Isotropy of six-dimensional quadratic forms over function fields of quadrics (Q1272455)

The authors study the question when an anisotropic \(6\)-dimensional quadratic form \(\phi\) over a field \(F\) of characteristic \(\neq 2\) becomes isotropic over the function field \(F(\psi)\) of another anisotropic form \(\psi\) of dimension \(\geq 2\). This is always the case if \(\psi\) is similar to a subform of \(\phi\) or if \(\phi\) contains a Pfister neighbor of some anisotropic \(3\)-fold Pfister form which in turn contains a subform similar to \(\psi\). In this case, the authors call the isotropy of \(\phi\) over \(F(\psi)\) standard. It has been shown by the reviewer [\textit{D. Hoffmann}, Commun. Algebra 22, 1999-2014 (1994; Zbl 0804.11031)], \textit{A. Laghribi} [Formes quadratiques de dimension 6, Math. Nachr. (to appear)] and the authors [\textit{O. T. Izhboldin} and \textit{N. Karpenko}, Isotropy of virtual Albert forms over function fields of quadrics, Math. Nachr. (to appear)] that the isotropy is always standard except possibly when \(\dim\psi =4\), the signed discriminants \(d_{\pm}\psi\) and \(d_{\pm}\phi\) are nontrivial and the index of the even part of the Clifford algebra \(C_0(\phi)\) is two. In the present paper, the authors show that in the remaining case the isotropy is also standard except possibly if in addition \(d_{\pm}\psi\neq d_{\pm}\phi\) and \(\text{ind}(C_0(\phi)\otimes_FC_0(\psi))=2\), in which case the authors give counterexamples to standard isotropy. However, it is not shown whether this necessary condition for nonstandard isotropy is also sufficient in general. Let \(X\) be a product of two quadrics associated to two \(4\)-dimensional quadratic forms over \(F\). Of crucial importance in the proofs is the determination of the kernel of the restriction map \(H^3F\to H^3F(X)\), where \(H^3K\) denotes the third Galois cohomology group of a field \(K\) with coefficients modulo \(2\), and of the torsion part of the second Chow group \(\text{CH}^2(X)\).