Algebraic Complex Multiplication

Abstract

This chapter contains the first fundamental theory of complex multiplication. When an abelian variety has a sufficiently large ring of endomorphisms, then the Frobenius endomorphism of the variety mod p can be represented as the reduction mod p of an element in that ring, which is, say, the ring of integers in a number field K. If π is that element, then a basic theorem gives the ideal factorization of π in D_K. We have followed Shimura-Taniyama for the proof of this result. On the other hand, Shimura in his book [Sh 1] gave a formulation in terms of ideles, and suggested that one could give a proof for this more general form, directly from the factorization theorem. We have carried out this approach.