Weakly Hyperbolic Involutions

TL;DR

This paper provides a new proof of Pfister's Local-Global Principle for quadratic forms and involutions, extending it to preorderings and exploring the relation between nilpotence and torsion.

Contribution

It introduces a novel proof of the principle and extends it to signatures at preorderings, also analyzing the link between nilpotence and torsion in algebraic structures.

Findings

Extended the Local-Global Principle to preorderings.

Established a quantitative relation between nilpotence and torsion.

Provided new insights into involutions and quadratic forms.

Abstract

Pfister's Local-Global Principle states that a quadratic form over a (formally) real field is weakly hyperbolic (i.e. represents a torsion element in the Witt ring) if and only if its total signature is zero. This result extends naturally to the setting of central simple algebras with involution. The present article provides a new proof of this result and extends it to the case of signatures at preorderings. Furthermore the quantitative relation between nilpotence and torsion is explored for quadratic forms as well as for central simple algebras with involution.