Related papers: Le d\'efaut d'approximation forte dans les groupes…

Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group…

Given a finite extension $K/k$ of number fields and a smooth quasi-projective variety $X$ over $K$. If the abelianized fundamental group of $X$ is trivial, we prove that there is a natural identification between Brauer-Manin sets of $X$ and…

We attach to any commutative ring R a subgroup of the Brauer group of R, called the Brauer-Galois group of R. Its elements are the classes of the Azumaya R-algebras which can be represented, via Brauer equivalence, by a Galois extension of…

Greither and Pareigis have established a connection between Hopf Galois structures on a Galois extension $L/K$ with Galois group $G$, and the regular subgroups of the group of permutations on $G$, which are normalized by $G$. Byott has…

Let $K/F$ be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified $p$-adic representation $\rho$ of the absolute Galois group of $K$ is determined…

Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic…

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite…

This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological…

Given an associative $\mathbb{C}$-algebra $A$, we call $A$ strongly rigid if for any pair of finite subgroups of its automorphism groups $G, H,$ such that $A^G\cong A^H$, then $G$ and $H$ must be isomorphic. In this paper we show that a…

Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $\rho_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $\rho_A$ encodes the Galois action on the torsion points…

Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a hypothesis on T, we establish a correspondence…

Let $\k$ be a (topological) field of characteristic 0. Using a Drinfeld associator $\Phi$, a representation $\Phi(\rho)$ of the braid group over the field $\k((h))$ of Laurent series can be associated to any representation $\rho$ of a…

Given a flat, finite group scheme G finitely presented over a base scheme we introduce the notion of ramified Galois cover of group G (or simply G-cover), which generalizes the notion of G-torsor. We study the stack of G-covers, denoted…

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant each…

If k is an arbitrary field, we construct a category of k-1-motives in which every commutative algebraic k-group G has a dual object $G^{\vee}$. When k is a local field of arbitrary characteristic, we establish Pontryagin duality theorems…

Let i be a homomorphism of the multiplicative group into a connected reductive algebraic group over C. Let G^i be the centralizer of the image i. Let LG be the Lie algebra of G and let L_nG (n integer) be the summands in the direct sum…

We give a method to produce faithful representations of the groups $G(n,m)=\langle X, Y \ \vert \ X^m = Y^n \rangle$ in $\mathrm{GL}_2(\mathbb{C}[t^{\pm 1}, q^{\pm 1}])$. These groups are Garside groups and the Garside normal forms of…

Standard results from non-abelian cohomology theory specialize to a theory of torsors and stacks for cosimplicial groupoids. The space of global sections of the stack completion of a cosimplicial groupoid $G$ is weakly equivalent to the…

Discrete subfactors include a particular class of infinite index subfactors and all finite index ones. A discrete subfactor is called local when it is braided and it fulfills a commutativity condition motivated by the study of inclusion of…

Let $G$ be a connected Lie group, $C$ be the maximal compact connected subgroup of the center of $G$, and let Aut$(G)$ denote the group of Lie automorphisms of $G$, viewed, canonically, also as a subgroup of GL$(\frak G)$, where $\frak G$…