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Published October 2019 | Submitted
Journal Article Open

On conjectural rank parities of quartic and sextic twists of elliptic curves

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 the previous results of Hadian and Weidner on the behavior of p-Selmer ranks under p-twists.

Additional Information

© 2019 World Scientific Publishing Co Pte Ltd. Received 26 April 2017; Accepted 11 April 2019; Published: 12 June 2019. The author would like to thank Majid Hadian for providing mentorship on this project, as well as for his collaboration on a previous paper [3] which inspired this one. The author would also like to thank the Caltech SFP Office for partially supporting his research on this project.

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August 22, 2023
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