Preface

Summary

Algebraic curves over finite fields and their function fields have been and are still a source of great fascination for number theorists and geometers alike, ever since the seminal work of Hasse and Weil in the 1930s and 1940s. Many important and fruitful ideas have arisen out of this area, where number theory and algebraic geometry meet, and these developments have even spawned a new subject called arithmetic algebraic geometry which now has a broad appeal.

For a long time, the study of algebraic curves over finite fields and their function fields was the province of pure mathematicians. But then, in a series of three papers in the period 1977-1982, Goppa found stunning applications of algebraic curves over finite fields, and especially of those with many rational points, to coding theory. This created a much stronger interest in the area and attracted new groups of researchers such as coding theorists and algorithmically inclined mathematicians. An added incentive was provided by the invention of elliptic-curve cryptosystems in 1985. Algebraic geometry over finite fields is a flourishing subject nowadays which produces exciting research and is immensely relevant for applications.

There has been tremendous research activity focused on algebraic curves over finite fields and their function fields in the last five years. Important theoretical advances were achieved, such as new techniques of constructing algebraic curves over finite fields with many rational points, or equivalently global function fields with many rational places, and improved lower bounds on A (q), the crucial quantity in the asymptotic theory of the number of F_q-rational points on algebraic curves over the finite field F_q of order q.