Abstract. Given a family of varieties X → P n over a number field, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.

We study the problem of counting the number of varieties in families which have a rational point. We give conditions on the singular fibres that force very few of the varieties in the family to contain a rational point, in a precise quantitative sense. This generalises and unifies existing results in the literature by Serre, Browning-Dietmann, Bright-Browning-Loughran, Graber-Harris-Mazur-Starr, et al.

We study the distribution of abelian extensions of bounded discriminant of a number field k which fail the Hasse norm principle. For example, we classify those finite abelian groups G for which a positive proportion of G-extensions of k fail the Hasse norm principle. We obtain a similar classification for the failure of weak approximation for the associated norm one tori. These results involve counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which we achieve using tools from harmonic analysis, building on work of Wright.

Abstract. We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.

We consider the problem of counting the number of varieties in a family over a number field which contain a rational point. In particular, for products of Brauer-Severi varieties and closely related counting functions associated to Brauer group elements. Using harmonic analysis on toric varieties, we provide a positive answer to a question of Serre on such counting functions in some cases. We also formulate some conjectures on generalisations of Serre's problem.