On the Bombieri–Lang conjecture over finitely generated fields

The strong Bombieri–Lang conjecture postulates that, for every variety X of general type over a field k finitely generated over Q, there exists an open subset U ⊂ X such that U(K) is finite for every finitely generated extension K/k. The weak Bombieri–Lang conjecture postulates that, for every positive dimensional variety X of general type over a field k finitely generated over Q, the rational points X (k) are not dense. Furthermore, Lang conjectured that every variety of general type X over a field of characteristic 0 contains an open subset U ⊂ X such that every subvariety of U is of general type, this statement is usually called geometric Lang conjecture. We reduce the strong Bombieri–Lang conjecture to the case k = Q. Assuming the geometric Lang conjecture, we reduce the weak Bombieri–Lang conjecture to k = Q, too.