Some Ree and Suzuki curves are not Galois covered by the Hermitian curve

TL;DR

This paper investigates the Galois coverings of Hermitian curves by Ree and Suzuki curves, showing specific non-coverings and analyzing the spectrum of genera of Galois subcovers, thus advancing understanding of maximal curves over finite fields.

Contribution

It proves that certain Suzuki and Ree curves are not Galois covered by Hermitian curves and characterizes the genera of Galois subcovers, providing new insights into the structure of maximal curves.

Findings

$ ext{S}_8$ is not Galois covered by $ ext{H}_{64}$

$ ext{R}_3$ is not Galois covered by $ ext{H}_{27}$

Some Galois subcovers of $ ext{R}_3$ are not subcovers of $ ext{H}_{27}$

Abstract

The Deligne-Lusztig curves associated to the algebraic groups of type $^2A_2$, $^2B_2$, and $^2G_2$ are classical examples of maximal curves over finite fields. The Hermitian curve $\mathcal H_q$ is maximal over $\mathbb F_{q^2}$, for any prime power $q$, the Suzuki curve $\mathcal S_q$ is maximal over $\mathbb F_{q^4}$, for $q=2^{2h+1}$, $h\geq1$ and the Ree curve $\mathcal R_q$ is maximal over $\mathbb F_{q^6}$, for $q=3^{2h+1}$, $h\geq0$. In this paper we show that $\mathcal S_8$ is not Galois covered by $\mathcal H_{64}$. We also give a proof for an unpublished result due to Rains and Zieve stating that $\mathcal R_3$ is not Galois covered by $\mathcal H_{27}$. Furthermore, we determine the spectrum of genera of Galois subcovers of $\mathcal H_{27}$, and we point out that some Galois subcovers of $\mathcal R_3$ are not Galois subcovers of $\mathcal H_{27}$.