Binary quadratic forms and ray class groups

TL;DR

This paper constructs an extended form class group related to binary quadratic forms and ray class fields of imaginary quadratic fields, providing explicit isomorphisms, algorithms, and descriptions of Galois groups.

Contribution

It introduces a new extended form class group using congruence subgroups and demonstrates its isomorphism to ray class field Galois groups, with practical algorithms and applications.

Findings

Constructed an extended form class group isomorphic to Galois groups.

Provided algorithms for class group computation and multiplication.

Described the maximal abelian extension in terms of these form 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 $\mathfrak{n}$.