Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties

Abstract

Let $N \geq 5$ be a prime number. Conrad, Edixhoven and Stein have conjectured that the rational torsion subgroup of the modular Jacobian variety $J_1(N)$ coincides with the 0-cuspidal class group. We prove this conjecture up to 2-torsion. To do this, we study certain ideals of the Hecke algebras, called the Eisenstein ideals, related to modular forms of weight 2 with respect to $\varGamma_1(N)$ that vanish at the 0-cusps.