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

TL;DR

This paper extends the verification of a conjecture relating rational torsion points and cuspidal divisor groups on modular Jacobians to new primes, using computational methods and providing a comprehensive list for primes up to 113.

Contribution

It proves the conjecture for additional primes and offers a general computational approach applicable to larger primes.

Findings

Confirmed the conjecture for p=97, 101, 109, 113

Provided a list of torsion groups for primes up to 113

Demonstrated a general method for larger primes

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.