Cohomological invariants for orthogonal involutions on degree 8 algebras

Abstract

Abstract. Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division-central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of non isomorphic involutions with trivial relative Arason invariant.The discriminant and the Clifford algebra are classical invariants of quadratic forms over a field F of characteristic different from 2. Up to similarity, the discriminant classifies quadratic forms of dimension 2, while the even part of the Clifford algebra classifies forms of dimension 4. In [2], Arason defined an invariant e 3 , for even-dimensional quadratic forms with trivial discriminant and split Clifford algebra, which has values in H 3 (F, µ 2 ). Again, this invariant is classifying in dimension 8, that is for forms similar to a 3-fold Pfister form. The main purpose of this paper is to study the same question for orthogonal involutions on central simple algebras of degree 8.To see the relation between quadratic forms and involutions, recall that every (nondegenerate) quadratic form q on an F -vector space V defines an adjoint involution ad q on the endomorphism algebra End F V , and every orthogonal involution on End F V has the form ad q for some (nondegenerate) quadratic form q on V , uniquely determined up to a scalar factor, see [18, p. 1]. Therefore, orthogonal involutions on central simple algebras can be viewed as twisted forms (in the sense of Galois cohomology) of quadratic forms up to scalars. Analogues for orthogonal involutions of the discriminant and the (even) Clifford algebra were defined by Jacobson and Tits. By [18, (7.4) and § 15.B], the discriminant classifies orthogonal involutions on a given quaternion algebra, and the Clifford algebra classifies orthogonal involutions on a given biquaternion algebra.On the other hand, it was shown in [4, § 3.4] that the Arason invariant does not extend to orthogonal involutions if the underlying algebra is a degree 8 division algebra. In this paper, we define a relative Arason invariant for orthogonal involutions on a degree 8 (possibly division) central simple algebra A over F . Our construction, based on triality [18, § 42], is specific to this degree. Namely, we assign to any pair of involutions (σ, σ ′ ) on A with trivial discriminant and Clifford invariant, a degree 3 cohomology class e 3

The Arason invariant of orthogonal involutions of degree 12 and 8, and quaternionic subgroups of the Brauer group

“…Because (M 4 (Q), σ 0 ) has trivial discriminant and Clifford invariant, by [35,Th. 5.2] we may find λ, µ ∈ F × and an orthogonal involution ρ on Q such that…”

“…Hence, the formula given in [35,Th. 5.5] for algebras of degree 8 is actually valid in arbitrary degree.…”

“…If σ is isotropic, its anisotropic part has degree 8 and index 2. The explicit description of the Arason invariant in degree 8 given in [35,Th. 5.2] shows it is equal to e 3 (π) mod…”

“…for some orthogonal involution σ 0 with trivial discriminant and trivial Clifford invariant, and e 3 (σ) = e 3 (σ 0 ). If (a, b, c) is a symbol representing e 3 (σ), then by [35,Th. 5.2] we may assume that one of the slots, say a, is such that F ( √ a) splits Q, hence Q carries an orthogonal involution ρ with discriminant a.…”

“…It follows from the proof of Corollary 10.11 in [15], and Remark 4.10 in Barry's paper [2], that these invariants do not coincide with what we call Arason invariant in this paper. A systematic study of the relative and absolute Arason invariants for orthogonal involutions was recently initiated in [35], where the degree 8 case is studied in detail. In this paper, we continue with an investigation of absolute invariants in degree 12.…”