A counting argument for the geometric Bombieri-Lang conjecture on ramified covers of abelian varieties

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite maps to abelian varieties over function fields of characteristic 0. This generalizes the recent results of Xie-Yuan, which require either the hyperbolicity assumption or the non-isotriviality assumption. The proof builds upon their strategy for constructing entire curves, yet hinges crucially on a new counting argument and draws substantially on tools from algebraic geometry and Nevanlinna theory to overcome various technical difficulties.