CONSTRUCTION OF CLASS FIELDS OVER IMAGINARY QUADRATIC FIELDS AND APPLICATIONS

Abstract

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].