  Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $A$, $B$ be abelian
varieties over $K$ of good reduction. For any integer $m\geq 1$, we consider
the 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 a
generalization of the Galois symbol $K_2^M(K)/m\rightarrow H^2(K,\mu_m^{\otimes
2})$ 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 this
generalized Galois symbol by looking at the Tate duality pairing
$H^{2}(K,A[m]\otimes
B[m])\times\mathrm{Hom}_{G_{K}}(A[m],B^{\star}[m])\rightarrow\mathbb{Z}/m,$
where $B^\star$ is the dual abelian variety of $B$. Under this perfect pairing
we compute the exact annihilator of the image of the Galois symbol in terms of
an object of integral $p$-adic Hodge theory.
