On maximal curves that are not quotients of the Hermitian curve

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 $n\ge 5$.

References in corpus (1)

• On Weierstrass points and optimal curves

Cited by in corpus (1)

• A complete characterization of Galois subfields of the generalized Giulietti--Korchmáros function field