Statistics for p-ranks of Artin-Schreier covers

TL;DR

This paper investigates the distribution of p-ranks in Artin-Schreier covers over finite fields, analyzing both geometric and arithmetic limits, and introduces methods for counting isomorphism classes of such covers.

Contribution

It provides new statistical insights into p-ranks of Artin-Schreier covers in large field limits and develops counting techniques for isomorphism classes.

Findings

Distribution of p-ranks in geometric limit analyzed

Arithmetic variation of p-rank distribution studied

Counting methods for isomorphism classes developed

Abstract

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 $\infty$.