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Table of Contents
Intro
Contents
1 Introduction
Part I The 'Eigen' Construction
2 Eigenalgebras
2.1 A Reminder on the Ring of Endomorphisms of a Module
2.2 Construction of Eigenalgebras
2.3 First Properties
2.4 Behavior Under Base Change
2.5 Eigenalgebras Over a Field
2.5.1 Structure of the Scheme Spec T and of the T-Module M
2.5.2 System of Eigenvalues, Eigenspaces and Generalized Eigenspaces
2.5.3 Systems of Eigenvalues and Points of Spec T
2.6 The Fundamental Example of Hecke Operators Acting on a Space of Modular Forms
2.6.1 Complex Modular Forms and Diamond Operators
2.6.2 General Theory of Hecke Operators
2.6.3 Hecke Operators on Modular Forms
2.6.4 A Brief Reminder of Atkin-Lehner-Li's Theory (Without Proofs)
2.6.5 Hecke Eigenalgebra Constructed on Spaces of Complex Modular Forms
2.6.6 Galois Representations Attached to Eigenforms
2.6.7 Reminder on Pseudorepresentations
2.6.8 Pseudorepresentations and Eigenalgebra
2.7 Eigenalgebras Over Discrete Valuation Rings
2.7.1 Closed Points and Irreducible Components of Spec T
2.7.2 Reduction of Characters
2.7.3 The Case of a Complete Discrete Valuation Ring
2.7.4 A Simple Application: Deligne-Serre's Lemma
2.7.5 The Theory of Congruences
Congruences Between Two Submodules
Congruences in Presence of a Bilinear Product
Congruences and Eigenalgebras
2.8 Modular Forms with Integral Coefficients
2.8.1 The Specialization Morphism for Hecke Algebras of Modular Forms
2.8.2 An Application to Galois Representations
2.9 A Comparison Theorem
2.10 Notes and References
3 Eigenvarieties
3.1 Non-archimedean Fredholm's Theory
3.1.1 General Notions
3.1.2 Compact Operators
3.3.2 A Canonical Factorization of Everywhere Convergent Power Series
3.3.3 Adapted Pairs
3.4 Submodules of Bounded Slope
3.5 Links
3.6 The Eigenvariety Machine
3.6.1 Eigenvariety Data
3.6.2 Construction of the Eigenvariety
3.7 Properties of Eigenvarieties
3.8 A Comparison Theorem for Eigenvarieties
3.8.1 Classical Structures
3.8.2 A Reducedness Criterion
3.8.3 A Comparison Theorem
3.9 A Simple Generalization: The Eigenvariety Machine for Complexes
3.9.1 Data for a Cohomological Eigenvariety
3.9.2 Construction of the Cohomological Eigenvariety
Contents
1 Introduction
Part I The 'Eigen' Construction
2 Eigenalgebras
2.1 A Reminder on the Ring of Endomorphisms of a Module
2.2 Construction of Eigenalgebras
2.3 First Properties
2.4 Behavior Under Base Change
2.5 Eigenalgebras Over a Field
2.5.1 Structure of the Scheme Spec T and of the T-Module M
2.5.2 System of Eigenvalues, Eigenspaces and Generalized Eigenspaces
2.5.3 Systems of Eigenvalues and Points of Spec T
2.6 The Fundamental Example of Hecke Operators Acting on a Space of Modular Forms
2.6.1 Complex Modular Forms and Diamond Operators
2.6.2 General Theory of Hecke Operators
2.6.3 Hecke Operators on Modular Forms
2.6.4 A Brief Reminder of Atkin-Lehner-Li's Theory (Without Proofs)
2.6.5 Hecke Eigenalgebra Constructed on Spaces of Complex Modular Forms
2.6.6 Galois Representations Attached to Eigenforms
2.6.7 Reminder on Pseudorepresentations
2.6.8 Pseudorepresentations and Eigenalgebra
2.7 Eigenalgebras Over Discrete Valuation Rings
2.7.1 Closed Points and Irreducible Components of Spec T
2.7.2 Reduction of Characters
2.7.3 The Case of a Complete Discrete Valuation Ring
2.7.4 A Simple Application: Deligne-Serre's Lemma
2.7.5 The Theory of Congruences
Congruences Between Two Submodules
Congruences in Presence of a Bilinear Product
Congruences and Eigenalgebras
2.8 Modular Forms with Integral Coefficients
2.8.1 The Specialization Morphism for Hecke Algebras of Modular Forms
2.8.2 An Application to Galois Representations
2.9 A Comparison Theorem
2.10 Notes and References
3 Eigenvarieties
3.1 Non-archimedean Fredholm's Theory
3.1.1 General Notions
3.1.2 Compact Operators
3.3.2 A Canonical Factorization of Everywhere Convergent Power Series
3.3.3 Adapted Pairs
3.4 Submodules of Bounded Slope
3.5 Links
3.6 The Eigenvariety Machine
3.6.1 Eigenvariety Data
3.6.2 Construction of the Eigenvariety
3.7 Properties of Eigenvarieties
3.8 A Comparison Theorem for Eigenvarieties
3.8.1 Classical Structures
3.8.2 A Reducedness Criterion
3.8.3 A Comparison Theorem
3.9 A Simple Generalization: The Eigenvariety Machine for Complexes
3.9.1 Data for a Cohomological Eigenvariety
3.9.2 Construction of the Cohomological Eigenvariety