On maximal curves that are not quotients of the Hermitian curve

TL;DR

This paper proves that certain maximal algebraic curves over finite fields are not Galois quotients of Hermitian curves, extending previous results to new classes of curves and prime powers.

Contribution

It establishes that the curves ll-ll+1=Y^{ll^2-ll+1} are not Galois covered by the Hermitian curve for all ll>3, and similarly for generalized GK curves over ll=2.

Findings

ll-ll+1 curves are not Galois covered by Hermitian curves for ll>3

Generalized GK curves ll^n are not quotients of Hermitian curves for ll=2, n

Extends previous non-quotient results to broader classes of maximal curves

Abstract

For each prime power $\ell$ the plane curve $\mathcal X_\ell$ with equation $Y^{\ell^2-\ell+1}=X^{\ell^2}-X$ is maximal over $\mathbb{F}_{\ell^6}$. Garcia and Stichtenoth in 2006 proved that $\mathcal X_3$ is not Galois covered by the Hermitian curve and raised the same question for $\mathcal X_\ell$ with $\ell>3$; in this paper we show that $\mathcal X_\ell$ is not Galois covered by the Hermitian curve for any $\ell>3$. Analogously, Duursma and Mak proved that the generalized GK curve $\mathcal C_{\ell^n}$ over $\mathbb{F}_{\ell^{2n}}$ is not a quotient of the Hermitian curve for $\ell>2$ and $n\ge 5$, leaving the case $\ell=2$ open; here we show that $\mathcal C_{2^n}$ is not Galois covered by the Hermitian curve over $\mathbb{F}_{2^{2n}}$ for $n\geq5$.