Complex Multiplication

Abstract

Most elliptic curves over ℂ have only the multiplication-by-m endomor-phisms. An elliptic curve that possesses extra endomorphisms is said to have complex multiplication, or CM for short. Such curves have many special properties. For example, the endomorphism ring of a CM curve E is an order in a quadratic imaginary field K, and the j-invariant and torsion points of E generate abelian extensions of K. This is analogous to the way in which the torsion points of G _m (ℂ) = ℂ* generate abelian extensions of ℚ. An important result in the cyclotomic theory is the Kronecker-Weber Theorem, which says that every abelian extension of ℚ is contained in a cyclotomic extension. We will prove corresponding results for a quadratic imaginary field K. For example, we will show how to construct an elliptic curve E such that K(j(E)) is the Hilbert class field of K, and we will explain how to use the torsion points of E to generate the maximal abelian extension of K.