An Introduction to Arithmetic Geometry

Overview

For n >= 3, there are no nontrivial solutions in the integers to the equation xn + yn = zn. So how hard can it be to prove something like this? Well, as we are well aware, the answer is `very hard'. The basic tools used are found in arithmetic geometry.

Loosely speaking, arithmetic geometry is the study of the solutions of polynomial equations over finite fields, the integers, or algebraic number fields. However, in general to obtain a good idea of what is currently going on in the field, one needs a solid grounding in commutative algebra, algebraic geometry, and algebraic number theory. This is often a very daunting task. The textbook chosen for this course does a wonderful job of trying to bring novices into the fold without a two-year initiation period.

To quote from the author's preface:

An Invitation to Arithmetic Geometry tries to present in a unified manner, from the beginning, some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and to bring out the deep analogies between these topics. This book introduces the reader to arithmetic geometry by focusing primarily on the dimension one case (that is, curves in algebraic geometry; rings of dimension one in commutative algebra). Topics covered and interconnected include:
1. rings of integers, discriminant and ramification, ideal class group (in algebraic number theory)
2. localizations, Dedekind domains, discrete valuation rings (in commutative algebra)
3. affine and projective curves, the Riemann-Roch theorem (in algebraic geometry)
4. the zeta function of a curve over a finite field and the analog of the Riemann hypothesis (in arithmetic geometry).

The book is intended for a year-long course, so we will only cover a portion of it. But then there are always seminars...