Eisenstein ideal

In mathematics, the Eisenstein ideal is a certain ideal in the endomorphism ring of the Jacobian variety of a modular curve. It was introduced by Barry Mazur in 1977, in studying the rational points of modular curves. The endomorphism ring in question is closely associated with a Hecke algebra, and the name comes from the way the definition in detail follows the action of Hecke operators on Eisenstein series.

Let "N" be a positive integer, and define

:"J"₀("N") = "J"

be 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 classes, which gives the action on "J". There is also an involution "w". 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, B. "Modular curves and the Eisenstein ideal." Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 33--186 (1978).