The Eigenbook: Eigenvarieties, families of Galois representations, p-adic L-functions

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
3.1.3 Orthonormalizable and Potentially Orthonormalizable Banach Modules
3.1.4 Serre's Sufficient Condition for Being Orthonormalizable
3.1.5 Fredholm's Determinant of a Compact Endomorphism
3.1.6 Property (Pr)
3.1.7 Extension of Scalars
3.2 Everywhere Convergent Formal Series and Riesz's Theory
3.2.1 The ν-Valuation and ν-Dominant Polynomials
3.2.2 Euclidean Division
3.2.3 Good Zeros
3.2.4 A Piece of Resultant Theory
3.2.5 Riesz's Theory
3.3 Adapted Pairs
3.3.1 Strongly ν-Dominant Polynomials
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
3.9.3 Properties of the Cohomological Eigenvariety
3.9.4 Classical Structures and an Application to Reducedness
3.10 Notes and References
Part II Modular Symbols and L-Functions
4 Abstract Modular Symbols
4.1 The Notion of Modular Symbols
4.2 Action of the Hecke Operators on Modular Symbols
4.3 Reminder on Cohomology with Local Coefficients
4.3.1 Local Systems
4.3.2 Local Systems and Representations of the Fundamental Group
4.3.3 Singular Simplices
4.3.4 Homology with Local Coefficients
4.3.5 Cohomology with Local Coefficients
4.3.6 Relative Homology and Relative Cohomology
4.3.7 Formal Duality Between Homology and Cohomology
4.3.8 Cohomology with Compact Support and Interior Cohomology
4.3.9 Cup-Products and Cap-Products
4.3.10 Poincaré Duality
4.3.11 Singular Cohomology and Sheaf Cohomology
4.4 Modular Symbols and Cohomology
4.4.1 Right Action on the Cohomology
4.4.2 Modular Symbols and Relative Cohomology
4.4.3 Modular Symbols and Cohomology with Compact Support
4.4.4 Pairings on Modular Symbols
4.5 Notes and References
5 Classical Modular Symbols, Modular Forms, L-functions
5.1 On a Certain Monoid and Some of Its Modules
5.1.1 The Monoid S
5.1.2 The S-modules Pk and Vk
5.2 Classical Modular Symbols
5.2.1 Definition
5.2.2 The Standard Pairing on Classical Modular Symbols
5.2.3 Adjoint of Hecke Operators for the Standard Pairing
5.3 Classical Modular Symbols and Modular Forms
5.3.1 Modular Forms and Real Classical Modular Symbols
5.3.2 Modular Forms and Complex Classical Modular Symbols
5.3.3 The Involution ι, and How to Get Rid of the Complex Conjugation
5.3.4 The Endomorphism WN and the Corrected Scalar Product
5.3.5 Boundary Modular Symbols and Eisenstein Series
5.3.6 Summary
5.4 Applications of Classical Modular Symbols to L-functions and Congruences
5.4.1 Reminder About L-functions
5.4.2 Modular Symbols and L-functions
5.4.3 Scalar Product and Congruences
5.5 Notes and References
6 Rigid Analytic Modular Symbols and p-Adic L-functions
6.1 Rigid Analytic Functions and Distributions
6.1.1 Some Modules of Sequences and Their Dual
6.1.2 Modules of Functions over Zp
6.1.3 Modules of Convergent Distributions
6.2 Overconvergent Functions and Distributions
6.2.1 Semi-normic Modules and Fréchet Modules
6.2.2 Modules of Overconvergent Functions and Distributions
6.2.3 Integration of Functions Against Distributions
6.2.4 Order of Growth of a Distribution
6.3 The Weight Space
6.3.1 Definition and Description of the Weight Space
6.3.2 Local Analyticity of Characters
6.3.3 Some Remarkable Elements in the Weight Space
6.3.4 The Functions logp[k] on the Weight Space
6.3.5 The Iwasawa Algebra and the Weight Space
6.4 The Monoid S0(p) and Its Actions on Overconvergent Distributions
6.4.1 The Monoid S0(p)
6.4.2 Actions of S0(p) on Functions and Distributions
6.4.3 The Module of Locally Constant Polynomials and Its Dual
6.4.4 The Fundamental Exact Sequence for Overconvergent Functions
6.4.5 The Fundamental Exact Sequence for Overconvergent Distributions
6.5 Rigid Analytic and Overconvergent Modular Symbols
6.5.1 Definitions and Compactness of Up
6.5.2 Space of Overconvergent Modular Symbols of Finite Slope
6.5.3 Computation of an H0
6.5.4 The Fundamental Exact Sequence for Modular Symbols
6.5.5 Stevens's Control Theorem
6.6 The Mellin Transform
6.6.1 The Real Mellin Transform
6.6.2 The p-Adic Mellin Transform
6.6.3 Properties of the p-Adic Mellin Transform
6.6.4 The Mellin Transform over a Banach Algebra
6.7 Applications to the p-Adic L-functions of Non-critical Slope Modular Forms
6.7.1 Refinements
6.7.2 Construction of the p-Adic L-functions
6.7.3 Computation of the p-Adic L-functions at Special Characters
6.8 Notes and References
Part III The Eigencurve and its p-Adic L-Functions
7 The Eigencurve of Modular Symbols
7.1 Construction of the Eigencurve Using Rigid Analytic Modular Symbols
7.1.1 Overconvergent Modular Symbols Over an Admissible Open Affinoid of the Weight Space
7.1.2 The Restriction Theorem
7.1.3 The Specialization Theorem
7.1.4 Construction
7.2 Comparison with the Coleman-Mazur Eigencurve
7.2.1 The Coleman-Mazur Full Eigencurve C
7.2.2 The Cuspidal Eigencurve
7.2.3 Applications of Chenevier's Comparison Theorem
7.3 Points of the Eigencurve
7.3.1 Interpretations of the Points as Systems of Eigenvalues of Overconvergent Modular Symbols
7.3.2 Very Classical Points
7.3.3 Classical Points
7.3.4 Hida Classical Points
7.4 The Family of Galois Representations Carried by the Eigencurve
7.4.1 Construction of the Family of Galois Representations
7.4.2 Local Properties at l ≠p of the Family of Galois Representations
7.4.3 Local Properties at p of the Family of Galois Representations
7.5 The Ordinary Locus
7.6 Local Geometry of the Eigencurve
7.6.1 Clean Neighborhoods
7.6.2 Étaleness of the Eigencurve at Non-critical Slope Classical Points
7.6.3 Geometry of the Eigencurve at Critical Slope Very Classical Points
7.6.4 Critical Slope Eigenforms and Points on C
7.6.5 Complements on the Geometry of the Eigencurve at Classical Points
7.7 Global Properties of the Eigencurve
7.7.1 Integrality of Fredholm Determinants and Integral Models of the Eigencurves
7.7.2 Valuative Criterion of Properness
7.7.3 Open Questions
7.8 Notes and References
8 p-Adic L-Functions on the Eigencurve
8.1 Good Points and p-Adic L-Functions
8.1.1 Good Points on the Eigencurve
8.1.2 The p-Adic L-Function of a Good Point
8.1.3 Companion Points and p-Adic L-Functions
8.2 p-Adic L-Functions of an Overconvergent Eigenform
8.2.1 Definition
8.2.2 Classical Cuspidal Eigenforms of Non-critical Slope
8.2.3 Ordinary Eisenstein Eigenforms
8.2.4 Classical Eigenforms of Critical Slope
8.2.5 Classical Eigenforms of Weight 1
8.3 The 2-Variable p-Adic L-Function
8.4 Notes and References
9 The Adjoint p-Adic L-Function and the Ramification Locus of the Eigencurve
9.1 The L-Ideal of a Scalar Product
9.1.1 The Noether Different of T/R
9.1.2 Duality
9.1.3 The L-Ideal of a Scalar Product
9.2 Kim's Scalar Product
9.2.1 A Bilinear Product on the Space of Overconvergent Modular Symbols of Weight k
9.2.2 Interpolation of Those Scalar Products
9.3 The Cuspidal Eigencurve, the Interior Cohomological Eigencurves, and Their Good Points
9.4 Construction of the Adjoint p-Adic L-Function on the Cuspidal Eigencurve
9.5 Relation Between the Adjoint p-Adic L-Function and the Classical Adjoint L-Function
9.5.1 Scalar Product and Refinements
The Action of WNp on the Old Subspace
The Adjoint of Up on the Old Subspace
The Matrix of Peterson's Product on the Old Subspace
The Corrected Scalar Product on the Old Subspace
9.5.2 Adjoint L-Function and Peterson's Product
Definition of the Classical Adjoint L-Function
Hida's Formula
9.5.3 p-Adic and Classical Adjoint L-Function
10 Solutions and Hints to Exercises
Bibliography