Principal homogeneous spaces over function fields of p-adic curves
by
Claus Scheiderer
University of Duisburg

Let k be a p-adic field, and let X be a smooth projective irreducible curve over k, with function field K=k(X). Let G be a linear algebraic group over K, and let E be a principal homogeneous G-space over K. Assume that, for any closed point P of X, E is trivial over the completion K_P of K with respect to the discrete valuation given by P. We study the question of whether E is necessarily trivial in this situation.

Let O_k be the ring of integers of k, and let X --> Spec O_k be a flat projective morphism of schemes whose generic fibre is k-isomorphic to X, and such that X is regular and connected. We also investigate the above question under the stronger assumption that, for any codimension one point x of X, E is trivial over the completion K_x of K with respect to the discrete valuation given by x.

Date received: July 12, 1999