Arbitrarily large $p$-torsion in Tate-Shafarevich groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866909369603457024 |
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| author | Flynn, E. Victor Shnidman, Ari |
| author_facet | Flynn, E. Victor Shnidman, Ari |
| contents | We show that, for any prime $p$, there exist absolutely simple abelian varieties over $\mathbb{Q}$ with arbitrarily large $p$-torsion in their Tate-Shafarevich group. To prove this, we construct explicit $μ_p$-covers of Jacobians 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_08088 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Arbitrarily large $p$-torsion in Tate-Shafarevich groups Flynn, E. Victor Shnidman, Ari Number Theory Algebraic Geometry Primary 11G30, Secondary 11G10, 14H40 We show that, for any prime $p$, there exist absolutely simple abelian varieties over $\mathbb{Q}$ with arbitrarily large $p$-torsion in their Tate-Shafarevich group. To prove this, we construct explicit $μ_p$-covers of Jacobians 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. |
| title | Arbitrarily large $p$-torsion in Tate-Shafarevich groups |
| topic | Number Theory Algebraic Geometry Primary 11G30, Secondary 11G10, 14H40 |
| url | https://arxiv.org/abs/2209.08088 |