  We prove new results on the distribution of rational points on ramified
covers of abelian varieties over finitely generated fields $k$ of
characteristic zero. For example, given a ramified cover $\pi : X \to A$, where
$A$ is an abelian variety over $k$ with a dense set of $k$-rational points, we
prove that there is a finite-index coset $C \subset A(k)$ such that $\pi(X(k))$
is disjoint from $C$.
  Our results do not seem to be in the range of other methods available at
present; they confirm predictions coming from Lang's conjectures on rational
points, and also go in the direction of an issue raised by Serre regarding
possible applications to the Inverse Galois Problem. Finally, the conclusions
of our work may be seen as a sharp version of Hilbert's irreducibility theorem
for abelian varieties.

Number Theory;Algebraic Geometry