Recursive towers of curves over finite fields using graph theory

We give a new way to study recursive towers of curves over a finite field, defined á la Elkies from a bottom curve $X$ and a correspondence $\Gamma$ on $X$. A close examination of singularities leads to a necessary condition for a tower to be asymptotically good. Then, spectral theory on a directed graph, Perron–Frobenius theory and considerations on the class of $\Gamma$ in $\mathrm{NS}(X\times X)$ lead to the fact that, under some mild assumption, a recursive tower can have in some sense only a restricted asymptotic quality. Results are applied to the Bezerra–Garcia–Stichtenoth tower along the paper for illustration.