Galois Cohomology of Elliptic Curves

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. These notes will mainly discuss the simplest non-trivial example of the Iwasawa theory of E over the cyclotomic Zp-extension of F. However, we 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. We have also discussed in detail a number of numerical examples. The only changes made to the original notes have been to take modest account of the considerable progress which has been made in non-commutative Iwasawa theory in the intervening years. We have also included a short section on the deep theorems of Kato on the cyclotomic Iwasawa theory of elliptic curves.