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 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 pth roots of unity and f (x) has no k-rational roots.