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

Abstract

We construct a compactification of the moduli space of Drinfeld modules of rank r r and level N N as a moduli space of A A -reciprocal maps. This is closely related to the Satake compactification but not exactly the same. The construction involves some technical assumptions on N N that are satisfied for a cofinal set of ideals  N N . In the special case where A = F q [ t ] A=\mathbb {F}_q[t] and N = ( t n ) N=(t^n) , we obtain a presentation for the graded ideal of Drinfeld cusp forms of level N 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.