The purpose of this reviewarticle is to serve as an introduction and at the same time, as an invitation to the theory of towers of function fields over finite fields. More specifically, we treat here the case of explicit towers; i.e., towers where the function fields are given by explicit equations. The asymptotic behaviour of the genus and of the number of rational places in towers are important features for applications to coding theory and to cryptography (cf. Chapter 2).
The interest in solutions of algebraic equations over finite fields has a long history in mathematics, especially when the equations define a one-dimensional object (a curve or, equivalently, a function field). The major result of this theory is the Hasse-Weil theorem which gives in particular an upper bound for the number of rational points in terms of the genus of the curve and of the cardinality of the finite field.
The Hasse-Weil theorem is equivalent to the validity of Riemann’s Hypothesis for the Zeta function associated to the curve by E. Artin, in analogy with the classical situation in Number Theory. This upper bound of Hasse-Weil is sharp, and the curves attaining this bound are called maximal curves. Y. Ihara was the first to notice that the Hasse-Weil bound can be improved for curves of high genus, and he gave in particular an upper bound for the genus of maximal curves in terms of the cardinality of the finite field.