turkmath.org

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. One such method is by means of explicit recursive towers and has been successfully used in all cases, except over prime fields. The questions of finding good towers over prime fields resisted all attempts for several decades and lead to the common belief that such towers do not exist. In this talk I will try to give an overview of explicit recursive towers and present a recently discovered tower over prime fields (joint work with Christophe Ritzenthaler).