On the isotropy of low-dimensional forms over the function field of a quadric (Q2761456)

The following problem is important in the algebraic theory of quadratic forms over a field \(F\). Let \(\varphi, \psi\) be anisotropic quadratic forms over a field \(F\), under which conditions is \(\varphi_{F(\psi)}\) isotropic (here \(F(\psi)\) is the function field of the quadric \(\psi\).) (For forms \(\varphi\) of dimension \(\leq 6\) this problem is the subject of several papers). Let \(\varphi\) be a quadratic form over \(F\) of dimension \(\leq 8\) and assume \(\varphi\) contains no \(4\)-dimensional quadratic forms with trivial discriminant. Then the main result of the paper proves that if \(\varphi_{F(\psi)}\) is isotropic then there exists a homomorphism \(C_{0}(\psi)\to C_{0}(\varphi)\) of the corresponding Clifford algebras. NEWLINENEWLINENEWLINEThe proof of this result is based on an interesting generalisation of the index reduction formula (Merkuriev-Tignol). In order to state this result, the notion of \(G\)-forms is introduced. We refer to the paper for the definition; we mention here that in the case \(5\leq \dim \varphi \leq 8\), \(\varphi\) is a \(G\)-form if and only if there exists a \(4\)-dimensional subform \(\tau\) of \(\varphi\) such that \(\tau\) is similar to a 2-fold Pfister form. Let \(\varphi, \psi\) be anisotropic forms over \(F\), \(\varphi\) not a \(G\)-form, then \(C_0(\varphi)\) is either a simple \(F\)-algebra or the product of 2 simple algebras. Assume that \(C_0(\varphi)\) is at most a \(2\times 2\) matrix ring over a skew field \(D\) (i.e. \(D\) or \(M_2(D)\)), respectively at most the product of \(2\times 2\) matrix rings over a skew field \(D\) (i.e. \(D\times D\) or \(M_2(D)\times M_2(D)\)). If \(\varphi_{F(\psi)}\) is isotropic then there is a homomorphism \(C_0(\psi)\to C_0(\varphi)\). (In case \(C_0(\varphi)\) is a skewfield (respectively the product of 2 skewfields) the result is a consequence of the Merkuriev-Tignol index reduction theorem.) NEWLINENEWLINENEWLINEThe main result of the paper also yields the following (corollary 6.4): Let \(\varphi\) be a quadratic form over \(F\) with \(\dim \varphi\leq 8\). Then the following conditions are equivalent: NEWLINENEWLINENEWLINE(1) there exists a 3-fold Pfister form \(\psi\) such that \(\varphi_{F(\psi)}\) is isotropic. NEWLINENEWLINENEWLINE(2) the form \(\varphi\) contains a 5-dimensional Pfister neighbor. NEWLINENEWLINENEWLINEThe paper is very nice and of interest to people interested in quadratic forms as well as division algebras.