The Arason Invariant of Orthogonal Involutions of Degree 12 and 8, and Quaternionic Subgroups of the Brauer Group

Using the Rost invariant for torsors under $\Spin$ groups one may define an analogue of the Arason invariant for certain hermitian forms and orthogonal involutions. We calculate this invariant explicitly in various cases, and use it to associate to every orthogonal involution $\sigma$ with trivial discriminant and trivial Clifford invariant over a central simple algebra $A$ of even co-index an element $f_3(\sigma)$ in the subgroup $F^\times\cdot[A]$ of $H^3(F,\QZt)$. This invariant $f_3(\sigma)$ is the double of any representative of the Arason invariant $e_3(\sigma)\in H^3(F,\QZt)/F^\times\cdot[A]$; it vanishes when $\deg Aleq10$ and also when there is a quadratic extension of $F$ that simultaneously splits $A$ and makes $\sigma$ hyperbolic. The paper provides a detailed study of both invariants, with particular attention to the degree $12$ case, and to the relation with the existence of a quadratic splitting field. As a main tool we establish, when $\deg(A)=12$, an additive decomposition of $(A,\sigma)$ into three summands that are central simple algebras of degree $4$ with orthogonal involutions with trivial discriminant, extending a well-known result of Pfister on quadratic forms of dimension $12$ in $I^3F$. The Clifford components of the summands generate a subgroup $U$ of the Brauer group of $F$, in which every element is represented by a quaternion algebra, except possibly the class of $A$. We show that the Arason invariant $e_3(\sigma)$, when defined, generates the homology of a complex of degree $3$ Galois cohomology groups, attached to the subgroup $U$, which was introduced and studied by Peyre. In the final section, we use the results on degree $12$ algebras to extend the definition of the Arason invariant to trialitarian triples in which all three algebras have index at most $2$.

2010 Mathematics Subject Classification: 11E72, 11E81, 16W10.

Keywords and Phrases: Cohomological invariant, orthogonal group, algebra with involution, Clifford algebra.

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