Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 0 \pmod 3$

In this article we complete the work started in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known $\mathbb{F}_{q^2}$-maximal function field $Z_3$ having the third largest genus, for $q \equiv 0 \pmod 3$. The cases $q \equiv 2 \pmod 3$ and $q \equiv 1 \pmod 3$ have been in fact analyzed in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], respectively. As in the other two cases, the function field $Z_3$ 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, $Z_3$ 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, $\mathrm{Aut}(Z_3)$ is exactly the automorphism group inherited from the Hermitian function field, apart from the case $q=3$.