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 -maximal function field , which is realised as a Galois subfield of the Hermitian function field and has the third largest genus, for . This completes the work contained in [3] and [4], where the cases and , respectively, were studied. Like for these other two cases, the problem of determining the uniqueness of the function field , 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 we find that many different types of Weierstrass semigroups appear, and that the set of Weierstrass places contains also non--rational places. We also determine , which turns out to be exactly the automorphism group inherited from the Hermitian function field, apart from the case .