Preface

Summary

This is an introduction to the theory of algebraic curves over finite fields. There are three main themes. The first is a complete presentation of Bombieri's proof of the Riemann hypothesis in the function field case. The second is a full development of the theory of exponential sums in one variable from the point of view of Hasse and Weil. The third and most novel part is the theory of error correcting codes following the program outlined by Goppa. The new results in this last area have come to depend increasingly on many ideas from the theory of modular curves over finite fields and to some extent have motivated our overall presentation. We have included two introductory chapters, one on the basic notions about algebraic curves and the associated function fields, the other including a proof of the Riemann–Roch theorem. In an appendix we verify constructively how the singularities of a plane algebraic curve defined by a homogeneous polynomial in three variables can be transformed into ordinary singularities, i.e. points with distinct tangents, at the expense of increasing the field of constants. This is an essential step in Goppa's program of constructing error correcting codes on algebraic curves from their linear systems. This book fills a gap in the literature of modern number theory; it makes available for the first time all the known results about exponential sums which have applications in algebraic and analytic number theory.