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

Abstract

The Deligne-Lusztig curves associated to the algebraic groups of type ${}^2{A}_2$, ${}^2{B}_2$, and ${}^2{G}_2$ 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 $q=2^{2h+1}$, $h\ge 1$ and the Ree curve ${\mathcal{R}}_q$ is maximal over ${\mathbb{F}}_{q^6}$, for $q=3^{2h+1}$, $h\ge 0$. 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 We also give a proof for an unpublished result due to Rains and Zieve stating that ${\mathcal{R}}_3$ is not Galois covered by ${\mathcal{H}}_{27}$.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 ${\mathcal{H}}_{27}$.