A note on zero-cycles on bielliptic surfaces

We study the Chow group of zero-cycles $\text{CH}_0(S)$ of a bielliptic surface $S=(E_1\times E_2)/G$, where $E_1, E_2$ are elliptic curves and $G$ is a finite group acting on $E_1$ by translations and on $E_2$ by automorphisms such that $E_2/G\simeq\mathbb{P}^1$. We show that if $S$ is defined over an arbitrary field $k$ of characteristic not equal to $2,3$, then the kernel of the Albanese map $\text{alb}_S:\text{CH}_0(S)^{\text{deg}=0}\rightarrow \text{Alb}_S(k)$ is a torsion group of exponent $2^2\cdot|G|$ or $3^2\cdot|G|$, depending on the type of bielliptic surface. We also construct explicit examples over $p$-adic fields that illustrate that this kernel can have nontrivial elements obtained by push-forward from the abelian surface.