Linkage of quadratic Pfister forms

ABSTRACT

We study the necessary conditions for sets of quadratic n-fold Pfister forms to have a common (n−1)-fold Pfister factor. For any set S of n-fold Pfister forms generating a subgroup of $I_q^{n}F∕I_q^{n+1}F$ of order 2^s in which every element has an n-fold Pfister representative, we associate an invariant in $I_q^{n+1}F$ which lives inside $I_q^{n+s-1}F$ when the forms in S have a common (n−1)-fold Pfister factor. We study the properties of this invariant and compute it explicitly in a few interesting cases.

KEYWORDS:

• Linkage
• Pfister forms
• quadratic forms
• quaternion algebras

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 11E81 (primary)
• 11E04, 16K20 (secondary)

Acknowledgments

We thank Jean-Pierre Tignol, Adrian Wadsworth, and an anonymous referee for their comments on the manuscript.