  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
$V\rtimes \mathbb{F}_q^\times$. 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.

Algebraic Geometry;Number Theory