Classification of all Galois subcovers of the Skabelund maximal curves

TL;DR

This paper fully characterizes all Galois subcovers of certain maximal curves constructed by Skabelund, expanding the known catalog of maximal curves over finite fields with new genus examples.

Contribution

It provides a complete classification of Galois subcovers of Skabelund maximal curves, revealing new genera of maximal curves and advancing understanding of their structure.

Findings

Complete classification of Galois subcovers of Skabelund curves

New genera of maximal curves identified

Enhanced understanding of the structure of maximal curves

Abstract

In 2017 Skabelund constructed two new examples of maximal curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ as covers of the Suzuki and Ree curves, respectively. The resulting Skabelund curves are analogous to the Giulietti-Korchm\'aros cover of the Hermitian curve. In this paper a complete characterization of all Galois subcovers of the Skabelund curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ is given. Calculating the genera of the corresponding curves, we find new additions to the list of known genera of maximal curves over finite fields.