On Conjectural Rank Parities of Quartic and Sextic Twists of Elliptic Curves

TL;DR

This paper investigates how twisting affects the parity of Selmer ranks in abelian varieties, especially elliptic curves with specific j-invariants, providing a classification method for quartic and sextic twists under certain conjectures.

Contribution

It introduces a framework to classify conjectural rank parities of quartic and sextic twists of elliptic curves, extending previous work on Selmer rank behavior under twists.

Findings

Classifies conjectural rank parities of quartic and sextic twists.

Relates Selmer rank parities to elliptic curves with specific j-invariants.

Provides a finite computational method for classification.

Abstract

We study the behavior under twisting of the Selmer rank parities of a self-dual prime-degree isogeny on a principally polarized abelian variety defined over a number field, subject to compatibility relations between the twists and the isogeny. In particular, we study isogenies on abelian varieties whose Selmer rank parities are related to the rank parities of elliptic curves with $j$-invariant 0 or 1728, assuming the Shafarevich-Tate conjecture. Using these results, we show how to classify the conjectural rank parities of all quartic or sextic twists of an elliptic curve defined over a number field, after a finite calculation. This generalizes previous results of Hadian and Weidner on the behavior of $p$-Selmer ranks under $p$-twists.