A note on zero-cycles on bielliptic surfaces
Evangelia Gazaki
TL;DR
This work studies the Chow group of zero-cycles ${\rm CH}_0(S)$ for bielliptic surfaces ${S=(E_1\times E_2)/G}$ over fields ${k}$ with ${\rm char}(k)\neq 2,3$, establishing that the Albanese kernel ${T(S)}$ is torsion with exponent ${2^2\cdot|G|}$ when ${2|G|}$ and ${3^2\cdot|G|}$ otherwise. The proof reduces to Type ${1}$ or Type ${5}$ bielliptic surfaces via intermediate étale covers and pushes forward torsion information from the abelian surface ${X=E_1\times E_2}$ through bilinear constructions of zero-cycles ${z_{P,Q}}$, leveraging bilinearity and the properties of the Albanese map. A second main result constructs explicit bielliptic surfaces over ${p}$-adic fields with nontrivial ${T(S)}$, detected through the Brauer-Manin pairing and the structure of ${Br}(X)[n]/Br_1(X)[n]\simeq {\rm Hom}_{G_k}(E_1[n],E_2[n])$, including explicit ${E_1,E_2}$ over ${\mathbb Q}$ with specified reduction behavior. A corollary shows that when ${E_1,E_2}$ have good reduction, ${T(S)}$ is ${2}$-torsion (Type ${1}$) or ${3}$-torsion (Type ${5}$), with reduction data implying nontrivial ${A_0}$ in the reduction, highlighting the interplay between Chow groups, Albanese kernels, and Brauer groups in bielliptic geometry.
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.