On the proportion of locally soluble superelliptic curves

TL;DR

This paper studies the likelihood that superelliptic curves over the rationals have points everywhere locally, providing explicit proportions and bounds, notably showing a 96.94% probability for certain cubic curves of degree 6.

Contribution

It establishes that the proportion of locally soluble superelliptic curves is positive and explicitly computes this proportion for specific families, including a precise 96.94% for certain degree 6 curves.

Findings

Proportion of locally soluble superelliptic curves is positive.

Explicit rational functions for local solubility proportions over _p.

Exact 96.94% solubility proportion for specific degree 6 superelliptic curves.

Abstract

We investigate the proportion of superelliptic curves that have a $\mathbb{Q}_p$ point for every place $p$ of $\mathbb{Q}$. We show that this proportion is positive and given by the product of local densities, we provide lower bounds for this proportion in general, and for superelliptic curves of the form $y^3 = f(x,z)$ for an integral binary form $f$ of degree 6, we determine this proportion to be 96.94%. More precisely, we give explicit rational functions in $p$ for the proportion of such curves over $\mathbb{Z}_p$ having a $\mathbb{Q}_p$-point.