Arithmétique des groupes algébriques linéaires sur certains corps géométriques de dimension deux

Let k be an algebraically closed field of characteristic zero. Let K be either a function field in two variables over k or the fraction field of a 2-dimensional, excellent, strictly henselian local domain with residue field k. We show that linear algebraic groups over such a field K satisfy most properties familiar in the context of number fields: finiteness of R-equivalence, Hasse principle for complete homogeneous spaces.