The Eigenbook

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

by Bellaïche, Joël

Type Book
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This book discusses the p-adic modular forms, the eigencurve that parameterize them, and the p-adic L-functions one can associate to them. These theories and their generalizations to automorphic forms for group of higher ranks are of fundamental importance in number theory. For graduate students and newcomers to this field, the book provides a solid introduction to this highly active area of research. For experts, it will offer the convenience of collecting into one place foundational definitio…

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Title Statements

  • Full title :The Eigenbook : Eigenvarieties, families of Galois representations, p-adic l-functions / Joël Bellaïche.

Summary

This book discusses the p-adic modular forms, the eigencurve that parameterize them, and the p-adic L-functions one can associate to them. These theories and their generalizations to automorphic forms for group of higher ranks are of fundamental importance in number theory. For graduate students and newcomers to this field, the book provides a solid introduction to this highly active area of research. For experts, it will offer the convenience of collecting into one place foundational definitions and theorems with complete and self-contained proofs. Written in an engaging and educational style, the book also includes exercises and provides their solution.

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Edition Statements

  • Publication Date:2021.

Publication Statements

  • Publisher(s):Birkhäuser
  • Place of Publication:Cham, Switzerland
  • Language:eng
  • Publication Date:2021.

The Physical Object

  • Description:1 online resource

Edition Identifiers

  • ISBN:9783030772635, 3030772632
  • DOI:10.1007/978-3-030-77263-5
  • oclc:on1263864913
  • LC Classification:QA214
  • Dewey Decimal:512/.32

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 Appli…

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Subjects

Eigenfunctions Fonctions propres

Writers

Bellaïche, Joël

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