Unramified Brauer Group Research Articles

Unramified Brauer group of quotient spaces by finite groups

We give a general procedure to determine the unramified Brauer group of quotients of rational varieties by finite groups.

Norm tori of étale algebras and unramified Brauer groups

Let k be a field, and let L be an étale k-algebra of finite rank. If a∈k×\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a \\in {k^ \ imes }$$\\end{document}, let Xa be the affine variety defined by NL/k(x)=a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${N_{L/k}}(x) = a$$\\end{document}. Assuming that L has at least one factor that is a cyclic field extension of k, we give a combinatorial description of the unramified Brauer group of Xa.

Multiplicative Invariant Fields of Dimension ≤6

Multiplicative Invariant Fields of Dimension ≤6

Irrationality of Generic Quotient Varieties via Bogomolov Multipliers

Abstract The Bogomolov multiplier of a group is the unramified Brauer group associated with the quotient variety of a faithful representation of the group. This object is an obstruction for the quotient variety to be stably rational. The purpose of this paper is to study these multipliers associated with nilpotent pro-$p$ groups by transporting them to their associated Lie algebras. Special focus is set on the case of $p$-adic Lie groups of nilpotency class $2$, where we analyse the moduli space. This is then applied to give information on asymptotic behaviour of multipliers of finite images of such groups of exponent $p$. We show that with fixed $n$ and increasing $p$, a positive proportion of these groups of order $p^n$ have trivial multipliers. On the other hand, we show that by fixing $p$ and increasing $n$, log-generic groups of order $p^n$ have non-trivial multipliers. Whence quotient varieties of faithful representations of log-generic $p$-groups are not stably rational, applications in non-commutative Iwasawa theory are developed.

Unramified Brauer groups of conic bundle threefolds in characteristic two

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.

Conic bundle fourfolds with nontrivial unramified Brauer group

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 of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.

The unramified Brauer group of homogeneous spaces with finite stabilizer

We give formulas for calculating the unramified Brauer group of a homogeneous space X X of a semisimple simply connected group G G with finite geometric stabilizer F ¯ \bar F over a wide family of fields of characteristic 0 0 . When k 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 ) X(k_v) for every v v outside of an explicit finite set of nonarchimedean places of k k .

Stable rationality of quadric and cubic surface bundle fourfolds

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2,3) in P^2 x P^3 is not stably rational. Via projections onto the two factors, X is a cubic surface bundle over P^2 and a conic bundle over P^3, and we analyze the stable rationality problem from both these points of view. This provides another example of a smooth family of rationally connected fourfolds with rational and nonrational fibers. Finally, we introduce new quadric surface bundle fourfolds over P^2 with discriminant curve of any even degree at least 8, having nontrivial unramified Brauer group and admitting a universally CH_0-trivial resolution.

Degree three unramified cohomology groups

Let k be any field, G be a finite group. Let G act on the rational function field k(xg:g∈G) by k-automorphisms defined by h⋅xg=xhg for any g,h∈G. Denote by k(G)=k(xg:g∈G)G, the fixed subfield. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. The unramified Brauer group Brnr(C(G)) and the unramified cohomology Hnr3(C(G),Q/Z) are obstructions to the rationality of C(G) (see [14] and [5]). Peyre proves that, if p is an odd prime number, then there is a group G such that |G|=p12, Brnr(C(G))={0}, but Hnr3(C(G),Q/Z)≠{0}; thus C(G) is not stably C-rational [12]. Using Peyre's method, we are able to find groups G with |G|=p9 where p is an odd prime number such that Brnr(C(G))={0}, Hnr3(C(G),Q/Z)≠{0}.

Birational classification of fields of invariants for groups of order 128

Let G be a finite group acting on the rational function field C(xg:g∈G) by C-automorphisms h(xg)=xhg for any g,h∈G. Noether's problem asks whether the invariant field C(G)=k(xg:g∈G)G is rational (i.e. purely transcendental) over C. By Fischer's theorem, C(G) is rational over C when G is a finite abelian group. Saltman and Bogomolov, respectively, showed that for any prime p there exist groups G of order p9 and of order p6 such that C(G) is not rational over C by showing the non-vanishing of the unramified Brauer group: Brnr(C(G))≠0, which is an avatar of the birational invariant H3(X,Z)tors given by Artin and Mumford where X is a smooth projective complex variety whose function field is C(G). For p=2, Chu, Hu, Kang and Prokhorov proved that if G is a 2-group of order ≤32, then C(G) is rational over C. Chu, Hu, Kang and Kunyavskii showed that if G is of order 64, then C(G) is rational over C except for the groups G belonging to the two isoclinism families Φ13 with Brnr(C(G))=0 and Φ16 with Brnr(C(G))≃C2. Bogomolov and Böhning's theorem claims that if G1 and G2 belong to the same isoclinism family, then C(G1) and C(G2) are stably C-isomorphic. We investigate the birational classification of C(G) for groups G of order 128 with Brnr(C(G))≠0. Moravec showed that there exist exactly 220 groups G of order 128 with Brnr(C(G))≠0 forming 11 isoclinism families Φj. We show that if G1 and G2 belong to Φ16, Φ31, Φ37, Φ39, Φ43, Φ58, Φ60 or Φ80 (resp. Φ106 or Φ114), then C(G1) and C(G2) are stably C-isomorphic with Brnr(C(Gi))≃C2. Explicit structures of non-rational fields C(G) are given for each cases including also the case Φ30 with Brnr(C(G))≃C2×C2.

Noether's problem for groups of order 243

Let k be any field, G be a finite group. Let G act on the rational function field k(xg:g∈G) by k-automorphisms defined by h⋅xg=xhg for any g,h∈G. Denote by k(G)=k(xg:g∈G)G the fixed field. Noether's problem asks, under what situations, the fixed field k(G) will be rational (= purely transcendental) over k. According to the data base of GAP there are 10 isoclinism families for groups of order 243. It is known that there are precisely 3 groups G of order 243 (they consist of the isoclinism family Φ10) such that the unramified Brauer group of C(G) over C is non-trivial. Thus C(G) is not rational over C. We will prove that, if ζ9∈k, then k(G) is rational over k for groups of order 243 other than these 3 groups, except possibly for groups belonging to the isoclinism family Φ7.

Universal commutator relations, Bogomolov multipliers, and commuting probability

Let G be a finite p-group. We prove that whenever the commuting probability of G is greater than (2p2+p−2)/p5, the unramified Brauer group of the field of G-invariant functions is trivial. Equivalently, all relations between commutators in G are consequences of some universal ones. The bound is best possible, and gives a global lower bound of 1/4 for all finite groups. The result is attained by describing the structure of groups whose Bogomolov multipliers are nontrivial, and Bogomolov multipliers of all of their proper subgroups and quotients are trivial. Applications include a classification of p-groups of minimal order that have nontrivial Bogomolov multipliers and are of nilpotency class 2, a nonprobabilistic criterion for the vanishing of the Bogomolov multiplier, and establishing a sequence of Bogomolov's absolute γ-minimal factors which are 2-groups of arbitrarily large nilpotency class, thus providing counterexamples to some of Bogomolov's claims. In relation to this, we fill a gap in the proof of triviality of Bogomolov multipliers of finite almost simple groups.

On the unramified Brauer group of a homogeneous space

On the unramified Brauer group of a homogeneous space

Groupe de Brauer non ramifié des espaces homogènes à stabilisateur fini

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 semisimple simply connected linear group G′ with finite stabilizer G. We also give examples of the calculations that can be done with these formulas.

Unramified Brauer group of the moduli spaces of PGLr(ℂ)-bundles over curves

Abstract Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGLr (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.

The unramified Brauer group of norm one tori

Let k be a number field and K/k Galois. We transform the construction of the unramified Brauer group of the norm one torus RK/k1(Gm) into the construction of a special abelian extension over K. If k=Q and K/Q biquadratic, we explicitly construct the unramified Brauer group of RK/Q1(Gm).

The genus of a division algebra and the unramified Brauer group

Let \(D\) be a finite-dimensional central division algebra over a field \(K\). We define the genus \(\mathbf{gen}(D)\) of \(D\) to be the collection of classes \([D^{\prime }] \in \mathrm{Br}(K)\), where \(D^{\prime }\) is a central division \(K\)-algebra having the same maximal subfields as \(D\). In this paper, we describe a general approach to proving the finiteness of \(\mathbf{gen}(D)\) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of \(K\). This approach is then implemented in some concrete situations, yielding in particular an extension of the Stability Theorem of A. Rapinchuk and I. Rapinchuk (Manuscr. Math. 132:273–293, 2010) from quaternion algebras to arbitrary algebras of exponent two. We also consider an example where the size of the genus can be estimated explicitly. Finally, we offer two generalizations of the genus problem for division algebras: one deals with absolutely almost simple algebraic \(K\)-groups having the same isomorphism/isogeny classes of maximal \(K\)-tori, and the other with the analysis of weakly commensurable Zariski-dense subgroups.

Le groupe de Brauer non ramifié sur un corps global de caractéristique positive

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 BrV at its local points.

Unramified Brauer groups of finite and infinite groups

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 posed by Bass. An algorithm for computing the Bogomolov multiplier is developed.

Groups of order [formula omitted] and their unramified Brauer groups

We prove that if p is a prime, p>3, and G is a group of order p5 not belonging to the 10-th isoclinism family, then the unramified Brauer group of G is trivial.