On the distribution of rational points on ramified covers of abelian varieties

TL;DR

This paper establishes new distribution results for rational points on ramified covers of abelian varieties over fields of characteristic zero, supporting Lang's conjectures and relating to the Inverse Galois Problem.

Contribution

It proves that the rational points on ramified covers of abelian varieties are contained in a finite union of cosets, a novel result not accessible by existing methods.

Findings

Rational points on ramified covers avoid a finite-index coset in the abelian variety.

Supports predictions from Lang's conjectures on rational points.

Provides a sharp version of Hilbert's irreducibility theorem for abelian varieties.

Abstract

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.