No 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.

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.