Complex multiplication

From Academic Kids

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). David Hilbert is said to have remarked that the theory of complex multiplication is the most beautiful part of mathematics.

Indeed, it is no accident that

<math>e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007...<math>

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

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

is a unique factorization domain.

An example of an elliptic curve with complex multiplication is

C/Z[i

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 = 4X3aX,

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 elliptic curve with complex multiplication, in the sense that over the complex number field they 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.

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 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.

See also: abelian variety of CM-type, Lubin-Tate formal group, Drinfel'd shtuka.

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