Arbitrarily large p-torsion in Tate-Shafarevich groups

We show that, for any prime p, there exist absolutely simple abelian varieties over Q with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit µp-covers of Jacobians of curves of the form y p = x(x − 1)(x − a) which violate the Hasse principle. In the a...

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Bibliografische gegevens
Hoofdauteurs: Flynn, E, Shnidman, A, Fisher, T
Formaat: Journal article
Taal:English
Gepubliceerd in: Cambridge University Press 2024
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author Flynn, E
Shnidman, A
Fisher, T
author_facet Flynn, E
Shnidman, A
Fisher, T
author_sort Flynn, E
collection OXFORD
description We show that, for any prime p, there exist absolutely simple abelian varieties over Q with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit µp-covers of Jacobians of curves of the form y p = x(x − 1)(x − a) which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.
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spelling oxford-uuid:3b32ab8c-0117-4392-90ef-e21f8d3eb66e2024-12-16T10:05:48ZArbitrarily large p-torsion in Tate-Shafarevich groupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3b32ab8c-0117-4392-90ef-e21f8d3eb66eEnglishSymplectic ElementsCambridge University Press2024Flynn, EShnidman, AFisher, TWe show that, for any prime p, there exist absolutely simple abelian varieties over Q with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit µp-covers of Jacobians of curves of the form y p = x(x − 1)(x − a) which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.
spellingShingle Flynn, E
Shnidman, A
Fisher, T
Arbitrarily large p-torsion in Tate-Shafarevich groups
title Arbitrarily large p-torsion in Tate-Shafarevich groups
title_full Arbitrarily large p-torsion in Tate-Shafarevich groups
title_fullStr Arbitrarily large p-torsion in Tate-Shafarevich groups
title_full_unstemmed Arbitrarily large p-torsion in Tate-Shafarevich groups
title_short Arbitrarily large p-torsion in Tate-Shafarevich groups
title_sort arbitrarily large p torsion in tate shafarevich groups
work_keys_str_mv AT flynne arbitrarilylargeptorsionintateshafarevichgroups
AT shnidmana arbitrarilylargeptorsionintateshafarevichgroups
AT fishert arbitrarilylargeptorsionintateshafarevichgroups