Complex multiplication

This article is about certain endomorphism rings. For information about multiplication of complex numbers, see complex numbers.

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 (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 functions 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 fields 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.

Example

An example of an elliptic curve with complex multiplication is

C/Z[i]θ

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

Y² = 4X³ − aX,

having an order 4 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 such curves are all found as such quotients

complex plane/period lattice

in which some order in the ring of integers in an imaginary quadratic field takes the place of the Gaussian integers.

Abstract theory of endomorphisms

When the base field is a finite field, 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.

Kronecker and abelian extensions

Kronecker first postulated that the values of elliptic functions 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; 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.

Sample consequence

It is no accident that

$e^{\pi {\sqrt {163}}}=262537412640768743.99999999999925007\dots \,$

or equivalently,

$e^{\pi {\sqrt {163}}}=640320^{3}+743.99999999999925007\dots \,$

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

$\mathbf {Z} \left[{\frac {1+{\sqrt {-163}}}{2}}\right]$

is a unique factorization domain.

Here 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,

$e^{\pi {\sqrt {163}}}=12^{3}(231^{2}-1)^{3}+743.99999999999925007\dots \,$

an internal structure due to certain Eisenstein series, and with similar simple expressions for the other Heegner numbers.

See also

• Abelian variety of CM-type, higher dimensions
• Lubin–Tate formal group, local fields
• Drinfel'd shtuka, function field case

References

• Borel, A.; Chowla, S.; Herz, C. S.; Iwasawa, K.; Serre, J.-P. Seminar on complex multiplication. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. Lecture Notes in Mathematics, No. 21 Springer-Verlag, Berlin-New York, 1966
• Serge Lang, Complex multiplication. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 255. Springer-Verlag, New York, 1983. ISBN 0-387-90786-6
• Goro Shimura, Abelian varieties with complex multiplication and modular functions. Princeton Mathematical Series, 46. Princeton University Press, Princeton, NJ, 1998. ISBN 0-691-01656-9