Weierstrass semigroups on the Giulietti-Korchmáros curve

In this article we explicitly determine the structure of the Weierstrass semigroups H(P) for any point P of the Giulietti-Korchmáros curve X. We show that as the point varies, exactly three possibilities arise: one for the Fq2-rational points (already known in the literature), one for the Fq6∖Fq2-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 Fq6∖Fq2-rational point.