Multiplicative quadratic forms on algebraic varieties

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.