Comparing descent obstruction and Brauer-Manin obstruction for open varieties

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 torsors under a torus. As an appliaction, we prove that the semi-simple part of a connected linear algebraic group will satisfy strong approximation with Brauer-Manin obstruction if G iteself satisfies strong approximation with Brauer-Manin obstruction.The equivalence between descent obstruction and etale Brauer-Manin obstruction for smooth projective varieties is extended to smooth quasi-projective varieties, which provides the perspective to study integral points

The Grunwald problem and approximation properties for homogeneous spaces

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 as the case of constant groups $G$ in the more general problem of determining for which $K$-groups $G$ the variety $\mathrm{SL}_n/G$ has weak approximation. We show that away from an explicit set of bad places both problems have an affirmative answer for iterated semidirect products with abelian kernel. Furthermore, we give counterexamples to both assertions at bad places. These turn out to be the first examples of transcendental Brauer-Manin obstructions to weak approximation for homogeneous spaces.Comment: 18 pages. Final version. Accepted for publication in Annales de l'Institut Fourie

Le principe de Hasse pour les espaces homog\`enes : r\'eduction au cas des stabilisateurs finis (The Hasse principle for homogeneous spaces: reduction to the case of finite stabilizers)

Nous montrons, pour une grande famille de propri\'et\'es $P$ des espaces homog\`enes, que $P$ vaut pour tout espace homog\`ene d'un groupe lin\'eaire connexe d\`es qu'elle vaut pour les espaces homog\`enes de $\mathrm{SL}_n$ \`a stabilisateur fini. Nous r\'eduisons notamment \`a ce cas particulier la v\'erification d'une importante conjecture de Colliot-Th\'el\`ene sur l'obstruction de Brauer-Manin au principe de Hasse et \`a l'approximation faible. Des travaux r\'ecents de Harpaz et Wittenberg montrent que le r\'esultat principal s'applique \'egalement \`a la conjecture analogue (dite conjecture (E)) pour les z\'ero-cycles. We prove, for a wide family of properties $P$ of homogeneous spaces, that if $P$ is satisfied for homogeneous spaces of $\mathrm{SL}_n$ with finite stabilizers, then $P$ is satisfied for all homogeneous spaces of linear connected groups. In particular, we reduce to this particular case the verification of an important conjecture by Colliot-Th\'el\`ene on the Brauer-Manin obstruction to the Hasse principle and to weak approximation. Recent work by Harpaz and Wittenberg show that our main result can also be applied to the analog conjecture on zero-cycles (known as conjecture (E)).Comment: 30 pages, in French. V4: generalized Sections 2, 3 and 4 to perfect fields assuming that all algebraic groups are smooth (with practically no changes!); other minor change