Division algebras and quadratic forms over fraction fields of two-dimensional henselian domains

Abstract

Let $K$ be the fraction field of a two-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not $2$, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using methods of Parimala and Suresh, and we obtain the local-global principle for isotropy of quadratic forms of rank $5$ with respect to discrete valuations of $K$. The latter result is proved by making a careful study of ramification and cyclicity of division algebras over the field $K$, following Saltman’s methods. A key step is the proof of the following result, which answers a question of Colliot-Thélène, Ojanguren and Parimala: for a Brauer class over $K$ of prime order $q$ different from the characteristic of $k$, if it is cyclic of degree $q$ over the completed field $K_v$ for every discrete valuation $v$ of $K$, then the same holds over $K$. This local-global principle for cyclicity is also established over function fields of $p$-adic curves with the same method.