Toroidal compactifications of the moduli spaces of Drinfeld modules

TL;DR

This paper constructs toroidal compactifications of moduli spaces of Drinfeld modules using log geometry and uniformization techniques, providing a new geometric framework for these moduli spaces.

Contribution

It introduces a novel construction of toroidal compactifications of Drinfeld module moduli spaces via log schemes and iterated Tate uniformization, extending previous compactification methods.

Findings

Constructed toroidal compactifications as log regular schemes.

Identified regular toroidal compactifications among the constructed schemes.

Applied formal moduli and Tate uniformization in the construction process.

Abstract

We construct toroidal compactifications of the moduli spaces of Drinfeld $\mathbb{F}_q[T]$-modules of rank $d$ with level $N$ structure as moduli spaces of log Drinfeld modules of rank $d$ with level $N$ structure. The toroidal compactifications are log regular schemes associated to rational cone decompositions, and there are regular ones among them. To construct these toroidal compactifications, we blow up the Satake compactification of Pink and employ the theory of formal moduli and a process of iterated Tate uniformization.