Degrees of points with rational $j$-invariant on $X_{0}(n)$ and $X_{1}(n)$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909693286285312 |
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| author | Terao, Kenji |
| author_facet | Terao, Kenji |
| contents | We give a classification of the degrees of the points with rational $j$-invariant on the modular curves $X_{0}(n)$ and $X_{1}(n)$. The degrees which occur infinitely often are computed unconditionally, while those which occur finitely often are determined assuming a conjecture of Zywina. To achieve this, we define the notion of $\mathcal{H}$-closures of subgroups of $\operatorname{GL}_{2}(\widehat{\mathbb{Z}})$, and compute the $\mathcal{B}_{0}(n)$- and $\mathcal{B}_{1}(n)$-closures of images of Galois representations of elliptic curves defined over $\mathbb{Q}$. An application to computing the set of isolated $j$-invariants in $\mathbb{Q}$ is also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13199 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Degrees of points with rational $j$-invariant on $X_{0}(n)$ and $X_{1}(n)$ Terao, Kenji Number Theory 11G18, 11G05 We give a classification of the degrees of the points with rational $j$-invariant on the modular curves $X_{0}(n)$ and $X_{1}(n)$. The degrees which occur infinitely often are computed unconditionally, while those which occur finitely often are determined assuming a conjecture of Zywina. To achieve this, we define the notion of $\mathcal{H}$-closures of subgroups of $\operatorname{GL}_{2}(\widehat{\mathbb{Z}})$, and compute the $\mathcal{B}_{0}(n)$- and $\mathcal{B}_{1}(n)$-closures of images of Galois representations of elliptic curves defined over $\mathbb{Q}$. An application to computing the set of isolated $j$-invariants in $\mathbb{Q}$ is also given. |
| title | Degrees of points with rational $j$-invariant on $X_{0}(n)$ and $X_{1}(n)$ |
| topic | Number Theory 11G18, 11G05 |
| url | https://arxiv.org/abs/2507.13199 |