An invariant of recursive towers of curves over finite fields

Recursive towers of curves over finite fields provide effective lower bounds for the Ihara constant. From a geometric perspective, they allow an easy construction for sequences of curves whose number of rational points grows proportionally to their geometric genus. This asymptotic behavior is particularly valuable in coding theory, as it enables the construction of algebraic geometry codes that come close to the Singleton bound.

In this talk, we will begin with an overview of the theoretical and practical motivations behind recursive towers. We will then provide a precise definition of these objects and methods for studying them. Finally, we will discuss a key difficulty in this setting: the curves in a recursive tower typically become singular after just a few steps. We will explain how to lift these constructions in the smooth world and introduce an invariant defined in this smooth framework.