Weierstrass semigroups on the Skabelund maximal curve

Abstract

In 2017, D. Skabelund constructed a maximal curve over ${\mathbb{F}}_{q^4}$ as a cyclic cover of the Suzuki curve. In this paper we explicitly determine the structure of the Weierstrass semigroup at any point $P$ of the Skabelund curve. We show that its Weierstrass points are precisely the ${\mathbb{F}}_{q^4}$-rational points. Also we show that among the Weierstrass points, two types of Weierstrass semigroup occur: one for the ${\mathbb{F}}_q$-rational points, one for the remaining ${\mathbb{F}}_{q^4}$-rational points. For each of these two types its Apéry set is computed as well as a set of generators.