In mathematics, an

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic 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 functi ...

''A'' defined over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K'' is said to have CM-type if it has a large enough

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...

in its

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 ...

End(''A''). The terminology here is from

complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visibl ...

theory, which was developed for

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

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 In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

''d'' > 1. The problem is at a deeper level of abstraction, because it is much harder to manipulate

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. Functions of each type are infinitely differentiable, but complex ...

s of

several complex variable The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...

s. The formal definition is that :

\operatorname_\mathbb(A)

the

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...

of End(''A'') with the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

field Q, should contain a commutative subring of

2''d'' over Q. When ''d'' = 1 this can only be a

quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...

, and one recovers the cases where End(''A'') is an

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...

in an

imaginary quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...

. For ''d'' > 1 there are comparable cases for

CM-field In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field. The abbreviation "CM" was introduced by . Formal definition A number field ' ...

s, the complex

quadratic extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...

s of

totally real field In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyn ...

s. There are other cases that reflect that ''A'' may not be a

simple abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic 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 funct ...

(it might be a

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...

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 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. Given polynomials, with coefficients in a field ''K'', it may not be ...

which is in fact a

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

. The possible types of endomorphism ring have been classified, as rings with

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...

(the

Rosati involution In mathematics, a Rosati involution, named after Carlo Rosati, is an involution of the rational endomorphism ring of an abelian variety induced by a polarization. Let A be an abelian variety, let \hat = \mathrm^0(A) be the dual abelian variety, ...

), 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 C^''d'', one must take into account the

Riemann relations In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data: * A lattice Λ in a complex vector space Cg. * An alternating bilinear form α from Λ to the integers satisfying the following Riemann b ...

of abelian variety theory. The CM-type is a description of the action of a (maximal) commutative subring ''L'' of End_Q(''A'') on the holomorphic

tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

of ''A'' at the

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...

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 a simple kind applies, to show that ''L'' acts via a basis of

eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...

s; in other words ''L'' has an action that is via

diagonal matrices In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...

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 2''d'' of those, occurring in

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

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 was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multip ...

and

Yutaka Taniyama was a Japanese mathematician known for the Taniyama–Shimura conjecture. Contribution Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and r ...

compute the Hasse–Weil L-function of ''A'', in terms of the CM-type and a Hecke L-function with

Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of ''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ...

, having infinity-type derived from it. These generalise the results of

Max Deuring Max Deuring (9 December 1907 – 20 December 1984) was a German mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p. He worked also in analytic number theory. Deuring graduated f ...

for the elliptic curve case.

References

* {{DEFAULTSORT:Abelian Variety Of Cm-Type Abelian varieties Arithmetic geometry