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.

Definition

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. There is also a Fricke involution w. The Eisenstein ideal, in the subring of End generated as a ring by the Tl, 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 Γ0. The ring T of Hecke operators on the cusp forms is a quotient of T*, so Spec can be viewed as a subscheme of Spec. Similarly Spec contains a line isomorphic to Spec 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 in Spec.

Example