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.

Let $A$ be the product of an abelian variety and a torus defined over a number field $K$. Fix some prime number $\ell$. If $\unicode[STIX]{x1D6FC}\in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak{p}$ of $K$ such that the reduction $(\unicode[STIX]{x1D6FC}\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{p})$ is well-defined and has order coprime to $\ell$. This set admits a natural density. By refining the method of Jones and Rouse [Galois theory of iterated endomorphisms, Proc. Lond. Math. Soc. (3)100(3) (2010), 763–794. Appendix A by Jeffrey D. Achter], we can express the density as an $\ell$-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of $\ell$) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.

We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of ‘universal covers’ of rigid divisible commutative finite Morley rank groups.