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Visualizing elements of order $7$ in the Tate–Shafarevich group of an elliptic curve

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 August 2016

Tom Fisher
Tom Fisher*
Affiliation:
University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom email T.A.Fisher@dpmms.cam.ac.uk

Abstract

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We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order $7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.

MSC classification

Primary: 11G05: Elliptic curves over global fields 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields

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