A characterization of the Artin-Mumford curve

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 $H=(C_p\times C_p)⋊D_{p-1}$. 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 $\mathcal{M}$.