Can we recover an integral quadratic form by representing all its subforms?

TL;DR

This paper investigates whether quadratic forms over totally real number fields can be reconstructed from their subforms, revealing conditions under which subform representation guarantees the original form's representation.

Contribution

It establishes that indefinite forms can be recovered from their subforms, introduces a new characterization of positive definite decomposable forms, and generalizes a finiteness theorem for quadratic form representations over number fields.

Findings

Indefinite forms are determined by their subforms.

Counterexamples exist for positive definite indecomposable forms.

A finiteness property for representing classes of forms over number fields.

Abstract

Let $\mathfrak o$ be the ring of integers of a totally real number field. If $f$ is a quadratic form over $\mathfrak o$ and $g$ is another quadratic form over $\mathfrak o$ which represents all proper subforms of $f$, does $g$ represent $f$? We show that if $g$ is indefinite, then $g$ indeed represents $f$. However, when $f$ is positive definite and indecomposable, then there exists a $g$ which represents all proper subforms of $f$ but not $f$ itself. Along the way we give a new characterization of positive definite decomposable quadratic forms over $\mathfrak o$ and a number-field generalization of the finiteness theorem of representations of quadratic forms by quadratic forms over $\mathbb Z$ which asserts that given any infinite set $\mathscr S$ of classes of positive definite integral quadratic forms over $\mathfrak o$ of a fixed rank, there exists a finite subset $\mathscr S_0$ of $\mathscr S$ with the property that a positive definite quadratic form over $\mathfrak o$ represents all classes in $\mathscr S$ if and only if it represents all classes in $\mathscr S_0$.