Recursive towers of curves over finite fields using graph theory

Emmanuel Hallouin (1) , Marc Perret (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Emmanuel Hallouin, Marc Perret. Recursive towers of curves over finite fields using graph theory. Moscow Maths Journal, 2014, 14 (4), pp.773--806. ⟨hal-00967368⟩