Local and local-to-global Principles for zero-cycles on geometrically Kummer $K3$ surfaces

Let $X$ be a $K3$ surface over a $p$-adic field $k$ such that for some abelian surface $A$ isogenous to a product of two elliptic curves, there is an isomorphism over the algebraic closure of $k$ between $X$ and the Kummer surface associated to $A$. Under some assumptions on the reduction types of the elliptic curve factors of $A$, we prove that the Chow group $A_0(X)$ of zero-cycles of degree $0$ on $X$ is the direct sum of a divisible group and a finite group. This proves a conjecture of Raskind and Spiess and of Colliot-Th\'{e}l\`{e}ne and it is the first instance for $K3$ surfaces when this conjecture is proved in full. This class of $K3$'s includes, among others, the diagonal quartic surfaces. In the case of good ordinary reduction we describe many cases when the finite summand of $A_0(X)$ can be completely determined. Using these results, we explore a local-to-global conjecture of Colliot-Th\'{e}lene, Sansuc, Kato and Saito which, roughly speaking, predicts that the Brauer-Manin obstruction is the only obstruction to Weak Approximation for zero-cycles. We give examples of Kummer surfaces over a number field $F$ where the ramified places of good ordinary reduction contribute nontrivially to the Brauer set for zero-cycles of degree $0$ and we describe cases when an unconditional local-to-global principle can be proved, giving the first unconditional evidence for this conjecture in the case of $K3$ surfaces.