The following is joint work with Rachel Newton. In the spir it of work by Yongqi Liang\, we relate the arithmetic of rational points t o that of zero-cycles for the class of Kummer varieties over number fields . In particular\, if X is any Kummer variety over a number field k\, we sh ow that if the Brauer-Manin obstruction is the only obstruction to the exi stence of rational points on X over all finite extensions of k\, then the Brauer-Manin obstruction is the only obstruction to the existence of a zer o-cycle of any odd degree on X. Building on this result and on some other recent results by Ieronymou\, Skorobogatov and Zarhin\, we further prove a similar Liang-type result for products of Kummer varieties and K3 surface s over k.

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