Compactification of the finite Drinfeld period domain as a moduli space of ferns

Abstract

Let ${\mathbb{F}}_q$ be a finite field with $q$ elements and let $V$ be a vector space over ${\mathbb{F}}_q$ of dimension $n>0$. Let ${\Omega }_V$ be the Drinfeld period domain over ${\mathbb{F}}_q$. This is an affine scheme of finite type over ${\mathbb{F}}_q$, and its base change to ${\mathbb{F}}_q(t)$ is the moduli space of Drinfeld ${\mathbb{F}}_q[t]$-modules with level $(t)$ structure and rank $n$. In this thesis, we give a new modular interpretation to Pink and Schieder's smooth compactification $B_V$ of ${\Omega }_V$. Let $\hat{V}$ be the set $V\cup \{\infty \}$ for a new symbol $\infty$. We define the notion of a $V$-fern over an ${\mathbb{F}}_q$-scheme $S$, which consists of a stable $\hat{V}$-marked curve of genus $0$ over $S$ endowed with a certain action of the finite group $V⋊{\mathbb{F}}_q^{\times }$. Our main result is that the scheme $B_V$ represents the functor that associates an ${\mathbb{F}}_q$-scheme $S$ to the set of isomorphism classes of $V$-ferns over $S$. Thus $V$-ferns over ${\mathbb{F}}_q(t)$-schemes can be regarded as generalizations of Drinfeld ${\mathbb{F}}_q[t]$-modules with level $(t)$ structure and rank $n$. To prove this theorem, we construct an explicit universal $V$-fern over $B_V$. We then show that any $V$-fern over a scheme $S$ determines a unique morphism $S\to B_V$, depending only its isomorphism class, and that the $V$-fern is isomorphic to the pullback of the universal $V$-fern along this morphism. We also give several functorial constructions involving $V$-ferns, some of which are used to prove the main result. These constructions correspond to morphisms between various modular compactifications of Drinfeld period domains over ${\mathbb{F}}_q$. We describe these morphisms explicitly.