Rational points on $X_0(N)^*$ when $N$ is non-squarefree
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908477546299392 |
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| author | Hashimoto, Sachi Keller, Timo Fourn, Samuel Le |
| author_facet | Hashimoto, Sachi Keller, Timo Fourn, Samuel Le |
| contents | Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps or CM points when $N$ is large enough. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, we provide a complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$, when they are finite. In the process we identify exceptional rational points on $X_0(147)^*$ and $X_0(75)^*$ which were not known before. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00680 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational points on $X_0(N)^*$ when $N$ is non-squarefree Hashimoto, Sachi Keller, Timo Fourn, Samuel Le Number Theory Algebraic Geometry Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps or CM points when $N$ is large enough. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, we provide a complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$, when they are finite. In the process we identify exceptional rational points on $X_0(147)^*$ and $X_0(75)^*$ which were not known before. |
| title | Rational points on $X_0(N)^*$ when $N$ is non-squarefree |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2505.00680 |