On some extension of Gauss' work and applications (II)

TL;DR

This paper extends Gauss's classical theory of binary quadratic forms to relate primitive forms with ray class groups of imaginary quadratic fields, providing explicit isomorphisms to Galois groups of ray class fields.

Contribution

It introduces a new equivalence relation on quadratic forms and constructs explicit isomorphisms to ray class groups and Galois groups, extending classical composition theory.

Findings

Defines an equivalence relation on quadratic forms related to ray class groups

Establishes an explicit isomorphism between quadratic form classes and Galois groups

Generalizes Gauss's composition theory to ray class fields

Abstract

Let $K$ be an imaginary quadratic field of discriminant $d_K$, and let $\mathfrak{n}$ be a nontrivial integral ideal of $K$ in which $N$ is the smallest positive integer. Let $\mathcal{Q}_N(d_K)$ be the set of primitive positive definite binary quadratic forms of discriminant $d_K$ whose leading coefficients are relatively prime to $N$. We adopt an equivalence relation $\sim_\mathfrak{n}$ on $\mathcal{Q}_N(d_K)$ so that the set of equivalence classes $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ can be regarded as a group isomorphic to the ray class group of $K$ modulo $\mathfrak{n}$. We further present an explicit isomorphism of $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ onto $\mathrm{Gal}(K_\mathfrak{n}/K)$ in terms of Fricke invariants, where $K_\mathfrak{n}$ is the ray class field of $K$ modulo $\mathfrak{n}$. This would be certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet.