Binary quadratic forms and ray class groups

Abstract

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_{\mathfrak{n}}$ be the ray class field of $K$ modulo $\mathfrak{n}=N{\mathcal{O}}_K$. By using the congruence subgroup $\pm {\Gamma }_1(N)$, we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group $\mathrm{Gal}(K_{\mathfrak{n}}/K)$. We also present algorithms to find all form classes and show how to multiply two form classes. As an application, we describe $\mathrm{Gal}(K_{\mathfrak{n}}^{\mathrm{ab}}/K)$ in terms of these extended form class groups for which $K_{\mathfrak{n}}^{\mathrm{ab}}$ is the maximal abelian extension of $K$ unramified outside prime ideals dividing $\mathfrak{n}$.

Cited by in corpus (1)

• On some extension of Gauss' work and applications