We study the rational torsion subgroup of the modular Jacobian variety $J_0(N)$ 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 $p\geq 3$ when $3 \nmid N$, and for $p\geq 5$ when $3 \mid N$. 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.
"Eisenstein Ideals and the Rational Torsion Subgroups of Modular Jacobian Varieties II." Tokyo J. Math. 37 (2) 273 - 318, December 2014. https://doi.org/10.3836/tjm/1422452795