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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.19861 |
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| _version_ | 1866917436042772480 |
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| author | Shiga, Asuka |
| author_facet | Shiga, Asuka |
| contents | Let $\ell$ be an odd prime. We study the visibility theorem for certain elliptic curves over $\mathbb{Q}$ with additive reduction at $\ell$, and deduce the existence of nontrivial $\ell$-torsion in $\Sha(E^D/\mathbb{Q})$ for suitable quadratic twists $E^D$. As an application for $\ell=3$, we exhibit pairs of non-isomorphic elliptic curves with the same BSD invariants, Kodaira symbols, and minimal discriminants, whose Tate--Shafarevich groups are isomorphic and have nontrivial $3$-primary parts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19861 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nontrivial torsion in the Tate--Shafarevich group of elliptic curves via visibility and twists Shiga, Asuka Number Theory Let $\ell$ be an odd prime. We study the visibility theorem for certain elliptic curves over $\mathbb{Q}$ with additive reduction at $\ell$, and deduce the existence of nontrivial $\ell$-torsion in $\Sha(E^D/\mathbb{Q})$ for suitable quadratic twists $E^D$. As an application for $\ell=3$, we exhibit pairs of non-isomorphic elliptic curves with the same BSD invariants, Kodaira symbols, and minimal discriminants, whose Tate--Shafarevich groups are isomorphic and have nontrivial $3$-primary parts. |
| title | Nontrivial torsion in the Tate--Shafarevich group of elliptic curves via visibility and twists |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.19861 |