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arXiv · 2404.18808

Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 1 \pmod 3$

Peter Beelen·Maria Montanucci·Lara Vicino
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Abstract

In this article we continue the work started in arXiv:2303.00376v1, explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known $\mathbb{F}_{q^2}$-maximal function field $Y_3$ having the third largest genus, for $q \equiv 1 \pmod 3$. This function field arises as a Galois subfield of the Hermitian function field, 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, $Y_3$ has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of $\mathbb{F}_{q^2}$-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, $\mathrm{Aut}(Y_3)$ is exactly the automorphism group inherited from the Hermitian function field, apart from small values of $q$.

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Peter Beelen, Maria Montanucci, Lara Vicino. 2024-04-29. Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 1 \pmod 3$. https://doi.org/10.1016/j.ffa.2025.102701

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