Singular values of principal moduli

TL;DR

This paper proves that singular values of principal moduli generate specific class fields over imaginary quadratic fields and constructs primitive generators for these fields using Hasse's generators.

Contribution

It provides a simple proof that singular values of principal moduli generate ray class fields and constructs primitive generators for arbitrary moduli over imaginary quadratic fields.

Findings

Singular values of principal moduli generate ray class fields.

Constructs primitive generators for ray class fields of arbitrary moduli.

Provides a simplified proof of known class field generation results.

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 $\Gamma_0(N)^\dag$ 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.