A Modular Interpretation of BBGS Towers

TL;DR

This paper generalizes Elkies' theorem to provide a detailed modular interpretation of rank-m Drinfeld modular curves, linking previous constructions of explicit towers and their equations.

Contribution

It introduces a generalized Elkies' theorem that describes the modular interpretation of equations for rank-m Drinfeld modular curves with m≥2.

Findings

Established a link between existing modular curve constructions and their equations.

Provided a detailed modular interpretation for rank-m Drinfeld modular curves.

Extended Elkies' theorem to higher rank cases.

Abstract

In 2000, based on his procedure for constructing explicit towers of modular curves, Elkies deduced explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. In 2015, Bassa, Beelen, Garcia, and Stichtenoth constructed a celebrated (recursive and good) tower (BBGS-tower for short) of curves and outlined a modular interpretation of the defining equations. Soon after that, Gekeler studied in depth the modular curves coming from sparse Drinfeld modules. In this paper, to establish a link between these existing results, we propose and prove a generalized Elkies' Theorem which tells in detail how to directly describe a modular interpretation of the equations of rank-m Drinfeld modular curves with m>=2.