The weighted moduli space of binary sextics

Abstract

We use the weighted moduli height as defined in [8] 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 equationwhere H(f ) is the minimal naive height of the curve as defined in [9]. Based on the weighted moduli height h we create a database of genus two curves defined over Q with small h and show that for small such height (h < 5) about 30% of points are fine moduli points.

On hyperelliptic curves of genus 3

“…where the product is taken over all the primes p ∈ O K . In [1] was proved that this is a well-defined height in a weighted projective space. The weighted moduli height of a minimal tuple is simply h(p) = max{|J i | 1 i }.…”

“…where the product is taken over all the primes p ∈ O K . In [1] was proved that this is a well-defined height in a weighted projective space. The weighted moduli height of a minimal tuple is simply h(p) = max{|J i | w (Q) is an absolute minimum tuple if and only if there is no λ ∈ C \ {0} such that λ i ∈ Z and λ i |J i , for all i = 2, .…”

“…Another benefit is computational: one has to store powers of invariants when dealing with H 3 , but this is not necessary when dealing with the weighted moduli space. The case of g = 2 treated in [1] illustrates how things become much easier when using the weighted moduli space instead of the regular moduli space.…”

“…The main idea of this paper is to use the approach from [1] for binary sextics and to study the weighted moduli space of binary octavics, including their twists and the weighted moduli height. We will follow the definition of the weighted moduli height as in [1]. There are a few things that are slightly different for binary octavics from the binary sextics, as we will see in the coming sections.…”

“…The main idea of this paper is to use the approach from [1] for binary sextics and to study the weighted moduli space of binary octavics, including their twists and the weighted moduli height. We will follow the definition of the weighted moduli height as in [1].…”