Regularity of quotients of Drinfeld modular schemes

TL;DR

This paper extends the theory of level structures on Drinfeld modules to more general torsion modules, proving that certain quotients of the associated moduli schemes remain regular, which generalizes classical modular curve results.

Contribution

The authors introduce a new notion of level N for Drinfeld modules, prove the regularity of certain quotient moduli schemes, and identify subgroups like $ ext{Gamma}_0$, $ ext{Gamma}_1$, and Hecke-related subgroups as examples.

Findings

The moduli functor with level N is representable by a regular affine scheme.

Certain subgroup quotients of the moduli scheme are regular.

Examples include generalizations of classical modular groups.

Abstract

Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I \subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module. We extend this to that of level $N$, where $N$ is a finitely generated torsion $A$-module. The case where $N=(I^{-1}/A)^d$, where $d$ is the rank of the Drinfeld module,coincides with the structure of level $I$. The moduli functor is representable by a regular affine scheme. The automorphism group $\mathrm{Aut}_{A}(N)$ acts on the moduli space. Our theorem gives a class of subgroups for which the quotient of the moduli scheme is regular. Examples include generalizations of $\Gamma_0$ and of $\Gamma_1$. We also show that parabolic subgroups appearing in the definition of Hecke correspondences are such subgroups.