Quaternions and universal quadratic forms over number fields

TL;DR

This paper explores quadratic forms over totally real number fields using quaternion rings, proving universality and representation properties through geometric and algebraic methods.

Contribution

It introduces a quaternion-based approach to analyze quadratic forms over number fields, establishing universality and representation results for specific forms.

Findings

Proved that a specific quadratic form is universal over (\u03b6_7+(_7^{-1})

Showed certain quadratic forms represent all totally positive multiples of special elements

Used geometry of numbers to find elements of small norm in quaternion ideals

Abstract

We study quadratic forms over totally real number fields by using an associated ring of quaternions. We examine some properties of residue class rings of these quaternions and use geometry of numbers to prove that certain ideals of the ring of quaternions contain elements of a small norm. We prove that $x^2+y^2+z^2+w^2+xy+xz+xw$ is universal over $\mathbb{Q}(\zeta_7+\zeta_7^{-1})$ and that $x^2+xy+y^2+z^2+zw+w^2$ represents all totally positive multiples of certain special elements.