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
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. Expand
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. AfterExpand
On Brauer groups of double covers of ruled surfaces
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 finiteExpand
Brauer--Manin obstructions on genus-2 K3 surfaces
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 explicitExpand
A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold
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. TheseExpand
Rational points and derived equivalence
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 ofExpand
Unramified Brauer Classes on Cyclic Covers of the Projective Plane
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 fromExpand
Derived equivalences and rational points of twisted K3 surfaces
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 theExpand
Brauer Groups on K3 Surfaces and Arithmetic Applications
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 toExpand
Moduli spaces of sheaves on K3 surfaces and Galois representations
  • S. Frei
  • 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 theExpand
...
1
2
3
4
...

References

SHOWING 1-10 OF 95 REFERENCES
On the Brauer-Manin obstruction for cubic surfaces
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 casesExpand
The Brauer group of cubic surfaces
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 VExpand
The Hasse problem for rational surfaces.
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 $* andExpand
The Brauer-Manin obstruction on Del Pezzo surfaces of degree 2
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 isExpand
The Brauer-Manin obstruction and Sha[2]
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 illustratedExpand
On the computation of the Picard group for K3 surfaces
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 moduleExpand
A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces
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 groupExpand
K3 surfaces with Picard number one and infinitely many rational points
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 highExpand
The Brauer–Manin obstruction on del Pezzo surfaces of degree 2 branched along a plane section of a Kummer surface
  • A. Logan
  • 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 theoreticalExpand
Tate-Shafarevich groups and K3 surfaces
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-ManinExpand
...
1
2
3
4
5
...