Hasse principle violations in twist families of superelliptic curves

Conditionally on the abc conjecture, we generalize the 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 DOUBLE-STRUCK CAPITAL Q-rational roots. We also show unconditionally that a curve defined by C : ypN = f(x) (for p prime and N > 1) 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.