  Given a prime $p$ and $q$ a power of $p$, we study the statistics of
$p$-ranks of Artin--Schreier covers of given genus defined over $\mathbb{F}_q$,
in the large $q$-limit. We refer to this problem as the geometric problem. We
also study an arithmetic variation of this problem, and consider
Artin--Schreier covers defined over $\mathbb{F}_p$, letting $p$ go to infinity.
Distribution of $p$-ranks has been previously studied for Artin--Schreier
covers over a fixed finite field as the genus is allowed to go to infinity. The
method requires that we count isomorphism classes of covers that are unramified
at $\infty$.
