Albanese Maps and Fundamental Groups of Varieties With Many Rational Points Over Function Fields

We investigate properties of the Albanese map and the fundamental group of a complex projective variety with many rational points over some function field and prove that every linear quotient of the fundamental group of such a variety is virtually abelian, as well as that its Albanese map is surjective, has connected fibres, and has no multiple fibres in codimension one.