| 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. |
