Hilbert properties of varieties, rational points, and dynamical systems

Abstract

An integral variety has the Hilbert property if its rational points are not thin. Corvaja–Zannier showed that a smooth projective integral variety with the Hilbert property over a finitely generated field k of characteristic 0 admits no non-trivial étale covers, motivating the refined notion of the “weak Hilbert property”. Conjecturally, every smooth projective integral k-variety with a dense set of k-rational points should have the weak Hilbert property – a question originally posed by Corvaja–Zannier. This extends to quasi-projective varieties by replacing rational points with near-integral points on arithmetic models. This thesis provides new evidence for this conjecture in the quasi-projective setting. We prove that the Hilbert property and weak Hilbert property for arithmetic schemes are stable under products, generalizing results for varieties by Bary-Soroker–Fehm–Petersen and Corvaja–Demeio–Javanpeykar–Lombardo–Zannier, and other persistence properties. We also prove the conjecture for all algebraic groups, extending known results for linear algebraic groups and abelian varieties. Combined with a conjecture of Campana, Corvaja–Zannier’s question predicts that a variety with a dense set of rational points over a number field satisfies the integral weak Hilbert property even after removing a closed subscheme of codimension at least 2. We verify this “punctured” conjecture for all linear algebraic groups