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UBC Theses and Dissertations

Distribution of integral points on varieties Coccia, Simone


One of the guiding principles in Diophantine geometry is that, if an algebraic variety contains "many" integral points, then there is a geometric reason explaining their abundance. In this thesis we will focus on two geometric notions of abundance for integral points, namely Zariski density and the Hilbert Property, the latter being a generalization of Hilbert's irreducibility theorem to arbitrary algebraic varieties. We will focus on the case of complements of anticanonical divisors in smooth del Pezzo surfaces, proving that the integral points are always potentially dense and that the Hilbert Property holds, potentially, when such a complement is simply connected. We will also discuss joint work of Dragos Ghioca and the author that led to the proof of the Drinfeld modules analogue of Siegel's theorem on the finiteness of integral points.

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