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 A₀ (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\'el\`ene 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 A₀ (X) can be completely determined. Using these results, we explore a local-to-global conjecture of Colliot-Th\'elene, 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.