On the proportion of locally soluble 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.

64 pages. Accepted for publication in Finite Fields and Their Applications

References in corpus (1)

• Many cubic surfaces contain rational points