On Kummer extensions with one place at infinity

TL;DR

This paper explicitly describes the Weierstrass semigroup at the infinity place of Kummer curves, determining key properties and conditions for symmetry, and characterizes certain maximal Castle curves.

Contribution

Provides an explicit description of the Weierstrass semigroup at infinity for Kummer extensions with gcd condition, including Frobenius number, multiplicity, and symmetry conditions.

Findings

Explicit description of $H(Q_)$ for Kummer extensions

Determination of Frobenius number and multiplicity in specific cases

Conditions for $H(Q_)$ to be symmetric and characterization of maximal Castle curves

Abstract

Let $K$ be the algebraic closure of $\mathbb{F}_{q}$. We provide an explicit description of the Weierstrass semigroup $H(Q_\infty)$ at the only place at infinity $Q_{\infty}$ of the curve $\mathcal{X}$ defined by the Kummer extension with equation $y^m=f(x)$, where $f(x)\in K[x]$ is a polynomial satisfying $\gcd (m, \text{deg} f)=1$. As a consequence, we determine the Frobenius number and the multiplicity of $H(Q_{\infty})$ in some cases, and we discuss sufficient conditions for the Weierstrass semigroup $H(Q_{\infty})$ to be symmetric. Finally, we characterize certain maximal Castle curves of type $(\mathcal{X}, Q_{\infty})$.