We present a method for calculating the Brauer group of a surface given by a diagonal equation in projective space. For diagonal quartic surfaces with coefficients in Q we determine the Brauer groups over Q and Q(i).

Brauer classes of a global field can be represented by cyclic algebras. Effective constructions of such algebras and a maximal order therein are given for Fq(t), excluding cases of wild ramification. As part of the construction, we also obtain a new description of subfields of cyclotomic function fields.

For an abelian variety A over an algebraically closed non-archimedean field of residue characteristic p, we show that there exists a perfectoid space which is the tilde-limit of lim ← −[p] A.Our proof also works for the larger class of abeloid varieties.

In this paper we study the existence of rational points for the family of K3 surfaces over Q given byWhen the coefficients are ordered by height, we show that the Brauer group is almost always trivial, and find the exact order of magnitude of surfaces for which there is a Brauer-Manin obstruction to the Hasse principle. Our results show definitively that K3 surfaces can have a Brauer-Manin obstruction to the Hasse principle that is only explained by odd order torsion.

We prove the following special case of Mazur's conjecture on the topology of rational points. Let E be an elliptic curve over Q with j-invariant 1728. For a class of elliptic pencils which are quadratic twists of E by quartic polynomials, the rational points on the projective line with positive rank fibres are dense in the real topology. This extends results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic polynomials.For the proof, we investigate a highly singular rational curve on the Kummer surface K associated to a product of two elliptic curves over Q, which previously appeared in publications by Mestre, Kuwata-Wang and Satgé. We produce this curve in a simpler manner by finding algebraic equations which give a direct proof of rationality. We find that the same equations give rise to rational curves on a class of more general surfaces extending the Kummer construction. This leads to further applications apart from Mazur's conjecture, for example the existence of rational points on simultaneous twists of superelliptic curves.Finally, we give a proof of Mazur's conjecture for the Kummer surface K without any restrictions on the j-invariants of the two elliptic curves.