Related papers: Good reduction of the Brauer-Manin obstruction

We discuss the role of primes of good reduction in the existence of the Brauer--Manin obstruction to weak approximation for varieties defined over number fields. Following Bright and Newton, we give some necessaries conditions on the…

We relate the Brauer group of a smooth variety over a p-adic field to the geometry of the special fibre of a regular model, using the purity theorem in \'etale cohomology. As an illustration, we describe how the Brauer group of a smooth del…

Let $X$ be a K3 surface defined over a number field $k$, with principal complex multiplication by a CM field $E$. We find explicit bounds, in terms of $k$ and $E$, on the size of the transcendental Brauer group…

Suppose $X$ is a torsor under an abelian variety $A$ over a number field. We show that any adelic point of $X$ that is orthogonal to the algebraic Brauer group of $X$ is orthogonal to the whole Brauer group of $X$. We also show that if…

This article focuses on smooth, projective, and geometrically integral varieties $X$ defined over a number field $k$ with torsion-free geometric Picard groups. We establish an isomorphism between the Brauer groups of $X$ and its symmetric…

We consider the Brauer-Manin obstruction to the existence of integral points on affine surfaces defined by $x^2 - ay^2 = P(t)$ over a number field. We enumerate the possibilities for the Brauer groups of certain families of such surfaces,…

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a…

Let $L$ be a number field and let $E/L$ be an elliptic curve with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$. We use class field theory and results of Skorobogatov and Zarhin to…

For a curve over a global field we consider for which integers d the d-primary part of the Brauer group can obstruct the existence of rational points. We give examples showing it is possible that there is a d-primary obstruction for…

Following Bright and Newton, we construct an explicit K$3$ surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.

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 set of generators for the geometric Picard…

Unlike the classical Brauer group of a field, the Brauer-Grothendieck group of a singular scheme need not be torsion. We show that there exist integral normal projective surfaces over a large field of positive characteristic with…

We study the Brauer groups of affine surfaces that are complements of singular hyperplane sections of smooth cubic surfaces over a field $k$ of characteristic $0$. We determine the Brauer group over the algebraic closure as a Galois module…

A torsor under a k-group scheme G on a variety X over a number field k imposes a descent obstruction against the existence of rational points on X. We discuss the finite descent obstruction, that is for all such torsors under finite…

Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether…

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur…

We prove that any open subset $U$ of a semi-simple simply connected quasi-split linear algebraic group $G$ with ${codim} (G\setminus U, G)\geq 2$ over a number field satisfies strong approximation by establishing a fibration of $G$ over a…

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between…

We determine the odd order torsion subgroup of the Brauer group of diagonal quartic surfaces over the field of rational numbers. We show that a non-constant Brauer element of odd order always obstructs weak approximation but never the Hasse…

On a projective variety defined over a global field, any Brauer--Manin obstruction to the existence of rational points is captured by a finite subgroup of the Brauer group. We show that this subgroup can require arbitrarily many generators.