相关论文: The \'{E}tale Homotopy Type and Obstructions to th…

In this paper we propose to use a relative variant of the notion of the \'{e}tale homotopy type of an algebraic variety in order to study the existence of rational points on it. In particular, we use an appropriate notion of homotopy fixed…

In 2010, Poonen gave the first example of failure of the local-global principle that cannot be explained by Skorobogatov's \'etale Brauer-Manin obstruction. Motivated by this example, we show that the Brauer-Manin obstruction detects…

The Brauer-Manin obstruction is used to explain the failure of the local-global principle for algebraic varieties. In 1999 Skorobogatov gave the first example of a variety that does not satisfy the local-global principle which is not…

Building upon our arithmetic duality theorems for 1-motives, we prove that the Manin obstruction related to a finite subquotient $\Be (X)$ of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors…

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…

G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles. Analogous obstruction…

Questions related to Brauer-Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces of tori over a number field are well-studied, generally using arithmetic duality theorems, starting with works of Sansuc and…

We study an analogue of the Brauer-Manin obstruction to the local-global principle for embedding problems over global fields. We will prove the analogues of several fundamental structural results. In particular we show that the (algebraic)…

We give a geometric criterion to check the validity of the integral Tate conjecture for one-cycles on a smooth projective variety that is separably rationally connected in codimension one, and to check that the Brauer-Manin obstruction is…

We study the homotopy type of the simplicial set of continuous semi-algebraic simplexes of an algebraic variety defined over a real closed field, which we will call the real homotopy type. We prove an analogue of the theorem of Artin-Mazur…

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur…

The \'etale homotopy groups of schemes as defined by Artin and Mazur have the disadvantage of being homotopy invariant only in characteristic zero. This and other related problems led to the definition of the tame topology which is coarser…

The \'etale Brauer\textendash Manin obstruction is equivalent to each other among Weil restrictions of an arbitrarily given quasi-projective algebraic variety defined over a number field.

We show that even within a class of varieties where the Brauer--Manin obstruction is the only obstruction to the local-to-global principle for the existence of rational points (Hasse principle), this obstruction, even in a stronger, base…

We study cohomological obstructions to the existence of global conserved quantities. In particular, we show that, if a given local variational problem is supposed to admit global solutions, certain cohomology classes cannot appear as…

Nous montrons comment associer \`a une gerbe d\'efinie sur un corps de nombres une obstruction de Brauer-Manin mesurant, comme dans le cas des vari\'et\'es, le d\'efaut d'existence d'une section globale. Ceci nous conduit \`a une…

An \'etale homotopy type $T(X, z)$ associated to any pointed locally fibrant connected simplicial sheaf $(X, z)$ on a pointed locally connected small Grothendieck site $(\mc{C}, x)$ is studied. It is shown that this type $T(X, z)$…

Let $K$ be a field of characteristic $\ne 2$ and let $X$ be the affine variety over $K$ defined by the equation $$ X:\ a_0x_0^2 + \cdots + a_nx_n^2 = 1 $$ where $n\ge 0$ and $a_i\in K$. In this paper we compute the lowest mod 2 \'{e}tale…

We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of "adjoint logic" in which the discretization and codiscretization modalities are characterized using a judgmental formalism of "crisp…

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…