Odd order obstructions to the Hasse principle on general K3 surfaces.

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 Q together with a 3-torsion Brauer class α 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, α) 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 α. 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 Q3 and local solubility at all other primes. [ABSTRACT FROM AUTHOR]