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ON SELMER RANK PARITY OF TWISTS

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  28 September 2016

MAJID HADIAN  and
MATTHEW WEIDNER Open the ORCID record for MATTHEW WEIDNER [Opens in a new window]
MAJID HADIAN*
Affiliation:
California Institute of Technology, Department of Mathematics, Pasadena, CA 91125, USA email hadian@caltech.edu
MATTHEW WEIDNER
Affiliation:
California Institute of Technology, Department of Mathematics, Pasadena, CA 91125, USA email mweidner@caltech.edu
*
✉hadian@caltech.edu
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Abstract

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In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

Keywords

elliptic curvesAbelian varietiesSelmer rank parityp-twist

MSC classification

Primary: 11G05: Elliptic curves over global fields
Secondary: 11G10: Abelian varieties of dimension $> 1$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 3 , June 2017 , pp. 316 - 330
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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