Ribet's conjecture for Eisenstein maximal ideals

2021/11/15 by Banerjee, Debargha, Kumar, Narasimha, Majumdar, Dipramit
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2111.07747

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 X₀(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.

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