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 \operatorname{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 .