Galois subcovers of the Hermitian curve in characteristic $p$ with respect to subgroups of order $p^2$

Abstract

A (projective, geometrically irreducible, non-singular) curve $\mathcal{X}$ defined over a finite field ${\mathbb{F}}_{q^2}$ is maximal if the number $N_{q^2}$ of its ${\mathbb{F}}_{q^2}$-rational points attains the Hasse-Weil upper bound, that is $N_{q^2}=q^2+2\mathfrak{g}q+1$ where $\mathfrak{g}$ is the genus of $\mathcal{X}$. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order $p^2$ where $p$ is the characteristic of ${\mathbb{F}}_{q^2}$. Doing so we also determine the ${\mathbb{F}}_{q^2}$-isomorphism classes of such curves and describe their full ${\mathbb{F}}_{q^2}$-automorphism groups.