Related papers: Local-global principles for 1-motives

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…

For algebraic stacks over number fields, we define their Brauer-Manin sets, Brauer-Manin pairings, and extend the descent theory of Colliot-Th\'el\`ene and Sansuc. By extending Sansuc's exact sequence, we show the torsionness of Brauer…

We describe descent on families of torsors of a constant torus. A recent result of Browning and Matthiesen then implies that the Brauer--Manin obstruction controls the Hasse principle and weak approximation when the ground field is the…

We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and to the weak approximation for zero-cycles on certain fibrations over a smooth curve or over the projective space. The principal novelty is that…

A powerful method pioneered by Swinnerton-Dyer allows one to study rational points on pencils of curves of genus 1 by combining the fibration method with a sophisticated form of descent. A variant of this method, first used by Skorobogatov…

We give formulas for calculating the unramified Brauer group of a homogeneous space $X$ of a semisimple simply connected group $G$ with finite geometric stabilizer $\bar F$ over a wide family of fields of characteristic 0. When $k$ is a…

The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that…

We wish to give a short elementary proof of S. Saito's result that the Brauer-Manin obstruction for zero-cycles of degree 1 is the only one for curves, supposing the finiteness of the Tate-Shafarevich-group $\sha^1(A)$ of the Jacobian…

We show that certain Tate--Shafarevich groups are unramified which enables us to give an obstruction to the Hasse principle for torsors under tori over p-adic function fields.

We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the…

In 2011, V\`arilly-Alvarado and the last author constructed an Enriques surface $X$ over $\mathbb{Q}$ with an \'etale-Brauer obstruction to the Hasse principle and no algebraic Brauer-Manin obstruction. In this paper, we show that the…

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.

Let $k$ be a number field. We construct homogeneous spaces of $SL_{n,k}$ with finite nilpotent non-abelian stabilizers for which the Brauer-Manin obstruction does not explain the failure of strong approximation (resp. the failure of the…

We generalise a result of Heath-Brown and Skorobogatov to show that a certain class of varieties over a number field $k$ satisfies Weak Approximation and the Hasse Principle, provided there is no Brauer-Manin obstruction.

We investigate the "ramified descent problem": which adelic points of a smooth geometrically connected variety $X$ defined over a number field $K$ can be approximated by points that lift to a (twist of a) given ramified cover? We show that…

We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning…

Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(\mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three…

We study Brauer-Manin obstructions to the Hasse principle and to weak approximation with special regard to effectivity questions.

The aim of this note is to revisit the question of local-global principles for embeddings of etale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and sufficient…

We provide a relation between Brauer-Manin obstruction and descent obstruction for torsors over open varieties under a connected linear algebraic group or a group of multiplicative type is given. Such a relation is further refined for…