Ribet's conjecture for Eisenstein maximal ideals

TL;DR

This paper extends Ribet's conjecture to non-rational Eisenstein maximal ideals, demonstrating that all such ideals are 'cuspidal' under certain hypotheses, thereby broadening the understanding of modular curve Jacobians.

Contribution

It proves Ribet's conjecture for non-rational Eisenstein maximal ideals, expanding previous results limited to rational ideals.

Findings

Ribet's conjecture holds for non-rational Eisenstein maximal ideals under certain conditions.

The result generalizes previous proofs limited to rational ideals.

Supports the broader applicability of the conjecture in modular forms theory.

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.