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 that annihilate the Eisenstein series. It was introduced by Barry Mazur (1977), 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).

Let N be a rational prime, and define

J₀(N) = J

as the Jacobian variety of the modular curve

X₀(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 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_l, is generated as an ideal by the elements

T_l − l - 1

for all l not dividing N, and by

w + 1.

References

• Mazur, Barry (1977), "Modular curves and the Eisenstein ideal", Publications Mathématiques de l'IHÉS (47): 33–186, ISSN 1618-1913, MR 488287 
• Mazur, Barry; Serre, Jean-Pierre (1976), "Points rationnels des courbes modulaires X₀(N) (d'après A. Ogg)", Séminaire Bourbaki (1974/1975), Exp. No. 469, Lecture Notes in Math. 514, Berlin, New York: Springer-Verlag, pp. 238–255, MR 0485882