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 (Formula presented.) -torsion of the modular Jacobian variety, where (Formula presented.) is an Eisenstein maximal ideal. In many cases, the dimension of the (Formula presented.) -torsion is 2, in other words, a multiplicity one theorem holds. © 2015, Springer-Verlag Berlin Heidelberg