Composition law and complex multiplication

Abstract

Let K be an imaginary quadratic field of discriminant dK, and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let QN(dK) be the set of primitive positive definite binary quadratic forms of discriminant dK whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼n on QN(dK) so that the set of equivalence classes QN(dK)/∼n can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of QN(dK)/∼n onto Gal(Kn/K) in terms of Fricke invariants, where Kn denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet.