A characterization of the Artin-Mumford curve

TL;DR

This paper characterizes the Artin-Mumford curve over finite fields by showing that any curve with a specific automorphism group structure and genus is birationally equivalent to it.

Contribution

It proves a uniqueness result for the Artin-Mumford curve based on its automorphism group and genus over finite fields.

Findings

The automorphism group of the Artin-Mumford curve is isomorphic to H.

Any curve with genus (p-1)^2 and automorphism group containing H is birationally equivalent to the Artin-Mumford curve.

The result characterizes the Artin-Mumford curve among algebraic curves over finite fields.

Abstract

Let $\mathcal{M}$ be the Artin-Mumford curve over the finite prime field $\mathbb{F}_p$ with $p>2$. By a result of Valentini and Madan, $\mbox{Aut}_{\mathbb{F}_p}(\mathcal{M})\cong H$ with $H=(C_p\times C_p)\rtimes D_{p-1}$. We prove that if $\mathcal{X}$ is an algebraic curve of genus $g=(p-1)^2$ such that $\mbox{Aut}_{\mathbb{F}_p}(\mathcal{X})$ contains a subgroup isomorphic to $H$ then $\mathcal{X}$ is birationally equivalent over $\mathbb{F}_p$ to the Artin-Mumford curve $\mathcal{M}$.