On Some Open Cases of a Conjecture of Conrad, Edixhoven and Stein

Davide De Leo; Michael Stoll

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arXiv · 2505.10777

On Some Open Cases of a Conjecture of Conrad, Edixhoven and Stein

Davide De Leo·Michael Stoll

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Abstract

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.

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Davide De Leo, Michael Stoll. 2025-08-01. On Some Open Cases of a Conjecture of Conrad, Edixhoven and Stein. https://arxiv.org/abs/2505.10777

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