A finer Tate duality theorem for local Galois symbols

Let K be a finite extension of Qp. Let A, B be abelian varieties over K with good reduction. For any integer m≥1, we consider the Galois symbol K(K;A,B)/m→H2(K,A[m]⊗B[m]), where K(K;A,B) is the Somekawa K-group attached to A,B. This map is a generalization of the Galois symbol K2M(K)/m→H2(K,μm⊗2) of the Bloch-Kato conjecture, where K2M(K) is the Milnor K-group of K. In this paper we give a geometric description of the image of this generalized Galois symbol by looking at the Tate duality pairing H2(K,A[m]⊗B[m])×HomGK(A[m],B⋆[m])→Z/m, where B⋆ is the dual abelian variety of B. Under this perfect pairing we compute the exact annihilator of the image of the Galois symbol in terms of an object of integral p-adic Hodge theory. In this way we generalize a result of Tate for H1. Moreover, our result has applications to zero cycles on abelian varieties.