Form class groups and class fields of CM-fields

TL;DR

This paper constructs class groups of binary quadratic forms over totally real fields and relates them to ray class groups of CM-fields, also describing associated class fields via Hilbert modular functions.

Contribution

It introduces a new method to explicitly realize ray class groups of CM-fields using quadratic forms over totally real fields and constructs related class fields through modular functions.

Findings

Class groups of quadratic forms over $F$ are isomorphic to ray class groups of $K$.

Explicit class fields of the reflex field are constructed using Hilbert modular functions.

The approach generalizes classical complex multiplication theory to higher degree fields.

Abstract

Let $F$ be a totally real number field of class number one, and let $K$ be a CM-field with $F$ as its maximal real subfield. For each positive integer $N$, we construct a class group of certain binary quadratic forms over $F$ which is isomorphic to the ray class group of $K$ modulo $N$. Assuming further that the narrow class number of $F$ is one, we construct a class field of the reflex field of $K$ in terms of the singular values of Hilbert modular functions.