Explicit Modular Towers

Abstract

We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as \(\{X_0(l^n)\}_{n>1}\). We illustrate the method by giving equations for eight towers with various geometric features. We conclude by observing that such towers are all of a specific recursive form, and speculate that perhaps every tower of this form that attains the Drinfeld-Vladut bound is modular.