Brauer groups on K3 surfaces and arithmetic applications

Abstract

For a prime $p$, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to sublattices of index p of the transcendental lattice T_S of S; we classify these lattices up to isomorphism using Nikulin's discriminant form technique. We then study geometric realizations of p-torsion Brauer elements as Brauer-Severi varieties in a few cases via projective duality. We use one of these constructions for an arithmetic application, giving new kinds of counter-examples to weak approximation on K3 surfaces of degree two.

40 pages; changes in exposition

References in corpus (1)

• Transcendental obstructions to weak approximation on general K3 surfaces

Cited by in corpus (6)

• Cubic fourfolds fibered in sextic del Pezzo surfaces
• Hyperkähler fourfolds and Kummer surfaces
• Effective bounds for Brauer groups of Kummer surfaces over number fields
• Extremal rays and automorphisms of holomorphic symplectic varieties
• A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold
• Brauer-Manin obstructions on degree 2 K3 surfaces