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

  • The Eisenstein ideal can also be defined for higher weight modular forms. Suppose that T is the full Hecke algebra generated by Hecke operators Tn acting on the 2-dimensional space of modular forms of level 1 and weight 12.This space is 2 dimensional, spanned by the Eigenforms given by the Eisenstein series E12 and the modular discriminant Δ. The map taking a Hecke operator Tn to its eigenvalues,τ) gives a homomorphism from T into the ring Z×'Z. The image is the set of pairs with c'' and d congruent mod 691 because of Ramanujan's congruence σ11 ≡ τ mod 691. The Hecke algebra of Hecke operators acting on the cusp form Δ is just isomorphic to Z'''. If we identify it with Z then the Eisenstein ideal is.