In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, complex multiplication (CM) is the theory of
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. I ...
s ''E'' that have an
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 a ...
larger than the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. Put another way, it contains the theory of
elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those ...
s with extra symmetries, such as are visible when the
period lattice is the
Gaussian integer lattice or
Eisenstein integer
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
:z = a + b\omega ,
where and are integers and
:\omega = \f ...
lattice.
It has an aspect belonging to the theory of
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined b ...
s, because such elliptic functions, or
abelian functions of
several complex variables
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 ...
, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
, allowing some features of the theory of
cyclotomic fields to be carried over to wider areas of application.
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.
There is also the
higher-dimensional complex multiplication theory of
abelian varieties ''A'' having ''enough'' endomorphisms in a certain precise sense, roughly that the action on the
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 ...
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 su ...
of ''A'' is a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of one-dimensional
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
.
Example of the imaginary quadratic field extension
Consider an imaginary quadratic field
.
An elliptic function
is said to have complex multiplication if there is an algebraic relation between
and
for all
in
.
Conversely, Kronecker conjectured – in what became known as the ''
Kronecker Jugendtraum
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analog ...
'' – that every abelian extension of
could be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication. To this day this remains one of the few cases of
Hilbert's twelfth problem
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analog ...
which has actually been solved.
An example of an elliptic curve with complex multiplication is
:
where Z
'i''is the
Gaussian integer ring, and ''θ'' is any non-zero complex number. Any such complex
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as
:
for some
, which demonstrably has two conjugate order-4
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s sending
:
in line with the action of ''i'' on the
Weierstrass elliptic function
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by ...
s.
More generally, consider the lattice Λ, an additive group in the complex plane, generated by
. Then we define the Weierstrass function of the variable
in
as follows:
:
and
:
:
Let
be the derivative of
. Then we obtain an isomorphism of complex Lie groups:
:
from the complex torus group
to the projective elliptic curve defined in homogeneous coordinates by
:
and where the point at infinity, the zero element of the group law of the elliptic curve, is by convention taken to be
.
If the lattice defining the elliptic curve is actually preserved under multiplication by (possibly a proper subring of) the ring of integers
of
, then the ring of analytic automorphisms of
turns out to be isomorphic to this (sub)ring.
If we rewrite
where
and
, then
:
This means that the
j-invariant of
is an
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of th ...
– lying in
– if
has complex multiplication.
Abstract theory of endomorphisms
The ring of endomorphisms of an elliptic curve can be of one of three forms: the integers Z; 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 number field; or an order in a definite
quaternion algebra over Q.
When the field of definition is a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
, there are always non-trivial endomorphisms of an elliptic curve, coming from the
Frobenius map, so every such curve has ''complex multiplication'' (and the terminology is not often applied). But when the base field is a number field, complex multiplication is the exception. It is known that, in a general sense, the case of complex multiplication is the hardest to resolve for the
Hodge conjecture.
Kronecker and abelian extensions
Kronecker first postulated that the values of
elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those ...
s at torsion points should be enough to generate all
abelian extensions for imaginary quadratic fields, an idea that went back to
Eisenstein in some cases, and even to
Gauss. This became known as the ''
Kronecker Jugendtraum
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analog ...
''; and was certainly what had prompted Hilbert's remark above, since it makes explicit
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is cre ...
in the way the
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
do for abelian extensions of the
rational number field
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 rat ...
, via
Shimura's reciprocity law
In mathematics, Shimura's reciprocity law, introduced by , describes the action of ideles of imaginary quadratic fields on the values of modular functions at singular moduli. It forms a part of the Kronecker Jugendtraum, explicit class field ...
.
Indeed, let ''K'' be an imaginary quadratic field with class field ''H''. Let ''E'' be an elliptic curve with complex multiplication by the integers of ''K'', defined over ''H''. Then the
maximal abelian extension of ''K'' is generated by the ''x''-coordinates of the points of finite order on some Weierstrass model for ''E'' over ''H''.
Many generalisations have been sought of Kronecker's ideas; they do however lie somewhat obliquely to the main thrust of the
Langlands philosophy
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic ...
, and there is no definitive statement currently known.
Sample consequence
It is no accident that
:
or equivalently,
:
is so close to an integer. This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of
modular forms, and the fact that
: