4 Ideles in field extensions
318(14)
5.5 S-class groups and holomorphy domains
332(9)
5.6 Ray class groups 1: Ideal- and divisor-theoretic approach
341(9)
5.7 Ray class groups 2: Idelic approach
350(12)
5.8 Ray class characters
362(7)
6 Global class field theory
369(72)
6.1 Cohomology of the idele groups
369(13)
6.2 The global norm residue symbol
382(9)
6.3 p-extensions in characteristic p
391(7)
6.4 The global reciprocity law
398(18)
6.5 Global class fields
416(11)
6.6 Special class fields and decomposition laws
427(8)
6.7 Power residues
435(6)
7 Functional equations and Artin L functions
441(114)
7.1 Gauss sums and L functions of number fields
441(17)
7.2 Further analytic tools
458(10)
7.3 Proof of the functional equation for L functions of number fields
468(12)
7.4 The functional equation for L functions of function fields
480(15)
7.5 Representation theory 1: Basic concepts
495(9)
7.6 Representation theory 2: Class functions and induced characters
504(7)
7.7 Artin conductors
511(10)
7.8 Artin L functions
521(23)
7.9 Prime decomposition and density results
544(11)
Bibliography 555(4)
Subject Index 559(4)
List of Symbols 563
Franz Halter-Koch is professor emeritus at the University of Graz, Graz, Austria. He is the author of Ideal Systems (Marcel Dekker,1998), Quadratic Irrationals (CRC, 2013), co-author of Non-Unique Factorizations (CRC 2006), and An Invitation to Algebraic Numbers and Algebraic Functions (CRC Press, 2020).
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