Géométrie et algèbre effectives

Good Recursive Towers over Prime Fields Exist

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 will be the emphasis of this talk. The first explicit examples were found by Garcia--Stichtenoth over quadratic finite fields in 1995. Shortly after followed the discovery of good towers over cubic finite fields in 2005 (Bezerra--Garcia-Stichtenoth) and all nonprime finite fields in 2013 (Bassa--Beelen--Garcia--Stichtenoth). 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 the landscape of explicit recursive towers and present a recently discovered tower over prime fields (joint work with Christophe Ritzenthaler).