WEAK APPROXIMATION FOR TORI OVER p-ADIC FUNCTION FIELDS

We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field, the main focus being weak approximation of rational points. We construct a 9-term Poitou–Tate type exact sequence for tori over a field as above (and also a 12-term sequence for finite modules). Like in the number field case, part of the sequence can then be used to analyze the defect of weak approximation for a torus. We also show that the defect of weak approximation is controlled by a certain subgroup of the third unramified cohomology group of the torus. 0. Introduction This paper is the companion piece to [15], containing investigations concerning local-global questions for tori defined over the function field K of a curve over a finite extension of Qp. As recalled in the introduction of [15], our project has been motivated by the recent awakening of interest in local-global principles for group schemes defined over fields of cohomological dimension strictly greater than 2, as documented in the work of Harbater, Hartmann and Krashen [16] as well as Colliot-Thélène, Parimala and Suresh ([7], [8]). In [15] two of us are studying the Hasse principle for torsors under tori over a field K as above, the main tool being a global duality theorem. Presently our main concern is weak approximation for tori. As in the classical case over number fields, this necessitates going beyond duality and establishing a Poitou–Tate type exact sequence for tori over K. Here is our first main result: Theorem 0.1. (= Theorem 2.9) Let k be a finite extension of Qp, and T a torus defined over the function field of a smooth proper k-curve X. There is an exact sequence of topological groups 0 −−−−→ H(K,T )∧ −−−−→ P (T )∧ −−−−→ H (K,T ) y H(K,T ) ←−−−− P(T ) ←−−−− H(K,T ) y H(K,T ) −−−−→ P(T )tors −−−−→ (H (K,T )∧) D −−−−→ 0. In this sequence, T ′ denotes the dual torus of T (i.e. the torus whose character group is the cocharacter group of T ) and the groups P(T ) are certain restricted products of local cohomology groups to be defined in Section 2. Furthermore, the subscript ‘tors’ stands for the torsion subgroup, the subscript ∧ indicates the inverse limit of mod n quotients for all n > 0, and the superscript D means continuous dual with values in Q/Z. In Section 2 we shall also construct a 12-term exact sequence of similar type for finite group schemes over K. Date: January 28, 2014. 1 2 DAVID HARARI, CLAUS SCHEIDERER AND TAMÁS SZAMUELY As in the number field case, part of the sequence can be used to analyze the defect of weak approximation for a torus. The question here is whether for T as above the group of points T (K) is dense in the topological product of the groups T (Kv), where Kv denotes the completion of K with respect to the discrete valuation associated with a closed point v ∈ X , equipped with its natural topology. The answer is yes when the torus is K-rational, but in general there is an obstruction, as shown by the following analogue of a classical result of Voskresenskii ([30]; see also [9] and [25]). Theorem 0.2. (= Theorem 3.3 b)) For K and X as above, denote by X the set of closed points of X. For a K-torus T let T (K) denote the closure of T (K) in the topological product of the T (Kv) for all v ∈ X . There is an exact sequence