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This paper is a continuation to \cite{Gazaki2017}. For every integer $n\geq1$, we consider the generalized Galois symbol $K(k;G_1,G_2)/n\xrightarrow{s_n}H^2(k,G_1[n]\otimes G_2[n])$, where $k$ is a finite extension of$\mathbb{Q}_p$, $G_1,G_2$ are semi-abelian varieties over $k$ and$K(k;G_1,G_2)$ is the Somekawa K-group attached to $G_1, G_2$. Under some mildassumptions, we describe the exact annihilator of the image of $s_n$ under theTate duality perfect pairing, $H^2(k,G_1[n]\otimes G_2[n])\timesH^0(k,Hom(G_1[n]\otimes G_2[n],\mu_n))\rightarrow\mathbb{Z}/n$. An importantspecial case is when both $G_1, G_2$ are abelian varieties with splitsemistable reduction. In this case we prove a finiteness result, which gives anapplication to zero-cycles on abelian varieties and products of curves.

A Tate duality theorem for local Galois symbols II; The semi-abelian case