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

Matthew Weidner

Computer Laboratory, University of Cambridge, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK

Work completed while the author was at the Department of Mathematics, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA.

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https://doi.org/10.1142/S1793042119501057Cited by:0 (Source: Crossref)

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.

Keywords:

AMSC: 11G05

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