  In this paper, we prove the geometric Bombieri-Lang conjecture for projective
varieties which have finite morphisms to abelian varieties of trivial traces
over function fields of characteristic 0. The proof is based on the idea of
constructing entire curves in the pre-sequel "Partial heights, entire curves,
and the geometric Bombieri-Lang conjecture." A new ingredient is an explicit
description of the entire curves in terms of Lie algebras of abelian varieties.
