Eisenstein ideal
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 , in studying the rational points of modular curves. An Eisenstein prime is a prime in the support of the Eisenstein ideal (this has nothing to do with primes in the Eisenstein integers). Definition. Let "N" be a rational prime, and define as the Jacobian variety of the modular curve There are endomorphisms "T" 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". There is also a Fricke involution "w" (and Atkin–Lehner involutions if "N" is composite). The Eisenstein ideal, in the (unital) subring of End("J") generated as a ring by the "T", is generated as an ideal by the elements for all "l" not dividing "N", and by Geometric definition. Suppose that "T"* is the ring generated by the Hecke operators acting on all modular forms for Γ("N ) (not just the cusp forms). The ring T" of Hecke operators on the cusp forms is a quotient of "T"*, so Spec("T") can be viewed as a subscheme of Spec("T"*). 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. The Eisenstein ideal is the ideal defining the intersection of the Eisenstein line with Spec("T") in Spec("T"*).