Hasse principle violations in twist families of superelliptic curves

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.