Hyperbolicity of certain involutions over the function field of a quadric (Q1605096)

Let \(F\) be a field of characteristic \(\neq 2\) and let \((A,\sigma)\) be a central simple algebra with involution \(\sigma\) over \(F\). The involution is called hyperbolic if there exists an idempotent \(e\in A\) with \(\sigma(e)=1-e\). Let \(\varphi\) be a quadratic form over \(F\) and denote by \(F(\varphi)\) the function field of the projective quadric defined by the equation \(\varphi=0\). Well known results imply that if there exists a homomorphism of algebras with involution \((C_0(\varphi),\sigma_0)\to(A,\sigma)\), where \(C_0(\varphi)\) denotes the even part of the Clifford algebra of \(\varphi\) and \(\sigma_0\) the standard involution on \(C_0(\varphi)\), then \((A\otimes_FF(\varphi),\sigma_{F(\varphi)})\) is hyperbolic. The author constructs an example which shows that the converse generally fails, but he gives an affirmative answer in the case of orthogonal \(\sigma\) under the assumption that either the degree of \(A\) is \(2\) or \(4\), or \(A\) is split and \(\dim\varphi\leq 4\) or \(\dim\varphi=5\) and \(\varphi\) is a Pfister neighbor. The second main result characterizes those forms \(\psi\) such that \((A,\sigma)\) is hyberbolic over \(F(\psi)\) in the case where \(A\) is a division algebra of degree \(8\), \(\sigma\) is orthogonal, and \(A\) decomposes completely, i.e. \(A\) can be written as a product of three quaternion algebras each of them being stable under the action of \(\sigma\). In the case where \(A\) does not decompose completely, a partial result is given.