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… 
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21 Citations
Background Citations
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21 Citations

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