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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2507.19462 |
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| _version_ | 1866918104420843520 |
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| author | Lee, Meghan |
| author_facet | Lee, Meghan |
| contents | An isolated point of degree $d$ is a closed point on an algebraic curve which does not belong to an infinite family of degree $d$ points that can be parameterized by some geometric object. We provide an algorithm to test whether a rational, non-CM $j$-invariant gives rise to an isolated point on the modular curve $X_0(n)$, for any $n \in \mathbb{Z}^+$, using key results from Menendez and Zywina. This work is inspired by the prior algorithm of Bourdon et al. which tests whether a rational, non-CM $j$-invariant gives rise to an isolated point on any modular curve $X_1(n)$. From the implementation of our algorithm, we determine that the set of $j$-invariants corresponding to isolated points on $X_1(n)$ is neither a subset nor a superset of those corresponding to isolated points on $X_0(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19462 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isolated $j$-invariants arising from the modular curve $X_0(n)$ Lee, Meghan Number Theory An isolated point of degree $d$ is a closed point on an algebraic curve which does not belong to an infinite family of degree $d$ points that can be parameterized by some geometric object. We provide an algorithm to test whether a rational, non-CM $j$-invariant gives rise to an isolated point on the modular curve $X_0(n)$, for any $n \in \mathbb{Z}^+$, using key results from Menendez and Zywina. This work is inspired by the prior algorithm of Bourdon et al. which tests whether a rational, non-CM $j$-invariant gives rise to an isolated point on any modular curve $X_1(n)$. From the implementation of our algorithm, we determine that the set of $j$-invariants corresponding to isolated points on $X_1(n)$ is neither a subset nor a superset of those corresponding to isolated points on $X_0(n)$. |
| title | Isolated $j$-invariants arising from the modular curve $X_0(n)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.19462 |