# Failure of the Hasse principle on general $K 3$ surfaces

@article{Hassett2013FailureOT, title={Failure of the Hasse principle on general \$K 3\$ surfaces}, author={Brendan Hassett and Anthony V{\'a}rilly-Alvarado}, journal={Journal of the Institute of Mathematics of Jussieu}, year={2013}, volume={12}, pages={853 - 877} }

Abstract We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general $K 3$ surface $X$ of degree $2$ over $ \mathbb{Q} $, together with a 2-torsion Brauer class $\alpha $ that is unramified at every finite prime, but ramifies at real points of $X$. With motivation from Hodge theory, the pair $(X, \alpha )$ is constructed from a double cover of ${ \mathbb{P} }^{2} \times { \mathbb{P} }^{2} $ ramified over a… Expand

#### 32 Citations

Odd order obstructions to the Hasse principle on general K3 surfaces

- Computer Science, Mathematics
- Math. 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. Expand

Brauer–Manin obstructions on degree 2 K3 surfaces

- Mathematics
- 2017

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… Expand

On Brauer groups of double covers of ruled surfaces

- Mathematics
- 2013

Let $$X$$X be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic different from $$2$$2. The main result of this paper is a finite… Expand

Brauer--Manin obstructions on genus-2 K3 surfaces

- Mathematics
- 2017

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

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

- Mathematics
- 2020

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

Rational points and derived equivalence

- Mathematics
- Compositio Mathematica
- 2021

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of… Expand

Unramified Brauer Classes on Cyclic Covers of the Projective Plane

- Mathematics
- 2017

Let \( {X} \rightarrow \mathbb{P}^{2}\) be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of characteristic diffierent from… Expand

Derived equivalences and rational points of twisted K3 surfaces

- Mathematics
- 2015

Using a construction of Hassett--V\'arilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the… Expand

Brauer Groups on K3 Surfaces and Arithmetic Applications

- Mathematics
- 2017

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… Expand

Moduli spaces of sheaves on K3 surfaces and Galois representations

- Mathematics
- Selecta Mathematica
- 2020

We consider two K3 surfaces defined over an arbitrary field, together with a smooth proper moduli space of stable sheaves on each. When the moduli spaces have the same dimension, we prove that if the… Expand

#### References

SHOWING 1-10 OF 95 REFERENCES

On the Brauer-Manin obstruction for cubic surfaces

- Mathematics
- 2010

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is of order two or four. This covers the vast majority of the cases… Expand

The Brauer group of cubic surfaces

- Mathematics
- 1993

1. Let V be a non-singular rational surface defined over an algebraic number field k . There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V… Expand

The Hasse problem for rational surfaces.

- Mathematics
- 1975

Let ̂ be a family of varieties F, for example the family of all non-singular cubic surfaces; and let ^"* be the family of all pairs (F, k) such that k is an algebraic number field and F is in $* and… Expand

The Brauer-Manin obstruction on Del Pezzo surfaces of degree 2

- Mathematics
- 2007

This paper explores the computation of the Brauer-Manin obstruction on Del Pezzo surfaces of degree 2, with examples coming from the class of “semi-diagonal” Del Pezzo surfaces of degree 2. It is… Expand

The Brauer-Manin obstruction and Sha[2]

- Mathematics
- 2007

We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in magma. This is illustrated… Expand

On the computation of the Picard group for K3 surfaces

- Mathematics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2011

Abstract We present a method to construct examples of K3 surfaces of geometric Picard rank 1. Our approach is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module… Expand

A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces

- Mathematics
- 2006

Let k be a field finitely generated over the field of rational numbers, and Br(k) the Brauer group of k. For an algebraic variety X over k we consider the cohomological Brauer–Grothendieck group… Expand

K3 surfaces with Picard number one and infinitely many rational points

- Mathematics
- 2005

In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron-Severi group over an algebraic closure of the base field, is high… Expand

The Brauer–Manin obstruction on del Pezzo surfaces of degree 2 branched along a plane section of a Kummer surface

- Mathematics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2008

Abstract This paper discusses the Brauer–Manin obstruction on double covers of the projective plane branched along a plane section of a Kummer surface from both the practical and the theoretical… Expand

Tate-Shafarevich groups and K3 surfaces

- Computer Science, Mathematics
- Math. Comput.
- 2010

This paper explores a topic taken up recently by Logan and van Luijk, finding nontrivial 2-torsion elements of the Tate-Shafarevich group of the Jacobian of a genus-2 curve by exhibiting Brauer-Manin… Expand