Transcendental obstructions to weak approximation on general K3 surfaces

Abstract

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.

24 pages, 3 figures, Magma scripts included at the end of the source file.

References in corpus (3)

• On the Brauer group of diagonal quartic surfaces
• On the computation of the Picard group for $K3$ surfaces
• Transcendental Brauer-Manin obstruction on a pencil of elliptic curves

Cited by in corpus (23)

• Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
• Brauer groups on K3 surfaces and arithmetic applications
• Quadric surface bundles over surfaces
• Cubic fourfolds containing a plane and a quintic del Pezzo surface
• Transcendental Brauer groups of products of CM elliptic curves
• Failure of the Hasse principle for Enriques surfaces
• On Brauer groups of double covers of ruled surfaces
• Effective bounds for Brauer groups of Kummer surfaces over number fields
• Failure of the Hasse principle on general K3 surfaces
• Extremal rays and automorphisms of holomorphic symplectic varieties
• Residual categories of quadric surface bundles
• On the Picard number of K3 surfaces over number fields
• An example of a Brauer-Manin obstruction to weak approximation at a prime with good reduction
• Some non-special cubic fourfolds
• Spaces of sections of quadric surface fibrations over curves
• Vector bundles of rank four and A_3 = D_3
• Effective computation of Picard groups and Brauer-Manin obstruction of degree two K3 surfaces over number fields
• Rationality in families of threefolds
• Integral Hodge classes on fourfolds fiberd by quadric bundles
• Rational distances from given rational points in the plane
• Brauer-Manin obstructions on degree 2 K3 surfaces
• A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold
• Transcendental Brauer elements via descent on elliptic surfaces