Compactification of Drinfeld Moduli Spaces as Moduli Spaces of $A$-Reciprocal Maps and Consequences for Drinfeld Modular Forms

TL;DR

This paper introduces a new compactification of Drinfeld moduli spaces using $A$-reciprocal maps, providing insights into Drinfeld modular forms and their dimensions, with potential generalizations beyond specific cases.

Contribution

It constructs a compactification of Drinfeld moduli spaces via $A$-reciprocal maps and derives a presentation and dimension formula for Drinfeld cusp forms in a special case.

Findings

Constructed a compactification of Drinfeld moduli spaces.

Presented a presentation for the graded ideal of cusp forms.

Derived a dimension formula for cusp forms of any weight.

Abstract

We construct a compactification of the moduli space of Drinfeld modules of rank $r$ and level $N$ as a moduli space of $A$-reciprocal maps. This is closely related to the Satake compactification, but not exactly the same. The construction involves some technical assumptions on $N$ that are satisfied for a cofinal set of ideals $N$. In the special case $A={\mathbb F}_q[t]$ and $N=(t^n)$ we obtain a presentation for the graded ideal of Drinfeld cusp forms of level $N$ and all weights and can deduce a dimension formula for the space of cusp forms of any weight. We expect the same results in general, but the proof will require more ideas.