On some Galois covers of the Suzuki and Ree curves

TL;DR

This paper determines the automorphism groups of two new families of maximal curves over finite fields, explores their Galois coverings, and computes genera of their subcovers, enriching the classification of maximal curves.

Contribution

It fully characterizes the automorphism groups of the curves, shows they are not Galois covered by Hermitian curves, and computes genera of many subcovers, providing new examples.

Findings

Automorphism groups of the curves are fully determined.

The curves are not Galois covered by Hermitian curves.

New genera for maximal curves are identified.

Abstract

We determine the full automorphism group of two recently constructed families $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ of maximal curves over finite fields. These curves are covers of the Suzuki and Ree curves, and are analogous to the Giulietti-Korchm\'aros cover of the Hermitian curve. We also show that $\tilde{\mathcal{S}}_q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}_{q^4}$, and $\tilde{\mathcal{R}}_q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}_{q^6}$. Finally, we compute the genera of many Galois subcovers of $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$; this provides new genera for maximal curves.