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2013 | Book

On p-adic Galois Representations Read chapter Read first chapter

Elliptic Curves, Hilbert Modular Forms and Galois Deformations

Authors: Laurent Berger, Gebhard Böckle, Lassina Dembélé, Mladen Dimitrov, Tim Dokchitser, John Voight

Publisher: Springer Basel

Book Series : Advanced Courses in Mathematics - CRM Barcelona

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year.

The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory.

The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed.

The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients.

The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed.

The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.

Table of Contents

Frontmatter

Galois Deformations

Frontmatter
On p-adic Galois Representations
Abstract
These are the notes for my part of the CRM course p-adic Galois Representations and Global Galois Deformations. My aim was to give a short introduction to the p-adic Hodge theory necessary for formulating the local conditions imposed on deformations of p-adic representations.
Laurent Berger
Deformations of Galois Representations
Abstract
These lecture notes give an introduction to deformations of Galois representations with an eye toward the application of this theory in the proof of the Serre conjecture [29, 30] by Khare–Wintenberger. There exist several other surveys such as [18, 23, 37, 40]. We nevertheless hope that with the above scope in mind and by the arrangement and detail of the material presented we can add something useful to the existing literature. Clearly, we claim no originality in the material presented and all errors are to be blamed on the present author.
Gebhard Böckle

Hilbert Modular Forms

Frontmatter
Arithmetic Aspects of Hilbert Modular Forms and Varieties
Abstract
Hilbert modular forms and varieties are the natural generalization of elliptic modular forms and curves, when the ground field of rational numbers is replaced by a totally real number field. The aim of these notes is to present the basics of their arithmetic theory and to describe some of the recent results in the area. A special emphasis will be put on the following two subjects: images of Galois representations associated to Hilbert modular forms and cohomology of Hilbert modular varieties with integral coefficients.
Mladen Dimitrov
Explicit Methods for Hilbert Modular Forms
Abstract
The study of modular forms remains a dominant theme in modern number theory, a consequence of their intrinsic appeal as well as their applications to a wide variety of mathematical problems. This subject has seen dramatic progress during the past half-century in an environment where both abstract theory and explicit computation have developed in parallel. Experiments will remain an essential tool in the years ahead, especially as we turn from classical contexts to less familiar terrain.
Lassina Dembélé, John Voight

Elliptic Curves

Frontmatter
Notes on the Parity Conjecture
Abstract
The main purpose of these notes is to prove, in a reasonably self-contained way, that finiteness of the Tate–Shafarevich group implies the parity conjecture for elliptic curves over number fields. Along the way, we review local and global root numbers of elliptic curves and their classification, and we end by discussing some peculiar consequences of the parity conjecture.
Tim Dokchitser
Metadata
Title
Elliptic Curves, Hilbert Modular Forms and Galois Deformations
Authors
Laurent Berger
Gebhard Böckle
Lassina Dembélé
Mladen Dimitrov
Tim Dokchitser
John Voight
Copyright Year
2013
Publisher
Springer Basel
Electronic ISBN
978-3-0348-0618-3
Print ISBN
978-3-0348-0617-6
DOI
https://doi.org/10.1007/978-3-0348-0618-3

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