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

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 $q$.

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• Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q\equiv 1(\mathrm{mod}3)$