On the proportion of locally soluble superelliptic curves

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.