Tata Institute of Fundamental Research Publications
Volume: 2; 2010; 103 pp; Softcover
MSC: Primary 11;
Print ISBN: 978-81-8487-023-7
Product Code: TIFR/2.R
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Galois Cohomology of Elliptic Curves: Second EditionShare this page
J. Coates; R. Sujatha
A publication of the Tata Institute of Fundamental Research
The genesis of these notes was a series of four lectures
given by the first author at the Tata Institute of Fundamental
Research. It evolved into a joint project and contains many
improvements and extensions on the material covered in the original
Let \(F\) be a finite extension of \(q\), and \(E\) an elliptic curve defined over \(F\). The fundamental idea of the Iwasawa theory of elliptic curves, which grew out of Iwasawa's basic work on the ideal class groups of cyclotomic fields, is to study deep arithmetic questions about \(E\) over \(F\) via the study of coarser questions about the arithmetic of \(E\) over various infinite extensions of \(F\). At present, we only know how to formulate this Iwasawa theory when the infinite extension is a \(p\)-adic Lie extension for a fixed prime number \(p\). These notes will mainly discuss the simplest non-trivial example of the Iwasawa theory of \(E\) over the cyclotomic \(zp\)-extension of \(F\). However, the authors also make some comments about the Iwasawa theory of \(E\) over the field obtained by adjoining all \(p\)-power division points on \(E\) to \(F\). They discuss in detail a number of numerical examples, which illustrate the general theory beautifully. In addition, they outline some of the basic results in Galois cohomology which are used repeatedly in the study of the relevant Iwasawa modules.
The only changes made to the original notes: The authors take modest account of the considerable progress which has been made in non-commutative Iwasawa theory in the intervening years. They also include a short section on the deep theorems of Kato on the cyclotomic Iwasawa theory of elliptic curves.
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Table of Contents
Table of Contents
Galois Cohomology of Elliptic Curves: Second Edition
Mathematicians interested in algebraic number theory.