This graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory.
Assuming a first graduate course in algebra and number theory, the book begins with an introduction to group and Galois cohomology. Local fields and local class field theory, including Lubin-Tate formal group laws, are covered next, followed by global class field theory and the description of abelian extensions of global fields. The final part of the book gives an accessible yet complete exposition of the Poitou-Tate duality theorems. Two appendices cover the necessary background in homological algebra and the analytic theory of Dirichlet L-series, including the Čebotarev density theorem.
Based on several advanced courses given by the author, this textbook has been written for graduate students. Including complete proofs and numerous exercises, the book will also appeal to more experienced mathematicians, either as a text to learn the subject or as a reference.
Author(s): David Harari
Series: Universitext
Publisher: Springer
Year: 2020
Language: English
Pages: 352
Preface
Notation and Conventions
Contents
Part I Group Cohomology and Galois Cohomology: Generalities
1 Cohomology of Finite Groups: Basic Properties
1.1 The Notion of G-Module
1.2 The Category of G-Modules
1.3 The Cohomology Groups HGA
1.4 Computation of Cohomology Using the Cochains
1.5 Change of Group: Restriction, Corestriction, the Hochschild–Serre …
1.6 Corestriction; Applications
1.7 Exercises
2 Groups Modified à la Tate, Cohomology of Cyclic Groups
2.1 Tate Modified Cohomology Groups.
2.2 Change of Group. Transfer
2.3 Cohomology of a Cyclic Group
2.4 Herbrand Quotient
2.5 Cup-Products
2.6 Cup-Products for the Modified Cohomology
2.7 Exercises
3 p-Groups, the Tate–Nakayama Theorem
3.1 Cohomologically Trivial Modules
3.2 The Tate–Nakayama Theorem
3.3 Exercises
4 Cohomology of Profinite Groups
4.1 Basic Facts About Profinite Groups
4.2 G-Modules
4.3 Cohomology of a Discrete G-Module
4.4 Exercises
5 Cohomological Dimension
5.1 Definitions, First Examples
5.2 Properties of the Cohomological Dimension
5.3 Exercises
6 First Notions of Galois Cohomology
6.1 Generalities
6.2 Hilbert's Theorem 90 and Applications
6.3 Brauer Group of a Field
6.4 Cohomological Dimension of a Field
6.5 C1
6.6 Exercises
Part II Local Fields
7 Basic Facts About Local Fields
7.1 Discrete Valuation Rings
7.2 Complete Field for a Discrete Valuation
7.3 Extensions of Complete Fields
7.4 Galois Theory of a Complete Field for a Discrete Valuation
7.5 Structure Theorem; Filtration of the Group of Units
7.6 Exercises
8 Brauer Group of a Local Field
8.1 Local Class Field Axiom
8.2 Computation of the Brauer Group
8.3 Cohomological Dimension; Finiteness Theorem
8.4 Exercises
9 Local Class Field Theory: The Reciprocity Map
9.1 Definition and Main Properties
9.2 The Existence Theorem: Preliminary Lemmas and the Case of a p-adic Field
9.3 Exercises
10 The Tate Local Duality Theorem
10.1 The Dualising Module
10.2 The Local Duality Theorem
10.3 The Euler–Poincaré Characteristic
10.4 Unramified Cohomology
10.5 From the Duality Theorem to the Existence Theorem
10.6 Exercises
11 Local Class Field Theory: Lubin–Tate Theory
11.1 Formal Groups
11.2 Change of the Uniformiser
11.3 Fields Associated to Torsion Points
11.4 Computation of the Reciprocity Map
11.5 The Existence Theorem (the General Case)
11.6 Exercises
Part III Global Fields
12 Basic Facts About Global Fields
12.1 Definitions, First Properties
12.2 Galois Extensions of a Global Field
12.3 Idèles, Strong Approximation Theorem
12.4 Some Complements in the Case of a Function Field
12.5 Exercises
13 Cohomology of the Idèles: The Class Field Axiom
13.1 Cohomology of the Idèle Group
13.2 The Second Inequality
13.3 Kummer Extensions
13.4 First Inequality and the Axiom of Class Field Theory
13.5 Proof of the Class Field Axiom for a Function Field
13.6 Exercises
14 Reciprocity Law and the Brauer–Hasse–Noether Theorem
14.1 Existence of a Neutralising Cyclic Extension
14.2 The Global Invariant and the Norm Residue Symbol
14.3 Exercises
15 The Abelianised Absolute Galois Group of a Global Field
15.1 Reciprocity Map and the Idèle Class Group
15.2 The Global Existence Theorem
15.3 The Case of a Function Field
15.4 Ray Class Fields; Hilbert Class Field
15.5 Galois Groups with Restricted Ramification
15.6 Exercises
Part IV Duality Theorems
16 Class Formations
16.1 Notion of Class Formation
16.2 The Spectral Sequence of the Ext
16.3 The Duality Theorem for a Class Formation
16.4 P-Class Formation
16.5 From the Existence Theorem to the Duality Theorem for a p-adic Field
16.6 Complements
16.7 Exercises
17 Poitou–Tate Duality
17.1 The P-Class Formation Associated to a Galois Group with Restricted Ramification
17.2 The Groups PSM
17.3 Statement of the Poitou–Tate Theorems
17.4 Proof of Poitou–Tate Theorems (I): Computation of the Ext Groups
17.5 Proof of the Poitou–Tate Theorems (II): Computation of the Ext with Values in IS and End of the Proof
17.6 Exercises
18 Some Applications
18.1 Triviality of Some of the Sha
18.2 The Strict Cohomological Dimension of a Number Field
18.3 Exercises
Appendix A Some Results from Homological Algebra
A.1 Generalities on Categories
A.2 Functors
A.3 Abelian Categories
A.4 Categories of Modules
A.5 Derived Functors
A.6 Ext and Tor
A.7 Spectral Sequences
Appendix B A Survey of Analytic Methods
B.1 Dirichlet Series
B.2 Dedekind ζ Function; Dirichlet L-Functions
B.3 Complements on the Dirichlet Density
B.4 The First Inequality
B.5 Class Field Theory in Terms of Ideals
B.6 Proof of the Čebotarev Theorem
Index
