Composition laws of binary quadratic forms and isolations of quadratic forms

TL;DR

This paper investigates the concept of isolations in quadratic forms, proving non-existence results for binary and ternary cases and relating discriminants of quaternary isolations to class group properties.

Contribution

It establishes new non-existence results for certain quadratic form isolations and links composition laws to these properties.

Findings

No binary isolation of unary quadratic forms exists.

No ternary isolation of binary quadratic forms exists.

Discriminants of quaternary isolations are squares if class group conditions are met.

Abstract

A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an isolation of $f$. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binary quadratic form has no element of order $4$, then the discriminant of any quaternary isolation of it, if exists, is a square of an integer. The composition laws of primitive binary quadratic forms play an essential role in the proofs of the results.