Quotient curves of the GK curve

TL;DR

This paper investigates the quotient curves of the GK curve, an $F_{q^2}$-maximal curve with a large automorphism group, by computing genera of curves Galois-covered by it, expanding the known spectrum of maximal curves.

Contribution

It computes the genera of many curves Galois-covered by the GK curve, providing new entries in the spectrum of $F_{q^2}$-maximal curves.

Findings

New genera values for curves Galois-covered by the GK curve

Expansion of the spectrum of $F_{q^2}$-maximal curves

Identification of quotient curves with specific properties

Abstract

For every $q=l^3$ with $l$ a prime power greater than 2, the GK curve $X$ is an $F_{q^2}$-maximal curve that is not $F_{q^2}$-covered by any $F_{q^2}$-maximal Deligne-Lusztig curve. Interestingly, $X$ has a very large $F_{q^2}$-automorphism group with respect to its genus. In this paper we compute the genera of a large variety of curves that are Galois-covered by the GK curve, thus providing several new values in the spectrum of genera of $F_{q^2}$-maximal curves.