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 three 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 Q 3 and local solubility at all other primes. 1 Brauer-Manin obstructions to the Hasse principle (see §6 for details):the constant classes Br 0 (X) := im (Br(k) → Br(X)) , and the algebraic classes Br 1 (X) := ker Br(X) → Br(X) .
Elements ofBr(X) that are not algebraic are called transcendental. Brauer classes of odd order, whether algebraic or transcendental, had previously only been known to obstruct weak approximation on K3 surfaces [Pre13, IS15]. Ieronymou and Skorobogatov [IS15] showed that elements of odd order can never obstruct the Hasse principle on smooth diagonal quartics in P 3 Q . Their work prompted them to ask if there exists a locally soluble K3 surface over a number field with an odd-order Brauer-Manin obstruction to the Hasse principle [IS15, p. 183]. Since then, Skorobogatov and Zarhin [SZ16] have shown that such classes cannot obstruct the Hasse principle on Kummer surfaces. Recently, Corn and Nakahara [CN17] gave a positive answer to Ieronymou and Skorobogatov's question, by showing that a 3-torsion algebraic class can obstruct the Hasse principle on a degree 2 K3 surface over Q. Our main result shows that 3-torsion transcendental Brauer classes can obstruct the Hasse principle on K3 surfaces. Theorem 1.1. There exists a K3 surface Y over Q of degree 2, together with a class α in Br Y [3] such that Y (A Q ) = ∅ and Y (A Q ) {α} = ∅.Moreover, Pic Y ∼ = Z, and hence Br 1 (Y )/ Br 0 (Y ) = 0. In particular, there is no algebraic Brauer-Manin obstruction to the Hasse principle on Y .
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.
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