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Let $A$ be an abelian variety defined over a field $k$. In this paper wedefine a filtration $F^{r}$ of the group $CH_{0}(A)$ and prove an isomorphism$\frac{K(k;A,...,A)}{\Sym}\otimes\mathbb{Z}[\frac{1}{r!}]\simeqF^{r}/F^{r+1}\otimes\mathbb{Z}[\frac{1}{r!}]$, where $K(k;A,...,A)$ is theSomekawa K-group attached to $r$-copies of the abelian variety $A$.\\ In thespecial case when $k$ is a finite extension of $\mathbb{Q}_{p}$ and $A$ hassplit 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$CH_{0}(A)\times Br(A)\rightarrow\mathbb{Q}/\Z$.

On a Filtration of CH_{0} for an Abelian Variety A