A descent problem for quadratic forms
type: Item
instance of: scholarly article
title: A descent problem for quadratic forms
zbMATH Open document ID: 0858.11024
DOI: 10.1215/S0012-7094-95-08006-5
published in: Duke Mathematical Journal
publication date: 1997-03-25T00:00:00Z
review text: For an anisotropic quadratic form \(q\) over a field \(\mathbb{F}\) of characteristic \(\neq 2\), let \(K=F(q)\) be the function field of the projective hypersurface \(X\) of equation \(q (x) = 0\). Suppose a quadratic form \(\varphi\) over \(K\) has all second residues at codimension 1 points of \(X\) equal to zero. The author considers the conjecture that, if \(\dim\varphi < (1/2) \dim q\), then \(\varphi\) is defined over the field \(F\). This has been proved here when \(\dim \varphi \leq 5\), and partially in dimension 6 (for \(\varphi\) an Albert form) and 8 (for \(\varphi\) a form similar to a Pfister form). As an immediate corollary the author obtains all known bounds on the dimensions of quadratic forms of height 2 established earlier by M. Knebusch, R. W. Fitzgerald, J. Hurrelbrink and U. Rehmann, and the author.
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 850803
zbMATH Keywords: unramified Witt ring
zbMATH Keywords: anisotropic quadratic form
zbMATH Keywords: function field
zbMATH Keywords: Albert form
zbMATH Keywords: Pfister form
zbMATH Keywords: dimensions of quadratic forms of height two
author: Bruno Kahn
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 850803