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

2025/05/16 by D. De Leo, Michael Stoll, De Leo, Davide +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Finite Group Theory Research #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2505.10777

Abstract

Let \( p ≥ 5 \) be a prime. In 2003 Conrad, Edixhoven, and Stein conjectured that the rational torsion subgroup of the modular Jacobian \( J₁(p) \) 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 \( J₁(p)(ℚ)_tors \) for every prime up to \( p ≤ 113 \). However, our method is general and can be applied to larger primes.

Citations

• The rational cuspidal divisor class group of X₀(N)

Cited by

• Prime order torsion on elliptic curves over number fields. Part I: Asymptotics

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