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

Abstract

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.

arXiv comment50 pages. We changed the title and the structure of the article, particularly in Section 1 and 3; we also fixed some typos