The automorphism group of torsion points of an elliptic curve over a field of characteristic $\ge 5$
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| Format: | Preprint |
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2025
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| _version_ | 1866908329023897600 |
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| author | Im, Bo-Hae Kim, Hansol |
| author_facet | Im, Bo-Hae Kim, Hansol |
| contents | For a field $\mathbb{K}$ of characteristic $p\ge5$ containing $\mathbb{F}_{p}^{\operatorname{alg}}$ and the elliptic curve $E_{s,t}: y^{2} = x^{3} + sx + t$ defined over the function field $\mathbb{K}\left(s,t\right)$ of two variables $s$ and $t$, we prove that for a non-negative positive integer $e$ and a positive integer $N$ which is not divisible by $p$, the automorphism group of the normal extension $\mathbb{K}\left(s,t\right)\left(E_{s,t}\left[p^{e} N\right]\right)$ over $\mathbb{K}\left(s,t\right)$ is isomorphic to $\left(\mathbb{Z}/p^{e}\mathbb{Z}\right)^{\times} \times \operatorname{SL}_{2} \left(\mathbb{Z}/N\mathbb{Z}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14859 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The automorphism group of torsion points of an elliptic curve over a field of characteristic $\ge 5$ Im, Bo-Hae Kim, Hansol Number Theory 14H52 For a field $\mathbb{K}$ of characteristic $p\ge5$ containing $\mathbb{F}_{p}^{\operatorname{alg}}$ and the elliptic curve $E_{s,t}: y^{2} = x^{3} + sx + t$ defined over the function field $\mathbb{K}\left(s,t\right)$ of two variables $s$ and $t$, we prove that for a non-negative positive integer $e$ and a positive integer $N$ which is not divisible by $p$, the automorphism group of the normal extension $\mathbb{K}\left(s,t\right)\left(E_{s,t}\left[p^{e} N\right]\right)$ over $\mathbb{K}\left(s,t\right)$ is isomorphic to $\left(\mathbb{Z}/p^{e}\mathbb{Z}\right)^{\times} \times \operatorname{SL}_{2} \left(\mathbb{Z}/N\mathbb{Z}\right)$. |
| title | The automorphism group of torsion points of an elliptic curve over a field of characteristic $\ge 5$ |
| topic | Number Theory 14H52 |
| url | https://arxiv.org/abs/2504.14859 |