An algebraic function field in one variable, over a given field K, is an extension of K of transcendence degree one. These field extensions are naturally associated to algebraic curves over the given field and as such have been studied in algebraic geometry since the 19th century. On the other hand, when the ground field K is a finite field, the arithmetic of a function field in one variable over K shows remarkable analogies with the arithmetic of a number field, i.e., a finite extension of the field of rational numbers. These analogies have provided important conjectures and results in number theory.
In this book we have an exposition of the theory of function fields in one variable from the algebraic point of view, leaving to an appendix a dictionary that translates some of the results of the algebraic theory to the algebraic geometry language.
Thus, from the algebraic approach, we find an exhaustive exposition of the elements of the theory of function fields in one variable over an arbitrary ground field (places, valuations, divisors, genus, Galois extensions of function fields, ramification, derivations and differentials) with self-contained proofs of some major results, e.g., duality, the Riemann-Roch and Weierstrass gap theorems and the Riemann-Hurwitz genus formula. The exposition includes the basic examples to illustrate the theory, namely the rational function field K(x), quadratic extensions of K(x), and one chapter is devoted to a detailed study of elliptic and hyperelliptic function fields and some cyclic extensions of the rational function field.
Next, the author focuses on the important case of function fields over a finite field proving that the zeta function of one such field has an Euler product expansion, satisfies a functional equation and an analogue of the Riemann hypothesis (proved using Bombieri’s approach) and as a consequence the Hasse-Weil theorem estimating the number of places of such a field, giving also some improvements (by Serre and Ihara) on the Hasse-Weil bound, with a view towards some applications.
As an application of the theory, three chapters of the book are devoted to the study of codes (subspaces of the vector space of n-tuples with entries in a finite field Fq), starting with a brief introduction in Chapter 2 and then proceeding to define the algebraic geometry codes associated to given divisors. This theory was pioneered by V. D. Goppa in the 1970s. The resulting codes are well-behaved and include the classical Reed-Solomon and BCH codes, now reinterpreted as algebraic geometry codes associated to divisors in the rational function field Fq(x). Moreover, the theory of function fields (or equivalently, the theory of algebraic curves over Fq) is used to obtain bounds on the parameters of these algebraic geometry (Goppa) codes (dimension, minimum distance and weight).
The book is carefully written, the concepts are well motivated and plenty of examples help to understand the ideas and proofs and so it can be used as a textbook for an introductory course on the (classical) arithmetic of function fields with an application to coding theory.
This is the second edition of a book originally published in 1993 (Springer Universitext). The changes consist, mainly, in the addition of exercises at the end of every chapter and a new chapter on the asymptotic theory of function fields over finite fields, allowing the author to give a complete proof of the Tsfasman-Vladut-Zink theorem that was just sketched on the first edition.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org