# Brauer Groups on K3 Surfaces and Arithmetic Applications

@article{McKinnie2017BrauerGO, title={Brauer Groups on K3 Surfaces and Arithmetic Applications}, author={Kelly L. McKinnie and Justin Sawon and Sho Tanimoto and Anthony V{\'a}rilly-Alvarado}, journal={arXiv: Algebraic Geometry}, year={2017}, pages={177-218} }

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…

## 21 Citations

Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

- Mathematics
- 2021

Given a smooth projective variety over a number eld and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class.…

Contractions of hyper-K\"ahler fourfolds and the Brauer group

- Mathematics
- 2021

We study the geometry of exceptional loci of birational contractions of hyperKähler fourfolds that are of K3-type. These loci are conic bundles over K3 surfaces and we determine their classes in the…

Neighbors and arithmetic of isogenous K3 surfaces

- Mathematics
- 2022

A BSTRACT . We use lattice theory to study the isogeny class of a K3 surface. Starting from isotropic Brauer classes, we construct isogenies via Kneser method of neighboring lattices. We also…

Brauer–Manin obstructions on degree 2 K3 surfaces

- MathematicsResearch in Number Theory
- 2018

We analyze the Brauer–Manin obstruction to rational points on the K3 surfaces over $${{\mathbb {Q}}}$$Q given by double covers of $${{\mathbb {P}}^{2}}$$P2 ramified over a diagonal sextic. After…

ARITHMETIC OF K 3 SURFACES

- Mathematics
- 2016

Being surfaces of intermediate type, i.e., neither geometrically rational or ruled, nor of general type, K3 surfaces have a rich yet accessible arithmetic theory, which has started to come into focus…

Odd order obstructions to the Hasse principle on general K3 surfaces

- MathematicsMath. Comput.
- 2020

It is proved 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 $\mathbb{Q}_3$ and local solubility at all other primes.

Moduli spaces of K3 surfaces and cubic fourfolds

- Mathematics
- 2019

This thesis is concerned with the Hodge-theoretic relation between polarized K3 surfaces of degree d and special cubic fourfolds of discriminant d, as introduced by Hassett. For half of the d, K3…

ABELIAN $n$ -DIVISION FIELDS OF ELLIPTIC CURVES AND BRAUER GROUPS OF PRODUCT KUMMER & ABELIAN SURFACES

- MathematicsForum of Mathematics, Sigma
- 2017

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$ . In 2008, Skorobogatov and Zarhin showed that the Brauer group…

ARITHMETIC OF K3 SURFACES DRAFT LECTURE NOTES

- Mathematics
- 2015

Example 1.2 (K3 surfaces of degrees 4, 6, and 8). Let X be a smooth complete intersection of type (d1, . . . , dr) in Pk , i.e., X ⊆ P has codimension r and X = H1∩· · ·∩Hr, where Hi is a…

A transcendental Brauer–Manin obstruction to weak approximation on a Calabi–Yau threefold

- MathematicsResearch in Number Theory
- 2022

In this paper we investigate the $$\mathbb {Q}$$ Q -rational points of a class of simply connected Calabi–Yau threefolds, which were originally studied by Hosono and Takagi in the context of mirror…

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