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...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Cambridge University Press
2024
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| _version_ | 1826316766904582144 |
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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. |
| 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 |