Let $A$ be an abelian variety defined over a field $k$. In this paper we define a descending filtration $\{F^{r}\}_{r\geqslant 0}$ of the group $\mathit{CH}_{0}(A)$ and prove that the successive quotients $F^{r}/F^{r+1}\otimes \mathbb{Z}[1/r!]$ are isomorphic to the group $(K(k;A,\dots ,A)/Sym)\otimes \mathbb{Z}[1/r!]$, where $K(k;A,\dots ,A)$ 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 $\mathit{CH}_{0}(A)\otimes \mathbb{Z}[\frac{1}{2}]\rightarrow \text{Hom}(Br(A),\mathbb{Q}/\mathbb{Z})\otimes \mathbb{Z}[\frac{1}{2}]$, induced by the pairing $\mathit{CH}_{0}(A)\times Br(A)\rightarrow \mathbb{Q}/\mathbb{Z}$.

We compare two known definitions for a relative family of effective zero cycles: one based on traces of functions and one based on norms of functions. In characteristic zero we show that both definitions agree. In the general setting we show that the norm map on functions can be expanded to a norm functor between certain categories of line bundles, thereby giving a third approach to families of zero cycles.