Complex multiplication
Encyclopedia
In mathematics
, complex multiplication is the theory of elliptic curve
s E that have an endomorphism ring
larger than the integer
s; and also the theory in higher dimensions of abelian varieties
A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space
at the identity element
of A is a direct sum
of one-dimensional modules
). Put another way, it contains the theory of elliptic function
s with extra symmetries, such as are visible when the period lattice is the Gaussian integer
lattice
or Eisenstein integer
lattice.
It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables
, are then 'very special' functions satisfying extra identities and taking evaluable special values at particular points. It has also turned out to be a central theme in algebraic number theory
, allowing some features of the theory of cyclotomic field
s to be carried over to wider areas of application.
David Hilbert
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.
where Z[i] is the Gaussian integer
ring, and θ is any non-zero complex number. Any such complex torus
has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as
having an order 4 automorphism
sending
in line with the action of i on the Weierstrass elliptic functions.
This is a typical example of an elliptic curve with complex multiplication. Over the complex number field
such curves are all found as such quotients
in which some order in the ring of integers
in an imaginary quadratic field
takes the place of the Gaussian integers.
, there are always non-trivial endomorphisms of an elliptic curve; so the complex multiplication case is in a sense typical (and the terminology isn't 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
.
first postulated that the values of elliptic function
s at torsion points should be enough to generate all abelian extension
s 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; and was certainly what had prompted Hilbert's remark above, since it makes explicit class field theory
in the way the roots of unity do for abelian extensions of the rational number field
. 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.
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
is a unique factorization domain
.
Here
satisfies α² = α − 41. In general, S[α] denotes the set of all polynomial
expressions in α with coefficients in S, which is the smallest ring containing α and S. Because α satisifies this quadratic equation, the required polynomials can be limited to degree one.
Alternatively,
an internal structure due to certain Eisenstein series, and with similar simple expressions for the other Heegner number
s.
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...
, complex multiplication is the theory of 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 E that have an 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...
larger than the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s; and also the theory in higher dimensions of abelian varieties
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 having enough endomorphisms in a certain precise sense (it roughly means that the action on the 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....
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...
of A is a direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
of one-dimensional modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
). Put another way, it contains the theory of elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
s with extra symmetries, such as are visible when the period lattice is the Gaussian integer
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
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...
or Eisenstein integer
Eisenstein integer
In mathematics, Eisenstein integers , also known as Eulerian integers , are complex numbers of the formz = a + b\omega \,\!where a and b are integers and...
lattice.
It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
, are then 'very special' functions satisfying extra identities and taking evaluable special values at particular points. It has also turned out to be a central theme 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,...
, allowing some features of the theory of cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s to be carried over to wider areas of application.
David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
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.
Example
An example of an elliptic curve with complex multiplication is- C/Z[i]θ
where Z[i] is the Gaussian integer
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
ring, and θ is any non-zero complex number. Any such complex torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as
- Y2 = 4X3 − aX,
having an order 4 automorphism
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 automorphisms of an object forms a group, called the automorphism...
sending
- Y → −iY, X → −X
in line with the action of i on the Weierstrass elliptic functions.
This is a typical example of an elliptic curve with complex multiplication. Over the complex number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
such curves are all found as such quotients
- complex plane/period lattice
in which some order in the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
in an imaginary 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...
takes the place of the Gaussian integers.
Abstract theory of endomorphisms
When the base field is a finite fieldFinite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
, there are always non-trivial endomorphisms of an elliptic curve; so the complex multiplication case is in a sense typical (and the terminology isn't 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
Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
.
Kronecker and abelian extensions
KroneckerLeopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
first postulated that the values of elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
s at torsion points should be enough to generate all abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
s for imaginary quadratic fields, an idea that went back to Eisenstein in some cases, and even to Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
. This became known as the Kronecker Jugendtraum; and was certainly what had prompted Hilbert's remark above, since it makes explicit class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
in the way the roots of unity do for abelian extensions of the rational number field
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...
. 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
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
is a unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
.
Here
satisfies α² = α − 41. In general, S[α] denotes the set of all polynomialPolynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
expressions in α with coefficients in S, which is the smallest ring containing α and S. Because α satisifies this quadratic equation, the required polynomials can be limited to degree one.
Alternatively,
an internal structure due to certain Eisenstein series, and with similar simple expressions for the other Heegner number
Heegner number
In number theory, a Heegner number is a square-free positive integer d such that the imaginary quadratic field Q has class number 1...
s.
See also
- Abelian variety of CM-typeAbelian variety of CM-typeIn 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. The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century...
, higher dimensions - Lubin–Tate formal group, local fieldLocal fieldIn mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s - Drinfel'd shtuka, function fieldFunction fieldFunction field may refer to:*Function field of an algebraic variety*Function field...
case
External links
- Complex multiplication from PlanetMath.orgPlanetMathPlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is hosted by the Digital...
- Examples of elliptic curves with complex multiplication from PlanetMath.orgPlanetMathPlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is hosted by the Digital...
- Galois Representations and Modular Forms by Kenneth A. RibetKenneth Alan RibetKenneth Alan "Ken" Ribet is an American mathematician, currently a professor of mathematics at the University of California, Berkeley. His mathematical interests include algebraic number theory and algebraic geometry....
(Bulletin of the American Mathematical SocietyBulletin of the American Mathematical SocietyThe Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society...
, Volume 32, Number 4, October 1995, Pages 375-402)