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...
Hoofdauteurs: | , , |
---|---|
Formaat: | Journal article |
Taal: | English |
Gepubliceerd in: |
Cambridge University Press
2024
|
_version_ | 1826316766904582144 |
---|---|
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. |
first_indexed | 2024-09-25T04:33:09Z |
format | Journal article |
id | oxford-uuid:3b32ab8c-0117-4392-90ef-e21f8d3eb66e |
institution | University of Oxford |
language | English |
last_indexed | 2025-02-19T04:28:02Z |
publishDate | 2024 |
publisher | Cambridge University Press |
record_format | dspace |
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 |