Conditionally on sparse 'strange' newforms, the largest prime order of rational torsion on elliptic curves over degree-d fields is at most 3d+1 for large even d and o(d) for odd d.
On Some Open Cases of a Conjecture of Conrad, Edixhoven and Stein
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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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Prime order torsion on elliptic curves over number fields. Part I: Asymptotics
Conditionally on sparse 'strange' newforms, the largest prime order of rational torsion on elliptic curves over degree-d fields is at most 3d+1 for large even d and o(d) for odd d.