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For an abelian variety $A$ over a field $k$ the author defined in\cite{Gazaki2015} a Bloch-Beilinson type filtration $\{F^r(A)\}_{r\geq 0}$ ofthe Chow group of zero-cycles, $\text{CH}_0(A)$, with successive quotientsrelated to a Somekawa $K$-group. In this article we show that this filtrationbehaves well with respect to isogeny, and in particular if $n:A\to A$ is themultiplication by $n$ map on $A$, then its push-forward $n_\star$ is given onthe quotient $F^r/F^{r+1}$ by multiplication by $n^r$. In the special case when$A=E_1\times\cdots\times E_d$ is a product of elliptic curves, we show thatthis filtration agrees with a filtration defined by Raskind and Spiess and withthe Pontryagin filtration previously considered by Beauville and Bloch. We alsoobtain some results in the more general case when $A$ is isogenous to a productof elliptic curves. When $k$ is a finite extension of $\mathbb{Q}_p$, usingJacobians of curves isogenous to products of elliptic curves, we give newevidence for a conjecture of Raskind and Spiess and Colliot-Th\'{e}l\`{e}ne,which predicts that the kernel of the Albanese map is the direct sum of adivisible group and a finite group.

Filtrations of the Chow group of zero-cycles on abelian varieties and behavior under isogeny