A Higher Dimensional Hilbert Irreducibility Theorem

Abstract: Assuming the weak Bombieri-Lang conjecture, we prove that a generalization of Hilbert’s irreducibility theorem holds for families of geometrically mordellic varieties (for instance, families of hyperbolic curves). As an application we prove that, assuming Bombieri-Lang, there are no polynomial bijections $\Q\times\Q\to\Q$, completing a strategy originally suggested by T. Tao.