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.

中文翻译：

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BBGS 塔的模块化诠释

2000 年，根据他构造显式模曲线塔的程序，埃尔基斯推导出了 2 阶 Drinfeld 模曲线的显式方程，该方程与 Garcia 和 Stichtenoth 构造的渐近最优曲线塔重合。2015 年，Bassa、Beelen、Garcia 和 Stichtenoth 构建了一个著名的（递归和良好的）曲线塔（简称 BBGS 塔），并概述了定义方程的模块化解释。不久之后，Gekeler 深入研究了来自稀疏 Drinfeld 模块的模块曲线。在本文中，为了在这些现有结果之间建立联系，我们提出并证明了一个广义 Elkies 定理，该定理详细说明了如何直接描述 m>=2 的秩为 m 的 Drinfeld 模曲线方程的模解释。