Proceedings of the Japan Academy, Series A, Mathematical Sciences

Multiplicative quadratic forms on algebraic varieties

Akinari Hoshi

Abstract

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

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

Dates
First available in Project Euclid: 18 May 2005

https://projecteuclid.org/euclid.pja/1116443656

Digital Object Identifier
doi:10.3792/pjaa.79.71

Mathematical Reviews number (MathSciNet)
MR1976359

Zentralblatt MATH identifier
1099.11017

Citation

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. https://projecteuclid.org/euclid.pja/1116443656

References

• Dickson, L. E.: Cyclotomy, higher congruences and Waring's problem. Amer. J. Math., 57, 391–424 (1935).
• Pfister, A.: Zur Darstellung von $-1$ als Summe von Quadraten in einem Körper. J. London Math. Soc., 40, 159–165 (1965).
• Pfister, A.: Multiplikative quadratische Formen. Arch. Math., 16, 363–370 (1965).
• Pfister, A.: Quadratic forms with applications to algebraic geometry and topology. London Mathematical Society Lecture Note Series, no. 217, Cambridge University Press, Cambridge (1995).
• Schafer, R. D.: An Introduction to Nonassociative Algebras. Dover Publications, Inc., New York (1995).
• Katre, S. A., and Rajwade, A. R.: Unique determination of cyclotomic numbers of order five. Manuscripta Math., 53, 65–75 (1985).
• Rajwade, A. R.: Squares. London Mathematical Society Lecture Note Series, no. 171, Cambridge University Press, Cambridge (1993).