Overview

This book is based on the material presented in four lectures given by J. Coates at the Tata Institute of Fundamental Research. The original notes were modified and expanded in a joint project with R. Sujatha. The book discusses some aspects of the Iwasawa theory of elliptic curves over algebraic fields. Let E be an elliptic curve defined over an algebraic number field F. The fundamental idea of the Iwasawa theory is to study deep arithmetic questions about E/F, via the study of coarser questions about the arithmetic of E over various infinite extensions of F. A precise formulation of this theory exists only when the infinite extension is a p-adic Lie extension for some fixed prime number p. It provides the only general method known at present for studying exact arithmetic formulae such as the Birch and Swinnerton-Dyer conjecture for elliptic curves. These notes mainly discuss the simplest non-trivial example of this theory when the infinite extension in question is the cyclotomic extension of F. Only algebraic (as opposed to analytic) aspects of the theory are considered in the notes.

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