Eisenstein ideals and the rational torsion subgroups of Modular Jacobian varieties II

Abstract

We study the rational torsion subgroup of the modular Jacobian variety J0(N) when N is square-free. We prove that the p-primary part of this group coincides with that of the cuspidal divisor class group for p ≥ 3 when 3 | N, and for p ≥ 5 when 3 | N. We further determine the structure of each eigenspace of such p-primary part with respect to the Atkin-Lehner involutions. This is based on our study of the Eisenstein ideals in the Hecke algebras.