#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 529-576

** Anne Quéguiner-Mathieu and Jean-Pierre Tignol **
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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