In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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. If the ...
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 ...
larger than the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s. Put another way, it contains the theory of
elliptic function
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...
s with extra symmetries, such as are visible when the
period lattice is the
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf ...
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 = \frac ...
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 by ...
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 functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...
, 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 ob ...
, allowing some features of the theory of
cyclotomic field
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...
s to be carried over to wider areas of application.
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
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
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular f ...
''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 is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
at the
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
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. As an example, the direct sum of two abelian groups A and B is anothe ...
of one-dimensional
modules.
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
Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity th ...
'' – 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
Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity t ...
which has actually been solved.
An example of an elliptic curve with complex multiplication is
:
where Z
'i''is the
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf ...
ring, and ''θ'' is any non-zero complex number. Any such complex
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
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 automorphism ...
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 is also referred to as ℘-functions and they are usually denoted by the s ...
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
In mathematics, Felix Klein's -invariant or function is a modular function of weight zero for the special linear group \operatorname(2,\Z) defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic a ...
of
is an
algebraic number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
– 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:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
in an
imaginary quadratic number field; or an order in a definite
quaternion algebra
In mathematics, a quaternion algebra over a field (mathematics), field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension (vector space), dimension 4 ove ...
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 (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
, there are always non-trivial endomorphisms of an elliptic curve, coming from the
Frobenius map
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class that includes finite fields. The endomorphism m ...
, 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
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
In simple terms, the Hodge conjectur ...
.
Kronecker and abelian extensions
Kronecker first postulated that the values of
elliptic function
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...
s at torsion points should be enough to generate all
abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galoi ...
s for imaginary quadratic fields, an idea that went back to
Eisenstein in some cases, and even to
Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
. This became known as the ''
Kronecker Jugendtraum
Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity th ...
''; 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 credit ...
in the way the
roots of unity
In mathematics, a root of unity 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 number theory, the theory of group char ...
do for abelian extensions of the
rational number field, via
Shimura's reciprocity law.
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, and there is no definitive statement currently known.
Sample consequence
It is no accident that
Ramanujan's constant
In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, the ring of algebraic integers of \Q\left ...
, the
transcendental number
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . ...
:
or equivalently,
:
is an
almost integer, in that it is
very close to an
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of
modular forms
In mathematics, a modular form is a holomorphic function on the Upper half-plane#Complex plane, complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the Group action (mathematics), group action of the ...
, and the fact that
: