Generalized Artin-Mumford curves over finite fields

Abstract

Let ${\mathbb{F}}_q$ be the finite field of order $q=p^h$ with $p>2$ prime and $h>1$, and let ${\mathbb{F}}_{\bar{q}}$ be a subfield of ${\mathbb{F}}_q$. From any two $\bar{q}$-linearized polynomials $L_1,L_2\in {\overline{\mathbb{F}}}_q[T]$ of degree $q$, we construct an ordinary curve ${\mathcal{X}}_{(L_1,L_2)}$ of genus $(q-1{)}^2$ which is a generalized Artin-Schreier cover of the projective line ${\mathbb{P}}^1$. The automorphism group of ${\mathcal{X}}_{(L_1,L_2)}$ over the algebraic closure ${\overline{\mathbb{F}}}_q$ of ${\mathbb{F}}_q$ contains a semidirect product $\Sigma ⋊\Gamma$ of an elementary abelian $p$-group $\Sigma$ of order $q^2$ by a cyclic group $\Gamma$ of order $\bar{q}-1$. We show that for $L_1\ne L_2$, $\Sigma ⋊\Gamma$ is the full automorphism group $\mathrm{Aut}({\mathcal{X}}_{(L_1,L_2)})$ over ${\overline{\mathbb{F}}}_q$; for $L_1=L_2$ there exists an extra involution and $\mathrm{Aut}({\mathcal{X}}_{(L_1,L_1)})=\Sigma ⋊\Delta$ with a dihedral group $\Delta$ of order $2(\bar{q}-1)$ containing $\Gamma$. Two different choices of the pair $\{L_1,L_2\}$ may produce birationally isomorphic curves, even for $L_1=L_2$. We prove that any curve of genus $(q-1{)}^2$ whose ${\overline{\mathbb{F}}}_q$-automorphism group contains an elementary abelian subgroup of order $q^2$ is birationally equivalent to ${\mathcal{X}}_{(L_1,L_2)}$ for some separable $\bar{q}$-linearized polynomials $L_1,L_2$ of degree $q$. We produce an analogous characterization in the special case $L_1=L_2$. This extends a result on the Artin-Mumford curves, due to Arakelian and Korchmáros.

References in corpus (2)

• Ordinary algebraic curves with many automorphisms in positive characteristic
• On Abelian Automorphism Groups of Mumford Curves