The rational torsion subgroup of $J_0(N)$

The main result of this paper is to determine the structure of the rational torsion subgroup of the modular Jacobian variety $J_0(N)$ for any positive integer $N$ up to finitely many primes. More precisely, we prove that the prime-to-$2n$ part of the rational torsion subgroup of $J_0(N)$ is equal to that of the rational cuspidal divisor class group of $X_0(N)$, where $n$ is the largest perfect square dividing $3N$. As the rational cuspidal divisor class group of $X_0(N)$ is already computed in [28], it determines the structure of the rational torsion subgroup of $J_0(N)$ up to primes dividing $2n$.