On Kummer extensions with one place at infinity

2022/08/20 by Mendoza, Erik A. R. · 3 citations
#11R58 #14H55 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2208.09729

Abstract

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

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