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 a Bloch–Beilinson type filtration {Fr(A)}r≥0 of the Chow group of zero-cycles, CH0(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→A is the multiplication by n map on A, then its pushforward n⋆ is given on the quotient Fr∕Fr+1 by multiplication by nr. In the special case when A=E1×⋯×Ed 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 Qp, 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.