Effectivity of Brauer-Manin obstructions on surfaces

德利涅-芒福德堆叠上的下降障碍与布饶尔-马林障碍

Descent and Brauer-Manin Obstructions on Deligne-Mumford Stacks

该研究在具有拟射粗模空间的有限型光滑分离德利涅-芒福德堆叠上，证明下降障碍包含于布饶尔-马林障碍，且下降障碍与 étale 布饶尔-马林障碍、迭代下降障碍一致，布饶尔-马林障碍也与迭代布饶尔-马林障碍一致。

数域与函数域上0-循环的Brauer–Manin障碍的乘积

Product of Brauer--Manin obstruction for 0-cycles over number fields and function fields

本文证明光滑簇乘积上1次0-循环的Brauer–Manin障碍等价于各因子同时存在该障碍，并将类似结果推广到$\mathbb{C}((t))$-曲线的函数域上的光滑簇。

Brauer-Manin obstructions to integral points

We study Brauer-Manin obstructions to integral points on open subsets of the projective plane.

Varietes de descente, gerbes et obstruction de Brauer-Manin (Descent varieties, gerbes and Brauer-Manin obstruction)

Nous montrons comment associer à une gerbe définie sur un corps de nombres une obstruction de Brauer-Manin mesurant, comme dans le cas des variétés, le défaut d'existence d'une section globale. Ceci nous conduit à une généralisation de la dualité de Tate-Poitou au cas non-abélien. We find a way to associate to a gerbe defined over a number field a Brauer-Manin obstruction, which measures, as for varieties, the defect to the existence of a global section. This leads us to a generalization of Tate-Poitou duality in the non-abelian case.

Brauer-Manin obstruction for Erdős-Straus surfaces

We study the failure of the integral Hasse principle and strong approximation for the Erdős-Straus conjecture using the Brauer-Manin obstruction.

Brauer-Manin obstruction for zero-cycles on certain varieties

We investigate the question of whether the existence of a family of local zero-cycles of degree $d$ orthogonal to the Brauer group implies the non-emptiness of the Brauer-Manin set for certain varieties. We provide various examples of Brauer-Manin obstruction to the existence of zero-cycles of appropriate degrees.

A note on the Brauer group and the Brauer-Manin set of a product

We generalize the results of Skorobogatov and Zarhin considering the commutativity of Brauer groups (and Brauer-Manin sets) with taking product of two varieties, by relaxing the condition that varieties are projective.

非分歧上同调与Brauer--Manin配对

Unramified cohomology and Brauer--Manin pairing

本文证明局部域上光滑射影曲面非分歧上同调的Tate模在适当条件下消失，并应用于Brauer--Manin配对的核问题，通过建立Bloch公式实现。

特征p下p-主Brauer-Manin障碍的二分法

Dichotomy for the Brauer-Manin obstruction in characteristic p

研究特征p全局函数域上光滑射影几何整变种的p-主Brauer-Manin障碍，证明当单幂Brauer群为零且Picard概形无p-挠时，仅有限个位可能相关；否则几乎所有位都可能相关。

The Brauer-Manin obstruction on a general diagonal quartic surface

We prove that, on a sufficiently general diagonal quartic surface, there is a non-trivial Brauer group but no Brauer-Manin obstruction to the existence of rational points.

需要任意多 Brauer 类的 Brauer-Manin 障碍

Brauer-Manin obstructions requiring arbitrarily many Brauer classes

本文研究定义在整体域上的射影簇，证明捕获 Brauer-Manin 障碍的 Brauer 群有限子群可能需要任意多个生成元。

簇上的有理点与Brauer-Manin障碍

Rational points on varieties and the Brauer-Manin obstruction

本讲义介绍了有理点存在性的Brauer-Manin障碍，重点探讨理论与计算之间的相互作用。

Insufficiency of the Brauer-Manin obstruction applied to etale covers

Let k be any global field of characteristic not 2. We construct a k-variety X such that X(k) is empty, but for which the emptiness cannot be explained by the Brauer-Manin obstruction or even by the Brauer-Manin obstruction applied to finite etale covers.

Singular curves and the etale Brauer-Manin obstruction for surfaces

We construct a smooth and projective surface over an arbitrary number field that is a counterexample to the Hasse principle but has the infinite etale Brauer-Manin set. We also construct a surface with a unique rational point and the infinite etale Brauer-Manin set. The key new ingredient is the arithmetic of singular projective curves.

Order 5 Brauer-Manin obstructions to the integral Hasse principle on log K3 surfaces

We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle.

The Brauer-Manin obstruction for sections of the fundamental group

We establish Grothendieck's section conjecture for an open subset of the Reichardt-Lind curve, and introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension.

Pathologies of the Brauer-Manin obstruction

We construct a conic bundle over an elliptic curve over a real quadratic field that is a counterexample to the Hasse principle not explained by the étale Brauer-Manin obstruction. We also give simple examples of threefolds with the same property that are families of 2-dimensional quadrics, and discuss some other examples and general properties of the Brauer-Manin obstruction.

具有Brauer-Manin障碍的2次对角del Pezzo曲面

Diagonal del Pezzo of degree 2 with a Brauer-Manin obstruction

该研究针对按高度排序的2次对角del Pezzo曲面，推导了存在Hasse原理的Brauer-Manin障碍的这类曲面数量的渐近公式。

Obstructions de Brauer-Manin entières sur les espaces homogènes à stabilisateurs finis nilpotents

Let $k$ be a number field. We construct homogeneous spaces of $SL_{n,k}$ with finite nilpotent non-abelian stabilizers for which the Brauer-Manin obstruction does not explain the failure of strong approximation (resp. the failure of the integral Hasse principle).

Brauer--Manin 障碍源于正特征中 Kodaira 消失定理的失效

Brauer--Manin obstruction from failure of Kodaira vanishing in positive characteristic

研究正特征全局域上 Kodaira 消失定理失效的光滑射影曲面的 Brauer-Manin 集，特别应用于 Raynaud 曲面。

好约化素数在Brauer–Manin障碍中的作用

The role of primes of good reduction in the Brauer--Manin obstruction

本研究探讨好约化素数在数域代数簇弱逼近的Brauer–Manin障碍中的作用，给出素理想分歧指数的相关必要条件，并以多个K3曲面的超越Brauer–Manin障碍实例佐证结果。

Brauer-Manin pairing, class field theory and motivic homology

For a smooth proper variety over a $p$-adic field, the Brauer group and abelian fundamental group are related to the higher Chow groups by the Brauer-Manin pairing and the class field theory. We generalize this relation to smooth (possibly non-proper) varieties, using the motivic homology and the tame version of Wiesend's ideal class group. Several examples are discussed.

On the Brauer-Manin Obstruction Applied to Ramified Covers

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 explained by the Brauer-Manin obstruction. He did so by applying the Brauer-Manin obstruction to étale covers of the variety, and thus defining a finer obstruction. In 2008 Poonen gave the first example of failure of the local-global principle which cannot be explained for by Skorobogatov's étale-Brauer obstruction. However, Poonen's construction was not accompani…

对称积的无分歧 Brauer 群及其 Brauer-Manin 障碍

Unramified Brauer groups of symmetric products and the Brauer-Manin obstructions

研究光滑、射影且几何积分的变种X，其几何Picard群为 torsion-free，建立X与对称积的Brauer群同构，探讨Brauer-Manin障碍对Hasse原理和弱逼近及理点障碍的关系。

Integral Brauer-Manin obstructions for sums of two squares and a power

We use Brauer-Manin obstructions to explain failures of the integral Hasse principle and strong approximation away from infinity for the equation x^2+y^2+z^k=m with fixed integers k>=3 and m. Under Schinzel's hypothesis (H), we prove that Brauer-Manin obstructions corresponding to specific Azumaya algebras explain all failures of strong approximation away from infinity at the variable z. Finally, we present an algorithm that, again under Schinzel's hypothesis (H), finds out whether the equation has any integral solutions.

The Brauer group and the Brauer-Manin set of products of varieties

Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $\mathbb Q$, and let $\ov X$ and $\ov Y$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of $\Br(\ov X)\oplus \Br(\ov Y)$ has finite index in the Galois invariant subgroup of $\Br(\ov X\times\ov Y)$. This implies that the cokernel of the natural map $\Br(X)\oplus\Br(Y)\to\Br(X\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer-Manin set of the product…

Applications of the fibration method for zero-cycles to the Brauer-Manin obstruction to the existence of zero-cycles on certain varieties

We study the Brauer-Manin obstruction to the existence of zero-cycles of degree $d$ on certain classes of varieties over number fields. We generalise existing results in the literature and prove some results about fibrations over the projective line, where the geometric Brauer group of the generic fibre is not assumed to be finite. The idea is to assume that the Brauer-Manin obstruction to the Hasse principle is the only one for certain fibres and then deduce analogous results for zero-cycles.

Cubic points on cubic curves and the Brauer-Manin obstruction on K3 surfaces

We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the Brauer-Manin obstruction is not the only obstruction to the Hasse principle for K3 surfaces.

Non-invariance of the Brauer-Manin obstruction for surfaces

In this paper, we study the properties of weak approximation with Brauer-Manin obstruction and the Hasse principle with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields. We assume a conjecture of M. Stoll. For any nontrivial extension of number fields $L/K,$ we construct two kinds of smooth, projective, and geometrically connected surfaces defined over $K.$ For the surface of the first kind, it has a $K$-rational point, and satisfies weak approximation with Brauer-Manin obstruction off $\infty_K,$ while its base change by $L$ does not so off $\infty_L.$ F…

On the Brauer-Manin obstruction for cubic surfaces

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is of order two or four. This covers the vast majority of the cases when this group is non-zero. Our approach is to associate a Brauer class with every Galois invariant double-six. We show that all order two Brauer classes may be obtained in this way. We also recover Sir Peter Swinnerton-Dyer's result that $\Br(S)/\Br(\bbQ)$ may take only five values.

代数堆上的Brauer-Manin 障碍

The Brauer-Manin obstruction on algebraic stacks

本文研究代数堆上的Brauer-Manin集和配对，扩展了Colliot-Thélène和Sansuc的降理论，证明了局部由变分线性群商的堆的Brauer群的 torsion 性，并展示了Brauer-Manin障碍与其它上同调障碍的一致性。

Towards the Brauer-Manin obstruction on varieties fibred over the projective line

Recently Dasheng Wei proved that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for 0-cycles of degree 1 on some fibrations over the projective line defined by bi-cyclic normic equations. In the present paper, we prove the exactness of the global-to-local sequence for Chow groups of 0-cycles of such varieties, which signifies that the Brauer-Manin obstruction is also the only obstruction to weak approximation for 0-cycles of arbitrary degree. Our main theorem also generalizes several existing results.

On the Brauer-Manin obstruction for zero-cycles on curves

We wish to give a short elementary proof of S. Saito's result that the Brauer-Manin obstruction for zero-cycles of degree 1 is the only one for curves, supposing the finiteness of the Tate-Shafarevich-group $\sha^1(A)$ of the Jacobian variety. In fact we show that we only need a conjecturally finite part of the Brauer-group for this obstruction to be the only one. We also comment on the situation in higher dimensions

Sous-groupe de Brauer invariant pour un groupe algébrique connexe quelconque

In this paper, for a smooth variety equiped with an action of a connected algebraic group (not necessary linear), we introduce the notion of invariant Brauer sub-group and the notion of invariant étale Brauer-Manin obstruction. Then we prove that this obstruction is equivalent to the étale Brauer-Manin obstruction. This extends the main notion and the key result of [C17]. As a corollary, this also extends a result of B. Creutz.

Non-invariance of weak approximation with Brauer-Manin obstruction for surfaces

In this paper, we study the property of weak approximation with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields. For any nontrivial extension of number fields L/K, assuming a conjecture of M. Stoll, we construct a smooth, projective, and geometrically connected surface over K such that it satisfies weak approximation with Brauer-Manin obstruction off all archimedean places, while its base change to L fails. Then we illustrate this construction with an explicit unconditional example.

Insufficiency of the Brauer-Manin obstruction for Enriques surfaces

In 2011, Vàrilly-Alvarado and the last author constructed an Enriques surface $X$ over $\mathbb{Q}$ with an étale-Brauer obstruction to the Hasse principle and no algebraic Brauer-Manin obstruction. In this paper, we show that the nontrivial Brauer class of $X_{\bar{\mathbb{Q}}}$ does not descend to $\mathbb{Q}$. Together with the results of Vàrilly-Alvarado and the last author, this proves that the Brauer-Manin obstruction is insufficient to explain all failures of the Hasse principle on Enriques surfaces. The methods of this paper build on the ideas in several recent papers by the last autho…

Persistence of the Brauer-Manin obstruction on cubic surfaces

Let $X$ be a cubic surface over a global field $k$. We prove that a Brauer-Manin obstruction to the existence of $k$-points on $X$ will persist over every extension $L/k$ with degree relatively prime to $3$. In other words, a cubic surface has nonempty Brauer set over $k$ if and only if it has nonempty Brauer set over some extension $L/k$ with $3\nmid[L:k]$. Therefore, the conjecture of Colliot-Thélène and Sansuc on the sufficiency of the Brauer-Manin obstruction for cubic surfaces implies that $X$ has a $k$-rational point if and only if $X$ has a $0$-cycle of degree $1$. This latter statement…

Markoff型三次曲面上整点的Brauer-Manin障碍

Brauer-Manin obstruction for integral points on Markoff-type cubic surfaces

该研究延续相关工作，探讨Markoff型三次曲面整点的Brauer-Manin障碍，构造可由该障碍解释的强逼近反例并给出计数结果，还提出无法用代数Brauer-Manin障碍解释的整点Hasse原理反例。

Transcendental Brauer-Manin obstruction on a pencil of elliptic curves

This note gives an explicit example of transcendental Brauer-Manin obstruction to weak approximation. It has two features which the only previously known example of such obstruction did not have: the class in the Brauer group which is responsible for the obstruction is divisible, and the underlying algebraic variety is an elliptic surface (a K3 surface).

The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields

We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.

非等变曲线 over 全局函数域上的 Brauer-Manin 障碍

The Brauer-Manin obstruction for nonisotrivial curves over global function fields

本文研究了非等变曲线 over 全局函数域上有理点集与 Brauer-Manin 障碍的关系，证明二者相等。

Brauer-Manin obstructions on degree 2 K3 surfaces

We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over $\mathbb{Q}$ given by double covers of $\mathbb{P}^2$ ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou …

On the Brauer-Manin obstruction for degree four del Pezzo surfaces

We show that, for every integer $1 \leq d \leq 4$ and every finite set $S$ of places, there exists a degree $d$ del Pezzo surface $X$ over ${\mathbb Q}$ such that ${\rm Br}(X)/{\rm Br}({\mathbb Q}) \cong {\mathbb Z}/2{\mathbb Z}$ and the Brauer-Manin obstruction works exactly at the places in $S$. For $d = 4$, we prove that in all cases, with the exception of $S = \{\infty\}$, this surface may be chosen diagonalizably over ${\mathbb Q}$.

The Brauer-Manin obstruction for constant curves over global function fields

Let $\mathbb{F}$ be a finite field and $C,D$ smooth, geometrically irreducible proper curves over $\mathbb{F}$ and set $K = \mathbb{F}(D)$. We consider Brauer-Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve $C \otimes_\mathbb{F} K$. In particular, we show that Brauer-Manin is the only obstruction to weak approximation and the Hasse principle in the case that the genus of $D$ is less than that of $C$. We also show that we can identify the points corresponding to non-constant maps $D \to C$ using Frobenius descents.

Bad reduction of the Brauer-Manin obstruction

We relate the Brauer group of a smooth variety over a p-adic field to the geometry of the special fibre of a regular model, using the purity theorem in étale cohomology. As an illustration, we describe how the Brauer group of a smooth del Pezzo surface is determined by the singularity type of its reduction. We then relate the evaluation of an element of the Brauer group to the existence of points on certain torsors over the special fibre; we use this to describe situations when the evaluation is constant, and situations when the evaluation is surjective. In the latter case, we describe how thi…

半整Brauer-Manin障碍与二次超曲面轨形对

Semi-integral Brauer-Manin obstruction and quadric orbifold pairs

本文研究Campana点和Darmon点半整点的局部-整体原则，提出介于有理点与整点之间的半整Brauer-Manin障碍，并在二次超曲面轨形族中确定其状态及定量度量其与整障碍的差异。

奇异K3曲面上的超越Brauer-Manin障碍

Transcendental Brauer-Manin obstructions on singular K3 surfaces

本文研究具有复乘的椭圆曲线乘积的Kummer曲面Y的Brauer群，并利用这些曲面构造了弱逼近的超越Brauer-Manin障碍的新例子。

An example of a Brauer-Manin obstruction to weak approximation at a prime with good reduction

Following Bright and Newton, we construct an explicit K$3$ surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.

Markoff型Wehler K3曲面的Brauer-Manin障碍

Brauer-Manin obstruction for Wehler K3 surfaces of Markoff type

本研究针对Markoff型Wehler K3曲面整数点的Brauer-Manin障碍展开，构造了违背整数Hasse原理的族并给出强逼近的反例，还研究了仿射曲面有理点的相关问题。

Brauer-Manin obstruction for Markoff surfaces

Ghosh and Sarnak have studied integral points on surfaces defined by an equation x^2+y^2+z^2-xyz= m over the integers. For these affine surfaces, we systematically study the Brauer group and the Brauer-Manin obstruction to the integral Hasse principle. We prove that strong approximation for integral points on any such surface, away from any finite set of places, fails, and that, for m\neq 0, 4, the Brauer group does not control strong approximation.

对于分类堆栈的étale Brauer-Manin 障碍

The étale Brauer-Manin obstruction for classifying stacks

本文研究了分类堆栈BG的强逼近问题，证明了étale Brauer-Manin障碍是强逼近的主要障碍。

Diagonal quartic surfaces with a Brauer-Manin obstruction

In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer-Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

Evidence for the Dynamical Brauer-Manin Criterion

Let f:X->X be a morphism of a variety over a number field K. We consider local conditions and a "Bruaer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X, i.e., the intersection of the orbit of P with V is empty. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P with V; this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.

堆栈曲线的 Brauer-Manin 障碍

The Brauer-Manin obstruction for stacky curves

本文证明 Brauer-Manin 障碍是亏格为 1/2 的堆栈曲线强逼近的唯一障碍，并计算其光滑固有整数模型的 Brauer-Manin 障碍。