Linked triples of quaternion algebras

Alexander S. Sivatski

Vol. 268 (2014), No. 2, 465–476

DOI: 10.2140/pjm.2014.268.465

Abstract

Let $F$ be a field of characteristic different from $2$, and let $Q_1,Q_2,Q_3$ be quaternion algebras over $F$ such that any element in $Br\left(F\right)$ generated by $Q_1,Q_2,Q_3$ has index at most $2$. For the triple $\left\{Q_1,Q_2,Q_3\right\}$ we construct a certain invariant, lying in $I^3\left(F\right)$, which is a $4$-fold Pfister form, provided the algebras $Q_1,Q_2,Q_3$ have a common slot. Among other results we prove that the algebras $Q_1,Q_2,Q_3$ have a common slot if and only if the torsion of the group ${CH}^2\left(X_1\times X_2\times X_3\right)$ is zero, where $X_i$ is the projective conic associated with the algebra $Q_i$.

Keywords

quaternion algebra, quadratic form, Chow group

Mathematical Subject Classification 2010

Primary: 16K50

Secondary: 19D45

Milestones

Received: 11 July 2012

Revised: 6 November 2012

Accepted: 13 November 2012

Published: 21 June 2014