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 II - p-Adic L-functions and Zeta Element
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
One of the most fruitful developments in the last couple of decades, has been the use of K-theory in the study of special values of L-functions of modular forms. This link is manifested in several ways. The first is through the work of Beilinson, who constructed zeta-elements living in K2 of a modular curve. He managed to prove that their image under the regulator map yielded the residue of the L-series at s = 0. In an orthogonal direction, Coates and Wiles found a non-archimedean approach to relating the L-function of a CM elliptic curve at s = 1, with a certain Euler system of elliptic units. They were then able to prove the first concrete theorems in the direction of the Birch and Swinnerton-Dyer conjecture.
It was Kato who realised the approach of Beilinson and that of Coates-Wiles could be combined, with spectacular results. He noticed that the zeta-elements considered by Beilinson satisfied norm-compatibility relations, reminiscent of those satisfied by elliptic units. Moreover, he devised a purely p-adic method whereby these elements could be used to study L-values. Underlying his discoveries was the ground-breaking work of Perrin-Riou in the early nineties, which extended the local Iwasawa theory used by Coates and Wiles to a very general framework (required for modular forms and non-CM elliptic curves). We shall review some of the highlights of their work, and in Chapters III, V and VI we will generalise it to modular symbols and Λ-adic families of modular forms.
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- Elliptic Curves and Big Galois Representations , pp. 31 - 49Publisher: Cambridge University PressPrint publication year: 2008