On the cohomological dimension of some pro-p-extensions above the cyclotomic Z_p-extension of a number field

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On the cohomological dimension of some

pro-p-extensions above the cyclotomic Z_p-extension of a number field

Julien Blondeau, Philippe Lebacque, Christian Maire

To cite this version:

Julien Blondeau, Philippe Lebacque, Christian Maire. On the cohomological dimension of some pro- p-extensions above the cyclotomic Z_p-extension of a number field. Moscow Mathematical Journal, Independent University of Moscow 2013, 13, pp.601-619. �hal-00854964�

ON THE COHOMOLOGICAL DIMENSION OF SOME PRO-p-EXTENSIONS ABOVE THE CYCLOTOMIC

Zp-EXTENSION OF A NUMBER FIELD

JULIEN BLONDEAU, PHILIPPE LEBACQUE AND CHRISTIAN MAIRE

Abstract. Let Ke^TS be the maximal pro-p-extension of the cyclotomic Z^p- extension K^cyc of a number field K, unramified outside the places aboveS and totally split at the places aboveT.LetGe^T_S = Gal(eK^T_S/K).

In this work we adapt the methods developed by Schmidt in [Sch3] in order to show that the groupGe^TS = Gal(eK^TS/K)is of cohomological dimension 2 provided the finite setS is well chosen. This groupGe^TS is in fact mild in the sense of Labute [La]. We compute its Euler characteristic, by studying the Galois cohomology groupsHⁱ(Ge^T_S,Fp),i= 1,2. Finally, we provide new situations where the groupGe^T_S is a free pro-p-group.

1. Introduction Fix a prime p >2 and an algebraic closureQof Q.

LetK be a number field with signature(r1, r2)and letK^cyc be the cyclotomic Zp-extension of K. Let Γ = Gal(K^cyc/K) ' Zp. In our work, we study pro-p- extensions Lof K^cyc with restricted ramification where some given places split.

LetS andT be two disjoint finite sets of finite places ofK.Denote byKe^T_S the maximal pro-p-extension of K^cyc,unramified outside the places above S and to- tally split at those aboveT(unramified outsideS, T-split). PutGe^T_S = Gal(Ke^T_S/K).

It is a quotient of GS∪Plp= Gal(KS∪Plp/K), whereKS∪Plp is the maximal pro-p- extension ofKunramified outsideS∪Pl_p. It follows thatGe^T_Sis a finitely generated pro-p-group. We later show that it admits a finite presentation.

We remark that if v is a place of K not dividing p which is not ramified in L/K^cycthenL_v= K^cyc_v . Thus it is sufficient to consider finite subsetsS of places of Kand T of p-adic places Plp ofK such thatS∩T =∅.

The case where T is empty and S contains the p-adic places of K is already well-known. The pro-p-extension Ke^T_S is indeed the maximal pro-p-extension of K unramified outside S, and the group GeS := Gal(KS/K) has cohomological dimension2 (see [NSW]). Moreover, ifS does not contain all the places above p, Schmidt proved in the series of papers [Sch1], [Sch2], [Sch3] that one can achieve the cohomological dimension2property forG_S by adding toSa finite set of tame places. He also computed its Euler characteristic.

In this article, we show:

Date: June 20, 2013.

2010Mathematics Subject Classification. 11R34, 11R37.

Key words and phrases. Mild pro-p-groups, Galois cohomology, restricted ramification, cy- clotomicZ^pextension.

1

Theorem 1.1. Let K be a number field not containing thep^th roots of unity.

Let S⁰ andT be two finite sets of places ofKwithS⁰∩T =∅. Then there exists a finite set S of finite places ofK such that

(i) S⁰ ⊂S and(S−S⁰)∩Plp =∅;

(ii) the pro-p-group Ge^T_S has cohomological dimension at most 2;

(iii) χ(Ge^T_S) = r₁ +r₂ −P

v∈Sp[K_v : Qp] where χ(Ge^T_S) stands for the Euler characteristic associated to the cohomology groupsH^∗(Ge^T_S,Fp).

Remark 1.2. As noticed by Schmidt in [Sch3], one can be more precise and ask S to avoid a fixed set of places ofKwith zero Dirichlet density. For instance, one can choose the places of S in the set of degree one places of K/Q.

LetH^T_S = Gal(Ke^T_S/K^cyc)andX_S^T =H^T_S/[H^T_S,H^T_S]. In the case whereT =∅, we denote H_S :=H^∅_S and X_S := H_S/[H_S,H_S]. The Zp[[Γ]]-module X_S corresponds to the inverse limit of the p-S-class group alongK^cyc/Kand the homology groups H_i(H_S,Fp)areFp[[Γ]]-modules. Letr_S denote theFp[[Γ]]-rank ofX_S.Recall that rS = 0 conjecturally (the Ferrero-Washington theorem assures that it is true for abelian extensions ofQand the works of Schneps [Schn] and Gillard [Gi] guarantee that it holds also for abelian extensions of imaginary quadratic fields).

Using the Cebotarev density theorem, one can prove that by adding a finite set S of well-chosen tame finite places, the groupH_S becomes a freeFp[[Γ]]-module.

Corollary 1.3. There is a finite set S of places of K such that S∩Plp =∅ and such that H₂(H_S,Fp)'Fp[[Γ]]^r1+r2+rS.

Now we suppose that T = Pl_p and we consider H⁰_S = Gal(Ke^T_S/K^cyc). When S =∅, the Zp[[Gal(K^cyc/K)]]-module X⁰ :=H⁰_∅/[H⁰_∅,H⁰_∅]is well-known. It is the inverse limit of the p-groups of p-classes along K^cyc/K. Conjecturally again, the coinvariants X_Γ⁰ are finite: it is a conjecture of Gross (see [?]).

Again, adding well-chosen primes, we obtain the following.

Corollary 1.4. There is a finite setS of places ofK, satisfyingS∩Plp =∅, such that H₂(H⁰_S,Fp)'Fp[[Γ]]^r1+r2+r.

The proof of Theorem 1.1 requires information on the relations of the group Ge^T_S and this can be done by considering the cup product (see [NSW]§III.9)

H¹(Ge^T_S,Fp)×H¹(Ge^T_S,Fp)→H²(Ge^T_S,Fp).

We first need to compute H¹(Ge^T_S,Fp).This is classical and is done through class field theory. Then we need to estimate the second cohomology groupH²(Ge^T_S,Fp).

This is done by considering the Shafarevich kernel X(Ge^T_S) which mesures the obstruction of the relations of H²(Ge^T_S,Fp) to being local, and one gets control of this group with the use of a Kummer group Ve_S^T/K^×p (to be defined in§3.1).

These estimations which are the core of our work are gathered in the following theorem.

Theorem 1.5. Let S, T be disjoint sets of places of a number field K such that T ⊂ P l_p and K_v contains the p^th roots of unity for any v ∈ S −P l_p. Then we have

(i) d_p(Ge^T_S) =−(r₁+r₂+|T|−1+θ(K))+|Pl_p−S_p|+P

v∈S_p([K_v :Qp] +θ(K_v))+

|S₀|+dp(Ve_S^T/K^×p);

(ii) d_pX(Ge^T_S)≤d_p(Ve_S^T/K^×p);

(iii) d_pH²(Ge^T_S)≤d_p(Ve_S^T/K^×p) +|S₀|+|Pl_p−(S_p∪T)|+P

v∈Spθ(K_v)−θ(K) + θ(K, S);

where θ(K, S), θ(Kv), θ(K)∈ {0,1} (see §2.1 for a precise definition).

As a consequence of these computations, we obtain new situations where the group Ge^T_S is a free pro-p-group.

Corollary 1.6. Suppose the following three conditions hold:

(i) Kcontains the p^th roots of unity, (ii) |S|= 1 andS∪T =P lp;

(iii) There is no non-trivialp-extension of K, unramified outsideT and totally split at S.

ThenGe^T_S is a free pro-p-group with d(Ge^T_S) = 1−(r₁+r₂) + [K_v0 :Qp]generators, where v0 is the place of S.

Remark 1.7. A fieldKsatisfying the conditions of Corollary 1.6 for the placev₀ is v0-rational according to the work of Jaulent and Sauzet [JSa].

Corollary 1.8. Suppose the three conditions are true:

(i) The fieldsK and Kv do not contain the p^th-roots of unity;

(ii) S∪T =P l_p;

(iii) there is no non-trivial unramified outside T and S-split p-extension of K⁰= K(ζ_p).

Then Ge^T_S is a free pro-p-group with d(Ge^T_S) = 1−(r₁ +r₂) +P

v∈S[K_v : Qp] generators.

The paper is organised as follow. In section 2 we make precise the notation and the context of our work. Section 3 is devoted to the estimations of the groups Hⁱ(Ge^T_S,Fp), fori= 1and2,and of a certain Shafarevich kernelX(Ge^T_S).For that purpose, we study a Kummer groupVe_S^T.Section 4 contains technical prerequisites for the proof of our main theorem 1.1, contained in Section 5.

Acknowledgement: The second author would like to thank warmly Alexander Schmidt for his hospitality in Heidelberg and for very useful discussions in recent years.

2. The arithmetic context

2.1. First notation. Fix a prime number p >2 and a number fieldK,an alge- braic closureQ ofQ.Put G = Gal(Q/K).

- µ_p is the set of p^th roots of unity inQ, ζ_p is a generator ofµ_p. - µ(K) denotes the set ofp^∞-th roots of unity contained inK.

- LetK^× denoteZp⊗K^×.

- Unless otherwise stated, all the considered sets of places are made of finite places.

3

- LetM/L be a field extension,Q a set of places ofL, R a set of places of M. If no confusion is possible,Qdenotes also the set of places QM of M lying above the places ofQ, andR denotes also the set R_L of places ofL lying underR.

- Plp ={v, v|p}is the set ofp-adic places ofKand, if S is a set of places of K, putS_p :=S∩Pl_p and S₀ :=S−S_p.

- IfS is a set of places of a fieldL, letθ(L, S) be1if S=∅and Lcontains thep^th roots of unity, and0 otherwise. Putθ(L) :=θ(L,∅).

- If M is a Zp-module, denote by _pM =M/pM the cokernel of M →^p M, byM[p]its kernel and by d_pM := dim_Fp(M/pM) thep-rank of M. - IfGis a pro-p-group and ifi≥0, let us denote byHⁱ(G)the cohomology

groupHⁱ(G,Fp). Moreover, ifGadmits a finite presentation, putd(G) :=

d_pH¹(G,Fp),r(G) =d_pH²(G,Fp),χ₂(G) = 1−d(G) +r(G)and (as soon as it makes sense)χ(G) =X

i≥0

(−1)ⁱd_pHⁱ(G,Fp).

- IfGis an abelian group, we put G^∗ = Hom(G,Q/Z).

- If S and T are sets of places of K, let K^T_S denote the maximal pro-p- extension ofKunramified outsideS and where the places ofT split com- pletely. We putKS = K^∅_S and K^T = K^T_∅.The Galois group Gal(K^T_S/K) is denoted byG^T_S and we omit in the same way∅ in the notation.

- If L/K is a Galois extension (possibly infinite) of Galois group G and v is a place ofK, we denote by G_v the decomposition group inside G of a fixed place ofL abovev.

2.2. Local setting. Fix a finite placev of Kand let K_v be the maximal pro-p- extension of Kv. LetGv := Gal(Kv/Kv)be the absolute pro-pGalois group ofKv

and let Iv be its inertia subgroup.

Denote by

- Frobv the arithmetic Frobenius ofGv/Iv,

- K^nr_v := K_v^Iv the maximal unramified pro-p-extension of K_v,

- K^cycv the cyclotomicZp-extension ofKv(in particular, ifv-p,K^cycv = K^nr_v ), - K^cr_v := K^nr_v K^cycv , the maximal cyclotomically ramified pro-p-extension of

K_v,

- Ie_v = Gal(K_v/K^cycv K^nr_v ), He_v = Gal(K_v/K^cycv ), G^cr_v = Gal(K^cr_v /K_v) and G^cyc_v = Gal(K^cyc_v /K_v).

Definition 2.1. A pro-p-extensionL_v/K_v is said to be - locally cyclotomic ifLv ⊂K^cycv ;

- cyclotomically ramified ifL_v ⊂K^cr_v .

For v|p, remark that if L_v/K_v is cyclotomically ramified, the Galois group Gal(L_v/K_v) is a quotient of G^cr_v 'Z²_p.

Lemma 2.2. The extensionK^cr_v /Kv is contained in a pro-p-free extensionFv/Kv

Proof. Ifv-p, it is obvious because K^cyc_v = K^nr_v andG^cr_v 'Zp.

For v|p, we take F_v to be the extension K_vQp/K_v (here p >2,so that Qp/Qp

is pro-p-free).

The normic local groups. Let K^ab_v be the maximal abelian pro-p-extension ofK_v and let Uv be the group of units ofKv.

Denote by K_v^×:= lim

←n

K^×_v/(K^×_v)^p

n

the pro-p-completion ofK^×_v and define U_v = Zp⊗U_v.

The extension K^ab_v /K^nr_v is generated, via the Artin map, by the local units U_v and the extensionK^ab_v /K^cyc_v by the elementsN_vofK_vwhich are norms inK^cyc_v /K_v. The extensionK^ab_v K^nr_v /K^cycv is therefore generated by the subgroupUe_v :=N_v∩ U_v of the group of units: they are the units which are norms in K^cyc_v /K_v. It is the group of the locally cyclotomic units.

Thus:

- U_v 'µ(K_v)if v-p, otherwiseU_v 'µ(K_v)×Z^[Kp ^v:Qp]; - N_v =U_v if v-p, otherwiseN_v'µ(Kv)×Z^[Kp ^v:Qp];

- Ue_v =U_v 'µ(Kv) if v-p, otherwise, Ue_v 'µ(Kv)×Z^[Kp ^v:Qp]−1. K^ab_v

K^cr_v ^Uev

Kv^cyc Nv

K^nr_v

Uv

Kv

2.3. Global setting. From now on, we fixSandT two finite sets of finite places of Ksatisfying:

• For any place v of S not dividing p, the complete field Kv contains the p^th roots of unity;

• S∩T =∅;

• T ⊂Pl_p.

Recall that Sp=S∩Plp and S0=S−Sp.

Definition 2.3. We denote by Ke^T_S the pro-p-extension of K, maximal for the following conditions:

• Ke^T_S/Kis cyclotomically ramified outside S;

• Ke^T_S/Kis locally cyclotomic at all placesv of T. Put Ge^T_S = Gal(Ke^T_S/K).

Remark 2.4. Ke^T_S/K is a subextension of the maximal pro-p-extension K_Σ of K unramified outside Σ = Plp∪S0.

5

Remark 2.5. The field Ke^T_S contains the cyclotomic Zp-extension K^cyc of K and Ke^T_S is the maximal pro-p-extension ofK^cyc unramified outside the places above S and totally split at the places above T.

One can describe the abelianization Ge^{T ,ab}_S of Ge^T_S by using the Artin map. In- deed, it induces an isomorphism Ge^{T ,ab}_S ' J/K^×Ue_S^T, where J = J_K is the re- stricted product over all places v of K of the groups K^×_v with respect to the groups U_v and

Ue_S^T =



 Y

v /∈S₀∪Pl_p

U_v



 Y

v∈T

N_v

!



Y

v∈Pl_p−(S_p∪T)

Ue_v



= Y

v /∈S∪T

Ue_v Y

v∈T

N_v.

2.4. The Labute and Schmidt works. In [La], Labute gave for the first time examples of groups GS of cohomological dimension2 withS∩Plp =∅.

These results come from the study of the Lie algebra obtained considering the descending centralp-series of GS.In certain good situations, the relations in this algebra can be deduced from the first terms of the relations ofG_S which were due to Koch [Ko]. In this case, the group GS is said to be mild.

In the series of articles [Sch2], [Sch3], Schmidt gives interpretations of Labute’s work, introducing the k(π,1) property for GS (and G^T_S) groups. He shows in particular that, after enlarging the setS, the groupsG_S have finite cohomological dimension, and he proves a Riemann existence theorem. In our setting, we use the following result (already partially present in [La]).

Theorem 2.6 (Schmidt [Sch2]). Let Gbe a pro-p-group of finite type, not pro-p- free. Suppose that H¹(G) =U ⊕V with

(i) ∀χ₁, χ₂∈V, χ₁∪χ₂= 0∈H²(G);

(ii) H¹(G)⊗V ^{∪ ////}H²(G). Then cd(G) = 2.

Remark 2.7. When the previous hypotheses are satisfied, the group G is mild, and the Poincaré series P(t) associated toG is(1−d(G)t+r(G)t²)⁻¹.

We are going to apply Theorem 2.6 to the groups Ge^T_S.

2.5. The pro-p-groups in the context of Iwasawa theory. LetGbe a pro- p-group,Ha closed normal subgroup of Gsuch thatG/H= Γ'Zp. Denote by Λ =Zp[[Γ]]the Iwasawa algebra associated with the pro-p-groupΓ and byΩthe integral local ring Ω =Fp[[Γ]] := lim

←

U

Fp[Γ/U].

If M is a Λ-module of finite type, denote by ρ_Λ(M) its Λ-rank and by ρ_Ω its Ω-rank.

We recall that the module M is torsion if, for example, the setM_Γ of coinvari- ants is a torsion Zp-module.

2.5.1. The case where G is pro-p-free. In this case, the normal subgroupH of G is pro-p-free too. PutX =X_∅.The following result is well known:

Proposition 2.8. If Gis pro-p-free, thenX is a freeΛ-module of rankd(G)−1.

Proof. See [NSW], Proposition 5.6.11 and Corollary 5.6.12.

2.5.2. The case where G has cohomological dimension 2. The case where G has cohomological dimension at most2has been also widely studied, certainly because the group G_S has cohomological dimension 2 as soon as S contains the p-adic places. We recall the following fact:

Proposition 2.9. Suppose that G has cohomological dimension at most 2. Put r =ρΩ(X). Then

(i) the Λ-module H2(H,Zp) is free of rank t, with t=ρ(X) +χ(G);

(ii) the Ω-module H₂(H,Fp) is free of rank t⁰, with t⁰ =t+r.

Proof. This comes from Proposition 5.6.7 of [NSW] for the module H₂(H,Fp).

Notice that Ωhas projective dimension1.

2.6. Known results. The groups Ge^T := Ge^T_∅ have been studied by several au- thors: Jaulent and Soriano [JSo], Jaulent and Maire [JM2], Assim [As], et al.

The groupsGeS :=Ge^∅_S have been studied by Salle in [Sa].

In these situation, it turns out that the groups coming into play admit a finite presentation. An estimation of the Euler characteristic is also given. It has to be noted that this estimation gives rise to situations where Ge^T_S is not analytic (see [JM1] and [JM3]).

Theorem 2.10 (Salle [Sa], Jaulent-Maire [JM1]). We still suppose p >2.

(i) d(Ge_S) andr(Ge_S) are finite and

χ2(GeS)≤r1+r2+θ(K, S)− X

v∈Sp

[Kv :Qp],

whereθ(K, S) = 1ifS is empty andKcontains thep^throots of unity, θ(K, S) = 0 otherwise.

(ii) If T = Plp, d(Ge^T) andr(Ge^T) are finite and χ2(Ge^T)≤r1+r2+θ(K),

where θ(K) = 1 ifK contains the p^th roots of unity, 0 otherwise.

3. The groups Hⁱ(Ge^T_S)

3.1. Kummer groups and Shafarevich kernel. Throughout this paragraph, S andT are again two finite sets of finite places ofKsatisfying the conditions of

§2.3.

Definition 3.1. We consider Ve_S^T ={x∈ K^××, x∈ J^pQ

v /∈S∪T Ue_vQ

v∈TN_v}.

This Kummer group will play a central role in the following. The next propo- sition is important, because it shows that, taking a suitableS,one can ensure the triviality of Ve_S^T/K^×p.

Proposition 3.2. (i) The group Ve_S^T/K^×p is finite.

(ii) If S ⊂S⁰, then Ve_S^T0/K^×p ,→Ve_S^T/K^×p.

(iii) If K(ζp) has no non-trivial p-extension unramified outside Plp and totally split at S then Ve_S^T/K^×p={1}.

(iv) There is a finite set S of finite places of K not dividing p such that Ve_S^T/K^×p ={1} for any T ⊂P lp.

7

Proof. Points (i) and (ii) are clear.

(iii) follows from Kummer theory. Let x ∈ Ve_S^T. Suppose that Kcontains the p^th-roots of unity. Then the extension K(√^p

x)/K is cyclic of degree dividing p, unramified outsidePlp and totally split at S. ThereforeK(√^p

x)/Kis trivial, thus x∈K^p.

If now ζp∈/ K, the elementx is for the same reason ap^th power, but this time in K(ζp).As the degree ofK(ζp)/K is prime top, we obtain by taking the norm that x is also ap^th power in K.

(iv) Using Cebotarev density theorem, one can consider a finite set S of places ofK containing places whose Frobenius morphisms generate the maximal abelian p-elementary extension of K(ζp) unramified outside Plp. By (iii), such a set S

satisfies the desired property.

Let us state the following result which will be used later in Section 3.

Corollary 3.3. LetS andT be two finite sets of places satisfying the conditions of

§2.3.Then there is a set Σof tame places ofKsuch that, for any placep∈S∪Σ, Ve_S∪Σ−{p}^T /K^×p={1}.

Proof. We take two disjoint finite sets S1 and S2 of tame places satisfying (iv) (using Cebotarev density theorem). By (ii) one deduces the desired result with

Σ =S1∪S2.

Before stating the first important result of this work, consider the following Shafarevich kernel.

Definition 3.4.

Let

X(Ge^T_S) = ker



H²(Ge^T_S)→

∗

M

v∈S

H²(G_v)⊕ M

v∈Pl_p−(S_p∪T)

H²(G^cr_v )



,

where the first direct sum is taken over the set S, deprived of a place in the case where K contains the p^th roots of unity and S is not empty.

3.2. The main result.

Theorem 3.5. Let S, T be disjoint sets of places of a number field K such that T ⊂ P lp and Kv contains the p^th roots of unity for any v ∈ S −P lp. Then we have

(i) dp(Ge^T_S) =−(r₁+r2+|T|−1+θ(K))+|Pl_p−S_p|+P

v∈Sp([Kv :Qp] +θ(Kv))+

|S₀|+d_p(Ve_S^T/K^p);

(ii) dpX(Ge^T_S)≤dp(Ve_S^T/K^×p);

(iii) d_pH²(Ge^T_S)≤d_p(Ve_S^T/K^×p) +|S₀|+|Pl_p−(S_p∪T)|+P

v∈Spθ(K_v)−θ(K) + θ(K, S).

We remark that, under these conditions, the dimension of each of the H²(Gv) and H²(G^cr_v )appearing in Definition 3.4 is smaller than 1.

One deduces immediately the following estimation:

Corollary 3.6.

χ₂(Ge^T_S)≤r₁+r₂− X

v∈Sp

[K_v :Qp] +θ(K, S).

Proof. (of Theorem 3.5)

(i) - This is a classical computation. We start with the following exact sequence coming from class field theory:

1 ^//Ve_S^T/K^{×p //}V^T/K^{×p //}Q

v /∈T U_vQ

v∈TK^×_v/(Ue_S^T Q

vU_v^p) ^//pGe^T,ab_S

pG^T,ab where V^T = Q

v /∈T U_vQ

v∈TK^×_vJ^p

∩ K^×. We then compute the dimensions using the exact sequence:

1 ^//E_K^T/(E_K^T)^{p //}V^T/K^{×p //}G^T,ab[p] ^//1, whereE^T_K is the group ofT-units ofK.

(ii) and (iii) - In the commutative diagramm (see in particular Paragraph 2.1 and Defintion 3.4 for notation)

H²(Ge^T_S) ^//

ϕ=⊕vϕv

((

H²(G)_{ _}

M

v

H²(Ge^T_S,v) ^//

∗

M

v

H²(G_v)

the morphism ϕfactors as follow:

- for v ∈ T, ϕ_v : H²(Ge^T_S,v) → H²(G^cycv ) → H²(G_v). As G^cycv ' Zp, ϕ_v is zero.

- for v /∈ S ∪T, ϕ_v : H²(Ge^T_S,v) → H²(G^cr_v ) → H²(G_v). As G^cr_v is a subextension of a pro-p-free extension (see Lemma 2.2),ϕ_v has also a null image.

Therefore, it follows that ker

H²(Ge^T_S)→H²(G)

= ker H²(Ge^T_S)→

∗

M

v∈S

H²(Gv)

!

and thus X(Ge^T_S) is a subgroup ofker

H²(Ge^T_S)→H²(G)

. One deduces (ii) from the following lemma.

Lemma 3.7. We have

X(Ge^T_S) ^{ //}

Ve_S^T/K^×p∗

.

The local and global inflation-restriction sequences give rise to the commutative diagramm:

9

H¹(G)/H¹(Ge^T_S) ^{ //}

φ

H¹(Gal(Q_{ _}/Ke^T_S))^{GeTS ////}

ker

H²(Ge^T_S)→H²(G)

Γ(S, T) ^{ //}∆(S, T) ^//// M

v /∈S∪T

H²(G^cr_v ) where

Γ(S, T) = M

v /∈S∪T

H¹(Gv)/H¹(G^cr_v )⊕M

v∈T

H¹(Gv)/H¹(G^cyc_v ), and where

∆(S, T) = M

v /∈S∪T

H¹(G^cr_v )^Gv⊕M

v∈T

H¹(G^cyc_v )^Gv.

Notice that the bottom right morphism is indeed surjective because for any v /∈S∪T, the morphism H²(G^cr_v )→H²(Gv)has a null image (again, K^cr_v /Kv is a subextension of pro-p-free extension, cf. lemma 2.2).

Then we have

M

v /∈S∪T

H²(G^cr_v ) = M

v∈Plp−(Sp∪T)

H²(G^cr_v ).

Thus the kernel of the right vertical arrow is exactlyX(Ge^T_S). By the snake lemma, we obtain

X(Ge^T_S) ^{ //}cokerφ . In order to relate it with

Ve_S^T/K^×p∗

,we need the following result.

Proposition 3.8 (Salle [Sa]).

H¹(Gv)/H¹(G^cr_v )'(Iev/Ie_v^p[Gv,Gv])^∗.

It comes from the fact that G_v/Ie_v 'Z²p together with the following lemma:

Lemma 3.9. LetGbe a pro-p-group of finite type andHa closed normal subgroup of Gsuch that G/H 'Z^tp. Then G^p∩H ⊂H^p[G,G].

Proof. Recall thatG^p denotes the closure inGof the subgroup ofGgenerated by the elementsg^p,g∈G. Letx∈G^p∩H. As[G,G]is open inG, there existsy∈G such that x≡y^p mod [G,G], in particular x≡y^p mod H and y^p = 1 inG/H.

As this last quotient is p-torsion free, we havey∈H, that is,x∈H^p[G,G].

We come back to the proof of Lemma 3.7. Taking the dual, we obtain ker Y

v /∈S∪T

Ie_v/Ie_v^p[G_v,G_v]Y

v∈T

He_v/He_v^p[G_v,G_v]→ker

pG^ab →_pGe^{T ,ab}_S

!

X(Ge^T_S)^∗.

By class field theory, one has ker

pG^ab →_pGe^{T ,ab}_S

' K^×J^pUe_S^T/K^×J^p, and

Y

v /∈S∪T

Iev/Ie_v^p[Gv,Gv]

! Y

v∈T

Hev/He_v^p[Gv,Gv]

!

'Ue_S^T/(Ue_S^T)^p.

We obtain that(cokerφ)^∗'Ue_S^T∩(K^×J^p)/(Ue_S^T)^p 'Ve_S^T/K^×p, the last isomorphism coming from a local-global principle for p^th powers together with the fact that Ue_S^T ∩ J^p= (Ue_S^T)^p. Lemma 3.7 follows.

(iii) of Theorem 3.5 follows from the following inequalities:

r(Ge^T_S) ≤

∗

X

v∈S

θ(K_v) +d_p ker

H²(Ge^T_S)→H²(G)

≤

∗

X

v∈S

θ(K_v) +d_pX(Ge^T_S) +|Pl_p−(S_p∪T)|

(the ∗ meaning that the sum is deprived of the contribution of one place in the case whereµ_p ⊂KandSis not empty), the last inequality coming from the exact sequence

0−→X(Ge^T_S)−→ker

H²(Ge^T_S)→H²(G)

−→ M

v∈Plp−(Sp∪T)

H²(G^cr_v ).

Then Lemma 3.7 and

∗

X

v∈S

θ(Kv) =|S₀|+ X

v∈Sp

θ(Kv)−θ(K) +θ(K, S)

give the desired result.

3.3. Some consequences.

3.3.1. On the freeness of Ge^T_S. Thanks to Theorem 3.5, we see that the triviality of Ve_S^T/K^×p implies that the global relations of Ge^T_S are indeed local, i.e. that we have an injection:

H²(Ge^T_S),→

∗

M

v∈S₀

H²(G_v)M M

v∈Plp−(Sp∪T)

H²(G^cr_v ).

It allows us to give new situations where the group Ge^T_S is free (other thanZp).

We thus obtain Corollaries 1.6 and 1.8.

Example 3.10. Leta≡1 mod 3be an integer andθa root ofX³+aX+ 1. Let K = Q(θ, ζ₃) (here p= 3). There is a place v₀ of Kabove 3 with residual degree 4.Let v₁ be the second place of Kabove 3. PutS ={v₀} and T ={v₁}.

It can be checked (with GP PARI [PA] for instance) that, for some values of a, there is no non trivial 3-extension ofK unramified outside T (e.g. take a= 1, a= 4...) In that cases,Ge^T_S is isomorphic to the free pro-3-group with2generators.

3.3.2. On the analytic structure of the groups Ge^T_S. Let G be a pro-p-group with rank d(G) at least 3. If G is p-adic analytic, we have r(G) > d(G)²/4 (see [Se], Annexe 3).

Thus, we immediately obtain the following.

Corollary 3.11. Suppose that d(Ge^T_S)≥3 and that α:=r1+r2+θ(K, S)− X

v∈Sp

[Kv :Qp]≥0.

If d(Ge^T_S)≥2 + 2√

α, then the pro-p-group Ge^T_S is not p-adic analytic.

11

3.3.3. The splitting ofGe^{T ,ab}_S .

Definition 3.12. Let Ke^{T ,el}_S /K denote the maximal abelian p-elementary pro-p- extension of K contained inKe^T_S/K. Thep-extensionKe^T,el_S /Kcorresponds by Ga- lois theory to the quotient _pGe^T,ab_S =Ge^T,ab_S /(Ge^{T ,ab}_S )^p.

Corollary 3.13. Suppose that Ve_S^T/K^×p is trivial. Let Σ be a set of finite places of K with norm congruent to1 mod p.Thend(Ge^T_S∪Σ) =d(Ge^T_S) +|Σ|and all the places of Σare ramified in Ke^{T ,el}_Σ /Ke^T,el_S .

Proof. Indeed, for any tame place v, d(Ge^T_S∪{v}) ≤ d(Ge^T_S) + 1 with equality if and only if v is ramified in Ke^T,el_S∪{v}/K. As Ve_S∪{v}^T /K^×p ⊂ Ve_S^T/K^×p = {1}, the formula (i) of Theorem 3.5 shows that v is ramified in Ke_Σ^T/K. The relation d(Ge^T_S∪Σ) =d(Ge^T_S) +|Σ|follows immediately.

4. Preparatory results 4.1. A technical result. We first prove the following:

Proposition 4.1. LetS⁰andT be two finite sets of finite places ofKwithS⁰∩T =

∅. There exists a finite set S of finite places of Ksuch that (i) S∩T =∅, S⁰⊂S and (S−S⁰)∩Pl_p=∅;

(ii) for any place v∈Sp, the decomposition group of v in Ke^{T ,el}_S /Kis of maxi- mal rank, i.e. 1 + [K_v :Qp] +θ(K_v);

(iii) For any placev∈Plp−(Sp∪T), the decomposition group ofv inKe^{T ,el}_S /K is of maximal rank, i.e 2;

(iv) any place v∈S₀ =S−S_p is ramified in Ke^{T ,el}_S /K;

(v) for any place v ∈ S∪(Pl_p−T), there is a cyclic subgroup H_v of Ge^{T ,el}_S of order p such that H_v ∩I_v ={1}, where I_v is the inertia group of v in Ge^T,el_S .

Proof. (i)-(iv). PutP = Plp−T. It is sufficient to make sure that J/(J^pK^× Y

v /∈S∪T

Ue_v Y

v∈T

N_v)→ J/(J^pK^× Y

v /∈S∪T∪P

Ue_v Y

v∈P

K_v^×Y

v∈T

N_v),

is of maximal rank, i.e. that J^pK^× Y

v /∈S∪T

Ue_v Y

v∈T

N_v

!

∩ Y

v∈P

K^×_v = Y

v∈Sp

K^×_v^p Y

v∈P−Sp

K^×_v^pUe_v.

According to Corollary 3.3 (forT andS⁰), there is a finite setΣof tame places such that, for any placevofS⁰∪Σ, we haveVe_S^T0∪Σ−{v}/K^×p ={1}. PutS =S⁰∪Σ.

The elementaryp-extensionKe^{T ,el}_S /Krealizes the quotients_pG^ab_v for anyv∈Sand is cyclotomically ramified for v∈Plp−(Sp∪T) with maximalp-rank (thanks to Corollary 3.13, we know that all the tame places of S are ramified in Ke^{T ,el}_S /K).

(v) It is clear for any place v of Plp−T. For a tame place v ∈ S0, we have I_v 'Z/pZ. It is thus also clear ifPl_p−T 6=∅. Otherwise, we have to ensure that dpGe^{T ,el}_S ≥2, which can also be done enlargingS if necessary.

Remark 4.2. Condition (ii) can be weakened. We will see later that it is sufficient to make sure that the decomposition group of v in K^{T ,el}_S contains a certain cyclic extension of degree p (see Section 5.3.1)

4.2. Gras-Munnier criterium. In order to obtain the cohomological dimension less than or equal to 2 for the group Ge^T_S, we need to add tame places to S.

Corollary 3.13 is not sufficient. The extensions ramified at the new places, i.e. at v ∈Σ−S,have to live above K.More precisely, one looks for places v of K for which there exists an extensionL/Kof degreep,totally ramified atv and totally split at T.

This problem has been considered and solved by Gras and Munnier in [GM]

(see also the book of Gras [Gr], Chapter V, Corollary 2.4.4). One can find it partially in the work of Schmidt [Sch2] (when Cl^T_K[p] ={1}).

IfSandT are two finite sets of places ofK withS∩T =∅, letK^{T ,el}_S denote the maximal abelian elementary p-extension of K, unramified outside S and totally split atT.We apply the Gras-Munnier criterium to exhibit places vfor which the extensions K^T,el_{v}/Khave degree pand are totally ramified atv.

Proposition 4.3 (Gras-Munnier, [GM]). LetT be a finite set of places ofKsuch that Cl^T_K[p] = {1}, where Cl^T_K stands for the group of T-classes of K. Let v be a place of K not dividing p. Then K^{T ,el}_{v}/K is non trivial (in fact, it is cyclic of order p) if and only if v splits totally in K⁰(p^p

E_K,T)/K, where K⁰ = K(ζ_p) and where E_K,T denotes the group of T-units of K. In that case the extensionK^{T ,el}_{v}/K is totally ramified at v.

4.3. Schmidt’s elements. Throughout this paragraph, let S, T again be two disjoint sets of places, withT ⊂P l_p.LetTbe a finite set of places ofKsuch that

- T ⊂T etT∩Plp =T∩Plp; - Cl^T_K[p] ={1}.

For such T,we have

K^×/(K^×pE_K,T)

u ω //L

v /∈TZ/pZ,

the isomorphism being induced by the valuation. This is used to define Schmidt’s elements for T.

Definition 4.4 (Schmidt’s elements). For v /∈ T, let sv ∈ K^× be such that ω(s_v) equals 1 at v and 0 elsewhere: v(s_v) ≡ 1 mod p, v⁰(s_v) ≡ 0 mod p, for v⁰ ∈ {v} ∪/ T.

The key point is the next proposition:

Proposition 4.5. Let S, T,T and Schmidt’s elements(sv) be as above. Let qbe a place of K not dividing p.

a) The extensionK^T,el_{q} /Kis a subextension ofKe^{T ,el}_S∪{q}/K.It is non-trivial if and only if q is totally split inK⁰(p^p

EK,T)/K. In this case, the extension K^T,el_{q} /K is totally ramified at q.

b) Let p be a place of K prime toq ( p may divide p). Then p splits totally in K^T,el_{q} /Kif and only if qsplits totally in K⁰(√^p

sp,p^p

EK,T)/K⁰(p^p EK,T).

13

Proof. a) The first point is clear, whereas the second follows from the triviality of Cl^T_K[p]: this is the Gras-Munnier argument.

b) By class field theory, the place p splits totally in K^T,el_{q} /Kif and only if the morphism

J/J^pK^×Q

v∈TK^×_v Q

v /∈T∪{q}U_v ^////J/J^pK^×Q

v∈T∪{p}K^×_v Q

v /∈T∪{q}U_v is injective, that is, if and only if

K^×_p/(J^pK^× Y

v∈T

K^×_v Y

v /∈T∪{q}

U_v)∩ K^×_p ={1}.

We remark that U_p is in the denominator. Thus the triviality of this quotient is equivalent to the existence of an element α∈K^× such that

(i) p(α)≡1 mod p;

(ii) for anyv /∈T∪ {p},v(α)≡0 mod p;

(iii) α∈(K^×_q)^p.

The element sp satisfies these properties as soon as q splits totally in the ex- tension K⁰(√^p

s_p,p^p

E_K,T)/K⁰(p^p E_K,T).

Conversely, the existence of suchαimplies thatω(α/s_q) = 0,α∈s_q(K^×)^pE_K,T and q is totally split inK⁰(√^p

sp,p^p

EK,T)/K⁰(p^p

EK,T).

Corollary 4.6. Suppose the conditions of Proposition 4.5 hold and letqbe a place satisfying (a).

Put Γ_q= Gal(K^T,el_{q} /K). Then Γ_q is a quotient of Ge^{T ,el}_S∪{q} and the morphism H¹(Ge^{T ,el}_S )⊕H¹(Γq)−→H¹(Ge^{T ,el}_S∪{q})

induced by the inflation map is an isomorphism.

Proof. By class field theory, one knows that d(Ge^T,el_S∪{q}) ≤ d(Ge^{T ,el}_S ) + 1. On the other hand, the condition on q implies that the extension K^T,el_{q} /K is cyclic of degree p and is ramified atq. Thus K^T,el_{q} ∩Ke^T,el_S = K and Ke^{T ,el}_S∪{q} = K^T,el_{q} Ke^T,el_S .

The results follows immediately.

5. Proof of Theorem 1.1

The proof relies strongly on the work of Schmidt in [Sch2]. Put K⁰ = K(ζp).

5.1. On Schmidt’s conditions. In this paragraph again, letS be a finite set of finite places whose tame places p satisfy Np= 1 mod p, let T ⊂P l_p, and letT be a finite set of places of Ksuch that

- T ⊂T and T∩Pl_p=T∩Pl_p; - Cl^T_K[p] ={1}.

Definition 5.1. Let S = {p₁,· · ·,p_m}, T and T three finite sets of places of K satisfying the previous conditions. Let Σ be a finite set of places of K with Σ∩(T∪S∪Pl_p) =∅.

We say that the set Σ ={q₁,· · · ,q_n}satisfies Schmidt’s conditions with respect to the sets S andT (and T) if, for any q∈Σand anyQ|q, Qprime of K⁰ above q, we have:

(i) for any a = 1, . . . , n, Frob_Qa ∈/ I_pa, where I_pa is the inertia group p_a in Ke^{T ,el}_S /K.

(ii) for a6=b, Q_a splits in K⁰(√^p

s_pb)/K⁰;

(iii) for a= 1, . . . , n,Q_a decomposes totally in K⁰(p^p

E_K,T)/K;

(iv) for a= 1, . . . , n, the placeQ_a is inert in K⁰(√^p

s_pa)/K⁰; (v) for a < b, Q_b is totally split inK^{T ,el}_{q

a}/K;

(vi) for a < b, Q_b is totally split inK⁰(√^p

s_qa)/K.

Remark 5.2. The considered tame places v ∈ S are such that Kv contains the p^th roots of unity.

The next proposition, which can (mostly) be found in [Sch3], is of crucial importance:

Proposition 5.3. Let S, T,T be as above.

(a) LetΣ ={q₁,· · ·,q_n}(possibly empty) be a set satisfying Schmidt’s condi- tions with respect toS andT. Then the fieldsK⁰(p^p

EK,T),K⁰(√^p

sp1),· · · , K⁰(√^p

spm),Ke^{T ,el}_S K⁰ andK⁰(√^p

sq1),· · ·,K⁰(√^p

sqn), are linearly disjoint over K⁰.

(b) There exists a setΣof cardinality|S|=m satisfying Schmidt’s conditions with respect toS and T.

Proof. See the proof of [Sch3], Theorem6.1.Let us briefly prove the two points.

(a) By construction and Kummer theory, the fields K⁰(p^p

E_K,T), K⁰(√^p

s_p1),· · · , K⁰(√^p

spm) and K⁰(√^p

sq1),· · ·,K⁰(√^p

sqn) are linearly disjoint over K⁰. Let L de- note their compositum. Because µ_p is not contained in K, the Galois group Gal(k(µ_p)/k)acts non trivially by conjugation onGal(L/K⁰)and this action shows that for any subfield L⁰ of Lstrictly containing K⁰,L⁰/Kis not an abelian exten- sion, so that L⁰∩Ke^T,el_S = K⁰.Therefore (a) holds.

(b) We takeQ₁, . . . ,Q_n recursively using Cebotarev density theorem. The asser- tions (ii)−(iv)and(vi) are made possible because of the fact that the extensions as linearly disjoint. For (i) one needs in addition Proposition 4.1(v) and for the

point (v) one needs (the proof of) Corollary 4.6.

5.2. The choice of tame places. Let T ⊂Pl_p and S⁰ two disjoints finite sets of Kas in Theorem 1.1.

• We first consider a set Sold satisfying Proposition 4.1 for S⁰ and T. In particular, we haveVe_S^T

old/K^×p ={1}.

• Then we choose a setT⁰ of tame places ofK, such thatCl^T_K[p] = 0, where T=T∪T⁰.

• Let nowS_new be a set of places ofKwith cardinalitym=|S_old|satisfying Schmidt’s conditions with respect toSoldetT. LetS be the resulting set S=S_old∪S_new.

Thanks to (iii) of Definition 5.1 and Corollary 4.6, the inflation map induces an isomorphism

H¹(Ge^T,el_S )'H¹(Ge^{T ,el}_S

old)⊕ M

q∈Snew

H¹(Γq).

15