Abelian variety of CM-type
Encyclopedia
In mathematics
, an abelian variety
A defined over a field
K is said to have CM-type if it has a large enough commutative
subring
in its endomorphism ring
End(A). The terminology here is from complex multiplication
theory, which was developed for elliptic curve
s in the nineteenth century. One of the major achievements in algebraic number theory
and algebraic geometry
of the twentieth century was to find the correct formulations of the corresponding theory for abelian varieties of dimension
d > 1. The problem is at a deeper level of abstraction, because it is much harder to manipulate analytic function
s of several complex variables.
The formal definition is that
the tensor product
of End(A) with the rational number
field Q, should contain a commutative subring of dimension 2d over Z. When d = 1 this can only be a quadratic field
, and one recovers the cases where End(A) is an order in an imaginary quadratic field. For d > 1 there are comparable cases for CM-field
s, the complex quadratic extensions of totally real fields. There are other cases that reflect that A may not be a simple abelian variety (it might be a cartesian product
of elliptic curves, for example). Another name for abelian varieties of CM-type is abelian varieties with sufficiently many complex multiplications.
It is known that if K is the complex numbers, then any such A has a field of definition
which is in fact a number field. The possible types of endomorphism ring have been classified, as rings with involution (the Rosati involution), leading to a classification of CM-type abelian varieties. To construct such varieties in the same style as for elliptic curves, starting with a lattice
Λ in Cd, one must take into account the Riemann relations of abelian variety theory.
The CM-type is a description of the action of a (maximal) commutative subring L of EndQ(A) on the holomorphic tangent space
of A at the identity element
. Spectral theory
of a simple kind applies, to show that L acts via a basis of eigenvectors; in other words L has an action that is via diagonal matrices on the holomorphic vector fields on A. In the simple case, where L is itself a number field rather than a product of some number of fields, the CM-type is then a list of complex embeddings of L. There are 2d of those, occurring in complex conjugate
pairs; the CM-type is a choice of one out of each pair. It is known that all such possible CM-types can be realised.
Basic results of Goro Shimura
and Yutaka Taniyama
compute the Hasse-Weil L-function of A, in terms of the CM-type and a Hecke L-function with Hecke character
, having infinity-type derived from it. These generalise the results of Max Deuring
for the elliptic curve case.
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...
, an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
A defined over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K is said to have CM-type if it has a large enough commutative
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
subring
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
in its 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...
End(A). The terminology here is from complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
theory, which was developed for elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s in the nineteenth century. One of the major achievements in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
of the twentieth century was to find the correct formulations of the corresponding theory for abelian varieties of dimension
Dimension of an algebraic variety
In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V.For example, an algebraic curve has by definition dimension 1...
d > 1. The problem is at a deeper level of abstraction, because it is much harder to manipulate analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
s of several complex variables.
The formal definition is that
- EndQ(A),
the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of End(A) with the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
field Q, should contain a commutative subring of dimension 2d over Z. When d = 1 this can only be a quadratic field
Quadratic field
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...
, and one recovers the cases where End(A) is an order in an imaginary quadratic field. For d > 1 there are comparable cases for CM-field
CM-field
In mathematics, a CM-field is a particular type of number field K, so named for a close connection to the theory of complex multiplication. Another name used is J-field. Specifically, K is a quadratic extension of a totally real field which is totally imaginary, i.e...
s, the complex quadratic extensions of totally real fields. There are other cases that reflect that A may not be a simple abelian variety (it might be a cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of elliptic curves, for example). Another name for abelian varieties of CM-type is abelian varieties with sufficiently many complex multiplications.
It is known that if K is the complex numbers, then any such A has a field of definition
Field of definition
In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong...
which is in fact a number field. The possible types of endomorphism ring have been classified, as rings with involution (the Rosati involution), leading to a classification of CM-type abelian varieties. To construct such varieties in the same style as for elliptic curves, starting with a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
Λ in Cd, one must take into account the Riemann relations of abelian variety theory.
The CM-type is a description of the action of a (maximal) commutative subring L of EndQ(A) on the holomorphic tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
of A at the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
. Spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
of a simple kind applies, to show that L acts via a basis of eigenvectors; in other words L has an action that is via diagonal matrices on the holomorphic vector fields on A. In the simple case, where L is itself a number field rather than a product of some number of fields, the CM-type is then a list of complex embeddings of L. There are 2d of those, occurring in complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
pairs; the CM-type is a choice of one out of each pair. It is known that all such possible CM-types can be realised.
Basic results of Goro Shimura
Goro Shimura
is a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.Shimura was a colleague and a friend of Yutaka Taniyama...
and Yutaka Taniyama
Yutaka Taniyama
Yutaka Taniyama was a Japanese mathematician known for the Taniyama-Shimura conjecture.-Contribution:...
compute the Hasse-Weil L-function of A, in terms of the CM-type and a Hecke L-function with Hecke character
Hecke character
In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class ofL-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to...
, having infinity-type derived from it. These generalise the results of Max Deuring
Max Deuring
Max Deuring was a mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p...
for the elliptic curve case.