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

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 Fq²-maximal function field Y₃ having the third largest genus, for q ≡ 1 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, Y₃ has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of Fq²-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, Aut(Y₃) is exactly the automorphism group inherited from the Hermitian function field, apart from small values of q.