Open Access
April 2003 Multiplicative quadratic forms on algebraic varieties
Akinari Hoshi
Proc. Japan Acad. Ser. A Math. Sci. 79(4): 71-75 (April 2003). DOI: 10.3792/pjaa.79.71


In this note we extend Hurwitz-type multiplication of quadratic forms. For a regular quadratic space $(K^n, q)$, we restrict the domain of $q$ to an algebraic variety $V \subsetneq K^n$ and require a Hurwitz-type ``bilinear condition'' on $V$. This means the existence of a bilinear map $\varphi\colon K^n \times K^n \rightarrow K^n$ such that $\varphi(V \times V) \subset V$ and $q(\mathbf{X}) q(\mathbf{Y}) = q(\varphi(\mathbf{X}, \mathbf{Y}))$ for any $\mathbf{X}, \mathbf{Y} \in V$. We show that the $m$-fold Pfister form is multiplicative on certain proper subvariety in $K^{2^m}$ for any $m$. We also show the existence of multiplicative quadratic forms which are different from Pfister forms on certain algebraic varieties for $n = 4, 6$. Especially for $n = 4$ we give a certain family of them.


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Akinari Hoshi. "Multiplicative quadratic forms on algebraic varieties." Proc. Japan Acad. Ser. A Math. Sci. 79 (4) 71 - 75, April 2003.


Published: April 2003
First available in Project Euclid: 18 May 2005

zbMATH: 1099.11017
MathSciNet: MR1976359
Digital Object Identifier: 10.3792/pjaa.79.71

Primary: 11E04 , 11E25
Secondary: 11T22

Keywords: Dickson's system , Multiplicative quadratic forms , Pfister forms

Rights: Copyright © 2003 The Japan Academy

Vol.79 • No. 4 • April 2003
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