Abstract. In this talk we will explain Hilbert’s irreducibility theorem and give an introduction to the notion of the Hilbert property, which characterizes varieties with many rational points and allows for generalizations of Hilbert’s classical theorem. In fact, conjecturally, a simply connected smooth projective variety with a dense set of rational points should have the Hilbert property (up to enlarging the base field slightly). Corvaja–Zannier introduced the more refined weak Hilbert property and conjectured that any smooth projective variety with a dense set of rational points has the weak Hilbert property (up to enlarging the base field slightly), thereby getting rid of the simple-connectedness hypothesis. Their conjecture applies to a much larger class of varieties (including abelian varieties and Enriques surfaces). Our goal is to explain a further extension of these conjectures to quasi-projective varieties by working with their integral points. We present some evidence for this more general conjecture and investigate some properties of the new notion, in particular its persistence under products.
Before this lecture, from 4.45 pm, there will again be coffee and tea in room 1404.
