Complex Multiplication

Complex multiplication is the theory of elliptic curves whose endomorphism ring is strictly larger than the integers. For an elliptic curve E over the complex numbers, the endomorphism ring is either ℤ or an order in an imaginary quadratic field. The theory connects the arithmetic of imaginary quadratic fields to the geometry of elliptic curves, modular forms, and Galois representations. The j-invariant of an elliptic curve with complex multiplication is an algebraic integer, and the Hilbert class field of the associated quadratic field is generated by adjoining this j-invariant. This is the simplest case of the deep correspondence between the arithmetic of fields and the geometry of curves that drives much of modern number theory.

Complex multiplication is not a special property of a few curves. It is a structural principle: the elliptic curves with complex multiplication are the fixed points of the action of the modular group on the upper half-plane, and their j-invariants generate the abelian extensions of quadratic fields. The theory of complex multiplication was developed by Kronecker, Weber, and Deuring, and it remains a prototype for the Langlands program and the study of modular forms. The connection between the class number of an imaginary quadratic field and the number of isomorphism classes of elliptic curves with complex multiplication by that field is a direct manifestation of the structural equivalence between arithmetic and geometry.