Tata Institute of Fundamental Research 2010; 103 pp; softcover ISBN10: 818487023X ISBN13: 9788184870237 List Price: US$30 Member Price: US$24 Order Code: TIFR/2.R
 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 nontrivial 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 noncommutative 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. Table of Contents  Notation
 Basic results from Galois cohomology
 The Iwasawa theory of the Selmer group
 The Euler characteristic formula
 Numerical examples over the cyclotomic \(Zp\)extension of \(Q\)
 Numerical examples over \(Q(\mu p)\)
 Appendix
 Bibliography
