# Galois Cohomology of Elliptic Curves: Second Edition

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*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
lectures.

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.

A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka.

#### Readership

Mathematicians interested in algebraic number theory.