The weighted moduli spaces of sextics

TL;DR

This paper investigates the distribution of fine moduli points in the moduli space of genus two curves using weighted moduli height, establishing bounds and creating a database of curves with small height to analyze their properties.

Contribution

It introduces bounds on weighted moduli height for genus two curves and constructs a database to study the distribution of fine moduli points based on this height.

Findings

Approximately 30% of points with small height are fine moduli points.

Established an upper bound for weighted moduli height in terms of naive height.

Created a database of genus two curves over Q with small weighted moduli height.

Abstract

We use the weighted moduli height as defined in \cite{sh-h} to investigate the distribution of fine moduli points in the moduli space of genus two curves. We show that for any genus two curve with equation $y^2=f(x)$, its weighted moduli height $\mathfrak h (\mathfrak{p}) \leq 2^3 \sqrt{3 \cdot 5 \cdot 7} \, \cdot H(f)$, where $H(f)$ is the minimal naive height of the curve as defined in \cite{height}. Based on the weighted moduli height $\mathfrak h$ we create a database of genus two curves defined over $\mathbb Q$ with small $\mathfrak h$ and show that for small such height ($\mathfrak h < 5$) about 30% of points are fine moduli points.