On Weierstrass semigroups of maximal Fermat function fields

TL;DR

This paper explicitly determines the Weierstrass semigroup at all places of certain maximal Fermat function fields, revealing complex structures and a richer set of Weierstrass places than rational ones.

Contribution

It provides the first explicit description of Weierstrass semigroups at all places for specific maximal Fermat function fields, a problem previously unresolved.

Findings

Explicit Weierstrass semigroups at all places of $F_m$ for $m=(q+1)/2$ and $(q+1)/3$

Discovery of diverse types of Weierstrass semigroups in these fields

Identification of a richer set of Weierstrass places compared to rational places

Abstract

In this article we explicitly determine the Weierstrass semigroup at any place of some $\mathbb{F}_{q^2}$-maximal Fermat function fields $\mathcal{F}_m$, namely for $m=(q+1)/2$ and $m=(q+1)/3$. These famous function fields arise as Galois subfields of the Hermitian function field, and even though they have been intensively studied in the literature, the Weierstrass semigroup at every place is still not fully known. Surprisingly enough this problem is in fact quite involved and $\mathcal{F}_m$ has many different types of Weierstrass semigroups. Moreover, its set of Weierstrass places is much richer than its set of rational places.