Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q≡0(mod3)

In this article, we explicitly determine the Weierstrass semigroup at any place and the full automorphism group of a known $F_{q^2}$-maximal function field $Z_3$, which is realised as a Galois subfield of the Hermitian function field and has the third largest genus, for $q\equiv 0\left(\mathrm{mod}3\right)$. This completes the work contained in [3] and [4], where the cases $q\equiv 2\left(\mathrm{mod}3\right)$ and $q\equiv 1\left(\mathrm{mod}3\right)$, respectively, were studied. Like for these other two cases, the problem of determining the uniqueness of the function field $Z_3$, with respect to the value of its genus, is still open. The knowledge of the Weierstrass semigroups may be instrumental in finding a solution to this problem, as it happened to be the case for the function fields with the largest [11] and second largest genera [1], [7]. Similarly to what observed in [3] and [4], also in the case of $Z_3$ we find that many different types of Weierstrass semigroups appear, and that the set of Weierstrass places contains also non-$F_{q^2}$-rational places. We also determine $\mathrm{Aut}\left(Z_3\right)$, which turns out to be exactly the automorphism group inherited from the Hermitian function field, apart from the case $q=3$.