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9780521728669

Elliptic Curves and Big Galois Representations

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  • ISBN13:

    9780521728669

  • ISBN10:

    0521728665

  • Edition: 1st
  • Format: Paperback
  • Copyright: 2008-09-01
  • Publisher: Cambridge University Press

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Summary

The arithmetic properties of modular forms and elliptic curves lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula. Three main steps are outlined: the first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. Finiteness results for big Selmer groups are then established. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture. As the first book on the subject, the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases.

Table of Contents

Introduction
List of notations
Background
p-adic L-functions and Zeta-elements
Cyclotomic deformations of modular symbols
A user's guide to Hida theory
Crystalline weight deformations
Super Zeta-elements
Vertical and half-twisted arithmetic
Diamond-Euler characteristics: the local case
Diamond-Euler characteristics: the global case
Two-variable Iwasawa theory of elliptic curves
The primitivity of Zeta elements
Specialising the universal path vector
The weight-variable control theorem
Bibliography
Table of Contents provided by Publisher. All Rights Reserved.

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