Hasse principle violations in twist families of superelliptic curves

Abstract

Conditionally on the [Formula: see text] conjecture, we generalize the previous work of Clark and the author to show that a superelliptic curve [Formula: see text] of sufficiently high genus has infinitely many twists violating the Hasse Principle if and only if [Formula: see text] has no [Formula: see text]-rational roots. We also show unconditionally that a curve defined by [Formula: see text] (for [Formula: see text] prime and [Formula: see text]) has infinitely many twists violating the Hasse Principle over any number field [Formula: see text] such that [Formula: see text] contains the [Formula: see text]th roots of unity and [Formula: see text] has no [Formula: see text]-rational roots.