A Mass Formula For Artin--Schreier Curves Over Finite Fields

TL;DR

This paper derives a mass formula for Artin--Schreier curves over finite fields, providing explicit counts of isomorphism classes for small genus and odd characteristic, extending previous results to characteristic p=2.

Contribution

It extends existing mass formulas to odd prime characteristics and small genus, offering explicit counts of Artin--Schreier curves over finite fields.

Findings

Explicit counts of isomorphism classes for small genus

Mass formula valid for odd prime characteristic

Extension of previous results to characteristic p=2

Abstract

We study a mass formula for Artin--Schreier curves of genus $g$ defined over a finite field $k$ of characteristic $p$. For an odd prime $p$ and for small $g$, we determine the number of $k$-isomorphism classes of Artin-Schreier curves of genus $g$, weighted by the order of the centralizer of the Artin-Schreier involution in the automorphism group. This extends earlier results by several authors in characteristic $p=2$. Keywords: Artin-Schreier curve, finite field, automorphism, Mass formula, moduli space, arithmetic statistics