The index of an Eisenstein ideal and multiplicity one

Abstract

Mazur’s fundamental work on the Eisenstein ideal 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 \({\mathfrak {m}}\)-torsion of the modular Jacobian variety, where \({\mathfrak {m}}\) is an Eisenstein maximal ideal. In many cases, the dimension of the \({\mathfrak {m}}\)-torsion is 2, in other words, a multiplicity one theorem holds.