On the problem of descent for quadratic forms
type: Item
instance of: scholarly article
title: On the problem of descent for quadratic forms
zbMATH Open document ID: 0927.11022
DOI: 10.1007/PL00000399
author: Ahmed Laghribi
published in: Archiv der Mathematik
publication date: 1999-12-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 1346229
zbMATH Keywords: quadratic forms
zbMATH Keywords: function field of a quadratic form
zbMATH Keywords: unramified Witt ring
zbMATH Keywords: descent
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: pl00000399
OpenAlex ID: W2053250620
description: scientific article; zbMATH DE number 1346229