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

TL;DR

This paper explicitly determines the Weierstrass semigroup at all places and the automorphism group of a specific maximal function field with high genus, revealing complex semigroup structures and confirming the automorphism group matches that of the Hermitian field.

Contribution

It provides the first complete description of the Weierstrass semigroup and automorphism group for the maximal function field $Y_3$, advancing understanding of its algebraic structure.

Findings

$Y_3$ has diverse Weierstrass semigroups.

The set of Weierstrass places is richer than the set of rational places.

The automorphism group of $Y_3$ matches that of the Hermitian function field.

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 $q$.