Local-global principle for classical groups over function fields of $p$-adic curves

Abstract

Let $K$ be a local field with residue field $\kappa$ and $F$ the function field of a curve over $K$. Let $G$ be a connected linear algebraic group over $F$ of classical type. Suppose $\mathrm{char}(\kappa )$ is a good prime for $G$. Then we prove that projective homogeneous spaces under $G$ over $F$ satisfy a local-global principle for rational points with respect to discrete valuations of $F$. If $G$ is a semisimple simply connected group over $F$, then we also prove that principal homogeneous spaces under $G$ over $F$ satisfy a local-global principle for rational points with respect to discrete valuations of $F$.