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arXiv · 2303.00376

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

Peter Beelen·Maria Montanucci·Lara Vicino
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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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Peter Beelen, Maria Montanucci, Lara Vicino. 2023-03-01. Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus. https://doi.org/10.1016/j.ffa.2023.102300

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