Related papers: Vanishing of algebraic Brauer-Manin obstructions

Consider strong approximation for algebraic varieties defined over a number field $k$. Let $S$ be a finite set of places of $k$ containing all archimedean places. Let $E$ be an elliptic curve of positive Mordell-Weil rank and let $A$ be an…

In this paper, for a smooth variety equiped with an action of a connected algebraic group (not necessary linear), we introduce the notion of invariant Brauer sub-group and the notion of invariant \'etale Brauer-Manin obstruction. Then we…

Let X be a toric variety over a number field k with \kbar[X]^\times=\kbar^\times. Let W\subset X be a closed subset of codimension at least 2. We prove that X\setminus W satisfies strong approximation with algebraic Brauer--Manin…

We construct a finite subgroup of Brauer-Manin obstruction for detecting the existence of integral points on integral models of homogeneous spaces of linear algebraic groups of multiplicative type. As application, the strong approximation…

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 study the distribution of abelian extensions of bounded discriminant of a number field $k$ which fail the Hasse norm principle. For example, we classify those finite abelian groups $G$ for which a positive proportion of $G$-extensions of…

Ghosh and Sarnak have studied integral points on surfaces defined by an equation x^2+y^2+z^2-xyz= m over the integers. For these affine surfaces, we systematically study the Brauer group and the Brauer-Manin obstruction to the integral…

We give an affirmative answer to the Grunwald problem for new families of non-solvable finite groups G, away from the set of primes dividing |G|. Furthermore, we show that such G verify the condition (BM), that is, the Brauer-Manin…

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.

Given a group $G$ and a number field $K$, the Grunwald problem asks whether given field extensions of completions of $K$ at finitely many places can be approximated by a single field extension of $K$ with Galois group G. This can be viewed…

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 closed subvariety of an abelian variety $A$ over a global function field $k$ such that the base change of $A$ to an algebraic closure does not have any positive dimensional isotrivial quotient. We prove that every adelic point…

Let k be a number field and X a smooth projective k-variety. In this paper, we study the information obtainable from descent via torsors under finite k-group schemes on the location of the k-rational points on X within the adelic points.…

We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, equipped with the…

We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient…

Embeddings of maximal tori into classical groups over global fields of characteristic not 2 are the subject matter of several recent papers, with special attention to the Hasse principle. The present paper gives necessary and sufficient…

In this paper, we construct three kinds of Ch\^atelet surfaces, which have some given arithmetic properties with respect to field extensions of number fields. We then use these constructions to study the properties of weak approximation…

The \'etale Brauer\textendash Manin obstruction is equivalent to each other among Weil restrictions of an arbitrarily given quasi-projective algebraic variety defined over a number field.

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…

Classical descent theory of Colliot-Th\'el\`ene and Sansuc for rational points tells that, over a smooth variety $X$, the algebraic Brauer--Manin subset equals the descent obstruction subset defined by a universal torsor. Moreover, Harari…