Local-Global

. Let K be a complete discretely valued field with the residue field κ . Let F be the function field of a smooth, projective, geometrically integral curve over K and X be a regular proper model 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( κ ). Suppose that κ is an algebraically closed field or a finite field containing a primitive n th 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 × is a norm from the extension L/F if and only if it is a norm from the extensions L ⊗ F F ν /F ν for all discrete valuations ν of F .