In Barry Mazur's famous 1977 article on the Eisenstein ideal, he defined he Eisenstein ideal in the Hecke ring of J0(p) in a somewhat ad hoc manner and then went on to prove that this ideal is the annihilator of every imaginable possible "Eisenstein-like" object associated to J0(p).

For generalizations (e.g., to J0(N), where N is no longer prime) it is best to define the Eisenstein ideal in relation to the actual Eisenstein series that are lurking alongside the relevant space of cusp forms. This ideal will then again annihilate all of the Eisenstein-like objects that one can imagine, but it tends not to be the exact annihilator of most of the objects. Nonetheless, is is likely that one can prove that the Eisenstein ideal is the precise annihilator of the cuspidal subgroup of the Jacobian in the generalization. Although one can probably work out a proof by transcendental methods, the aim of my talk will be to explain the elements of an arithmetic proof.