Cohomological invariants for orthogonal involutions on degree 8 algebras (Q2883231)

For a field \(F\) of characteristic \(\neq 2\) non-degenerate 8-dimensional quadratic forms \(q\) over \(F\) with trivial discriminant and Clifford invariant are classified up to similarity by their Arason invariant \(e_3(q) \in H^3(F,\mu_2)\). As similarity classes of non-degenerate \(8\)-dimensional quadratic forms correspond to orthogonal involutions on the split central simple algebra \(M_8(F)\), it is tempting to try to extend the Arason invariant to orthogonal involutions on central simple algebras of degree \(8\). However this is impossible if the underlying algebra is a division algebra as shown by [\textit{E. Bayer-Fluckiger, R.\ Parimala} and \textit{A. Quéguiner-Mathieu}, Proc. Indian Acad. Sci., Math. Sci. 113, No. 4, 365--377 (2007; Zbl 1049.16011)].NEWLINENEWLINEFix a central simple algebra \(A\) of degree 8. In the paper under review a relative Arason invariant \(e_3(\sigma/\sigma')\) is defined for pairs of orthogonal involutions \(\sigma, \sigma'\) of trivial discriminant and Clifford invariant on \(A\), which takes values in the quotient \(H^3(F,\mu_2)/(F^\times \cdot [A])\). The definition is as follows: By triality there are \(8\)-dimensional quadratic forms \(\phi, \phi'\) (unique up to similarity) such that \((C_0(\phi),\text{can})=(A,\sigma)\times (A,\sigma)\) and \((C_0(\phi'),\text{can})=(A,\sigma')\times (A,\sigma')\). Then \(e_3(\sigma/\sigma')\) is defined as the class of \(e_3(\phi-\phi')\) in \(H^3(F,\mu_2)/(F^\times \cdot [A])\).NEWLINENEWLINEIf \(A\) is not division there exists a hyperbolic orthogonal involution \(\sigma_0\) on \(A\) (unique up to conjugacy) and the authors define an absolute Arason-invariant of \(\sigma\) by \(e_3(\sigma):=e_3(\sigma/\sigma_0)\in H^3(F,\mu_2)/(F^\times \cdot [A])\).NEWLINENEWLINEIt is shown that the absolute Arason-invariant for a non-divison algebra \(A\) detects hyperbolicity, i.e. \(e_3(\sigma)=0\) if and only if \(\sigma\) is hyperbolic, which happens if and only if \(\sigma\) is conjugate to \(\sigma_0\). However the relative Arason-invariant does not detect conjugacy. In fact the authors construct non-conjugate involutions \(\sigma, \sigma'\) on algebras \(A\) of index \(4\) and \(8\) such that \(e_3(\sigma/\sigma')=0\).NEWLINENEWLINEIt is shown in the paper that conjugacy of \(\sigma\) and \(\sigma'\) can be detected by yet another invariant \(e_4(\sigma/\sigma')\) which is defined only in case \(e_3(\sigma/\sigma')=0\). The latter condition is shown to be equivalent to the existence of \(\mu \in F^\times\) such that \(\phi - \langle \mu \rangle \phi'\) is similar to a \(4\)-Pfister form, where \(\phi\) and \(\phi'\) are as above. Moreover \(E_4:=\{e_4(\langle \langle \nu \rangle \rangle \phi) \mid \nu \in F^\times \text{ such that } \langle \nu \rangle \cdot [A] = 0 \}\) is a subgroup of \(H^4(F,\mu_2)\) and the class \(e_4(\sigma/\sigma')\) of \(e_4(\phi - \langle \mu \rangle \phi')\) in \(H^4(F,\mu_2)/E_4\) is well defined. With this definition of \(e_4(\sigma/\sigma') \in H^4(F,\mu_2)/E_4\) the authors show that two orthogonal involutions \(\sigma, \sigma'\) of trivial discriminant and Clifford invariant on \(A\) are conjugate if and only if both \(e_3(\sigma/\sigma')=0\) and \(e_4(\sigma/\sigma')=0\).NEWLINENEWLINEIn the end it is shown that for \(\text{ind}(A)\leq 4\) the element \(e_3(\sigma)\) is represented by a \(3\)-symbol in \(H^3(F,\mu_2)\) if and only if either \(\text{ind}(A)\leq 2\) or \(\text{ind}(A)=4\) and \((A,\sigma)\) decomposes as \((M_2(F),\text{ad}_{\langle \langle \lambda \rangle \rangle}) \otimes (Q_1,{ }^{-})\otimes (Q_2,{ }^{-})\) for some quaternion algebras \(Q_1, Q_2\) with canonical involution \({ }^{-}\) and some \(\lambda \in F^\times\).NEWLINENEWLINEA different approach to extend the Arason invariant to orthogonal involutions, which uses the Rost invariant for groups of type \(E_8\) and Spin-groups, was taken in [\textit{S. Garibaldi}, Contemp. Math. 493, 131--162 (2009; Zbl 1229.11063)]