Obstructions to arithmetic properties of rational points on varieties - Balestrieri, F (2017)
The aim of this thesis is to develop further the theory of obstruction sets, which is used in the study of the arithmetic behaviour of rational points on nice varieties over number fields. Let X be a nice variety over a number field k. In the first part of this thesis, we characterise in pure descent-type terms some inequivalent obstruction sets refining the inclusion X(Ak)ét Br ⊂ X(Ak)Br1 . More precisely, we apply ideas from papers by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. We then show that descent under unipotent linear algebraic groups over k does not give any obstruction at all. This allow us, among other things, to characterise the étale-Brauer set in terms of descent under (non-necessarily connected) reductive linear algebraic groups and to characterise descent under solvable linear algebraic groups in terms of a Brauer-type set. The second main part is motivated by the need of finding new obstruction sets, as the current finest one - the étale-Brauer set - is not sufficiently small to explain all the violations of the Hasse principle (there is a counterexample by Poonen for which there are no rational points but the étale-Brauer set is non-empty). We iterate the algebraic étale-Brauer set for any nice variety X over a number field k with πét1 (X̄) finite and we show that the iterated set coincides with the original algebraic étale-Brauer set. This provides some evidence towards the conjectures by Colliot-Thélène on the arithmetic of rational points on nice geometrically rationally connected varieties over k and by Skorobogatov on the arithmetic of rational points on K3 surfaces over k; moreover, it gives a partial answer to an algebraic analogue of a question by Poonen about iterating the descent set.