Given a number field k, the CCT over k is about giving birational conditions on morphisms of projective smooth k-varieties which imply surjectivity on the local rational points for almost all localization of k.

Among other things, CCT aimed at giving an "arithmetical proof" of the celebrated Ax-Kochen-Ershov principle. CCT was proved in an even stronger form by Denef, and Loughran-Skorobogatov-Smeets gave necessary and sufficient condition for Denef's result to hold. In this talk I will present generalizations of those results in several ways, e.g., no projectivity or smoothness is required, k can be the function field of a curve over a PAC field, etc. The point in my approach is to use special forms of the AKE.