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 $3$-torsion Brauer class $\alpha$ that is unramified at all primes except for $3$, but ramifies at all $3$-adic points of $Y$. Motivated by Hodge theory, the pair $(Y, \alpha )$ 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 $\alpha$. 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.