Recursive Towers over Finite Fields

Questions on solutions of polynomial equations over finite fields have a long history and occupy an important place in number theory. In this talk we will be interested in the particular case, where the equations define algebraic curves of large genus. Understanding the number of rational points on such curves has been an interesting question. As usual, extremal examples play an important role and in the past, various methods have been employed to construct high genus curves over finite fields with many rational points. I will try to give an overview of several of these methods. One particular construction is by means of explicit recursive towers and will be the emphasis of this talk. I will present a result (jointly with Beelen, Garcia and Stichtenoth) on towers over non-prime finite fields, and a result (jointly with Ritzenthaler) over prime fields.