The index of an Eisenstein ideal and multiplicity one

TL;DR

This paper generalizes Mazur's work on Eisenstein ideals from prime to square-free levels, computing their index and exploring the multiplicity one property in modular Jacobians.

Contribution

It extends the understanding of Eisenstein ideals to square-free levels and analyzes the dimension of torsion subgroups, establishing conditions for multiplicity one.

Findings

The index of Eisenstein ideals can be computed for square-free levels.

The dimension of m-torsion often equals 2, confirming multiplicity one in many cases.

The results generalize previous prime level cases to broader settings.

Abstract

Mazur's fundamental work on Eisenstein ideals of prime level has a variety of arithmetic applications. In this article, we generalize some of his work to square-free level. More specifically, we attempt to compute the index of an Eisenstein ideal and the dimension of the m-torsion of the modular Jacobian variety, where m is an Eisenstein maximal ideal. In many cases, the dimension of the m-torsion is 2, in other words, a multiplicity one theorem holds.