| The book divides naturally into several parts according to the level of the material, the background required of the reader, and the style of presentation with respect to details of proofs. For example, the first part, to Chapter 6, is undergraduate in level, the second part requires a background in Galois theory and the third some complex analysis, while the last parts, from Chapter 12on, are mostly at graduate level. A general outline of much of the material can be found in Tate’s colloquium lectures reproduced as an article in Inventiones [1974].
The first part grew out of Tate’s 1961 Haverford Philips Lectures as an attempt to write something for publication closely related to the original Tate notes which were more or less taken from the tape recording of the lectures themselves. This includes parts of the Introduction and the first six chapters. The aim of this part is to prove, by elementary methods, the Mordell theorem on the finite generation of the rational points on elliptic curves defined over the rational numbers. In 1970 Tate returned to Haverford to give again, in revised form, the original lectures of 1961 and to extend the material so that it would be suitable for publication. This led to a broader plan for the book.
The second part, consisting of Chapters 7 and 8, recasts the arguments used in the proof of the Mordell theorem into the context of Galois cohomology and descent theory. The background material in Galois theory that is required is surveyed at the beginnng of Chapter 7 for the convenience of the reader.
The third part, consisting of Chapters 9, 10, and 11, is on analytic theory. A background in complex analysis is assumed and in Chapter 10 elementary results on p-adic fields, some of which were introduced in Chapter 5, are used in our discussion of Tate’s theory of p-adic theta functions. This section is based on Tate’s 1972 Haverford Philips Lectures. |