A note on zero-cycles on bielliptic surfaces

TL;DR

This paper investigates the structure of zero-cycles on bielliptic surfaces, showing the torsion nature of the kernel of the Albanese map over fields of characteristic not 2 or 3, and provides explicit examples illustrating nontrivial elements.

Contribution

It determines the torsion properties of the kernel of the Albanese map for bielliptic surfaces over various fields and constructs explicit examples with nontrivial elements.

Findings

Kernel of Albanese map is torsion of exponent dividing 2^2·|G| or 3^2·|G|

Explicit examples over p-adic fields show nontrivial push-forward elements

Results depend on the type of bielliptic surface

Abstract

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.