Odd order obstructions to the Hasse principle on general K3 surfaces

Abstract

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface $Y$ of degree 2 over $\mathbb{Q}$ together with a three torsion Brauer class $\hat{I}\pm$ that is unramified at all primes except for 3, but ramifies at all 3-adic points of $Y$. Motivated by Hodge theory, the pair $(Y,\hat{I}\pm )$ is constructed from a cubic fourfold $X$ of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for $\hat{I}\pm$. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold $X$ (and hence the fibers) over ${\mathbb{Q}}_3$ and local solubility at all other primes.

22 pages; Magma scripts included as ancillary files in the arXiv distribution