Elliptic Curves, Hilbert Modular Forms and Galois Deformations

  • Laurent Berger
  • Gebhard Böckle
  • Lassina Dembélé
  • Mladen Dimitrov
  • Tim Dokchitser
  • John Voight

Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Galois Deformations

    1. Front Matter
      Pages 1-1
    2. Laurent Berger
      Pages 3-19
    3. Gebhard Böckle
      Pages 21-115
  3. Hilbert Modular Forms

    1. Front Matter
      Pages 117-117
    2. Lassina Dembélé, John Voight
      Pages 135-198
  4. Elliptic Curves

    1. Front Matter
      Pages 199-199
    2. Tim Dokchitser
      Pages 201-249

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.


Galois representations Hilbert modular forms elliptic curves

Authors and affiliations

  • Laurent Berger
    • 1
  • Gebhard Böckle
    • 2
  • Lassina Dembélé
    • 3
  • Mladen Dimitrov
    • 4
  • Tim Dokchitser
    • 5
  • John Voight
    • 6
  1. 1.UMPA-ENS LyonLyon Cedex 7France
  2. 2.Interdisciplinary Center for ScientificUniversität HeidelbergHeidelbergGermany
  3. 3.Warwick Mathematics InstituteUniversity of WarwickCoventryUnited Kingdom
  4. 4.UFR MathématiquesUniversité Lille 1 Cité ScientifiqueVilleneuve d'Ascq CedexFrance
  5. 5.Department of MathematicsUniversity of BristolBristolUnited Kingdom
  6. 6.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA

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