Zero-cycles and Brauer groups on certain Enriques surfaces

In this paper, we study the Chow group of zero-cycles ${\text{CH}}_0(S)$ for certain Enriques quotients $S$ of the Kummer surface associated to the product of two geometrically non-isogenous elliptic curves $E_1$ and $E_2$. We show that over fields of characteristic different from $2$ or $3$, the degree-zero subgroup $A_0(S)\subset {\text{CH}}_0(S)$ is 16-torsion. Over p-adic fields, we compute the transcendental Brauer group $\text{Br}(S)\mathrm{/}{\text{Br}}_1(S)$. Using the Brauer-Manin pairing, we construct examples of Enriques surfaces for which $A_0(S)$ attains nontrivial 2-torsion elements arising from push-forward from the abelian surface $E_1\times E_2$.