Weierstrass semigroups at totally ramified places of degree one on Kummer extensions

TL;DR

This paper explicitly describes the Weierstrass semigroup and gaps at totally ramified places of degree one on Kummer extensions, providing conditions for symmetry and constructing minimal generators.

Contribution

It offers a unified description of Weierstrass semigroups at totally ramified places, including explicit minimal generating sets and applications to specific curves.

Findings

Explicit description of gaps and semigroups at ramified places

Necessary and sufficient condition for semigroup symmetry

Construction of functions with pole divisors in the minimal generating set

Abstract

We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a Kummer extension defined by the affine equation $y^m = f(x)$ over $K$, an algebraic extension of $\mathbb{F}_q$, where $f(x)\in K(x)$. Our description takes a unified form for distinct totally ramified places of degree one. We then provide a necessary and sufficient condition for the Weierstrass semigroup at a totally ramified place of degree one to be symmetric. Furthermore, we investigate the minimal generating set of the Weierstrass semigroups at many totally ramified places of degree one. We not only explicitly describe the minimal generating set, but also construct functions whose pole divisors have coefficients lying in the set. Finally, we apply our results to specific Kummer extensions, including function fields of GGS curves and subcovers of the BM curve.