Kitaoka's Conjecture and sums of squares

TL;DR

This paper links the existence of certain universal quadratic forms over totally real fields to sums of squares properties, confirming Kitaoka's Conjecture for fields with odd discriminant.

Contribution

It establishes a connection between universal quadratic forms and sums of squares in totally real fields, proving Kitaoka's Conjecture for fields with odd discriminant.

Findings

Kitaoka's Conjecture holds for all fields with odd discriminant.

Universal quadratic forms relate to sums of squares in specific number fields.

Conditions involving units and the presence of √2 affect the form's universality.

Abstract

We connect the existence of a ternary classical universal quadratic form over a totally real number field $K$ with the property that all totally positive multiples of 2 are sums of squares (if $K$ does not contain $\sqrt 2$ or contains a nonsquare totally positive unit). In particular, we get that Kitaoka's Conjecture holds for all fields of odd discriminant.