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Abstract. We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field (a field of cohomological dimension 3). We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of the base field coming from a closed point of the curve. In the case of a torus we establish a perfect duality between the first Tate-Shafarevich group of the torus and the second Tate-Shafarevich group of the dual torus. Building upon the duality theorem, we show that the failure of the local-global principle for rational points on principal homogeneous spaces under tori is controlled by a certain subquotient of a third etale cohomology group. We also prove a generalization to principal homogeneous spaces of certain reductive group schemes in the case when the base curve has good reduction. IntroductionIn recent years there has been considerable interest in local-global principles for group schemes defined over fields of cohomological dimension strictly greater than 2. In particular, Harbater, Hartmann and Krashen, in a series of papers of which perhaps [18] is the most relevant to our present context, have used patching techniques to introduce and study an analogue of the Tate-Shafarevich group for certain linear algebraic groups over the function field of a curve defined over a complete discretely valued field. In [8] and the recent preprint [9], Colliot-Thélène, Parimala and Suresh have obtained local-global results over similar fields whose formulation shows a closer analogy with the classical theory over number fields.These developments motivate a systematic study of local-global questions for Galois cohomology over the function field K of a curve defined over a finite extension of Q p . In the present paper, except for the last section, we limit ourselves to tori over K and prove two kinds of results: duality theorems for Galois cohomology, and analogues of the by now classical results of Sansuc [31] regarding the Hasse principle for torsors under tori. In the last section we also prove a generalization to torsors under certain reductive groups. Concerning duality, a direct precursor of our theorems is the Artin-Verdier style statement of Scheiderer and van Hamel [32].Let us now review the main results of the paper. Given a torus T over a field K as above, setwhere the product is over closed points of the smooth proper curve X whose function field is K, and K v is the completion of K with respect to the discrete valuation coming from the point v. Note that this definition is different from that of [8] and [9] where all discrete rank 1 valuations of K are considered, and also from that of [18] where the authors complete the local rings of closed points of the special fibre of an integral model of K.

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