Statistics for p-ranks of Artin-Schreier covers

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 $\infty$.