Local-global principles for norm one tori over semi-global fields

Let K be a complete discretely valued field with the residue field.. Let F be the function field of a smooth, projective, geometrically integral curve over K and X be a regular propermodel of F such that the reduced special fibre X is a union of regular curves with normal crossings. Suppose that the graph associated to X is a tree (e.g. F = K(t)). Let L/ F be a Galois extension of degree n such that n is coprime to char(kappa). Suppose that kappa is an algebraically closed field or a finite field containing a primitive nth root of unity. Then we show that the local-global principle holds for the norm one torus associated to the extension L/F with respect to discrete valuations on F i.e. an element in F-x is a norm from the extension L/F if and only if it is a norm from the extensions L circle times(F) F-nu/F-nu for all discrete valuations nu of F.