Eva Bayer-Fluckiger, Raman Parimala
In this paper we introduce a method to obtain algebraic information using arithmetic one in the study of tori and their principal homogeneous spaces. In particular, using some results of the authors with Tingyu Lee, we determine the unramified Brauer groups of some norm one tori, and their torsors.
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We produce a partial compactification of the variety given by P(t)=N_{K/k}(\mathbf z) whose Brauer group coincides with the unramified Brauer group, where K is an étale k-algebra and P(t)\in k[t] is a nonconstant polynomial. Then we obtain a systematic method to compute the unramified Brauer group for all such varieties.
This is a textbook on arithmetic geometry with special regard to unramified Brauer groups of algebraic varieties. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, arithmetic and geometry of quadrics, Weil restriction of scalars, algebraic tori, an example of a stably rational non-rational variety, Brauer-Manin obstruction. All material is split into locally trivial problems with detailed hints.
Using a reduction of the Galois cohomology of a linear algebraic group $G$ to that of a certain finite subquotient, we give different formulas allowing the calculation of the unramified algebraic Brauer group of a homogeneous space $V=G\backslash G'$ with $G'$ semisimple and simply connected, both over a finite field and over an arbitrary field of characteristic $0$.
Let $X$ be a smooth projective curve over the complex numbers. We compute the Brauer group of the moduli stack of Bruhat-Tits group scheme $\mathcal{G}$-torsors on $X$. When $g(X) \geq 3$ we compute the Brauer group of the regularly stable locus of the coarse moduli space of semi-stable $\mathcal{G}$-torsors.
Classifying torsors of tori with Brauer groups
本文利用Mackey函子，通过基域有限扩张的Brauer群对代数环面的torsors进行分类，并将Blunk的结果推广至所有retract有理环面。
Unramified Brauer groups of homogeneous spaces with finite stabilisers and the Grunwald Problem
本文针对半单单连通群作用且具有有限几何稳定子的齐性空间，给出计算无分歧Brauer群的算法，并据此研究数域上弱逼近的Brauer-Manin障碍，从而保证超可解群Grunwald问题的有效性。
Splitting unramified Brauer classes by abelian torsors and the period-index problem
本文利用扭曲相对Picard簇，通过固定阿贝尔概型的挠子分裂射影簇上的Brauer类，并证明非分歧Brauer类的指标整除其周期的固定幂次。
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 $k$.
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。
Torsors of Generalized del Pezzo Tori and Brauer Groups
利用上同调 Mackey 函子，给出广义 del Pezzo 簇上环面 torsor 的显式 Brauer 群分类，推广了 Blunk 对六次 del Pezzo 曲面的 Brauer 理论描述。
Vertical unramified Brauer groups of Galois normic bundles
计算由 N_{K/k}(z)=P(x) 给出的 Galois 范数丛的垂直非分歧 Brauer 群，通过 Galois 群结构和 P(x) 的不可约因子给出组合公式。
We study the homogeneous irreducible Severi-Brauer varieties over an Abelian variety $A$. Such objects were classified by Brion, \cite{Bri}. Here we interpret that result within the context of cubical structures and biextensions for certain $\G_m$-torsors over finite subgroups of $A$. Our results can be seen as an instance of the theory developed by Breen, \cite{Breen:1983}, and Moret-Bailly, \cite{Moret-Bailly}.
Varieties with prescribed finite unramified Brauer groups and subgroups precisely obstructing the Hasse principle
针对数域上簇的Hasse原理阻碍问题，证明对任意非零有限阿贝尔群对$B_0\subset B$，存在射影线上的范丛簇$X$，其非分歧Brauer群同构于$B$且$B_0$是$B$中阻碍Hasse原理的最小子群。
Descent and étale-Brauer obstructions for 0-cycles
该研究定义了数域上代数簇0-闭链的下降集与平展-Brauer障碍集，证明其包含于Brauer-Manin集且与Suslin 0次奇异同调相容，将有理点算术研究工具迁移至0-闭链场景并推广Liang的策略到挠子，给出多类代数簇的应用。
Unramified Brauer group of quotient spaces by finite groups
本文提出了一个通用算法，用于计算有理簇在有限群作用下的商空间的无分歧布劳尔群。
Let D be a finite-dimensional central division algebra over a field K. We define the genus gen(D) of D to be the collection of classes in the Brauer group of K represented by central division K-algebras D' having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K.
We develop different formulas of algebraic and/or arithmetic nature allowing an explicit calculation, both over a finite field and over a field of characteristic 0, of the algebraic part of the unramified Brauer group of a homogeneous space G\G' under a semisiple simply connected linear group G' with finite stabilizer G. We also give examples of the calculations that can be done with these formulas.
We give a new proof of the theorem stating that for any connected linear algebraic group G over an algebraically closed field k of characteristic 0 and for any closed connected subgroup H of G, the unramified Brauer group of G/H vanishes.
Brauer groups of smooth loci in linear systems and torsors over Jacobians of plane curves
研究单连通光滑射影簇线性系统中光滑轨迹的 Brauer 群，在适当丰沛条件下证明其至多为 Z/2Z，并应用于计算通用光滑平面曲线相对雅可比上的 Tate-Shafarevich 群。
We classify all groups of order $p^5$ with non-trivial unramified Brauer groups. We show that if $p>3$, then there are precisely $\gcd (p-1,4)+\gcd (p-1,3)+1$ such groups.
We show that if $G_1$ and $G_2$ are isoclinic groups, then their Bogomolov multipliers are isomorphic.
En utilisant un théorème de Gabber sur les altérations, on démontre un résultat décrivant la partie de torsion première à $p$ du groupe de Brauer non ramifié d'une variété $V$ lisse et géométriquement intègre sur un corps global de caractéristique $p$ au moyen de l'évaluation des éléments de $\Br V$ sur ses points locaux. Using a theorem of Gabber on alterations, we prove a result describing the prime-to-$p$ torsion part of the unramified Brauer group of a smooth and geometrically integral variety $V$ over a global field of characteristic $p$ by evaluating the elements of $\Br V$ at its local …
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。
Let $X$ be an irreducible smooth complex projective curve of genus at least two. Let $N$ be a connected component of the moduli space of semistable principal ${\rm PGL}_r({\mathbb C})$- bundles over $X$; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of $N$ is trivial.
We establish a formula for computing the unramified Brauer group of tame conic bundle threefolds in characteristic 2. The formula depends on the arrangement and residue double covers of the discriminant components, the latter being governed by Artin-Schreier theory (instead of Kummer theory in characteristic not 2). We use this to give new examples of threefold conic bundles defined over the integers that are not stably rational over the complex numbers.
Let X --> P^2 be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of prime-to-p characteristic. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut(X/P^2) arise as pullbacks of certain Brauer classes on k(P^2) that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on k(P^2) and relate the kernel of the pullback map to the Picard group of X. If p = 2, we give a second construction, which works over any base field of characteristic not 2, that uses …
Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(\mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three cases. For $[K:k]=4$ and $P(t)$ irreducible over $k$ and split in $K$, we prove the Hasse principle and weak approximation. For $k=\mathbb{Q}$ with arbitrary $K$, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For $[K:k]=4$ and $P(t)$ irreducible over $k$, we determine the Brauer group of smooth proper mod…
We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P^3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen--Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P^2. We also prove the existence of universally CH_0-trivial resolutions for the general class…
We study the subgroup B_0(G) of H^2(G,Q/Z) consisting of all elements which have trivial restrictions to every Abelian subgroup of G. The group B_0(G) serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties V/G where V is a faithful complex linear representation of the group G. We prove that B_0(G) is trivial for finite simple groups of Lie type A_{\ell}.
Unramified Grothendieck-Serre for isotropic groups
本文证明了在无分歧环R上，各向同性群G的Grothendieck-塞雷猜想成立，并利用Brauer群纯洁性将问题简化到单连通群。
The unramified Grothendieck--Serre conjecture
本文证明非分歧正则局部环上约化群概形的一般平凡主齐性空间必为平凡，解决了该情形下的 Grothendieck--Serre 猜想。
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。
In this paper, we study the Nisnevich sheafification $\mathcal{H}^1_{\acute{e}t}(G)$ of the presheaf associating to a smooth scheme the set of isomorphism classes of $G$-torsors, for a reductive group $G$. We show that if $G$-torsors on affine lines are extended, then $\mathcal{H}^1_{\acute{e}t}(G)$ is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for $G$-torsors. We also identify the sheaf $\mathcal{H}^1_{\acute{e}t}(G)$ with the sheaf of $\mathbb{A}^1$-connected components of the classifying space ${\rm B}_{\acute{e}t}G$. This establishe…
The Bogomolov multiplier is a group theoretical invariant isomorphic to the unramified Brauer group of a given quotient space. We derive a homological version of the Bogomolov multiplier, prove a Hopf-type formula, find a five term exact sequence corresponding to this invariant, and describe the role of the Bogomolov multiplier in the theory of central extensions. A new description of the Bogomolov multiplier of a nilpotent group of class two is obtained. We define the Bogomolov multiplier within K-theory and show that proving its triviality is equivalent to solving a long-standing problem pos…
Let $k$ be any field, $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $h\cdot x_g=x_{hg}$ for any $h,g\in G$. Define $k(G)=k(x_g : g\in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is known that, if $\bC(G)$ is rational over $\bC$, then $B_0(G)=0$ where $B_0(G)$ is the unramified Brauer group of $\bC(G)$ over $\bC$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G)=0$. This result was disproved by Moravec for $p=3,5,7$ by computer computing. We will give a theoretic proof of the followin…
Unramified Grothendieck-Serre for simply-connected group schemes satisfying an isotropy condition via unipotent chains
论文证明了Grothendieck-Serre猜想的一个情况，研究了在特定条件下群方案的 torsor 三重性，引入了单因子 torsor 链的概念，并简化了准分裂无分支情况下的证明。
We study Grothedieck groups of triangulated categories using weight structures, weight complexes, and the corresponding pure (co)homological functors. We prove some general statements on $K_0$ of weighted categories and apply it to Voevodsky motives endowed with so-called Chow weight structures. We obtain certain "motivic substitutes" for smooth compactifications of smooth varieties over arbitrary perfect fields; this enables us to make certain unramified cohomology and Euler characteristic calculations that are closely related to results of T. Ekedahl and B. Kahn.
For a homogeneous space X of a connected algebraic group G (with connected stabilizers) over a field k of characteristic zero, we construct a canonical complex of Galois modules of length 3 and a canonical isomorphism between an hypercohomology group of this complex and an explicit subgroup of the Brauer group of X. This result is obtained as a consequence of a formula describing the "algebraic Brauer group of a torsor", and it generalizes recent results by Borovoi and van Hamel, by considering non-linear groups G and by taking into account some transcendental elements in the Brauer group of X…
Let k be a field, G a finite group embedded in the k-group SL(n). For k an algebraically closed field, Bogomolov gave a formula for the unramified Brauer group of the quotient SL(n)/G. We develop his method over any characteristic zero field. This purely algebraic method enables us to recover and generalize results of Harari and of Demarche over number fields, such as the triviality of the unramified Brauer group for k=Q and G of odd order. --- Soient k un corps et G un groupe fini plongé dans le k-groupe SL(n).Pour k algébriquement clos, Bogomolov a donné une formule pour le groupe de Brauer …
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。
Let $k$ be any field, $G$ be a finite group acing on the rational function field $k(x_g:g\in G)$ by $h\cdot x_g=x_{hg}$ for any $h,g\in G$. Define $k(G)=k(x_g:g\in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is known that, if $\bm{C}(G)$ is rational over $\bm{C}$, then $B_0(G)=0$ where $B_0(G)$ is the unramified Brauer group of $\bm{C}(G)$ over $\bm{C}$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G)=0$. This result was disproved by Moravec for $p=3,5,7$ by computer calculations. We will prove the following theorem.…
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…
This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified cohomology for the construction of nontrivial stable invariants of the special fiber. In particular, we find an explicit formula for the Brauer group of fourfolds fibered in quadrics of dimension two over a rational surface.
The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. We settle it in the case when $G$ is quasi-split and $R$ is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.
Malle's conjecture and Brauer groups of stacks
本文提出Malle猜想领头常数的栈版本猜想，并引入代数栈的非分歧Brauer群新概念，为研究数域判别式分布提供新框架。
We show that certain Tate--Shafarevich groups are unramified which enables us to give an obstruction to the Hasse principle for torsors under tori over p-adic function fields.
Abelianized Descent Obstruction for 0-Cycles
本文通过阿贝尔上同调理论，定义了0-循环的阿贝尔化降阶障碍，并证明其与Brauer-Mein障碍的相等性，同时探讨了其在射影理性连通变种和射影K3表面中的闭包性质。
Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. For Kummer varieties we show th…
The linear positive torsor presentation and arithmetic purity of strong approximation
本文定义线性正挠子表示概念，对数域上具该表示的代数簇建立带Brauer--Manin障碍的强逼近算术纯性，恢复光滑环面簇的对应性质并分类其有理点链连通类。
Combining work of Peyre, Colliot-Thélène and Voisin, we give the first example of a finite group $G$ such that the motivic class of its classifying stack $BG$ in Ekedahl's Grothendieck ring of stacks over $\mathbb{C}$ is non-trivial and $BG$ has trivial unramified Brauer group.
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…
We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C_1. In general, we show that a large (but in general proper) subgroup of the 2-torsion classes are given by the construction. Examples demonstrating applications to the arithmetic of hyperelliptic curves defined over number fields are given.
We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class, and there exist many such curves splitting the class. We show that this can be false when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class.
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 a smooth connected characterfree group. If k has characteristic zero and if X^c is a smooth compactification of X over k, we obtain a formula for the algebraic Brauer group of X^c. Several variants are obtained in positive characteristic p, including the finite field case and the global field case, where the formulae describe the prime-to-p part of the algebraic unramified Brauer group of X, without a…
A torsor under a k-group scheme G on a variety X over a number field k imposes a descent obstruction against the existence of rational points on X. We discuss the finite descent obstruction, that is for all such torsors under finite k-groups G, in view of a local-global interpolation property for sections of the fundamental group short exact sequence of X/k. There are applications to the Brauer-Manin obstruction, to the descent obstruction by torsors under linear groups, and to the birational version of Grothendieck's section conjecture over number fields. In particular, we obtain examples of …
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.
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…
Auel-Bigazzi-Böhning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as well. In this paper, we generalize their argument to arbitrary unramified mod $p$ étale motivic cohomology groups. We also see that the properness assumption on the variety $X$ can be dropped off by using the Suslin homology together with a certain tame subgroup of the unramified cohomology group.
We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class A that is unramified at every finite prime, but ramifies at real points of X. Motivated by Hodge theory, the pair (X,A) is constructed from a double cover of P^2 x P^2 ramified over a hypersurface of bi-degree (2,2).
We show that, in general, over a regular integral noetherian affine scheme X of dimension at least 6, there exist Brauer classes on X for which the associated division algebras over the generic point have no Azumaya maximal orders over X. Despite the algebraic nature of the result, our proof relies on the topology of classifying spaces of algebraic groups.
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