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

TL;DR

This paper investigates the distribution of point counts and zeta zeroes in various p-rank strata of Artin-Schreier covers over finite fields, revealing Gaussian behavior of zeroes and family-dependent point distributions.

Contribution

It provides a detailed analysis of how point counts and zeta zeroes distribute across different p-rank strata as genus increases, highlighting new insights into their probabilistic behavior.

Findings

Zeta zeroes tend to follow a Gaussian distribution.

Point counts vary significantly across different p-rank families.

Distribution patterns depend on the specific p-rank stratum considered.

Abstract

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 curves 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.