Maximal Eisenstein ideals and cuspidal subgroups of modular Jacobian varieties

TL;DR

This paper investigates the torsion subgroup of modular Jacobian varieties over specific fields, demonstrating that both this subgroup and the cuspidal subgroup are supported at maximal Eisenstein ideals as Hecke modules.

Contribution

It establishes the support of the torsion subgroup and cuspidal subgroup at maximal Eisenstein ideals within the Hecke algebra framework.

Findings

Torsion subgroup of J_0(N) is supported at maximal Eisenstein ideals.

Cuspidal subgroup shares the same support at Eisenstein ideals.

Results apply to the field generated by cuspidal points.

Abstract

In this paper, we study the torsion subgroup of $J_0(N)$ over the field generated by those points in the cuspidal group, where $N$ is an odd positive integer. We prove that, considered as Hecke modules, this group and the cuspidal subgroup are both supported at the maximal Eisenstein ideals.