Singular values of principal moduli

Abstract

Let g be a principal modulus with rational Fourier coefficients for a discrete subgroup of SL 2 ( R ) lying in between Γ ( N ) and Γ 0 ( N ) † for a positive integer N . Let K be an imaginary quadratic field. We introduce a relatively simple proof, without using Shimuraʼs canonical model, of the fact that the singular value of g generates the ray class field modulo N or the ring class field of the order of conductor N over K . Further, we construct a primitive generator of the ray class field K c of arbitrary modulus c over K from Hasseʼs two generators.