Statistics for ordinary Artin-Schreier covers and other $p$-rank strata

We study the distribution of the number of points and of the zeroes of the zeta function in different p-rank strata of Artin-Schreier covers over F-q when q is fixed and the genus goes to infinity. The p-rank strata considered include the ordinary family, the whole family, and the family of covers with p-rank equal to p - 1. While the zeta zeroes always approach the standard Gaussian distribution, the number of points over F-q has a distribution that varies with the specific family.