Odd order obstructions to the Hasse principle on general K3 surfaces

TL;DR

This paper demonstrates that odd order elements in the Brauer group of certain K3 surfaces can prevent the Hasse principle from holding, using a construction linked to cubic fourfolds and del Pezzo fibrations.

Contribution

It introduces a novel example of a K3 surface with a 3-torsion Brauer class obstructing the Hasse principle without needing a central simple algebra representation.

Findings

Constructed a K3 surface with a 3-torsion Brauer class obstructing the Hasse principle.

Linked the obstruction to the insolubility of an associated cubic fourfold over a9_3.

Showed the obstruction occurs despite local solubility at all other primes.

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 $\alpha$ 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.