Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q ≡ 1 (mod 3)

In this article we continue the work started in [3], explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known $F_{q^2}$-maximal function field $Y_3$ having the third largest genus, for $q\equiv 1\left(\mathrm{mod}3\right)$. 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 $F_{q^2}$-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, $\mathrm{Aut}\left(Y_3\right)$ is exactly the automorphism group inherited from the Hermitian function field, apart from small values of q.