Autour des formes quadratiques quasi-voisines

Abstract

In this article we study a generalization of the notion of Pfister neighbors. An anisotropic quadratic form $\phi$ over a field $F$ of characteristic not $2$ is called a quasi-Pfister neighbor when the anisotropic part $(\phi_{F(\phi)})_{\an}$ is $F(\phi)$-similar to an $F$-quadratic form $\psi$ where $F(\phi)$ denotes the function field of the projective quadric given by $\phi$. We prove the uniqueness of $\psi$ up to $F$-similarity for forms $\phi$ of dimension $\leq 8$, odd dimension and many others of large dimension, and in these cases we give a precise description of $\psi$.