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 $\mathfrak{n}$.