Advanced Courses in Mathematics - CRM Barcelona

Elliptic Curves, Hilbert Modular Forms and Galois Deformations

Authors: Berger, L., Böckle, G., Dembélé, L., Dimitrov, M., Dokchitser, T., Voight, J.

  • The book contains the first published notes on the recent developments and major changes in Galois deformation theory during the last decade (deformations of pseudo-representations, framed deformations, groupoids, etc.)
  • A survey on the parity conjecture is presented
  • Computational aspects of Hilbert modular forms are presented by the people responsible for the most powerful and widely spread algorithms available
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eBook 23,79 €
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  • ISBN 978-3-0348-0618-3
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Softcover 31,19 €
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  • ISBN 978-3-0348-0617-6
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About this Textbook

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 (5 chapters)

  • On p-adic Galois Representations

    Berger, Laurent

    Pages 3-19

    Preview Buy Chapter 30,19 €
  • Deformations of Galois Representations

    Böckle, Gebhard

    Pages 21-115

    Preview Buy Chapter 30,19 €
  • Arithmetic Aspects of Hilbert Modular Forms and Varieties

    Dimitrov, Mladen

    Pages 119-134

    Preview Buy Chapter 30,19 €
  • Explicit Methods for Hilbert Modular Forms

    Dembélé, Lassina (et al.)

    Pages 135-198

    Preview Buy Chapter 30,19 €
  • Notes on the Parity Conjecture

    Dokchitser, Tim

    Pages 201-249

    Preview Buy Chapter 30,19 €

Buy this book

eBook 23,79 €
price for Spain (gross)
  • ISBN 978-3-0348-0618-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover 31,19 €
price for Spain (gross)
  • ISBN 978-3-0348-0617-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Elliptic Curves, Hilbert Modular Forms and Galois Deformations
Authors
Series Title
Advanced Courses in Mathematics - CRM Barcelona
Copyright
2013
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel
eBook ISBN
978-3-0348-0618-3
DOI
10.1007/978-3-0348-0618-3
Softcover ISBN
978-3-0348-0617-6
Series ISSN
2297-0304
Edition Number
1
Number of Pages
XII, 249
Number of Illustrations and Tables
9 b/w illustrations, 2 illustrations in colour
Topics