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 7r : X-* 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 of A(k) such that 7r(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.

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

Zannier, U
2022

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 7r : X-* 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 of A(k) such that 7r(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.
2022
Settore MAT/03 - Geometria
Hilbert's irreducibility theorem; Kummer theory; Chebotarev density theorems; abelian varieties; Campana's conjectures; Lang's conjectures; rational points; ramified covers
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/126802
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