The rational torsion subgroup of J0(N)

Abstract

Let N be a positive integer and let J0(N) be the Jacobian variety of the modular curve X0(N). For any prime p≥5 whose square does not divide N, we prove that the p-primary subgroup of the rational torsion subgroup of J0(N) is equal to that of the rational cuspidal divisor class group of X0(N), which is explicitly computed in [33]. 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.