Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 1 \pmod 3$
Abstract
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 $\mathbb{F}_{q^2}$maximal function field $Y_3$ having the third largest genus, for $q \equiv 1 \pmod 3$. 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 wellknown 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 $\mathbb{F}_{q^2}$rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, $\mathrm{Aut}(Y_3)$ is exactly the automorphism group inherited from the Hermitian function field, apart from small values of $q$.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.18808
 arXiv:
 arXiv:2404.18808
 Bibcode:
 2024arXiv240418808B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 14H37;
 14H05
 EPrint:
 22 pages