No abstract

We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, R 5,5 . Such surfaces have a natural elliptic fibration induced by the fibration on R 5,5 . Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on R 5,5 . This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in detail with the cases for which the double cover is branched over the two reducible fibers of type I 5 and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios.

In [VAV11], Várilly-Alvarado and the last author constructed an Enriques surface X over Q with anétale-Brauer obstruction to the Hasse principle and no algebraic Brauer-Manin obstruction. In this paper, we show that the nontrivial Brauer class of X Q does not descend to Q. Together with the results of [VAV11], this proves that the Brauer-Manin obstruction is insufficient to explain all failures of the Hasse principle on Enriques surfaces.The methods of this paper build on the ideas in [CV14a,CV14b,IOOV]: we study geometrically unramified Brauer classes on X via pullback of ramified Brauer classes on a rational surface. Notably, we develop techniques which work over fields which are not necessarily separably closed, in particular, over number fields.

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.