Book contents
- Frontmatter
- Contents
- Introduction
- List of Notations
- Chapter I Background
- Chapter II p-Adic L-functions and Zeta Element
- Chapter III Cyclotomic Deformations of Modular Symbols
- Chapter IV A User's Guide to Hida Theory
- Chapter V Crystalline Weight Deformations
- Chapter VI Super Zeta-Elements
- Chapter VII Vertical and Half-Twisted Arithmetic
- Chapter VIII Diamond-Euler Characteristics: the Local Case
- Chapter IX Diamond-Euler Characteristics: the Global Case
- Chapter X Two-Variable Iwasawa Theory of Elliptic Curves
- Appendices
- Bibliography
- Index
Chapter I - Background
Published online by Cambridge University Press: 04 May 2010
- Frontmatter
- Contents
- Introduction
- List of Notations
- Chapter I Background
- Chapter II p-Adic L-functions and Zeta Element
- Chapter III Cyclotomic Deformations of Modular Symbols
- Chapter IV A User's Guide to Hida Theory
- Chapter V Crystalline Weight Deformations
- Chapter VI Super Zeta-Elements
- Chapter VII Vertical and Half-Twisted Arithmetic
- Chapter VIII Diamond-Euler Characteristics: the Local Case
- Chapter IX Diamond-Euler Characteristics: the Global Case
- Chapter X Two-Variable Iwasawa Theory of Elliptic Curves
- Appendices
- Bibliography
- Index
Summary
Although the study of elliptic curves can be traced back to the ancient Greeks, even today there remain surprisingly many unanswered questions in the subject. The most famous are surely the conjectures of Birch and Swinnerton-Dyer made almost half a century ago. Their predictions have motivated a significant portion of current number theory research, however they seem as elusive as they are elegant. Indeed the Clay Institute included them as one of the seven millenium problems in mathematics, and there is a million dollar financial reward for their resolution.
This book is devoted to studying the Birch, Swinnerton-Dyer (BSD) conjecture over the universal deformation ring of an elliptic curve. A natural place to begin is with a short exposition of the basic theory of elliptic curves, certainly enough to carry us through the remaining chapters. Our main motivation here will be to state the BSD conjecture in the most succinct form possible (for later reference). This seems a necessary approach, since the arithmetic portion of this work entails searching for their magic formula amongst all the detritus of Galois cohomology, i.e. we had better recognise the formula when it finally does appear!
After defining precisely what is meant by an elliptic curve E, we introduce its Tate module Tap(E) which is an example of a two-dimensional Galois representation. The image of the Galois group inside the automorphisms of Tap(E) was computed by Serre in the late 1960's.
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- Elliptic Curves and Big Galois Representations , pp. 7 - 30Publisher: Cambridge University PressPrint publication year: 2008