Hasse principle violations in twist families of superelliptic curves

TL;DR

This paper proves that certain superelliptic curves have infinitely many twists violating the Hasse Principle, contingent on the abc conjecture or unconditionally under specific conditions, extending previous work in the field.

Contribution

It generalizes prior results by establishing conditions under which superelliptic curves have infinitely many Hasse Principle violations, both conditionally and unconditionally.

Findings

Conditional on abc conjecture, high genus superelliptic curves have infinitely many Hasse Principle violations.

Unconditionally, certain superelliptic curves over number fields with roots of unity have infinitely many violations.

The work extends previous results to broader classes 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 $\mathbb Q$-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.