Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917575320928256 |
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| author | Banwait, Barinder S. Derickx, Maarten |
| author_facet | Banwait, Barinder S. Derickx, Maarten |
| contents | We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbović. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$ for all squarefree $d$ with $|d| < 800$, improving on the previously known range of $-5 < d < 26$. We use our computations to study the validity of a conjecture of Granville concerning how many twists of a given hyperelliptic curve admit a nontrivial rational point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14514 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville Banwait, Barinder S. Derickx, Maarten Number Theory 11G05 (primary), 11Y60, 11G15 (secondary) We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbović. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$ for all squarefree $d$ with $|d| < 800$, improving on the previously known range of $-5 < d < 26$. We use our computations to study the validity of a conjecture of Granville concerning how many twists of a given hyperelliptic curve admit a nontrivial rational point. |
| title | Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville |
| topic | Number Theory 11G05 (primary), 11Y60, 11G15 (secondary) |
| url | https://arxiv.org/abs/2401.14514 |