On Some Open Cases of a Conjecture of Conrad, Edixhoven and Stein
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916875497111552 |
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| author | De Leo, Davide Stoll, Michael |
| author_facet | De Leo, Davide Stoll, Michael |
| contents | Let \( p \geq 5 \) be a prime. In 2003 Conrad, Edixhoven, and Stein conjectured that the rational torsion subgroup of the modular Jacobian \( J_1(p) \) coincides with the rational cuspidal divisor class group. Using explicit computations in Magma, the open case \( p = 29 \) has been proven by Derickx, Kamienny, Stein, and Stoll in 2023. We extend these results to primes \( p = 97, 101, 109, \) and \( 113 \). In addition, we provide a list of the groups \( J_1(p)(\mathbb{Q})_{\text{tors}} \) for every prime up to \( p \leq 113 \). However, our method is general and can be applied to larger primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10777 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Some Open Cases of a Conjecture of Conrad, Edixhoven and Stein De Leo, Davide Stoll, Michael Number Theory Let \( p \geq 5 \) be a prime. In 2003 Conrad, Edixhoven, and Stein conjectured that the rational torsion subgroup of the modular Jacobian \( J_1(p) \) coincides with the rational cuspidal divisor class group. Using explicit computations in Magma, the open case \( p = 29 \) has been proven by Derickx, Kamienny, Stein, and Stoll in 2023. We extend these results to primes \( p = 97, 101, 109, \) and \( 113 \). In addition, we provide a list of the groups \( J_1(p)(\mathbb{Q})_{\text{tors}} \) for every prime up to \( p \leq 113 \). However, our method is general and can be applied to larger primes. |
| title | On Some Open Cases of a Conjecture of Conrad, Edixhoven and Stein |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.10777 |