Eisenstein ideal
Encyclopedia
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 operator
s on Eisenstein series
.
Let N be a positive integer, 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 an involution w. The Eisenstein ideal, in the (unital) subring of End(J) generated as a ring by the Tl, is generated as an ideal by the elements
for all l not dividing N, and by
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Eisenstein ideal is a certain ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in the endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
of the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
of a modular curve
Modular curve
In number theory and algebraic geometry, a modular curve Y 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 group of integral 2×2 matrices SL...
. It was introduced by Barry Mazur
Barry Mazur
-Life:Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959,...
in 1977, in studying the rational points of modular curves. The endomorphism ring in question is closely associated with a Hecke algebra
Hecke algebra
In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a one-parameter deformation of the group algebra of a Coxeter group....
, and the name comes from the way the definition in detail follows the action of Hecke operator
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 representations....
s on Eisenstein series
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...
.
Let N be a positive integer, and define
- J0(N) = J
as the Jacobian variety of the modular curve
- X0(N) = X.
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
Correspondence (mathematics)
In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings.* In general mathematics, correspondence is an alternative term for a relation between two sets...
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 Tl, is generated as an ideal by the elements
- Tl − l - 1
for all l not dividing N, and by
- w + 1.