Cover

Half-title

Title

Copyright

Contents

Preface

1 Overview of Modular Forms

1.1 Hecke Characters

1.1.1 Hecke characters of finite order

1.1.2 Arithmetic Hecke characters

1.1.3 A theorem of Weil

1.2 Introduction to Modular Forms

1.2.1 Modular forms

1.2.2 Abelian modular forms and abelian deformation

2 Representations of a Group

2.1 Group Representations

2.1.1 Coefficient rings

2.1.2 Topological and profinite groups

2.1.3 Nakayama's lemma

2.1.4 Semi-simple algebras

2.1.5 Representations of finite groups

2.1.6 Induced representations

2.1.7 Representations with coefficients in Artinian rings

2.2 Pseudo-representations

2.2.1 Pseudo-representations of degree 2

2.2.2 Higher degree pseudo-representations

2.3 Deformation of Group Representations

2.3.1 Abelian deformation

2.3.2 Non-abelian deformation

2.3.3 Tangent spaces of local rings

2.3.4 Cohomological interpretation of tangent spaces

3 Representations of Galois Groups and Modular Forms

3.1 Modular Forms on Adele Groups of GL(2)

3.1.1 Elliptic modular forms

3.1.2 Structure theorems on GL(A)

3.1.3 Maximal compact subgroups

3.1.4 Open Compact subgroups of GL(A) and Dirichlet characters

3.1.5 Adelic and classical modular forms

3.1.6 Hecke algebras

3.1.7 Fourier expansion

3.1.8 Rationality of modular forms

3.1.9 p-adic Hecke algebras

3.2 Modular Galois Representations

3.2.1 Hecke eigenforms

3.2.2 Galois representation of Hecke eigenforms

3.2.3 Galois representation with values in the Hecke algebra

3.2.4 Universal deformation rings

3.2.5 Local deformation ring

3.2.6 Taylor-Wiles systems

3.2.7 Taylor-Wiles system of Hecke algebras

3.2.8 Tangential dimensions of deformation rings

4 Cohomology Theory of Galois Groups

4.1 Categories and Functors

4.1.1 Categories

4.1.2 Functors

4.1.3 Representability

4.1.4 Abelian categories

4.2 Extension of Modules

4.2.1 Extension groups

4.2.2 Extension functors

4.2.3 Cohomology groups of complexes

4.2.4 Higher extension groups

4.3 Group Cohomology Theory

4.3.1 Cohomology of finite groups

4.3.2 Tate cohomology groups

4.3.3 Continuous cohomology for profinite groups

4.3.4 Inflation and restriction sequences

4.3.5 Applications to representation theory

4.4 Duality in Galois Cohomology

4.4.1 Class formation and duality of cohomology groups

4.4.2 Global duality theorems

4.4.3 Tate-Shafarevich groups

4.4.4 Local Euler characteristic formula

4.4.5 Global Euler characteristic formula

5 Modular L-Values and Selmer Groups

5.1 Selmer Groups

5.1.1 Definition

5.1.2 Motivic interpretation

5.1.3 Character twists

5.2 Adjoint Selmer Groups

5.2.1 Adjoint Galois representations

5.2.2 Universal deformation rings

5.2.3 Kahler differentials

5.2.4 Adjoint Selmer groups and differentials

5.3 Arithmetic of Modular Adjoint L-Values

5.3.1 Analyticity of adjoint L-functions

5.3.2 Rationality of adjoint L-values

5.3.3 Congruences and adjoint L-values

5.3.4 Gorenstein and complete intersection rings

5.3.5 Universal p-ordinary Hecke algebras

5.3.6 p-adic adjoint L-functions

5.4 Control of Universal Deformation Rings

5.4.1 Deformation functors of group representations

5.4.2 Nearly ordinary deformations

5.4.3 Ordinary deformations

5.4.4 Deformations with fixed determinant

5.5 Base Change of Deformation Rings

5.5.1 Various deformation rings

5.6 Hilbert Modular Hecke Algebras

5.6.1 Various Hecke algebras for GL(2)

5.6.2 Automorphic base change

5.6.3 An Iwasawa theory for Hecke algebras

5.6.4 Adjoint Selmer groups over cyclotomic extensions

5.6.5 Proof of Theorem 5.44

Bibliography

Subject Index

List of Statements

List of Symbols