Explicit modular towers

TL;DR

This paper presents a general method for explicitly constructing asymptotically optimal towers of modular curves, providing equations for specific examples and exploring their recursive structure and potential modularity.

Contribution

The paper introduces a universal recipe for constructing explicit modular towers and demonstrates it with concrete examples, highlighting their recursive form and possible modular nature.

Findings

Constructed eight explicit modular towers with diverse features

Identified a recursive pattern in all optimal towers

Speculated on the modularity of towers with the recursive form

Abstract

We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as {X_0(l^n): n=1,2,3,...}. 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.