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

Let $A$ be an abelian variety defined over a field $k$. In this paper we define a filtration $F^r$ of the group $C{H}_0(A)$ 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 $CH_{0}(A)\otimes\Z[\frac{1}{2}]\rightarrow \rm{Hom}(Br(A),\Q/\Z)\otimes\Z[\frac{1}{2}]$,inducedbythepairing$CH_{0}(A)\times Br(A)\rightarrow$\mathbb{Q}$/\Z$.