Generalized Artin-Mumford curves over finite fields

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 $L_1,L_2\in {\bar{\mathbb{F}}}_q[T]$ 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 ${\bar{\mathbb{F}}}_q$ of ${\mathbb{F}}_q$ contains a semidirect product $\mathrm{\Sigma }⋊\mathrm{\Gamma }$ of an elementary abelian $p$-group $\mathrm{\Sigma }$ of order $q^2$ by a cyclic group $\mathrm{\Gamma }$ of order $\bar{q}-1$. We show that for $L_1\mathrm{\ne }{L}_2$, $\mathrm{\Sigma }⋊\mathrm{\Gamma }$ is the full automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}({\mathcal{X}}_{(L_1,L_2)})$ over ${\bar{\mathbb{F}}}_q$; for $L_1=L_2$ there exists an extra involution and ${\rm Aut}($\mathcal{X}${(L_1,L_1)})=\Sigma \rtimes \Delta$withadihedralgroup$\Delta$oforder$2(\bar{q}-1)$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 ${\bar{\mathbb{F}}}_q$-automorphism group contains an elementary abelian subgroup of order $q^2$ is birationally equivalent to $$\mathcal{X}${(L_1, L_2)}$forsomeseparable$\bar{q}$-linearizedpolynomials$L_1,L_2$ofdegree$q$\mathrm{.}Weproduceananalogouscharacterizationinthespecialcase$L_1=L_2$. This extends a result on the Artin-Mumford curves, due to Arakelian and Korchm'aros.