This paper extends a result of Soumya Sankar where she showed that the proportion of Artin-Schreier curves which are ordinary is zero for odd characteristic and nonzero for characteristic two. Here we extend these counts to other p-corank (the difference between genus and p-rank). Theese counts are closely related to results of Pries and Zhu which studies the irreducible components of relevant moduli spaces. Some of our counts are suggestive about the cohomology of these moduli spaces.

In a recent paper, Gundlach showed that the generating function counting the number of etale G-extensions (where G is an abelian p-group) ordered by a specific type of conductor displayed nice rationality properties. Using Gundlach's framework, we can show that the proportion of Artin-Schreier curves which are ordinary ordered by this modified conductor had nice properties not seen in the counts by genus. We also study the p-coranks for other abelian wild covers.

Gundlach and Seguin showed how we can use Abrashkin's nilpotent Artin-Schreier theory to study etale G-extensions where G is a two-step nilpotent p-group. Inspired by their work, we use this to study two related problems. Firstly, can we study the asymptotics of local extensions? The second direction is to study the p-ranks of G-covers. In particular, if G is a p-group of nilpotency class less than p, there exists an ordinary cover if and only if G has exponent p.