Abstract. CCT is about sufficient conditions on a morphism f : X → Y of varieties over a number field k to be arithmetically surjective (a.s.), i.e., fv : X(kv) → Y (kv) is surjective for almost all places v of k. CCT was proved by Denef in a sharper form, and Loughran, Skorobogatov, Smeets gave equivalent condition on f to be a.s. I will explain all the terms, and give wide generalization of the above results, which in particular show that a.s. is equivalent to completely birational conditions on f. The question is open in positive characteristic.
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