The notion of ‘arithmetical surjectivity’ (a.s.) for dominant morphisms f of proper smooth varieties over number fields was introduced by Colliot-Thelene, and he made a precise

conjecture (CCT) relating a.s. to birational properties of the morphisms f. The CCT was proved in a sharper form by Denef (2019), and Loughran-Skorobogatov-Smeets gave a

characterization of a.s. (2020). I will present a new method of proof which allows generalizations/refinements of the above results by: First, allowing k to be any finitely generated base fields k with char(k)=0 (and beyond). Second, showing that a.s. is a fully birational property, i.e., a.s. depends only on properties of the function field extension defined by morphisms f. The method of proof also yields generalizations of the so called

zero-cycle surjectivity, considered/characterized over number fields by Gvirtz (2020).

NOTE: The problems are completely open in positive characteristic!