Singular values of principal moduli

Abstract

Let $g$ be a principal modulus with rational Fourier coefficients for a discrete subgroup of ${\mathrm{SL}}_2(\mathbb{R})$ between $\Gamma (N)$ or ${\Gamma }_0(N{)}^{†}$ for a positive integer $N$. Let $K$ be an imaginary quadratic field. We give a simple proof 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 $K$. Furthermore, we construct primitive generators of ray class fields of arbitrary moduli over $K$ in terms of Hasse's two generators.