Twisted Pfister forms

Abstract

Let $F$ be a field of characteristic $\ne 2$. In this paper we investigate quadratic forms $\phi$ over $F$ which are anisotropic and of dimension $2^n,n\ge 2$, such that in the Witt ring $WF$ they can be written in the form $\phi =\sigma -\pi$ where $\sigma$ and $\pi$ are anisotropic $n$- resp. $m$-fold Pfister forms, $1\le m<n$. We call these forms twisted Pfister forms. Forms of this type with $m=n-1$ are of great importance in the study of so-called good forms of height 2, and such forms with $m=1$ also appear in Izhboldin's recent proof of the existence of $n$-fold Pfister forms $\tau$ over suitable fields $F,n\ge 3$, for which the function field $F(\tau )$ is not excellent over $F$. We first derive some elementary properties and try to give alternative characterizations of twisted Pfister forms. We also compute the Witt kernel $W(F(\phi )/F)$ of a twisted Pfister form $\phi$. Our main focus, however, will be the study of the following problems: For which forms $\psi$ does a twisted Pfister form $\phi$ become isotropic over $F(\psi )$? Which forms $\psi$ are equivalent to $\phi$ (i.e., the function fields $F(\phi )$ and $F(\psi )$ are place-equivalent over $F$)? We also investigate how such twisted Pfister forms behave over the function field of a Pfister form of the same dimension which then leads to a generalization of the result of Izhboldin mentioned above.