On a filtration of CH0 for an abelian variety

Let A be an abelian variety defined over a field k. In this paper we define a descending filtration {F-r}(r >= 0) of the group CH0(A) and prove that the successive quotients F-r/Fr+1 circle times Z[1/r!] are isomorphic to the group (K (k; A, ... , A)/Sym) circle times Z[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 Q(p) and A has split multiplicative reduction, we compute the kernel of the map CH0(A)circle times Z[1/2] -> Hom(Br(A), Q/Z)circle times Z[1/2], induced by the pairing CH0(A) x Br(A) -> Q/Z.