Ribet's conjecture for Eisenstein maximal ideals

Abstract

According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form $X_0(N)$ for a {\it prime} number $N$. There is a recent interest to generalize the conjecture for arbitrary $N$ by Ribet, Ohta and Yoo. In this direction, Ribet conjectured that all the Eisenstein maximal ideals are "cuspidal". Hwajong Yoo proved the conjecture ( under certain hypothesis) provided that those ideals are {\it rational}. In this article, we show that ( under certain hypothesis), Ribet's conjecture is true for {\it non-rational} Eisenstein maximal ideals.