Torsion points and Galois representations on CM elliptic curves

We prove three theorems on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. The first theorem computes the Weber function field of the $N$-torsion subgroup of a CM elliptic curve over an arbitrary order as an explicit class field, generalizing the classical Main Theorem of Complex Multiplication. The second theorem determines the degrees in which such a CM elliptic curve has a rational point of order $N$. The third theorem bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. We give several applications.