On quadratic forms of height two and a theorem of Wadsworth

Abstract:

Let $\varphi$ and $\psi$ be anisotropic quadratic forms over a field $F$ of characteristic not $2$. Their function fields $F( \varphi )$ and $F(\psi )$ are said to be equivalent (over $F$) if $\varphi \otimes F(\psi )$ and $\psi \otimes F( \varphi )$ are isotropic. We consider the case where $\dim \varphi =2^n$ and $\varphi$ is divisible by an $(n-2)$-fold Pfister form. We determine those forms $\psi$ for which $\varphi$ becomes isotropic over $F(\psi )$ if $n\leq 3$, and provide partial results for $n\geq 4$. These results imply that if $F( \varphi )$ and $F(\psi )$ are equivalent and $\dim \varphi =\dim \psi$, then $\varphi$ is similar to $\psi$ over $F$. This together with already known results yields that if $\varphi$ is of height $2$ and degree $1$ or $2$, and if $\dim \varphi =\dim \psi$, then $F( \varphi )$ and $F(\psi )$ are equivalent iff $F( \varphi )$ and $F(\psi )$ are isomorphic over $F$.