Let f be a morphism of projective smooth varieties X, Y defined over the rationals. The conjecture by Colliot-Thélène under discussion gives (conjectural) sufficient conditions which imply that for almost all rational prime numbers p, the map f maps the p-adic points X(Q_p) surjectively onto Y(Q_p). The aim of the talk is to present some recent results by Denef, Skorobogatov et al

The dynamical Mordell-Lang conjecture in characteristic zero predicts that if f : X --> X is a map of algebraic varieties over a field K of characteristic zero, Y subset X is a closed subvariety and a in X(K) is a K-rational point on X, then the return set { n in N : f^n(a) in Y(K) } is a finite union of points and arithmetic progressions. For K a field of characteristic p > 0, it is necessary to allow for finite unions with sets of the form { […]

(Joint work with A. Chernikov)Let V subseteq C^3 be a complex variety of dimension 2.The Elekes-Szabo Theorem says that if V contains `too many' points on n x n x n Cartesian products then V has a special form: either V contains a cylinder over a curve or V is related to the graph of the multiplication of an algebraic group.In this talk we generalize the Elekes-Szabo Theorem to relations on strongly minimal sets interpretable in distal structures.