Maximal curves from subcovers of the GK-curve

TL;DR

This paper investigates Galois subcovers of the GK-curve, providing explicit equations, discovering new genera in the spectrum of maximal curves, and presenting infinitely many examples of maximal curves not covered by the Hermitian curve.

Contribution

It introduces explicit equations for certain quotients of the GK-curve and expands the known spectrum of genera of maximal curves, demonstrating new examples not Galois covered by the Hermitian curve.

Findings

Explicit equations for some GK-curve quotients.

New values in the spectrum of genera of maximal curves.

Infinitely many maximal curves not Galois covered by the Hermitian curve.

Abstract

For every $q=n^3$ with $n$ a prime power greater than $2$, the GK-curve is an $\mathbb F_{q^2}$-maximal curve that is not $\mathbb F_{q^2}$-covered by the Hermitian curve. In this paper some Galois subcovers of the GK curve are investigated. We describe explicit equations for some families of quotients of the GK-curve. New values in the spectrum of genera of $\mathbb F_{q^2}$-maximal curves are obtained. Finally, infinitely many further examples of maximal curves that cannot be Galois covered by the Hermitian curve are provided.