Pfister's Local--Global Principle and Systems of Quadratic Forms

TL;DR

This paper extends Pfister's local-global principle from single quadratic forms to systems of forms and algebras with involution, providing new analogues and generalizations of classical results in quadratic form theory.

Contribution

It introduces two analogues of Pfister's local-global principle for systems of quadratic forms, including nonsingular pairs, and generalizes a theorem to finite-dimensional algebras with involution.

Findings

Established two analogues of Pfister's local-global principle for systems of quadratic forms.

Proved a generalization of Pfister's theorem for finite-dimensional algebras with involution.

Abstract

Let $q$ be a unimodular quadratic form over a field $K$. Pfister's famous local--global principle asserts that $q$ represents a torsion class in the Witt group of $K$ if and only if it has signature $0$, and that in this case, the order of Witt class of $q$ is a power of $2$. We give two analogues of this result to systems of quadratic forms, the second of which applying only to nonsingular pairs. We also prove a counterpart of Pfister's theorem for finite-dimensional $K$-algebras with involution, generalizing a result of Lewis and Unger.