Arithmetic of rational points and zero-cycles on Kummer varieties

Let k be a number field. Jean-Louis Colliot-Thélène has conjectured that, for any smooth, projective, geometrically integral variety X over k, the Brauer-Manin obstruction governs the arithmetic of zero-cycles of degree 1 on X. In this direction, Yongqi Liang has shown that, for geometrically rationally connected varieties over k, sufficiency of the Brauer-Manin obstruction to weak approximation for rational points over all finite extensions of k implies sufficiency of the Brauer-Manin obstruction to weak approximation for zero-cycles of degree 1 over k. In this talk, I will discuss some work in progress with Rachel Newton where we extend Liang's result to geometrically Kummer varieties. In particular, we prove that, for Kummer varieties over k, sufficiency of the Brauer-Manin obstruction to the existence of a rational point over all finite extensions of k implies sufficiency of the Brauer-Manin obstruction to the existence of a zero-cycle of degree 1 over k.