In his study about almost rational varieties, Colliot-Thelene considered morphisms of projective smooth varieties defined over number fields k, and he calls such a morphism f: X->Y arithmetically surjective if for almost all places v of k, the induced map on rational points X(k_v)->Y(k_v) is surjective; here k_v is the completion of k at v. Then he asked what would be an equivalent birational condition for such a morphism to be arithmetically surjective.

The question has since been studied intensively, and various people have given their answers, among them there are the results of Denef, and results of Loughran, Skorobogatov, and Smeets, where both papers use very careful analysis of the so called splitness of fibers; Pop gave another answer, by considering ultraproducts of k_v, in the process, simplifying the proof and generalizing the result at the same time.

In this talk, I will discuss briefly all the approaches so far, and show how the question can be considered in the context of zero-cycles, instead of just rational points, and outline how an answer in the spirit of Pop can be obtained.