Some new examples in the theory of quadratic forms
type: Item
instance of: scholarly article
title: Some new examples in the theory of quadratic forms
zbMATH Open document ID: 0979.11025
author: O. T. Izhboldin
author: Nikita Karpenko
published in: Mathematische Zeitschrift
publication date: 2000-11-20T00:00:00Z
review text: The aim of this paper is to answer some questions in the algebraic theory of quadratic forms. In the first part of the paper, the authors obtain some important computations of the topological filtration on the Grothendieck group of some projective homogeneous varieties. In the second part of the paper, which is a consequence of the first one, the authors deduce some results on the isotropy problem and the structure of certain forms in \(I^2\) and \(I^3\) over a field of characteristic not \(2\), where \(I^n\) is the \(n\)-th power of the ideal of even-dimensional quadratic forms. Recall that the isotropy problem over function fields of quadrics is to classify the pairs of anisotropic quadratic forms \(\varphi\), \(\psi\) such that \(\varphi\) becomes isotropic over the function field \(F(\psi)\) of the projective quadric defined by \(\psi=0\). When \(\varphi\) is isotropic over \(F(\psi)\), the isotropy is called standard (in the sense of Definition 18.1) when one of the two following conditions is satisfied: \(\psi\) is a sub-form of \(\varphi\) (up to a scalar); or \(\varphi\) contains a Pfister neighbour form which is isotropic over \(F(\psi)\). The authors treat the isotropy in the case where \(\dim \varphi=6\). It was shown before that the isotropy is always standard when \(\dim \varphi \leq 6\), except for the case: \(\dim \varphi=6\), \(\dim \psi=4\), \(1\neq d_{\pm}\varphi \neq d_{\pm}\psi\neq 1\) and \(\text{ind}(C_0(\varphi))= \text{ind}(C_0(\psi)\otimes C_0(\varphi))=2\), where \(``d_{\pm}"\), \(``C_0"\) and ``ind'' mean respectively the signed discriminant, the even Clifford algebra and the Schur index. In this paper, the authors produce an example where the isotropy is not standard in the above exceptional case. After this, they construct an example of an \(8\)-dimensional quadratic form with trivial discriminant such that the Schur index of its Clifford algebra is \(4\) but the form is not isometric to a sum of two quadratic forms of dimension \(4\) wi…
reviewed by: Ahmed Laghribi
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 19E15
zbMATH DE Number: 1515376
zbMATH Keywords: quadratic form
zbMATH Keywords: Clifford algebra
zbMATH Keywords: projective quadric
zbMATH Keywords: Severi-Brauer variety
zbMATH Keywords: Grothendieck group and Chow group of a smooth algebraic variety
MaRDI profile type: Publication
DOI: 10.1007/S002090000095
description: scientific article; zbMATH DE number 1515376