On the proportion of locally soluble superelliptic curves

Abstract

We investigate the proportion of superelliptic curves that have a Qp point for every place p of 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 y3=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 Zp having a Qp-point.