Abundance of rational points

For a smooth algebraic variety X defined over a number field K, one could ask several questions about the abundance of its rational points. This thesis revolves, in particular, around the following three properties: Hilbert Property, weak approximation and strong approximation. The first concerns, more or less, the question of extending the Hilbert Irreducibility Theorem to an arbitrary X (in the sense that the parameters of the Theorem are allowed to vary through rational points of this variety), the interesting case being when X is non-rational, for otherwise one recovers precisely the original theorem of Hilbert. The other two concern the question of density of rational points of X in the adelic ones (possibly with some places removed). The adjective weak" is more commonly used when talking about proper varieties, and the adjective strong" is used otherwise. In the first original work that is part of this thesis, we prove that, under a technical assumption, a proper algebraic surface X, with Zariski-dense rational points, that is endowed with two or more genus 1 fibrations, has the Hilbert Property. This result generalizes an earlier result of Corvaja and Zannier, who proved the Hilbert Property for the Fermat surface x4 + y4 = z4 + w4. The technique used is similar to theirs, the main idea being that of transporting rational points around the surface using the elliptic fibers of the various fibrations. In the second part of the thesis, we prove that on an arbitrary homogeneous space X, under some technical assumptions, the étale-Brauer-Manin obstruction is the only one to strong approximation. This obstruction is obtained by applying the more classical Brauer-Manin obstruction on all finite étale torsors over X. The proof is basically a reduction to a theorem of Borovoi and Demarche, who proved that (again under technical assumptions) strong approximation up to Brauer{Manin obstruction holds on homogeneous spaces with connected stabilizers. In this part of the thesis we also prove a compatibility result, suggested to be true by work of Cyril Demarche, between Brauer pairing and the so-called abelianization map, for homogeneous spaces of the form G=H, with H connected and linear. Finally, in the third and last part of the thesis, we explore the problem of "ramified descent", or, in other words, the question of which adelic points of X may be lifted to (a desingularization of a twist of) a fixed geometrically integral and geometrically Galois cover φ : Y --> X, with commutative geometric Galois group (although in some parts of the work this commutativity assumption is not needed). The case where the cover is unramified is already well-studied, and, therefore, the interest lies in the ramified case (whence the terminology " amified descent"). We prove that a certain naturally defined descent set" provides an obstruction to Hasse principle and weak approximation on X (the main difficulty in proving this lies in showing that rational points that lie on the branch locus of ' are unobstructed). Moreover, in analogy with the classical unramied case, we construct a subgroup B' of the Brauer group of X such that the the descent set associated to ' lies in the Brauer{Manin set associated to B'. Interestingly enough, the transcendental part of B' may provide a non-trivial obstruction, contrary to what happens in the unramified case. It seems reasonable to expect that this B' is the only obstruction to the "ramified descent" problem.