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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2407.14322 |
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| _version_ | 1866914495620710400 |
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| author | Bourdon, Abbey Ryalls, Nina Watson, Lori D. |
| author_facet | Bourdon, Abbey Ryalls, Nina Watson, Lori D. |
| contents | In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine the least degree of a point on the modular curve $X_1(N)$ associated to any $E \in \mathcal{E}$. In the present work, we consider the cases where $\mathcal{E}$ is rational, i.e., contains an elliptic curve with rational $j$-invariant, or where $\mathcal{E}$ consists of elliptic curves with complex multiplication (CM). If $N=\ell^k$ is a power of a single prime, we give a complete characterization upon restricting to points of odd degree, and also in the case where $\mathcal{E}$ is CM. We include various partial results in the more general setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14322 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal torsion curves in geometric isogeny classes Bourdon, Abbey Ryalls, Nina Watson, Lori D. Number Theory 14G35, 11G05 In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine the least degree of a point on the modular curve $X_1(N)$ associated to any $E \in \mathcal{E}$. In the present work, we consider the cases where $\mathcal{E}$ is rational, i.e., contains an elliptic curve with rational $j$-invariant, or where $\mathcal{E}$ consists of elliptic curves with complex multiplication (CM). If $N=\ell^k$ is a power of a single prime, we give a complete characterization upon restricting to points of odd degree, and also in the case where $\mathcal{E}$ is CM. We include various partial results in the more general setting. |
| title | Minimal torsion curves in geometric isogeny classes |
| topic | Number Theory 14G35, 11G05 |
| url | https://arxiv.org/abs/2407.14322 |