Recursive towers of curves over finite fields using graph theory

TL;DR

This paper introduces a graph-theoretic approach to analyze recursive towers of curves over finite fields, focusing on their asymptotic properties and conditions for optimality, with applications to known towers.

Contribution

It presents a novel method using spectral graph theory and algebraic geometry to study the asymptotic behavior and optimality of recursive towers over finite fields.

Findings

A necessary condition for a tower to be asymptotically good.

Recursive towers not reaching the Drinfeld-Vladut bound cannot be optimal under mild assumptions.

Application of the method to the Bezerra-Garcia-Stichtenoth tower.

Abstract

We give a new way to study recursive towers of curves over a finite field, defined from a bottom curve $\Cun$ and a correspondence $\Cdeux$ on $\Cun$.In particular, we study their asymptotic behavior. A close examination of singularities leads to a necessary condition for a tower to be asymptotically good. Then, spectral theory on a directed graph and considerations on the class of $\Cdeux$ in $\NS (\Cun \times \Cun)$ lead to the fact that, under some mild assumptions, a recursive tower which does not reach Drinfeld-Vladut bound cannot be optimal in Tsfasmann-Vladut sense. Results are applied to the Bezerra-Garcia-Stichtenoth tower along the paper for illustration.