Realising the cup product of local Tate duality

We present an explicit description, in terms of central simple algebras, of a cup product map which occurs in the statement of local Tate duality for Galois modules of prime cardinality p. Given cocycles f and g, we construct a central simple algebra of dimension p^2 whose class in the Brauer group gives the cup product f\cup g. This algebra is as small as possible

Arithmetic of rational points and zero-cycles on products of Kummer varieties and K3 surfaces

Let k be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of k to that of zero-cycles over k for Kummer varieties over k. For example, for any Kummer variety X over k, we show that if the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rational points on X over all finite extensions of k, then the (2-primary) Brauer-Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on X over k. We also obtain similar results for products of Kummer varieties, K3 surfaces and rationally connected varieties

Evaluating the wild Brauer group

Classifying elements of the Brauer group of a variety X over a p-adic field according to the p-adic accuracy needed to evaluate them gives a filtration on Br X. We show that, on the p-torsion, this filtration coincides with a modified version of that defined by Kato's Swan conductor, and that the refined Swan conductor controls how the evaluation maps vary on p-adic discs. We give applications to the study of rational points on varieties over number fields.Comment: 41 pages; comments welcom

Number fields with prescribed norms

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100\%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.Comment: 35 pages, comments welcome

The Hasse norm principle for abelian extensions

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