Brauer groups on K3 surfaces and arithmetic applications

Kelly McKinnie, Justin Sawon, Sho Tanimoto, Anthony Várilly-Alvarado

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

14 Scopus citations

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 TS 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, accounted for by transcendental Brauer-Manin obstructions.

Original languageEnglish
Title of host publicationProgress in Mathematics
PublisherSpringer Basel
Pages177-218
Number of pages42
DOIs
StatePublished - 2017

Publication series

NameProgress in Mathematics
Volume320
ISSN (Print)0743-1643
ISSN (Electronic)2296-505X

Keywords

  • Brauer groups
  • K3 surfaces
  • Projective duality
  • Special cubic fourfolds
  • Weak approximation

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