  Let $A$ be an abelian variety defined over a field $k$. In this paper we
define a filtration $F^{r}$ of the group $CH_{0}(A)$ and prove an isomorphism
$\frac{K(k;A,...,A)}{\Sym}\otimes\mathbb{Z}[\frac{1}{r!}]\simeq
F^{r}/F^{r+1}\otimes\mathbb{Z}[\frac{1}{r!}]$, where $K(k;A,...,A)$ is the
Somekawa K-group attached to $r$-copies of the abelian variety $A$.\\ In the
special case when $k$ is a finite extension of $\mathbb{Q}_{p}$ and $A$ has
split multiplicative reduction, we compute the kernel of the map
$CH_{0}(A)\otimes\Z[\frac{1}{2}]\rightarrow
\rm{Hom}(Br(A),\Q/\Z)\otimes\Z[\frac{1}{2}]$, induced by the pairing
$CH_{0}(A)\times Br(A)\rightarrow\mathbb{Q}/\Z$.
