Hasse principle violations in twist families of superelliptic curves

2021/03/09 by Lori D. Watson, Watson, Lori D.
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Black Holes and Theoretical Physics

paper · pdf · doi:10.48550/arxiv.2103.05731

Abstract

Conditionally on the abc conjecture, we generalize previous work of Clark and the author to show that a superelliptic curve C: yⁿ = 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 pth roots of unity and f(x) has no k-rational roots.

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