Superelliptic curves with minimal weighted moduli height

Abstract

For a superelliptic curve $\mathcal{X}$, defined over $\mathbb{Q}$, let $\mathfrak{p}$ denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of $\mathcal{X}$ such that: i) the corresponding moduli point $\mathfrak{p}$ has minimal weighted height, ii) the equation of the curve has minimal coefficients. Part i) is accomplished by reduction of the moduli point which is equivalent with obtaining a representation of the moduli point $\mathfrak{p}$ with minimal weighted height, as defined in [5], and part ii) by the classical reduction of the binary forms.

The accepted version is updated. The paper will appear in Contemporary Math., (AMS), 2022

References in corpus (4)

• From hyperelliptic to superelliptic curves
• Rational points in the moduli space of genus two
• Semistable types of hyperelliptic curves
• Reduction theory of binary forms