Let $k$ be a field of characteristic zero and ${\bar k}$ an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\bar k}(X)$ for the function field of ${\bar X}=X\times_k{\bar k}$. If $X$ has a smooth $k$-point, the natural embedding of multiplicative groups ${\bar k}^*\hookrightarrow {\bar k}(X)^* $ admits a Galois-equivariant retraction.
In the first part of the paper, over local and then over global fields, equivalent conditions to the existence of such a retraction are given. They are expressed in terms of the Brauer group of $X$.
In the second part of the paper, we restrict attention to varieties which are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For $k$ local or global, for such a variety $X$, in many situations but not all, the existence of a Galois-equivariant retraction to ${\bar k}^*\hookrightarrow {\bar k}(X)^* $ ensures the existence of a $k$-rational point on $X$. For homogeneous spaces of linear algebraic groups, the technique also handles the case where $k$ is the function field of a complex surface.
In this article, we study obstructions to weak approximation for connected linear groups and homogeneous spaces with connected or abelian stabilizers over finite extensions of $\mathbb C((x,y))$ or function fields of curves over $\mathbb C((t))$. We show that for connected linear groups, the usual Brauer-Manin obstruction works as in the case of tori. However, this Brauer-Manin obstruction is not enough for homogeneous spaces, as shown by the examples we give. We then construct an obstruction using torsors under quasi-trivial tori that explains the failure of weak-approximation.
Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms
一个整数可以用每个p-adic整数环和实数上的二次型来表示，而不用整数上的这个二次型来表示。更一般地，局部-整体原理的这种失败可能发生在用另一个整二次型表示一个整二次型的情况下。我们表明，许多这样的例子可以占布劳尔-马宁障碍的存在的整数定义的计划上的积分点。对于几类线性代数群的齐性空间，证明了这种障碍是整点存在的唯一障碍。- Une forme quadratique entière peut être représentée par une autre forme quadratique entière sur tous les anneaux d'entiers p-adiques et sur les réels，sans l'être sur les entiers.在文学作品中有许多相似之处。我们的观点是，这两个问题中有一个部分是关于布劳尔-马宁式障碍物的解释。对于代数线性组空间的多个类型，这种障碍是一个整体点存在的障碍。
In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field $\mathbb{C}((x,y))$ of Laurent series in two variables over the complex numbers and over function fields of curves over $\mathbb{C}((t))$. We give examples that prove that the usual Brauer-Manin obstruction is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare t…
We reduce the question about whether the Brauer-Manin obstruction to weak approximation for homogeneous spaces is the only obstruction to the "simpler" question of the particular case of homogeneous spaces of $\mathrm{SL}_n$ with finite stabilizer. We also give unconditional results based on the few known results on finite stabilizers.
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 a connected algebraic group, under assumption of finiteness of a suitable Tate-Shafarevich group), provided that the geometric stabilizers of $X$ are connected. In this work we prove, under similar assumptions, that the étale-Brauer-Manin obstruction to strong approximation is the only one for homogeneous spaces with arbitrary stabilisers. We also deal with some related questions, concerning strong appr…
We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for homogeneous spaces with nilpotent stabilizer. We thus generalize recent results by Harpaz and Wittenberg on finite "hyper-solvable" stabilizers. In particular, this result is true for abelian stabilizers without any hypotheses on the ambient group, which generalizes classic results by Borovoi on the subject. We prove moreover that these spaces have the real approximation property. On montre que l'obstruction de Brauer-Manin est la seule obstruction au principe de Hasse pour les espaces homogènes à sta…
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 toric variety. We also prove a similar result of strong approximation with Brauer-Manin obstruction for a partial equivariant smooth compactification of a homogeneous space where all invertible functions are constant and the semi-simple part of the linear algebraic group is quasi-split. Some semi-abelian varieties of any given dimension where the complements …
Let $K$ be a global field of positive characteristic. We prove that the Brauer-Manin obstructions to the Hasse principle, to weak approximation and to strong approximation are the only ones for homogeneous spaces of reductive groups with reductive stabilizers. The methods involve abelianization techniques and arithmetic duality theorems for complexes of tori over K.
Let X be a homogeneous space of a quasi-trivial k-group G, with geometric stabilizer H, over a number field k. We prove that under certain conditions on the character group of H, certain algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation vanish, because the abelian groups where they take values vanish. When H is connected or abelian, these algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation are the only ones, so we prove the Hasse principle and weak approximation for X under certain conditions. As an application, we obtain new suffic…
Nous montrons, pour une grande famille de propriétés $P$ des espaces homogènes, que $P$ vaut pour tout espace homogène d'un groupe linéaire connexe dès qu'elle vaut pour les espaces homogènes de $\mathrm{SL}_n$ à stabilisateur fini. Nous réduisons notamment à ce cas particulier la vérification d'une importante conjecture de Colliot-Thélène sur l'obstruction de Brauer-Manin au principe de Hasse et à l'approximation faible. Des travaux récents de Harpaz et Wittenberg montrent que le résultat principal s'applique également à la conjecture analogue (dite conjecture (E)) pour les zéro-cycles. We pr…
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 number field, we use these formulas in order to study the Brauer-Manin obstruction to the Hasse principle and weak approximation. We prove in particular that the Brauer-Manin pairing is constant on $X(k_v)$ for every $v$ outside from an explicit finite set of non archimedean places of $k$.
We report on our investigations concerning algebraic and transcendental Brauer-Manin obstructions to integral points on complements of a hyperplane section in degree four del Pezzo surfaces. We discuss moreover two concepts of an obstruction at an archimedean place. Concrete examples are given of pairs of non-homogeneous quadratic polynomials in four variables representing (0,0) over Q and over Z_p for all primes p, but not over Z. By blow-up, these yield cubic polynomials in three variables all integral solutions of which satisfy a gcd condition.
Let C be the complex field and K=C((x,y)) or K=C((x))(y). Let G be a connected linear algebraic group over K. Under the assumption that the K-variety G is K-rational, i.e. that the function field is purely transcendant, it was proved that a principal homogeneous space of G has a rational point over K as soon as it has one over each completion of K with respect to a discrete valuation. In this paper we show that one cannot in general do without the K-rationality assumption. To produce our examples, we introduce a new type of obstruction. It is based on higher reciprocity laws on a 2-dimensional…
In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching. We give a sufficient condition for the vanishing of the obstruction, as well as examples were the obstruction is nontrivial or even infinite. A major tool is the notion of a flasque resolution …
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…
