On the problem of descent for quadratic forms
Let \(F\) be a field of characteristic \(\neq 2\) and let \(q\) be a quadratic form over \(F\), \(F(q)\) its function field and \(X\) its associated projective quadric. The unramified Witt ring \(W_{nr}F(q)\) consists of those elements of the Witt ring \(WF(q)\) whose images under the second residue homomorphisms at all codimension one points of \(X\) vanish. The question considered in this paper is as follows: Let \(\varphi\) be a form over \(F(q)\) such that \(\varphi\in W_{nr}F(q)\). Is \(\varphi\) defined over \(F\)? It is shown that this is the case if \(\dim q >16\) and either \(\dim\varphi\leq 6\) or for certain types of form \(\varphi\) with \(7\leq \dim\varphi\leq 10\). This extends results by \textit{B. Kahn} [Duke Math. J. 80, 139-155 (1995; Zbl 0858.11024)]. The methods of proof are similar to those employed in Kahn's paper. The proof is case by case and makes use of several deep theorems such as the isomorphisms \(e^n: H^nF\to I^nF/I^{n+1}F\) for \(n=3,4\) and Merkurjev's index reduction theorems.
scientific article; zbMATH DE number 1346229