Hasse principle violations in twist families of superelliptic curves

Abstract

Conditionally on the $abc$ conjecture, we generalize previous work of Clark and the author to show that a superelliptic curve $C:y^n=f(x)$ of sufficiently high genus has infinitely many twists violating the Hasse Principle if and only if $f(x)$ has no $\mathbb{Q}$-rational roots. We also show unconditionally that a curve defined by $C:y^{pN}=f(x)$ has infinitely many twists violating the Hasse Principle over any number field $k$ such that $k$ contains the $p$th roots of unity and $f(x)$ has no $k$-rational roots.