# Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus

## TL;DR

This paper explicitly determines the Weierstrass semigroup and automorphism group of a specific maximal curve with the third largest genus, revealing complex structures and exceptional behaviors in its Weierstrass points.

## Contribution

It provides the first explicit computation of the Weierstrass semigroup and automorphism group for this maximal curve, addressing an open problem and uncovering unique properties.

## Key findings

- The Weierstrass semigroup at any point is explicitly determined.
- The automorphism group matches that inherited from the Hermitian curve for most cases.
- The curve exhibits richer Weierstrass point structures than known maximal curves.

## Abstract

In this article we explicitly determine the Weierstrass semigroup at any point and the full automorphism group of a known $\mathbb{F}_{q^2}$-maximal curve $\mathcal{X}_3$ having the third largest genus. This curve arises as a Galois subcover of the Hermitian curve, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough $\mathcal{X}_3$ has many different types of Weierstrass semigroups and the set of its Weierstrass points is much richer than the set of $\mathbb{F}_{q^2}$-rational points, as instead happens for all the known maximal curves where the Weierstrass points are known. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, $\mathrm{Aut}(\mathcal{X}_3)$ is exactly the automorphism group inherited from the Hermitian curve, apart from small values of $q$.