Weierstrass semigroups on the Giulietti-Korchmáros curve

Abstract

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti-Korchmáros curve $\mathcal{X}$. We show that as the point varies, exactly three possibilities arise: One for the ${\mathbb{F}}_{q^2}$-rational points (already known in the literature), one for the ${\mathbb{F}}_{q^6}\setminus {\mathbb{F}}_{q^2}$-rational points, and one for all remaining points. As a result, we prove a conjecture concerning the structure of $H(P)$ in case $P$ is an ${\mathbb{F}}_{q^6}\setminus {\mathbb{F}}_{q^2}$-rational point. As a corollary we also obtain that the set of Weierstrass points of $\mathcal{X}$ is exactly its set of ${\mathbb{F}}_{q^6}$-rational points.