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

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.

34 pages

References in corpus (4)

• Semiample Bertini theorems over finite fields
• Distribution of zeta zeroes of Artin--Schreier curves
• The distribution of points on superelliptic curves over finite fields
• Distribution of zeta zeroes for abelian covers of algebraic curves over a finite field