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.