Selmer Ranks of twists of hyperelliptic curves and superelliptic curves

TL;DR

This paper investigates how the Selmer ranks of Jacobians of twists of hyperelliptic and superelliptic curves vary, providing conditions for infinitely many twists with a fixed Selmer rank, extending previous elliptic curve results.

Contribution

It generalizes earlier work on elliptic curves to hyperelliptic and superelliptic curves, establishing conditions for the existence of infinitely many twists with specified Selmer ranks.

Findings

Identifies sufficient conditions for infinitely many twists with given Selmer rank

Extends results from elliptic curves to hyperelliptic and superelliptic curves

Provides a framework for understanding Selmer rank variation in these families

Abstract

We study the variation of Selmer ranks of Jacobians of twists of hyperelliptic curves and superelliptic curves. We find sufficient conditions for such curves to have infinitely many twists whose Jacobians have Selmer ranks equal to $r$, for any given nonnegative integer $r$. This generalizes earlier results of Mazur-Rubin on elliptic curves.