The geometric Bombieri-Lang conjecture for varieties of maximal Albanese dimension

This is a joint work with Xinyi Yuan. Let K=k(B) the function field a variety B over a field k of characteristic 0. Let X be a projective variety over K. Assume that there is a finite morphism from X to an abelian variety A with trivial trace. We show that X(K) is contained in the algebraic special subset. In particular, if further X is of general type, then X(K) is not Zariski dense.