Let K be a p-adic field and M a finite continuous Galois module. Local Tate duality is a perfect duality between the Galois cohomology of M and the Galois cohomology of its dual module. In the special case when M is the module of the m-torsion points of an abelian variety A over K, Tate has a finer result. In this case the group H^1(K,M) has a significant subgroup, namely there is map from the K-rational points of A to H^1(K,M) induced by the Kummer sequence on A. Tate computed the orthogonal complement of A(K) under the duality pairing.

In this talk I will present an analogue for H^2 of this classical result. The "significant subgroup" in this case will be given by a Galois symbol map, similar to the classical Galois symbol of the Bloch-Kato conjecture. After introducing the set up and discussing some details of the main theorem, I will present some applications to zero cycles and to p-adic Hodge theory. Speaker(s): Evangelia Gazaki (University of Michigan)