Compactification of Drinfeld moduli spaces as moduli spaces of $A$-reciprocal maps and consequences for Drinfeld modular forms

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 where $A=\mathbb {F}_q[t]$ 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 similar results in general, but the proof will require more ideas.