Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of $\mathbb{Q}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912517668732928 |
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| author | Avci, Omer |
| author_facet | Avci, Omer |
| contents | Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(ζ_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers such as $3, 4, 5, 7$, or $11$, we obtain a sharper classification. For any abelian number field $K$, the torsion subgroup $E(K)_{\text{tors}}$ is a subgroup of $E(\mathbb{Q}^{\text{ab}})_{\text{tors}}$. Our methods provide tools to eliminate non-realized torsion structures from the list of possibilities for $E(K)_{\text{tors}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_15606 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of $\mathbb{Q}$ Avci, Omer Number Theory 11G05, 11R20, 14H52 Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(ζ_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers such as $3, 4, 5, 7$, or $11$, we obtain a sharper classification. For any abelian number field $K$, the torsion subgroup $E(K)_{\text{tors}}$ is a subgroup of $E(\mathbb{Q}^{\text{ab}})_{\text{tors}}$. Our methods provide tools to eliminate non-realized torsion structures from the list of possibilities for $E(K)_{\text{tors}}$. |
| title | Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of $\mathbb{Q}$ |
| topic | Number Theory 11G05, 11R20, 14H52 |
| url | https://arxiv.org/abs/2406.15606 |