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

TL;DR

This paper proves that certain maximal algebraic curves over finite fields with large automorphism groups are Galois-covered by Hermitian curves, and demonstrates the sharpness of the automorphism group size condition.

Contribution

It establishes a criterion linking the size of the automorphism group of a maximal curve to its Galois-covering relation with the Hermitian curve, including a counterexample showing the condition's sharpness.

Findings

Maximal curves with automorphism groups larger than 84(g-1) are Galois-covered by the Hermitian curve.

Existence of a maximal curve with automorphism group size exactly 84(g-1) not Galois-covered by the Hermitian curve.

The automorphism group size condition is proven to be optimal.

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-Korchm\'aros example shows provided that $q>8$ and $h\equiv 0\pmod{3}$. In this paper, we show that if an $\mathbb{F}$-maximal curve $\mathcal{X}$ of genus $g\geq 2$ 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 $\mathcal{H}_{72}$.