Brauer groups on K3 surfaces and arithmetic applications

TL;DR

This paper classifies p-torsion subgroups of the Brauer group on general complex K3 surfaces and explores their geometric realizations, leading to new counterexamples to weak approximation.

Contribution

It extends the classification of p-torsion Brauer groups on K3 surfaces and connects these to geometric constructions and arithmetic applications.

Findings

Classified p-torsion subgroups via lattice techniques.

Constructed geometric realizations of Brauer elements.

Provided new counterexamples to weak approximation.

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.