mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Eisenstein ideal is an

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

in the

endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a p ...

of the

Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian vari ...

of a

modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular grou ...

, consisting roughly of elements of the

Hecke algebra In mathematics, the Hecke algebra is the algebra generated by Hecke operators. Properties The algebra is a commutative ring. In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ''n'' coprime to the level acting on ...

Hecke operator In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic repr ...

s that annihilate the

Eisenstein series Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...

. 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 :''J''₀(''N'') = ''J'' as the Jacobian variety of the modular curve :''X''₀(''N'') = ''X''. There are endomorphisms ''T''_''l'' 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 class In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumfo ...

es, which gives the action on ''J''. There is also a

Fricke involution In mathematics, a Fricke involution is the involution of the modular curve ''X''0(''N'') given by τ → –1/''N''τ. It is named after Robert Fricke Karl Emanuel Robert Fricke (24 September 1861 – 18 July 1930) was a German math ...

''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''_''l'', is generated as an ideal by the elements : ''T''_''l'' − ''l'' - 1 for all ''l'' not dividing ''N'', and by :''w'' + 1.

Geometric definition

Suppose that ''T''* is the ring generated by the Hecke operators acting on all modular forms for Γ₀(''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''*).

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 ''T''_''n'' 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

''E''₁₂ and the

modular discriminant In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by th ...

Δ. The map taking a Hecke operator ''T''_''n'' to its eigenvalues (σ₁₁(''n''),τ(n)) gives a homomorphism from ''T'' into the ring Z×Z (where τ is the

Ramanujan tau function The Ramanujan tau function, studied by , is the function \tau : \mathbb \rarr\mathbb defined by the following identity: :\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z), where with , \phi is the Euler function, is the ...

and σ₁₁(''n'') is the sum of the 11th powers of the divisors of ''n''). The image is the set of pairs (''c'',''d'') with ''c'' and ''d'' congruent mod 691 because of Ramanujan's congruence σ₁₁(''n'') ≡ τ(n) 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 (691).

References

* *{{Citation , last1=Mazur , first1=Barry , author1-link=Barry Mazur , last2=Serre , first2=Jean-Pierre , author2-link=Jean-Pierre Serre , title=Séminaire Bourbaki (1974/1975), Exp. No. 469 , url=http://www.numdam.org/item?id=SB_1974-1975__17__238_0 , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, location=Berlin, New York , series= Lecture Notes in Math. , mr=0485882 , year=1976 , volume=514 , chapter=Points rationnels des courbes modulaires X₀(N) (d'après A. Ogg) , pages=238–255 Modular forms Abelian varieties