Theorem 3.18 generalizes in a natural way to the more general situation in which is replaced by an order . Then the Hilbert class field is replaced by the ray class field , which is a finite abelian extension of that is unramified outside (see Definition 3.13 above). There is an elliptic curve defined over whose endomorphism ring is , and the set of -conjugates of forms a set of representatives for . Moreover, the group of Theorem 3.15 acts simply transitively on , and the action of on the set of conjugates of is consistent with the Artin reciprocity map.
William 2007-05-25