Construction of class fields over imaginary quadratic fields and applications

Let K be an imaginary quadratic field, H_O the ring class field of an order O in K and K_(N) be the ray class field modulo N over K for a positive integer N. In this paper we provide certain general techniques of finding H_O and K_(N) by using the theory of Shimura's canonical models via his reciprocity law, from which we partially extend some results of Schertz (Remark 4.2), Chen-Yui (Remark 4.2, Corollary 4.4), Cox-McKay-Stevenhagen (Corollary 4.5) and Cais-Conrad (Remark 5.3). And, we further reilluminate the classical result of Hasse by means of such a method (Corollary 5.4), and discover how to get one ray class invariant over K from Hasse's two generators (Corollary 5.5) which is different from Ramachandra's invariant [K. Ramachandra, Some applications of Kronecker's limit formulas, Ann. Math. 80 (1964), 104-148].