Proceedings of the Japan Academy, Series A, Mathematical Sciences

Multiplicative quadratic forms on algebraic varieties

Akinari Hoshi

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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.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 79, Number 4 (2003), 71-75.

First available in Project Euclid: 18 May 2005

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Zentralblatt MATH identifier

Primary: 11E04: Quadratic forms over general fields 11E25: Sums of squares and representations by other particular quadratic forms
Secondary: 11T22: Cyclotomy

Multiplicative quadratic forms Pfister forms Dickson's system


Hoshi, Akinari. Multiplicative quadratic forms on algebraic varieties. Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no. 4, 71--75. doi:10.3792/pjaa.79.71.

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