L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

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L’INSTITUT FOURIER

Cyril DEMARCHE,

Giancarlo LUCCHINI ARTECHE & Danny NEFTIN

The Grunwald problem and approximation properties for homogeneous spaces

Tome 67, n^o3 (2017), p. 1009-1033.

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THE GRUNWALD PROBLEM AND APPROXIMATION PROPERTIES FOR HOMOGENEOUS SPACES

by Cyril DEMARCHE,

Giancarlo LUCCHINI ARTECHE & Danny NEFTIN (*)

Abstract. — Given a groupGand a number fieldK, the Grunwald problem asks whether given field extensions of completions of K at finitely many places can be approximated by a single field extension ofK with Galois groupG. This can be viewed as the case of constant groups Gin the more general problem of determining for whichK-groupsGthe variety SLn/Ghas weak approximation. We show that away from an explicit set of bad places both problems have an affirmative answer for iterated semidirect products with abelian kernel. Furthermore, we give counterexamples to both assertions at bad places. These turn out to be the first examples of transcendental Brauer–Manin obstructions to weak approximation for homogeneous spaces.

Résumé. — Pour un groupe G et un corps de nombres K, le problème de Grunwald est le suivant : étant données des extensions des complétés deKen un ensemble fini de places, peut-on les approcher de façon simultanée par une seule extension deK de groupe de GaloisG? Cela peut être interprété comme un cas particulier de la question plus générale de déterminer pour quelsK-groupesGla variété SLn/Gvérifie l’approximation faible. Nous démontrons qu’en dehors d’un ensemble explicite de mauvaises places, ces deux problèmes ont une réponse positive pour les groupes obtenus par des produits semi-directs itérés à noyau abélien. En outre, nous donnons des contre-exemples aux deux affirmations dans l’ensemble des mauvaises places. Ceux-ci sont par ailleurs les premiers exemples d’obstructions de Brauer–Manin transcendantes à l’approximation faible pour les espaces homogènes.

Keywords:Grunwald problem, Galois cohomology, homogeneous spaces, weak approximation, Brauer–Manin obstruction.

Math. classification:11R34, 14M17, 14G05, 11E72.

(*) The first author acknowledges the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005. The second author’s work was partially supported by the Fondation Mathématique Jacques Hadamard through the grant N^o ANR-10-CAMP-0151-02 in the “Programme des Investissements d’Avenir”.

The third author’s work was supported by the National Science Foundation under Award No. DMS-1303990 and the Israel Science Foundation under grant No. 577/15.

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1. Introduction

The Grunwald–Wang theorem has fundamental applications to the structure theory of finite dimensional semisimple algebras, cf. [12, Ch. 18], and provides an answer for abelian groups G to the more general Grunwald problem. The latter is an inverse Galois problem of increasing interest due to its recently studied connections with the regular inverse Galois problem and with weak approximation, cf. [3, 6].

Fix a number field K, let K_v denote the completion ofK at a place v, and Gal(K) denote its absolute Galois group. LetGbe a finite group, and S a finite set of places of K. The Grunwald problem then asks whether every prescribed local Galois extensionsL^(v)/K_v, v∈S, with embeddings Gal(L^(v)/Kv),→Gcan be approximated by a global extensionL/K with Galois groupG. More precisely:

Grunwald Problem. — Is the restriction map Hom(Gal(K), G)→ Y

v∈S

Hom(Gal(K_v), G)/∼

surjective?

Here φ₁∼φ₂ ifφ₁=gφ₂g⁻¹for some g∈G. Note that the quotient by

∼is necessary since the decomposition group of a placev of K is defined up to conjugation.

Families of groupsGand number fieldsK for which (G, K, S) has an affirmative answer to the Grunwald problem for everyS include: (1) abelian groups of odd order over every number field, by the Grunwald–Wang theorem [5, 16]; (2) solvable groups of order prime to the number of roots of unity inK, by Neukirch’s theorem [10]; and (3) groups with a generic extension overK, by Saltman’s theorem [13].

Recent results by Dèbes-Ghazi on the inverse Galois problem [3] and by Harari on weak approximation for homogeneous spaces [6], suggest that for every finite groupGthere exists a finite setT :=T(G, K) of “bad places” of Ksuch that the Grunwald problem has an affirmative answer for (G, K, S) for every set S that is disjoint from T. In fact, the existence of such a set T is implied by a conjecture of Colliot-Thélène on the Brauer–Manin obstruction for rationally connected varieties. The connection is done by considering the following more general version of the Grunwald problem for finiteK-groups G.

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Approximation property. — A K-group G has (weak) approxima- tionin a setS of places of Kif the natural restriction map

H¹(K, G)→ Y

v∈S

H¹(Kv, G),

is surjective. We shall say G has approximation away from T, if G has approximation in S for every finite set S of places of K that is disjoint fromT.

The approximation property for every finite S ⊂ ΩK is equivalent to weak approximation for certain homogeneous spaces (hence its name), see Section 2.5. Moreover, for constant groupsG, it is equivalent to a positive answer to the Grunwald problem for (G, K, S), see Section 2.3.

The existence of a finite set T of “bad places” away from which the approximation property holds is expected to hold for arbitrary finite K- groups, and is strongly related with arithmetic properties of homogeneous spaces. Moreover, the existence of such a set T for K-groups has been proved over arbitrary number fields K for: (1) abelian K-groups G by Wang, cf. [16]; (2) iterated semidirect productsG=A1o(A2o· · ·oAr) of abelianK-groups by Harari, cf. [6]; and (3) solvableK-groupsGof order prime to the number of roots of unity in an extension ofK splittingGby the second author (hereT =∅, cf. [8]).

Ever since Wang’s work on the subject, triples (G, K, S) having a negative answer to the Grunwald problem (and thus not having the approximation property) are known to exist. However, it is unclear what the set T =T(G, K) should be. In fact, there is no explicit description of a setT even for basic groups such as semidirect products of two abelianp-groups.

This paper gives both affirmative and negative answers to the Grun- wald problem and the approximation property, suggesting that the set T =T(G, K) can always be taken to be the union of the places ofKwhich divide the order of G and those which ramify in the minimal extension splittingG.

1.1. Main results

We give a precise description of the set T of “bad places” for iterated semidirect products of abelian groups, complementing Harari’s result [6, Thm. 1].

Theorem 1.1. — LetK be a number field andG be a finiteK-group which is an iterated semidirect productG=A1o(A2o· · ·oAr)of abelian

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K-groups. Let L/K be an extension splitting G, and T the set of places that either divide the order ofG, or ramify inL. ThenGhas approximation away fromT.

If G is constant, L=K and hence the only condition on placesv ∈ S is to be prime to the order of G. The Grunwald problem has then an affirmative answer for such (G, K, S). The family of iterated semidirect products of abelian groups contains the dihedral groups, the Heisenberg groups of orderp³, and the p-Sylow subgroups of the symmetric groups, cf. [17], of GLn(Fq), and of other classical groups over Fq whenqis prime top, cf. [7].

In an opposite scenario, whenSconsists of the primes ofK dividing the order ofGandGis constant, we give the following examples in which the approximation property doesn’t hold. Recall that abicyclic group is either cyclic or a direct product of two cyclic groups.

Theorem 1.2. — Let K be a number field and G a finite abelian p- group that is not bicyclic. Assume that G is a Galois group over some completion ofK (which a fortiori divides p) and let S be the finite set of places of K lying above p. Then there exists an abelian G-module A (of order a power ofp) such that if we consider E := AoG as a constant K-group and Kcontains sufficiently many roots of unity, the map

H¹(K, E)→ Y

v∈S

H¹(Kv, E),

is not surjective.

In contrast to Wang’s counterexamples, here the set S consists of the primes ofK dividing any given rational prime p, as requested in [3, Sec- tion 1.6] by Dèbes-Ghazi. Examples of constantp-groups that do not admit the approximation property at the places dividing a given prime p were also given over K =Q(µp) by the first author in [4, §6, Prop. 2], where an algebraic Brauer–Manin obstruction to the approximation property is given. However, in contrast with [6, §5], [4] and [9, §5.2], which study algebraic Brauer–Manin obstructions, Theorem 1.2 provides examples in which the approximation property doesn’t hold and the algebraic Brauer–

Manin obstruction vanishes, givingthe first examples of a transcendental Brauer–Manin obstruction to weak approximation on homogeneous spaces, see Example 5.4.

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1.2. Tame problems

The above results suggest that the answer to the Grunwald problem is affirmative away from a particular set of bad primes:

Tame approximation problem. — Does the approximation property hold for every finiteK-group Gand every finite setS of places ofK that are prime to its order and are unramified in an extension splittingG?

Here we do not consider the extension C/R to be ramified, hence the question aboutreal approximation(i.e. having the approximation property for the setSof archimedean places), asked years ago by Borovoi, is included in this last one.

Negative answers to these questions are unknown, and a complete affirmative answer is currently out of reach, as it implies a solution to the inverse Galois problem (see [6, §4]). In view of Theorem 1.1 and its proof, it seems reasonable to conjecture that the answer is affirmative for all solvable groups.

Acknowledgements

The authors would like to thank David Harari for being at the origin of this collaboration. The third author would like to thank Pierre Dèbes, David Saltman, Nguyêñ Duy Tân and Jack Sonn for helpful discussions.

2. Preliminaries

2.1. Global and local Galois groups

All throughout this text, K denotes a number field, ΩK is the set of places ofKand, forv∈Ω_K,K_vdenotes the completion ofKinv. For any Galois extension L/K we denote by Gal(L/K) the corresponding Galois group, and by Gal(K) the absolute Galois group. Throughout the text, we fix an embedding of Gal(K_v) in Gal(K) and identify it with its image for eachv∈ΩK.

For a finite placev∈Ω_K, letK_v^tr be the maximal tamely ramified extension ofKv, let Γv := Gal(K_v^tr/Kv) be the tame Galois group and Wv :=

Gal(K_v^tr) the wild ramification group. Recall that the tame inertia group Tv := Gal(K_v^tr/K_v^un) is a procyclic normal subgroup of Γv of order prime

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tov; Letτ_v be a generator ofT_v. The quotient Γv/T_v∼= Gal(K_v^un/K_v)∼= ˆZ is generated by a liftσv of the Frobenius automorphism.

We let σ_v ∈ Γ_v be a preimage of σ_v. Note that σ_v is defined up to conjugation and that its action by conjugation on Tv is equivalent to its action on roots of unity, that is,σvτvσ⁻¹_v =τ_v^q^v whereqv is the cardinality of the residue field ofKv, see [11, §7.5].

For archimedean v, letσv be the generator of Gal(Kv) and putτv = 1, so that Γv:= Gal(K_v) andT_v =hτvi={1}.

2.2. K-groups

A finiteK-groupGis a finite group scheme overK. SinceKis of charac- teristic 0, there is an equivalence of categories between finite group schemes overK and finite Gal(K)-groups. Identifying the two, we shall writeGfor the set of its geometric points. An extension L/K is said to split G if G×KL is a constant L-group, i.e. if the Galois group Gal(L) ⊂Gal(K) acts trivially onG. We also need the following notion of “bad places” for suchG:

Definition 2.1. — LetKbe a number field,Gbe a finiteK-group and L/K the minimal extension splittingG. We define the set of “bad places”

BadG ⊂ΩK as the union of the set of places dividing the order ofG and the places ramified inL/K.

Note that the minimal extension splitting G always exists. Indeed, the action of Gal(K) onGis a (continuous) morphism Gal(K)→Aut(G) and the minimal extension splittingGis given by the kernel of this morphism.

2.3. Cohomology

Recall [14, I, §5] that for a field K and aK-group G, the set H¹(K, G) is defined as the quotient of the set of 1-cocycles (also calledcrossed homomorphisms)

Z¹(K, G) :={a: Gal(K)→Gcontinuous|aστ=aσ^σa_τ,∀σ, τ∈Gal(K)}, by the equivalence

a∼b ⇔ ∃g∈Gsuch thataσ=gbσ^σg⁻¹, ∀σ∈Gal(K).

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Hence for aconstant K-group G, the set H¹(K, G) is the set of continuous group homomorphisms Gal(K)→Gmodulo conjugation inG. In particular, the Grunwald problem for (G, K, S) is equivalent to the approximation property forGconstant. IfL/Kis an extension splittingG, then Gal(L/K) acts onGand one may similarly define the set H¹(L/K, G), which embeds intoH¹(K, G) by inflation.

A class α∈H¹(K, G) is denoted by [a] if it corresponds to the class of the cocycle a. For α ∈ H¹(K, G), we denote by α_v its image under the restriction map Resv: H¹(K, G)→H¹(Kv, G).

2.4. Twisting

We recall briefly the basic properties of twisting. For further details see [14, §I.5]. Let Γ be a profinite group and G be a discrete Γ-group.

Assume thatGand Γ act both on the left on some objectX in a compatible way, that is, (^σ g·x) =^σg·^σxforσ∈Γ, g∈G,x∈X. Given a cocycle a∈Z¹(Γ, G), define a new action of Γ onX bytwisting the first action as follows:

σ∗x:=aσ·^σx.

Such a twisted object, denoted by_aX, still has an action of G. However, the latter is not necessarily compatible with the action of Γ. To fix this, one twists also the action of Γ onG. To do so it suffices to viewGas acting on itself by conjugation, and repeat the same construction, so that

σ∗g:=a_σ^σga⁻¹_σ .

This twist ofG, denoted by_aG, is a Γ-group which acts on_aX compatibly with the Γ-action.

This construction takes principal homogeneous spaces under Gto principal homogeneous spaces under _aG. More precisely, one has [14, I.5.3, Prop. 35bis]:

Proposition 2.2. — Leta∈Z¹(Γ, G)and letG⁰ =_aG. The map ta:Z¹(Γ, G⁰)→Z¹(Γ, G) :a⁰7→(σ7→a⁰_σaσ),

is a bijection which moreover induces a bijection τa: H¹(Γ, G⁰)→H¹(Γ, G),

sending the trivial element ofH¹(Γ, G⁰)to the class of ain H¹(Γ, G).

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An exact sequence of Γ-groups such as 1→H →E→G→1 gives rise to an exact sequence of pointed sets, cf. [14, I.5.5, Prop. 38]:

(2.1) H¹(Γ, H)→H¹(Γ, E)→H¹(Γ, G),

which means that the elements in H¹(Γ, E) falling onto the trivial element of H¹(Γ, G) are precisely those coming from H¹(Γ, H).

Let e∈Z¹(Γ, E) and let g be its image inZ¹(Γ, G). To study the fiber of the class [e]∈H¹(Γ, E), that is, the fiber above [g]∈H¹(Γ, G), one uses twisting. SinceE acts onE, onGand onH by conjugation, one may twist all three groups bye, getting an exact sequence

1→_eH→_eE→eG→1

and the following commutative diagram of pointed sets with exact se- quences:

H¹(Γ, H) //H¹(Γ, E) //

τ_e⁻¹

H¹(Γ, G)

τ_g⁻¹

H¹(Γ, H_e ) //H¹(Γ, E_e ) //H¹(Γ, G_g ).

Note that the twisted form of Gis denoted by _gGinstead of _eG, since E acts onGvia its own image inG. Since the lower row is exact and since the τ’s are bijections sending [g] and [e] to the trivial elements, we now know that the fiber over [g] is in bijection with the image of the set H¹(Γ, H_e ) in H¹(Γ, E_e ). Note also that in general there is no vertical arrow at the level ofH (as Proposition 2.2 applies only to twists by a group acting on itself by conjugation).

Two elements in H¹(Γ, H) are mapped under (2.1) to the same element of H¹(Γ, E) if and only if they lie in the same orbit of H⁰(Γ, G), [14, I.5.5, Prop. 39]. Here an elementg∈H⁰(Γ, G) =G^Γacts on H¹(Γ, H) by sending a class [a]∈H¹(Γ, H) to the class of the cocycle

(2.2) (g·a)_σ=ea_σ^σe⁻¹

wheree∈E is a preimage ofg.

2.5. Homogeneous spaces

Let us now recall the notion of weak approximation forK-varieties. LetX then be a (smooth, geometrically integral)K-variety such thatX(K)6=∅.

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Definition 2.3. — LetS⊂ΩK be a finite set of places of K. We say thatXhasapproximation inSifX(K)is dense in the productQ

SX(Kv).

We say that X has weak approximation away from T ⊂ Ω_K if X has approximation inSfor all finiteS⊂ΩKrT. Equivalently, one can demand X(K) to be dense in the product Q

ΩKrTX(Kv). The set T is usually refered to as the set of “bad places”.

We say thatX hasweak approximationif one can takeT =∅.

Let us briefly recall the Brauer–Manin obstruction to weak approximation, cf. [15, §5.1] for details. For X a (smooth, geometrically integral) K-variety, consider its unramified Brauer group BrunX, as well as the subgroup of its “algebraic” elements Brun,alX:= ker[BrunX→Brun(X×KK)].¯ Denote, forv∈ΩK, invv: BrK_v→Q/Zthe Hasse invariant map. Finally, denote byX(KΩ) the product Q

ΩKX(Kv) in whichX(K) embeds diago- nally. Then one can define the setX(KΩ)^Br^un (resp.X(KΩ)^Br^un,al) as the subset of all families of points (P_v)_v∈Ω_K∈X(K_Ω) such that

X

v∈ΩK

invv(α(Pv)) = 0, for allα∈BrunX (resp.α∈Brun,alX), whereα(Pv)∈BrKv denotes the evaluation ofαat the point Pv and the sum, which a priori is infinite, is actually finite for elements α∈ BrunX. The following inclusions then hold:

X(K)⊆X(K_Ω)^Br^un ⊆X(K_Ω)^Br^un,al ⊆X(K_Ω).

IfX(KΩ)^Br^un 6=X(KΩ) (resp.X(KΩ)^Br^un,al 6=X(KΩ)) one says that there is a Brauer–Manin obstruction (resp. an algebraic Brauer–Manin obstruction) to weak approximation. A Brauer–Manin obstruction that is not algebraic is called transcendental.

A conjecture by Colliot-Thélène [1, Introduction] says that the Brauer–

Manin obstruction to weak approximation should be the only obstruction for rationally connected varieties, that is,X(K) =X(K_Ω)^Br^un. Now, given a finiteK-groupG, one can always embed it into SLnfor somenand consider the homogeneous spaceX = SLn/G, which is unirational (since SLn

is itself a rational variety) and hence rationally connected. In [6, Thm. 1], Harari proved that ifGis an iterated semidirect product of abelian groups, then we do haveX(K) =X(K_Ω)^Br^un. The fact that this theorem implies the approximation property away from a finite setTof places as we claimed in §1 follows from the finiteness of Br_unX/BrK, cf. [2, Prop. 5.1(iii)], and from [6, §1.2] or [8, §1]:

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Proposition 2.4. — LetGa beK-group embedded intoSLn and put X := SLn/G. Let S ⊂ ΩK be a finite set of places of K. Then X has approximation inS if and only if the natural map

H¹(K, G)→ Y

v∈S

H¹(K_v, G), is surjective.

Note that this result proves in particular that approximation properties for varieties such asX = SL_n/G depend only onG, i.e. they are indepen- dent of the embedding ofGand even on the dimension ofX since there is no condition onn, justifying the definition of the approximation property in §1.

The tame approximation problem is thus equivalent to determining whetherG(orX = SL_n/G) has weak approximation away fromT = Bad_G. Note that a positive answer to this question does not necessarily imply a positive answer to Colliot-Thélène’s conjecture forX = SLn/G. Conversely, if the conjecture were true, we would only get the existence of a finite set TBr of bad places. However, not enough is known on the unramified Brauer group ofX in order to show that T_Br = BadG.

2.6. Poitou–Tate

We next recall the obstruction to weak approximation for finite abelian K-groups. Let Abe a finite abelianK-group and let ˆA= Hom(A,Gm) be its Cartier dual, which is also a finite abelianK-group.

Let K_vûn denote the maximal unramified extension and K_v^tr its maximal tamely ramified extension. Recall [14, II, §6] that for every finite place v such that Gal(K_vûn) acts trivially on A, the unramified cohomol- ogy H¹_un(Kv, A) is defined as the image of the group H¹(K_vûn/Kv, A) under inflation to H¹(K_v, A). One can consider then the restricted product Q

`

Ω_KH¹(Kv, A) with respect to these subgroups. It is well known that the product of the restriction maps Res_vsends H¹(K, A) intoQ`

ΩKH¹(K_v, A).

A classical result by Poitou and Tate [14, II.6.3] gives a perfect pairing

(2.3) Ya

Ω_K

H¹(K_v, A)×Ya

Ω_K

H¹(K_v,A)ˆ →Q/Z,

defined via local pairings and such that the image of H¹(K, A) is the orthog- onal complement of the image of H¹(K,A) for this pairing. A classic con-ˆ sequence of these is the following well-known proposition [11, Lem. 9.2.2].

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For a subsetSof ΩK, letX¹S(K, G) :={α∈H¹(K, G)|α_v= 1, ∀v6∈S}, and letX¹(K, G) :=X¹∅(K, G).

Proposition 2.5. — LetA be a finite abelian K-group and S ⊂ΩK

be a finite set of places. Then there is a pairing Y

v∈S

H¹(Kv, A)×X¹S(K,A)ˆ →Q/Z,

such that its right kernel is X¹(K,A)ˆ and its left kernel is the image of the restriction map

H¹(K, A)→ Y

v∈S

H¹(Kv, A)

In particular,Ahas approximation inSif and only ifX¹S(K,A) =ˆ X¹(K,A).ˆ We shall use the following property of X¹S(K,·) to descend to finite extensions:

Lemma 2.6. — LetAbe a finite abelianK-group. LetL/Kbe a Galois extension splittingA. Then for any finiteS⊂Ω_K,X¹S(K, A)is contained in the image ofH¹(L/K, A)by the inflation map.

Proof. — Since Gal(L) acts trivially on A, we have the following com- mutative diagram with exact rows:

1 //H¹(L/K, A) ^inf //H¹(K, A)

Res //H¹(L, A)

Y

v∈ΩK

H¹(Kv, A) ^Res // Y

w∈ΩL

H¹(Lw, A).

SinceLsplitsA, the H¹’s on the right hand side of the diagram are actually groups of homomorphisms. Restricting a classα∈X¹S(K, A) to H¹(L, A), we get a homomorphism Gal(L)→Athat is trivial everywhere locally ex- cept for the finitely manyw∈Ω_L that lie aboveS ⊂Ω_K. By Chebotarev’s density theorem it is the trivial morphism. Thus by exactness, α comes

from H¹(L/K, A).

3. Reocurrence

The main idea in proving Theorem 1.1 is to show that, given a finite K-group G of order n and a finite place v 6∈ BadG, there are infinitely

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many other placeswsuch that H¹(K_v, G)∼= H¹(K_w, G). In the particular case of a constant group, this infinite set of places will only depend onv andn. Hence, for a place v prime to n, a group Gof ordern will appear as a Galois group overKv if and only if it appears overKw.

Let us start by proving that, for finite places not in BadG, the set H¹(Kv, G) depends only on the tamely ramified part Γv = Gal(K_v^tr/Kv).

Recall that Γv is generated by a liftσv of the Frobenius and a generator of the inertia groupτ_v, as in §2.1. For every placev, we fix a choice ofσ_v andτv and keep it all throughout the text.

Lemma 3.1. — Let G be a finite K-group of order n and letv ∈Ω_K be a place outsideBadG. Denote by T_vⁿ the subgroup ofΓv generated by τ_vⁿ. Then there is an isomorphism H¹(Kv, G) ∼= H¹(Γv/T_vⁿ, G) given by inflation.

Proof. — The statement is trivial for archimedeanv, so we may assume that v is finite. Since the wild ramification subgroup W_v 6Gal(K_v) acts trivially onG, the inflation-restriction sequence gives:

1→H¹(Γ_v, G)−^inf−→H¹(K_v, G)−−→^Res H¹(W_v, G),

and H¹(W_v, G) is the set of morphisms W_v→Gup to conjugation. Since Wv is a pro-p group with p not dividing n, there are no such nontrivial morphisms and hence H¹(K_v, G)∼= H¹(Γ_v, G).

Consider now a classα∈H¹(Γv, G). Sincevis unramified in the minimal extension splittingG, T_v also acts trivially onG and henceαrestricts to a morphism (up to conjugation) fromTv to G. It is then evident that this morphism is trivial overT_vⁿ. The same inflation-restriction argument then gives

H¹(Kv, G)∼= H¹(Γv/T_vⁿ, G).

The following reocurrence result generalizes the statement given in the beginning of the section.

Proposition 3.2. — Let G be a finite K-group of order n, L/K a Galois extension splitting G and let v ∈ ΩK be either an archimedean place or a finite place which is unramified in L and does not divide n (in particular, v 6∈ BadG). Then there exist infinitely many finite places w6∈Bad_G for which:

(1) the decomposition groups ofv andwinGal(L/K)are conjugate;

(2) there is an epimorphism φ : Γ_w/T_wⁿ Γ_v/T_vⁿ given by φ(σ_w) = σv, φ(τw) = τv, which is moreover an isomorphism ifv is finite.

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The epimorphismφinduces a monomorphismφ^∗: H¹(K_v, G),→H¹(K_w, G), which is moreover an isomorphism ifvis finite.

Proof. — By Chebotarev’s density theorem applied to the Galois exten- sionL(µn)/K (which is unramified in v) there are infinitely many places w6∈BadGfor which the image of the decomposition group in Gal(L(µn)/K) is generated by the image of σw and is conjugate to the image of σv. In particular, we get that these groups are conjugate in Gal(L/K).

If v is finite, the images of σ_v and σ_w coincide in the quotient Gal(K(µn)/K) and hence their residue degrees qv, qw, respectively, are congruent modn. Moreover, recall that the group Γ_v/T_vⁿhas the following presentation:

hσv, τv|σvτvσ_v⁻¹=τ_v^q^v, τ_vⁿ = 1i.

Since q_w ≡ q_v modn, we get an isomorphism φ : Γ_w/T_wⁿ → Γ_v/T_vⁿ by settingφ(σw) =σvandφ(τw) =τv. Ifvis archimedean, the same definition gives an epimorphismφ: Γw/T_wⁿ→Gal(Kv) sinceσ²_v=τv= 1.

Let ρ ∈ Gal(K) be an element for which σw coincides with ρσvρ⁻¹ in Gal(L/K). Denote byρGthe Kv-group where σ∈Gal(Kv) acts on Gby g7→^ρσρ⁻¹gand note that this action coincides with that of Gal(Kw) onG.

Consider the maps

H¹(Γv/T_vⁿ, G)−→^ρ^∗ H¹(Γv/T_vⁿ, G_ρ ) ^φ

∗

−→H¹(Γw/T_wⁿ, G),

whereρ∗ is a canonical isomorphism defined at the level of cocycles by the identity on Γv/T_vⁿand by sendingg∈Gto^ρg. We abusively denote byφ^∗ the whole composition. Then evidentlyφ^∗ is an isomorphism if v is finite and injective as an inflation morphism ifv is archimedean. Recall finally that by Lemma 3.1, we have H¹(K_v, G)∼= H¹(Γ_v/T_vⁿ, G) for finitev and Γv/T_vⁿ = Gal(Kv) for archimedean v, hence the result.

We finally give a particular application necessary for the proof of Theo- rem 1.1:

Lemma 3.3. — Let G, L/K, v, w, φ and φ^∗ be as given by Proposi- tion 3.2. Letα∈H¹(Kv, G)and assume that there exists a classβ= [b]∈ H¹(K, G)such that βv =α,βw=φ^∗αand such that the group morphism Gal(L)→Gobtained by restriction ofbtoGal(L)is surjective. Denote by L⁰ the extension defined by the kernel of this morphism. Then the decomposition groups ofv andwinGal(L⁰/K)are conjugate.

We first note that L⁰/K is indeed Galois:

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Lemma 3.4. — Let G be a K-group and L/K be a Galois extension splittingG. Letb∈Z¹(K, G)and L⁰/Lbe the field fixed by the kernel of the restriction ofbto Gal(L). ThenL⁰ is Galois overK.

Proof. — Letτ∈Gal(L⁰) andσ∈Gal(K). Then b_{στ σ}⁻¹=bσ^σb_τ^{στ σ}⁻¹b⁻¹_σ .

SinceL/K is Galois and τ ∈Gal(L⁰)⊆Gal(L), we get στ σ⁻¹ ∈Gal(L), hence it acts trivially onb_σ. Asb_τ= 1 by definition ofL⁰, we getb_{στ σ}−1= 1, which proves thatστ σ⁻¹∈Gal(L⁰) and thusL⁰/K is Galois.

Proof of Lemma 3.3. — We first show that the decomposition groups ofvand win Gal(L⁰/K) are quotients of Γv/T_vⁿ and Γw/T_wⁿ. Indeed, this holds for archimedeanvsince Gal(Kv) = Γv/T_vⁿ. For finitevandw,L⁰/Kis tamely ramified sinceL/K is unramified andL⁰/Lis of degreenand hence prime tov andw. Since the restriction ofbtoTv andTwis a morphism, it must be trivial onT_vⁿ andT_wⁿ, i.e.T_vⁿ, T_wⁿ⊂Gal(L⁰), proving the claim.

Denote then (abusively) by σv, σw, τv, τw, the images in Gal(L⁰/K) of these elements. Recall that by the proof of Proposition 3.2, there exists an elementρ∈Gal(K) such thatσw coincides withρσvρ⁻¹ in Gal(L/K).

Also note thatρτvρ⁻¹ = τw = 1 in Gal(L/K). We claim that we may choose ρ so that b_ρ = 1. Note that b_χρ = b_χb_ρ for χ ∈ Gal(L). Since b is surjective when restricted to Gal(L), there exists χ∈Gal(L) such that b_χ = b⁻¹_ρ so that b_χρ = 1. Since χ ∈ Gal(L), we have that σ_w coincides with (χρ)σv(χρ)⁻¹ in Gal(L/K), proving the claim.

The isomorphism φ^∗ between H¹(Kv, G) and H¹(Kw, G) is given at the level of cocycles by the morphismsρ_∗ :G→G andφ: Γw/T_wⁿ →Γv/T_vⁿ, which satisfy ρ_∗(g) = ^ρg, φ(σw) = σv, and φ(τw) = τv. In particular, if α = [a] for a ∈ Z¹(Γ_v/T_vⁿ, G), then φ^∗α = [a⁰] with a⁰_σ

w = ^ρa_σ

v and a⁰_τ_w=^ρa_τ_v.

Now, since β_v =α and β_w = φ^∗α, we know that there exist g, g⁰ ∈ G such that

ρ(

gbσ_v^σ^vg⁻¹) =^ρa_σ_v =a⁰_σ_w =g⁰bσ_w^σ^wg⁰⁻¹,

ρ(

gbτ_vg⁻¹) =^ρa_τ_v =a⁰_τ_w=g⁰bτ_wg⁰⁻¹.

Sincebrestricted to Gal(L⁰/L) is an isomorphism by hypothesis, there are uniqueχ, χ⁰ ∈Gal(L⁰/L) such thatbχ =g,bχ⁰ =g⁰, so that

(3.1) ^ρb_χ^ρb_σ_v^ρσ^vb⁻¹_χ =bχ⁰bσ_w^σ^wb⁻¹_χ0,

ρb

χ^ρb_τ_v^ρb⁻¹_χ =bχ⁰bτ_wb⁻¹_χ0.

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Note now that sinceb_ρ= 1, we haveb_ρηρ−1 =^ρb_η for everyη∈Gal(L⁰/L), so in particular forη=χ, τv, andχ⁻¹. Noting that

ρGal(L⁰/L)ρ⁻¹⊆Gal(L⁰/L),

sinceL/K is Galois, that the restriction of bto Gal(L⁰/L) is a homomor- phism, and puttingρ₀:=χ⁰⁻¹ρχ∈Gal(L⁰/K), (3.1) givesb_ρ

0τ_vρ⁻¹₀ =b_τ_w and henceρ₀τ_vρ⁻¹₀ =τ_w in Gal(L⁰/L)⊂Gal(L⁰/K).

Finally, we claim thatρ0σvρ⁻¹₀ =σwin Gal(L⁰/K). Sinceχ, χ⁰∈Gal(L), we already know that σw and ρ0σvρ⁻¹₀ coincide on Gal(L/K) and hence act equally onG. Thus, on the one hand, (3.1) gives

bρ0^ρb_σ_v^σ^wb⁻¹_ρ₀ =bσw.

On the other hand, we have ρ₀σ_vρ⁻¹₀ = ψσ_w for some ψ ∈ Gal(L⁰/L).

Applyingb to this equality another direct computation gives bρ₀^ρb_σ_v^σ^wb⁻¹_ρ₀ =bψbσ_w.

From the two last equalities we get thatbψ= 1. Sincebis an isomorphism over Gal(L⁰/L), we getψ= 1, proving the claim. Thus conjugation byρ₀ sends the decomposition group ofv inL⁰/K to that ofw.

4. Weak approximation away from Bad_G

We now restate and prove Theorem 1.1 in the language of approximation properties. Recall that, given aK-group G, Bad_G denotes the set of bad places associated to it, cf. Definition 2.1.

Theorem 4.1. — LetK be a number field andG be a finiteK-group with weak approximation away fromBadG. Then every semidirect product E=AoGwithAan abelianK-group has weak approximation away from BadE.

Remark 4.2. — Note that this result does imply Theorem 1.1. Indeed, one can prove it by induction starting with the fact that the trivial group has weak approximation.

Proof of Theorem 4.1. — Consider a finite set of places S ⊂ ΩK not meeting Bad_E and local classes β_v ∈H¹(K_v, E) for v∈S. We will find a global classβ∈H¹(K, E) mapping to the βv’s in three steps:

Step 1. — Constructing a class β⁰ ∈H¹(K, E) with prescribed images in H¹(Kv, G) forvin S∪S⁰ for a suitableS⁰.

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We have the following commutative diagram of pointed sets with exact rows:

(4.1)

H¹(K, A)

//H¹(K, E)

//H¹(K, G)

ww

Y

v∈ΩK

H¹(K_v, A) // Y

v∈ΩK

H¹(K_v, E) // Y

v∈ΩK

H¹(K_v, G),

zz

where the arrows going left are set-theoretical sections induced by a fixed sectionG→E.

The βv’s give us images γv ∈ H¹(Kv, G) for v ∈ S. Let n = |E| and La Galois extension splittingE. Then, by Proposition 3.2 applied to the extensionL(µ_n)/K (which splits G), we know that for everyv ∈S there exist placesv⁰6∈S∪BadE for which there are inclusions

φ^∗_v: H¹(K_v, G),→H¹(K_v⁰, G).

Choose one suchv⁰for eachv∈Sand denote byS⁰the set of these places.

We may assume that all thev⁰ are different since we have an infinite choice at each time. Define thenγ_v⁰ :=φ^∗_v(γ_v)∈H¹(K_v⁰, G).

Since we’ve chosen these new places away from Bad_E ⊃Bad_G, we know by hypothesis that there exists a global classγ∈H¹(K, G) mapping onto γ_v for every v ∈ S ∪S⁰. Moreover, we may assume that the restriction of a cocycle c ∈ Z¹(K, G), representing γ, to Gal(L(µn)) is surjective:

Indeed by Chebotarev’s density theorem, there are infinitely many places w6∈S∪S⁰∪BadE totally split inL(µn), so in particular such that G is constant overKw; then, for each element g ∈Gwe may choose one such w and an unramified class γ_w ∈ H¹(K_w, G) represented by a morphism sendingσwtog, so that allw’s are distinct; then finally, adding these local conditions toS∪S⁰forcescto be surjective when restricted to Gal(L(µ_n)).

Let β⁰ be the image ofγ in H¹(K, E) via the section of diagram (4.1).

Viewing G as a subgroup of E, the class β⁰ is represented by the same cocyclec.

Step 2: Twisting. — We twist by c to look for a class in H¹(K, E) satisfying the necessary prescribed local conditions inS.

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Let us now twist diagram (4.1) by the cocycle c representing β⁰. This gives us the following new diagram with exact rows

(4.2)

H¹(K, A_c )

//H¹(K, E_c )

//H¹(K, G_c )

Y

v∈ΩK

H¹(K_v, A_c ) // Y

v∈ΩK

H¹(K_v, E_c ) // Y

v∈ΩK

H¹(K_v, G_c ).

For every cohomology classξin a set of diagram (4.1), we denote bycξits twisted image in the corresponding set of diagram (4.2).

Since the images of βv and β⁰ coincide on H¹(Kv, G) for v ∈S by construction and_cβ⁰ is trivial by the very definition of twisting, we know by exactness that_{c v}β ∈H¹(Kv, E_c ) comes from an elementαv∈H¹(Kv, A_c ).

It suffices then to find a global classα∈H¹(K, A_c ) mapping ontoαv for v∈ S to conclude. Indeed, the image of αin H¹(K, E_c ) would then map onto the_{c v}β ’s forv∈Sand hence its preimage in H¹(K, E) by the twisting morphism would map onto theβ_v’s as desired. Thus, in order to conclude, it will suffice by Proposition 2.5 to prove thatX¹S(K,_cAˆ) =X¹(K,_cAˆ).

Step 3: Proof of X¹S(K,_cAˆ) = X¹(K,_cAˆ). — Consider a class α in X¹S(K,_cAˆ), i.e. a class in H¹(K,_cAˆ) such that its image αv∈H¹(Kv,_cAˆ) is trivial for everyv 6∈S, so in particular forv⁰ ∈S⁰. We must show that αv= 0 forv∈S too.

Fix thenv∈S and its correspondingv⁰ ∈S⁰. Lemma 3.3 applies in this context to the local classesγv andγv⁰ =φ^∗_v(γv), the global classγand the extensionL(µn)/K. By the lemma the decomposition groups ofv and v⁰ in Gal(L⁰/K) are conjugate, whereL⁰ is the extension ofL(µn) given by the kernel ofcrestricted to Gal(L(µn)). Note moreover thatL⁰ splits_cAˆ. Indeed, since|A|dividesnwe have_cAˆ= Hom(_cA, µ_n) and sinceL⁰contains µn, then this amounts toL⁰ splitting_cA. The latter holds since the action ofσ∈Gal(K) ona∈_cAis given byc_σ^σac⁻¹_σ , where^σadenotes the action ofσonaas an element of A. Sincec is trivial over Gal(L⁰) andL⁰ clearly splitsA, we get then our claim. In particular, by Lemma 2.6, we know that the whole groupX¹S(K,_cAˆ) comes by inflation from H¹(L⁰/K,_cAˆ).

Our element α∈X¹S(K,_cAˆ) comes then from H¹(L⁰/K,_cAˆ) and hence α_v comes from H¹(L⁰^(v)/K_v,_cAˆ), whereL⁰^(v)denotes the (unique) extension ofKvinduced byL⁰/K. The same goes forv⁰. Since the decomposition

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groups ofvandv⁰are conjugate in Gal(L⁰/K), there is a canonical isomor- phism

H¹(L⁰^(v)/K_v,_cAˆ)−^∼→H¹(L⁰^(v

0)

/K_v⁰,_cAˆ),

which is compatible with the restrictions from H¹(L⁰/K,_cAˆ). But αis in X¹S(K,_cAˆ) and v⁰ 6∈ S, so thatα_v⁰ = 0 and hence α_v = 0 too. Since the same argument works for everyv ∈ S, we deduce that α ∈ X¹(K,_cAˆ),

proving the claim.

5. Counterexamples to weak approximation

In this section we show that Theorem 4.1 is sharp in the sense that one cannot expect to get approximation (or to solve the Grunwald problem) on the set of bad places.

Recall that a group is bicyclic if and only if it is cyclic or a direct product of two cyclic groups. For a finite groupGand a finiteG-moduleA, we let

X¹bic(G, A) := ker



H¹(G, A)→ Y

H∈bic(G)

H¹(H, A)



,

where bic(G) denotes the set of bicyclic subgroups of G.

Theorem 5.1. — Let Gbe a finite abelian group, K a number field, andSa finite set of places ofK. LetAbe a finiteG-module andE=AoG a constantK-group. Assume:

(1) Kcontains the exp(A)-th roots of unity;

(2) there existsv0∈S such thatGis a Galois group overKv₀; (3) S contains all the places ofKdividing the order ofG;

(4) X¹bic(G,A)ˆ 6= 0 (in particularGis not bicyclic).

Then the mapH¹(K, E)→Q

v∈SH¹(Kv, E)is nonsurjective.

Remark 5.2. — Note that, given the structure of local Galois groups recalled in Section 2.1, Condition (4) implies that the place v₀ in Condi- tion (2) must divide the order ofG.

Corollary 5.3. — Assume in addition thatKcontains theexp(E)-th roots of unity, then there is a transcendental Brauer–Manin obstruction to weak approximation onX := SLn/E.

Proof. — Indeed, Theorem 5.1 and Proposition 2.4 tell us that X does not have approximation in S and hence it doesn’t have weak approximation. Now, by [6, Thm. 1] the Brauer–Manin obstruction to weak approximation for such a variety is the only obstruction, whereas the formula