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Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $A$, $B$ be abelianvarieties over $K$ of good reduction. For any integer $m\geq 1$, we considerthe Galois symbol $K(K;A,B)/m\rightarrow H^2(K,A[m]\otimes B[m])$, where$K(K;A,B)$ is the Somekawa $K$-group attached to $A,B$. This map is ageneralization of the Galois symbol $K_2^M(K)/m\rightarrow H^2(K,\mu_m^{\otimes2})$ of the Bloch-Kato conjecture, where $K_2^M(K)$ is the Milnor $K$-group of$K$. In this paper we give a geometric description of the image of thisgeneralized Galois symbol by looking at the Tate duality pairing$H^{2}(K,A[m]\otimesB[m])\times\mathrm{Hom}_{G_{K}}(A[m],B^{\star}[m])\rightarrow\mathbb{Z}/m,$where $B^\star$ is the dual abelian variety of $B$. Under this perfect pairingwe compute the exact annihilator of the image of the Galois symbol in terms ofan object of integral $p$-adic Hodge theory.

A finer Tate duality theorem for local Galois symbols