${\mathbb{F}}_{p^2}$-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Abstract

Let $\mathbb{F}$ be the finite field of order $q^2$, $q=p^h$ with $p$ prime. It is commonly atribute to J.P. Serre the fact that any curve $\mathbb{F}$-covered by the Hermitian curve ${\mathcal{H}}_{q+1}:y^{q+1}=x^q+x$ is also $\mathbb{F}$-maximal. Nevertheless, the converse is not true as the Giulietti-Korchmáros example shows provided that $q>8$ and $h\equiv 0(\mathrm{mod}3)$. In this paper, we show that if an $\mathbb{F}$-maximal curve $\mathcal{X}$ of genus $g\ge 2$ where $q=p$ is such that $|Aut(\mathcal{X})|>84(g-1)$ then $\mathcal{X}$ is Galois-covered by ${\mathcal{H}}_{p+1}$. Also, we show that the hypothesis on the order of $Aut(\mathcal{X})$ is sharp, since there exists an $\mathbb{F}$-maximal curve $\mathcal{X}$ for $q=71$ of genus $g=7$ with $|Aut(\mathcal{X})|=84(7-1)$ which is not Galois-covered by the Hermitian curve ${\mathcal{H}}_{72}$.