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

For an abelian variety $A$ over a field $k$ the author defined in \cite{Gazaki2015} a Bloch-Beilinson type filtration $\{F^r(A){\}}_{r\ge 0}$ of the Chow group of zero-cycles, ${\text{CH}}_0(A)$, with successive quotients related to a Somekawa $K$-group. In this article we show that this filtration behaves well with respect to isogeny, and in particular if $n:A\to A$ is the multiplication by $n$ map on $A$, then its push-forward $n_{\star }$ is given on the quotient $F^r\mathrm{/}{F}^{r+1}$ 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 that this filtration agrees with a filtration defined by Raskind and Spiess and with the Pontryagin filtration previously considered by Beauville and Bloch. We also obtain some results in the more general case when $A$ is isogenous to a product of elliptic curves. When $k$ is a finite extension of ${\mathbb{Q}}_p$, using Jacobians of curves isogenous to products of elliptic curves, we give new evidence for a conjecture of Raskind and Spiess and Colliot-Thélène, which predicts that the kernel of the Albanese map is the direct sum of a divisible group and a finite group.