Pfister forms and a conjecture due to Colliot-Thélène in the mixed characteristic case

Abstract

let $R$ be a regular local ring of a mixed characteristic $(0,p)$ where $p\ne 2$ is a prime number. Suppose that the quotient ring $R/pR$ is also regular. Fix a non-degenerate Pfister form $Q(T_1,\dots ,T_{2^m})$ over $R$ and an invertible element $c$ in $R$. Then the equation $Q(T_1,\dots ,T_{2^m})=c$ has a solution over $R$ if and only if it has a solution over the fraction field $K$.