Zéro-cycles sur les espaces homogènes et problème de Galois inverse

  title={Z{\'e}ro-cycles sur les espaces homog{\`e}nes et probl{\`e}me de Galois inverse},
  author={Yonatan Harpaz and Olivier Wittenberg},
Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Th\'el\`ene, Sansuc, Kato and Saito on the image of the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable, we show that rational points of X are dense in the Brauer-Manin set. For finite supersolvable… Expand
The \'etale Brauer-Manin obstruction to strong approximation on homogeneous spaces
It is known that, under a necessary non-compactness assumption, the Brauer-Manin obstruction is the only one to strong approximation on homogeneous spaces $X$ under a linear group $G$ (or under aExpand
On the Malle conjecture and the Grunwald problem.
We contribute to the Malle conjecture on the number N (K, G, y) of finite Galois extensions E of some number field K of finite group G and of discriminant of norm |N K/Q (d E)| $\le$ y. We prove theExpand
The Grunwald problem and specialization of families of regular Galois extensions
We investigate specializations of infinite families of regular Galois extensions over number fields. The problem to what extent the local behaviour of specializations of one single regular GaloisExpand
Number fields with prescribed norms.
We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence ofExpand
The Massey vanishing conjecture for number fields
A conjecture of Min\'a\v{c} and T\^an predicts that for any n>2, any prime p and any field k, the Massey product of n Galois cohomology classes in H^1(k,Z/pZ) must vanish if it is defined. WeExpand
Compatibility of weak approximation for zero-cycles on products of varieties
Zero-cycles are conjectured to satisfy weak approximation with Brauer-Manin obstruction for proper smooth varieties defined over number fields. Roughly speaking, we prove that the conjecture isExpand
Le principe de Hasse pour les espaces homogènes : réduction au cas des stabilisateurs finis
Nous montrons, pour une grande famille de propriétés des espaces homogènes, qu’une telle propriété vaut pour tout espace homogène d’un groupe linéaire connexe dès qu’elle vaut pour les espacesExpand
Constructing abelian extensions with prescribed norms
Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of theExpand
Bad places for the approximation property for finite groups.
Given a number field $k$ and a finite $k$-group $G$, the Tame Approximation Problem for $G$ asks whether the restriction map $H^1(k,G)\to\prod_{v\in\Sigma}H^1(k_v,G)$ is surjective for every finiteExpand
The local dimension of a finite group over a number field
Let $G$ be a finite group and $K$ a number field. We construct a $G$-extension $E/F$, with $F$ of transcendence degree $2$ over $K$, that specializes to all $G$-extensions of $K_\mathfrak{p}$, whereExpand


Zéro-cycles sur les fibrations au-dessus d’une courbe de genre quelconque
Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k wereExpand
Groupe de Picard et groupe de Brauer des compactifications lisses d’espaces homogènes
This is a thoroughly revised version of math.AG/0502516v1 (24 Feb. 2005). Let k be a field of characteristic zero. Let Y=G/H, where G is a connected linear algebraic group over k and H is a connectedExpand
Complexes de groupes de type multiplicatif et groupe de Brauer non ramifié des espaces homogènes
Let k be a field, G a smooth connected linear algebraic group and X a homogeneous space of G over k, such that the geometric stabilizers are extensions of a smooth group of multiplicative type by aExpand
On the fibration method for zero-cycles and rational points
Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Th\'el\`ene, Sansuc, Kato and Saito in the 1980's. We prove that theseExpand
The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)
This is a survey on the ancient question : Let G be a reductive group over an algebraically closed field k and let V be a vector space over k with an almost free linear action of G on V. Let k(V)Expand
Rational connectedness and Galois covers of the projective line
Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely thereExpand
Number Theory and Algebraic Geometry: Valeurs d'un polynôme à une variable représentées par une norme
The aim of the paper is twofold. On the one hand, we study the Brauer group of a smooth and proper model of the k-variety given by P (t) = NormK/k(z), where P (t) is a polynomial, and NormK/k(z) isExpand
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 $K$Expand
The Brauer-Manin obstructions for homogeneous spaces with connected or abelian stabilizer.
More precisely, let X be an algebraic variety over a number field k. The variety X is called a counter-example to the Hasse principle, if X has a kv-poinl for any completion kv ofk, but has noExpand
Rational solutions of certain equations involving norms
Let k be an algebraic closure of k. In the case when P(t) has at most one root in k, the open subset of the affine variety (1) given by P(t)y~O is a principal homogeneous space under an algebraicExpand