In mathematics, the Eisenstein ideal is an ideal in the endomorphism ring of the Jacobian variety of a modular curve, consisting roughly of elements of the Hecke algebra of Hecke operators that annihilate the Eisenstein series.
It was introduced by Barry Mazur (1977), in studying the rational points of modular curves.
Let N be a rational prime, and define as the Jacobian variety of the modular curve There are endomorphisms Tl of J for each prime number l not dividing N. These come from the Hecke operator, considered first as an algebraic correspondence on X, and from there as acting on divisor classes, which gives the action on J.
The Eisenstein ideal, in the (unital) subring of End(J) generated as a ring by the Tl, is generated as an ideal by the elements for all l not dividing N, and by Suppose that T* is the ring generated by the Hecke operators acting on all modular forms for Γ0(N) (not just the cusp forms).
Similarly Spec(T*) contains a line (called the Eisenstein line) isomorphic to Spec(Z) coming from the action of Hecke operators on the Eisenstein series.