The rational torsion subgroup of $J_0(N)$

TL;DR

This paper proves that for most primes, the rational torsion subgroup of the Jacobian of modular curves matches the explicitly computed cuspidal divisor class group, extending known results to include prime 3 under certain conditions.

Contribution

It establishes the equality of the rational torsion subgroup and the cuspidal divisor class group for primes p ≥ 3 with specific conditions, generalizing previous results.

Findings

For primes p ≥ 5, the p-primary rational torsion equals the cuspidal divisor class group.

For p=3, the equality holds under additional divisibility conditions on N.

Explicit computation of the cuspidal divisor class group is used in the proof.

Abstract

Let $N$ be a positive integer and let $J_0(N)$ be the Jacobian variety of the modular curve $X_0(N)$. For any prime $p\ge 5$ whose square does not divide $N$, we prove that the $p$-primary subgroup of the rational torsion subgroup of $J_0(N)$ is equal to that of the rational cuspidal divisor class group of $X_0(N)$, which is explicitly computed in \cite{Yoo9}. Also, we prove the same assertion holds for $p=3$ under the extra assumption that either $N$ is not divisible by $3$ or there is a prime divisor of $N$ congruent to $-1$ modulo $3$.