Local-global principle for reduced norms over function fields of p-adic curves

Abstract

Let K be a (non-archimedean) local field and let F be the function field of a curve over K. Let D be a central simple algebra over F of period n and lambda is an element of F*. We show that if n is coprime to the characteristic of the residue field of K and D. (lambda) 0 in H-3 (F, mu(circle times)(n)2), then lambda is a reduced norm from D. This leads to a Hasse principle for the group SL1 (D), namely, an element lambda is an element of F* is a reduced norm from D if and only if it is a reduced norm locally at all discrete valuations of F.