SOME NOTES ON COHOMOLOGY OF

NUMBER FIELDS AND ANALYTIC

PRO-p-GROUPS

by

Christian Maire

Abstract. � In this paper, we are interested in the tame version of theFontaine-Mazur conjecture. After recalling the role of étale cohomologyin the context of this conjecture, we establish a relationship between itand the computation of the cohomological dimension of the pro-p-groupsGS that appear. We then look at this conjecture by viewing the pro-p-proup GS as a quotient of a Galois group with rami�cation at p and Sand give a connection between the conjecture studied here and a questionof Galois structure.

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. Galois representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3. Analytic pro-p-group and GS . . . . . . . . . . . . . . . . . . . . . . . . . 8

4. Cohomology of GS and étale cohomology. . . . . . . . . . . . . 11

5. The group GS and wild rami�cation. . . . . . . . . . . . . . . . . . 17

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2000 Mathematics Subject Classi�cation. � 11R37, 11R23.

I thank Farshid Hajir for his help and his interest in this work. I thank Denis Vögelfor having sent me his preprint [27].

2 CHRISTIAN MAIRE

1. Introduction

Let us �x an algebraic closure Q̄ of Q.Let K be a number �eld and let p be a prime number. Let Sp be the setof places of K above p and let S be an another �nite set of places of K.Denote by KS the maximal pro-p-extension of K unrami�ed outside S.Put GS = Gal(KS/K).In this paper, we are interested by the following conjecture due toFontaine and Mazur [6]:

Conjecture 1.1. � Suppose that the characteristic of the residue �eldkv of each place v of S is di�erent from p (in other words, S ∩ Sp = ∅).Then, every p-adic analytic quotient of GS is �nite.

The abelian version of this conjecture is controlled by Class �eld theory.More precisely, some classical calculations allow to show the following:

Proposition 1.2. � (i) The prop-p-group GS is �nitely generated.(ii) If S ∩ Sp = ∅, then Gab

S := GS/[GS,GS] is �nite.(iii) If S contains all places of K above p, then the Zp-rank of Gab

S is atleast r2 + 1, where (r1, r2) is the signature of K. Leopoldt's conjectureasserts the equality.

In the context of conjecture 1.1, this proposition is important for at leasttwo reasons. First, it shows that the conjecture 1.1 is a theorem in theabelian case. Second, as we will see later, the abelian case will be used forthe fundamental example issuing from geometry (see Proposition 2.8).

In the �rst part of this paper, we recall quickly the geometric origin ofthis conjecture. This part contains no new results. For more details, werefer to the papers of Kisin-Wortmann [11] and Kisin-Lehrer [10].In a second part, we develop, in the context of the conjecture 1.1, thep-adic analytic group point of view. In particular, we show how thequestion of the knowledge of the cohomological dimension of the groupsGS is connected to conjecture 1.1.In a third part, we go back to the recent results of Labute [13] andSchmidt [20], [21]. Recall that Labute gave for the �rst time some ex-amples of groups GS of cohomological dimension 2, for S prime to p.Schmidt extended the work of Labute by showing that the cohomologyof the pro-p-groups in question coincide with some étale cohomology.

3

Finally, in a last part, we compare the groups GS with the groups GS′ ,where S ′ = S ∪ Sp, Sp being the set of all places of K above p.More precisely, let H = Gal(KS′/KS) be the closed normal subgroup ofGS′ generated by all inertia groups of all places of S ′ \S. Then X := Hab

is a Zp[[GS]]-module. Following a work of Brumer, we give a relationshipbetween the cohomological dimension of GS and the structure of theZp[[GS]]-module X . In particular, we show how the examples of Labute,Schmidt and Vögel [27] are such that in each case, X is Zp[[GS]]-free.

2. Galois representations

Let K be a number �eld and GK := Gal(Q̄/K) be its absolute Galoisgroup.For a place v of K, denote by Kv the completion of K at v. Let G :=Gal(Kv/Kv) be the absolute Galois group of Kv and let Iv ⊂ Gv be theabsolute inertia group of Kv.

Let ρ : GK → Aut(V ) be a continuous p-adic representation of GK, whereV is a Qp-vector space of �nite dimension.For a place v of K, we denote by ρv the restriction of ρ to Gv. Therepresentation ρ is said unrami�ed (resp. potentially unrami�ed) at v ifρ(Iv) is trivial (resp. ρ(Iv) is �nite).

2.1. First observations. � Let us �rst remark the following:

Proposition 2.1. � The image ρ(GK) of ρ is virtually a �nitely gener-ated pro-p-group.

Proof. � By continuity, the image of ρ is a compact subgroup ofGlm(Qp), i.e. conjuguated to a closed subgroup of Glm(Zp). In particu-lar, ρ(GK) contains an open subgroup U which is a pro-p-group of �nitetype. ♦

Remark 2.2 (see [4]). � Let G be a �nitely generated pro-p-group.

Then for all i, Gpiis closed and open in G.

Take G = ρ(GK). By a �rst scalar extension, we can assume that G isa �nitely generated pro-p-group and then by a second scalar extension,one can replace G by Gpi

.

Next, recall a well-know result:

4 CHRISTIAN MAIRE

Proposition 2.3. � Let H be a compact subgroup of Glm(Qp). Then:

(i) The subgroup [H, H] is closed in H.(ii) If H is unipotent, then H is conjugated to a subgroup of the group

of the triangular matrices

1 ∗ · · · ∗0

. . . . . ....

.... . . 1 ∗

0 · · · 0 1

with diagonal entries

all 1. Hence, H is nilpotent of class at most m− 1.

As application of the previous proposition one obtains:

Corollary 2.4. � Suppose S �nite and such S ∩ Sp = ∅. Let H be aquotient of GS with H ⊂ Glm(Qp) and such that H is unipotent. ThenH is �nite.

Proof. � The group H is solvable. Moreover, the commutators seriesis a sequence of closed subgroups of H. The successive quotients areabelian and then by Proposition 1.2, one deduces that H is �nite. ♦

2.2. Hodge-Tate representations. � A representation ρ of GK isinteresting when it comes from �geometry�. If it is the case, the repre-sentation ρ satis�ed the following two properties. The �rst one is thatρ is unrami�ed outside a �nite S of places of K. The second one willconcern the places v above p. A consequence of works of Tsuji, De Jong... shows that for all v|p, ρv := ρ|Gv is potentially semi-stable (in the senseof Fontaine's theory). Hence, for all v|p, ρv is of Hodge-Tate and thenthe knowledge of the weights of Hodge-Tate iv of ρv is very important.Let Kv/Qp be a �nite extension and let V be a p-adic representationof Gv. Denote the Hodge-Tate ring by BHT := Cp[t, t

−1]. The pro�nitegroup Gv acts on Cp (by continuity) and on t via the cyclotomic characterχ:

∀σ ∈ Gv, σ · t = χ(σ)t.

By de�nition, the representation V is of Hodge-Tate type if

BHT⊗Qp

(BHT ⊗Qp V

)Gv ' BHT ⊗Qp V.

Let V be a Hodge-Tate representation of Gv. Then, DHT (V ) :=

(BHT ⊗ V )Gv has a grading corresponding to the powers of t. Theintegers iv for which grad−iv(DHT (V )) is not trivial are the weights of

5

Hodge-Tate of the representation V (with multiplicity corresponding tothe dimension over Kv of grad−iv(DHT (V ))). Hence, one has:

Cp ⊗Qp V ' Cp(i1,v)⊕ · · · ⊕ C(in,v),

isomorphism of Gv-modules, where Cp(i) := Cp ⊗Qp ti is of dimension 1over Cp with diagonal action of Gv.

The situation where all weights of Hodge-Tate are zero is well-known.First consider an unrami�ed representation V . Then V is Hodge-Tatewith zero weights (cf. [23], Theorem 1, III-31)). A result of Sen [24]shows that the converse is still true after extension of scalars:

Proposition 2.5. � Let v|p and let ρv : Gv → Aut(V ) be a Hodge-Taterepresentation. Then the following are equivalent:

(i) the Hodge-Tate weights of ρv are zero;(ii) the representation ρv is potentially unrami�ed.

Remark 2.6. � Let v|p. For a representation ρv of Gv, following Sen,one can de�ne the generalized Hodge-Tate weights. When the represen-tation is of Hodge-Tate type, these generalized weights are exactly theHodge-Tate weights (see [25]). In the general case, the triviality of thesegeneralized weights do not imply a priori the �niteness of ρv(Iv) (see forexample Mazur [17]).

Example 2.7. � Let σ be a generator of the cyclotomic Zp-extensionof Kv. Let ρv : Gv → Aut(V ) be a Hodge-Tate representation of Gv. Ifthe only root in Zp of the minimal polynomial Pσ ∈ Zp[X] of σ over V is1, then the Hodge-Tate weights of V are zero. Hence, if the action of σover the Hodge-Tate representation V is unipotent, then the Hodge-Tateweights of this representation are zero. This remark seems restrictive, butwhen the representation is global and geometric with zero Hodge-Tateweights (for all v|p), we will see that Im(ρ) is unipotent.

2.3. A fundamental example. � Recall that we have �xed a number�eld K and a prime p.Let X/K be a smooth projective variety over K.Let

H iet(X, Qp) := lim

n←H i

et(X, Z/pnZ),

6 CHRISTIAN MAIRE

be the ith etale cohomology group of X := X/K with coe�cients in Qp.Put:

hi,j(X) := H iet(X, Qp)(j) = H i

et(X, Qp)⊗Qp tj,

the twist of H iet(X, Qp) by Qp(j).

This group hi,j(X) is in fact a �nite Qp-vector space of dimension m onwhich acts the absolute Galois group GK of K. This action gives a Galoisrepresentation of GK:

ρi,j : GK → Glm(Qp) = Aut(hi,j(X)).

The main goal of this part is to recall why the following holds:

Proposition 2.8. � Suppose that the Galois representation

ρ : GK → Aut(H iet(X, Qp)(j))

is potentially unrami�ed outside p. Then, the image ρ(GK) is �nite.

Let v be a place of K. Recall that the representation ρi,j induces arepresentation ρi,j

v of Gv (in other words: ρi,jv = ρi,j

|Gv).

Denote by S the set of places of K at which: ρ is rami�ed; or X has badreduction. We add to S the places above p and in�nite places. This �nalset S is �nite.The crucial fact will be a variation of places v with `-adic cohomology,p-adic cohomology and isomorphisms of comparaison of the di�erent co-homology. We will show why ρ(GK) should be virtually unipotent andthen �nite (see corollary 2.4).

2.3.1. `-adic and p-adic representations. � First, let v|p. Then the p-adic representation ρi,j

v : Gv → Aut(hi,j(X)) is potentially semi-stable,Now let v |`, ` 6= p, v /∈ S be a place of K (X has good reduction at v).Denote by σv the geometric Frobenius at v . Let us consider the actionof σv on hi,0(X) and put Pv, the characteristic polynomial of σv comingfrom this action. One knows that Pv is in Q[X]. Its roots λk,v are some

integers such that |λk,v|σ equals qi/2v , where |.|σ is the absolute value of

the di�erent embeddings in C and where qv := N(v) = #OK/v. Hencethe eigenvalue of the action of σ on hi,0(X) are λk,vq

−jv . Their absolute

values are (independently of the embeddings): qi/2−jv .

The polynomial Pv can be obtained by an another calculation. For vnot in S, denote by H∗cris(Xv) the crystalline cohomology of Xv, whereXv := X ⊗ kv, X being a smooth model over the ring of integers OKv

of Kv. This group, which is a vector space over the maximal unrami�ed

7

sub-extension K0v of Kv/Qp, is equipped with an action of a Frobenius ϕv.

In [9], Katz and Messing have shown that the characteristic polynomialof ϕqv

v acting on H icris(Xv) is exactly Pv. Moreover, in this case, one can

compare the polygon attached to the Hodge-Tate weights of ρi,j and theNewton polygon of ϕqv

v acting on H icris(Xv)(j), (i.e. Pv for j = 0): these

polygons have the same extremities and the former is on or above thelatter. In particular:

Proposition 2.9. � If the sum of the weights of ρi,jv is zero, then i = 2j.

Proof. � As the extremities of the Hodge-Tate polygon are the same asthe extremities of Newton's polygon, one deduces that the determinantof ϕqv

v has a zero v-valuation. But, by using Pσ, this determinant is in Qwith absolute value q

i/2−jv . Hence i/2 = j. ♦

2.3.2. The situation where the weights of Hodge-Tate are zero. � Sup-pose now that the p-adic representation ρ : GK → Aut(hi,j(X)) is poten-tially unrami�ed for all places v dividing p. Then for all places v|p, theweights of Hodge-Tate of ρv : Gv → Aut(hi,j(X)) are zero. Hence, fromProposition 2.9, i = 2j. Now, there exists some Theorems of comparisonof cohomology. Typically, the following holds:

Theorem 2.10. � Let ` be a prime number and let v|`. Then theweights of Hodge-Tate {iv,`} of ρv,` : Gv → Aut(H i

et(X, Q`)) are not de-pending on v and on `.

And then, it is not di�cult to see that �nally the eigenvalues of σv (forv not in S) acting above hi,i/2(X) are algebraic integers with absolutevalue 1. These numbers are then some roots of the unity, and roots ofa polynomial of degree m. In particular, for a certain integer n, λn

k,v =1,∀k,∀v. By density, one deduces that for all g ∈ GK, (ρ(g)n − 1)m = 0.By a �rst scalar extension, one can assume that ρ is unrami�ed at p,that ρ(GK) is a pro-p-group and then that n is a power of p. By usingthe remark 2.2 and by after a second scalar extension L/K, we concludethat ρi,i/2(GL) is unipotent. And then, by Corollary 2.4, Proposition 2.8is proved.

Here is the philosophy of the Fontaine-Mazur conjecture. Let ρ : GK →Gln(Qp) be a continuous p-adic representation representation of GK. Sup-pose: (i) ρ is unrami�ed outside a �nite set S of places ofK; (ii) for all v|p,the representation ρv := ρ|Gv is potentially semi-stable. Then ρ should

8 CHRISTIAN MAIRE

come from �geometry�: ρ should correspond to the action of GK on asubquotient of an étale cohomology group, twisted by Qp(j). With theconjectures of Tate and of Serre-Grothendieck, the previous formalismsshould hold (See [11] for more details) giving a �nite image for ρ.

Conjecture 2.11 (The Fontaine-Mazur conjecture - Tame ver-sion)Let K be a number �eld and S be a �nite set of places of K. Everypotentially semi-stable p-adic representation of GS with weights of Hodge-Tate all equal to zero (for all v|p) has a �nite image.

Classicaly, by a change base and by Proposition 2.5, to study this con-jecture, one assumes (S, p) = 1, i.e. S ∩ Sp = ∅.

3. Analytic pro-p-group and GS

3.1. On analytic pro-p-groups. � The references for this part are:Lazard[14], Dixon, Du Sautoy, Segal, Mann [4], Lubotzky ...

De�nition 3.1. � A topological group G is p-adic analytic if G has astructure of p-adic analytic manifold for which the morphism G × G →G : (x, y) 7→ xy−1 is analytic.

Example 3.2. � A p-adic analytic group G is of dimension 0 if andonly if G is �nite.A p-adic analytic group G is of dimension 1 if and only if there exists anopen subgroup U of G isomorphic to Zp.

Thanks to the following fact, the theory of p-adic analytic groups isrelated to the discussion of the previous section.

Theorem 3.3 ([4] Theorem 7.19). � Every p-adic analytic group isisomorphic to a closed subgroup of Glm(Zp), for a certain m.

We can give a reformulation of Conjecture 1.1 in terms of dimension ofthe analytic manifold.

Conjecture 3.4 (Cr). � Let K be a number �eld and S be a �nite setof places of K. Let KS be the maximal pro-p-extension of K unrami�edoutside S. Put GS = Gal(KS/K). Assume that for all places v in S, kv

is of characteristic di�erent from p. Let r > 0 be an integer. Then thereis no p-adic analytic quotient of GS of dimension r.

9

Immediatly, it comes:

Proposition 1. � Conjecture C1 is true.

Proof. � After scalars extension, it is exactly (ii) of Proposition 1.2. ♦

3.2. Uniformly powerful groups. � Let G be a pro-p-group of �nitetype. Denote by Pi(G) the subgroups of the p-lower central series of G :

P1(G) = G, Pi+1(G) = Pi(G)p[G, Pi(G)],

which is a sequence of closed subgroups of G.

Denote by G1 the quotient G/Gp[G,G] = G/P2(G) and by dp(G) the p-rankof G, i.e. the dimension of G1 over Fp.

De�nition 2. � The pro-p-group G is uniformly powerful if:

(i) G/Gp (for p = 2, G/G4) is abelian.(ii) for all i, #Pi(G)/Pi+1(G) = #G/P2(G)

The study of p-adic analytic group can be done via the uniformly powerfulpro-p-groups:

Theorem 3.5. � Let G be a p-adic analytic group. Then G has an openuniformly powerful subgroup.

If G is uniformly powerful, Lazard has proved that the cohomology groupsH i(G,Fp) are isomorphic to the exterior product

∧iH1(G,Fp). Hence,for a uniformly powerful pro-p-group G, the p-rank is equal to the coho-mological dimension cd(G) of G.Note that a uniformly powerful pro-p-group has no torsion. Conse-quently, archimedean places play no role for conjecture 1.1.

Corollary 3.6. � Let G be a uniformly powerful group of dimension 2.Then, there exists a surjective morphism: Gab � Zp.

Proof. � Indeed, in this case, one has: dpH1(G,Fp) = 2dpH

2(G,Fp) = 2.The short exact sequence

1 −→ Fp −→ Qp/Zpp−→ Qp/Zp −→ 1,

gives1 = dpH

1(G,Fp)− dpH2(G,Fp) ≤ rkZpGab.

♦

10 CHRISTIAN MAIRE

Corollary 3.7. � Let G be a p-adic analytic group of dimension 2.There exists an open subgroup U of G, for which the Zp-rank of Uab is nottrivial.

Corollary 3.8. � Conjecture C2 is true.

Proof. � Let L/K be a Galois extension in KS/K, where S ia �nite setof places of K all prime to p. Suppose that the group G := Gal(L/K) isp-adic analytic of dimension 2. By corollary 3.7, there exists an open sub-group U of G such that Uab has a non trivial Zp-rank which is impossibleby Proposition 1.2. ♦

In some situations, GS can be equipped with a certain action. By com-paring this one with the structure of uniformly powerful pro-p-groups,one has some situations that corroborate the conjecture 1.1. Typically:

Proposition 3.9. � Let K be a CM �eld with a maximal totally realsub�eld K+. Let p > 2. To simplify, assume that K does not contain thep-roots of the unity µp. Denote by d(GS) = d+ + d− the decomposition ofthe p-rank of GS following the action of Gal(K/K+). If d− > 0, then GS

is not uniformly powerful.

Proof. � It is a direct application of a result of Wingberg [28] plusCorollary 3.7. ♦

3.3. Just in�nite pro-p-groups.�

De�nition 3.10. � A pro-p-group G is just in�nite if every proper quo-tient of G is in�nite.

Example 3.11. � The pro-p-group Zp is just in�nite.

It is well known that every in�nite pro-p-group has a just in�nite quotient.The following result of Engler, Haran, Kochloukova and Zalesskii allowsus to give a nice application in our context.

Theorem 3.12 ([5]). � Let G be a pro-p-group of cohomological dimen-sion n and let N be a closed distinguished p-adic analytic subgroup ofdimension k ≥ 1. Then G/N is virtually of cohomological dimensionn− k.

Remark 3.13. � A pro-p-group G is virtually of cohomological dimen-son d, if there exists an open subgroup U of G, with cohomological di-mension cd(U) = d.

11

Proposition 3.14. � Suppose that all places in S are prime to p. Let Gbe a quotient of GS. Suppose that G is in�nite of cohomological dimensionat most 3. If G is not just in�nite, then G is not p-adic analytic.

Proof. � Suppose that G is p-adic analytic. The conjectures (C1) and(C2) being true, one can assume that cd(G) = 3. The pro-p-group Gbeing not just in�nite, there exists a closed and distinguished subgroupN such that G/N is in�nite. The analytic subgroup N is of cohomologicaldimension k ∈ {1, 2, 3}. By Theorem 3.12, the quotient G/N has avirtually conhomological dimension 3 − k ≤ 2, i.e. 1 or 2 because G/Nis in�nite, which contradicts (C1) and (C2). ♦

Corollary 3.15. � Let S ′ ( S be two �nite sets of places of K, with(S, p) = 1. Suppose: (i) #Gab

S′ < #GabS ; (ii) GS′ is in�nite; (iii) the coho-

mological dimension of GS is at most 3. Then GS is not p-adic analytic.

Proof. � The inequality #GabS′ < #Gab

S shows that at least one place ofS\S ′ is rami�ed in KS/K. So KS 6= KS′ and consequently GS′ is a properquotient of GS. In particular, GS is not just in�nite. By Proposition 3.14,GS is not p-adic analytic. ♦

4. Cohomology of GS and étale cohomology

Let S be a �nite set of places of K. Recall that KS is the maximalpro-p-extension of K unrami�ed outside S; put GS = Gal(KS/K). Thefollowing is well-known:

Theorem 4.1. � Suppose that S contains all the places above p. Forp > 2, the cohomological dimension of GS is 1 or 2.

For p = 2, see the work of Schmidt [22]. Recall the fact that the strictcohomological dimension is related to Leopoldt's conjecture.When S contains no places above p, Labute, in [13], has recently givensome examples for which the cohomological dimension of GS is 2. For themixed case, i.e S ∩ Sp 6= ∅, see [16] and [18].

Remark 4.2. � Suppose S prime to p.

(i) Then GS can not be free.(ii) If cd(GS) = 2, then the Principal Ideal Theorem shows that the

strict cohomological dimension of GS is 3.

12 CHRISTIAN MAIRE

In the context of the Fontaine-Mazur conjecture, it seems natural to askthe following question:

Question 4.3. � Suppose S ∩ Sp = ∅. Is the pro-p-group GS virtuallyof �nite cohomological dimension ?

4.1. The results of Schmidt. � In [20], Schmidt expands on thework of Labute. Schmidt proved that the examples of the groups GS

given by Labute have a cohomology related to the étale cohomology ofsome certain schemes.

Let U be an open subgroup of GS. Denote by KU the maximal sub�eldof KS �xed by U. Put Spec OU \ S := Spec(OKU

) \ SKU, where SKU

isthe set of places of KU above the places of S.Let M be a torsion GS-module. We consider M as the constant sheaf overSpec OU \ S. Denote by H i

et(Spec OU \ S, M) the ith étale cohomologygroup of the sheaf M and put

H iet(X(KS), M) := lim

U→H i

et(Spec OU \ S, M),

the inductive limit being on open subgroups U of GS with⋂U

U = {1}.

The spectral sequence H i(GS, (Hjet(X(KS), M)) =⇒ H i+j

et (Spec OK \S, M), shows the existence of a morphism

φi(M) : H i(GS, M) → H iet(Spec OK \ S, M).

As shown in [20], the study of this morphism is related to the case whereM = Fp. Hence we de�ne:

De�nition 4.4. � The pro-p-group GS is cohomologically étale if forall i ≥ 0, the morphism φi := φi(Fp) is an isomorphism.

Remark 4.5. � When φi is an isomorphism for all i, following Schmidtin [21], the scheme Spec OK \ S is called K(π, 1).

Question 4.6. � Suppose the pro-p-group GS in�nite. In which cases,GS is cohomologically étale?

Questions 4.3 and 4.6 are related by the following Theorem:

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Theorem 4.7 (SGA 4, [21]). � Suppose GS cohomologically étale.For p = 2, suppose moreover that K totally imaginary. Then thecohomological dimension of GS is at most 3. Moreover, GS is of cohomo-logical dimension at most 2 when S is not empty or when KS/K doesnot contain the pth roots of unity.

Proof. � The �rst part of this Theorem can be found in SGA [8] (Propo-sition 6.1). The second part is a calcultation of Schmidt in [21]. ♦

From now on, we assume p > 2 or K totally imaginary.

4.2. Some consequences of a spectral sequence. � First recalltwo lemmas:

Lemma 4.8. � One has:

H1et(X(KS), Fp) = 1 .

Proof. � The group H1et(X(KS), Fp) classi�es the Galois étale covers of

degree p of X(KS). By maximality of X(KS), this group is trivial. ♦

Hence, the morphism φ1 is an isomorphism. When GS is in�nite, one hasmore:

Lemma 4.9 (Schmidt, [21]). � If GS is in�nite, then

H3et(X(KS), Fp) = 1.

These lemmas applied to the spectral sequence

H i(GS, (Hjet(X(KS), Fp)) =⇒ H i+j

et (Spec OK \ S, Fp)

allow us to show:

Proposition 4.10. � Assume GS in�nite. Then, the following long ex-act sequence holds:

H2(GS, Fp)� � // H2

et(SpecOK \ S, Fp) // H2et(X(KS), Fp)

GS

��H1(GS, H2

et(X(KS), Fp))

����

H3et(SpecOK \ S, Fp)oo H3(GS, Fp)oo

H4(GS, Fp)

14 CHRISTIAN MAIRE

One could introduce the notion of pro-p-group virtually cohomologicallyétale. But in fact, it is not necessary:

Corollary 4.11. � (i) Suppose that the pro-p-group GS is virtually co-homologically étale. Then GS is cohomologically étale.(ii) If GS is cohomologically étale, then for all open subgroup U of GS, Uis cohomologically étale.

Proof. � (i) The pro-p-group GS being virtually cohomologically étale,it means that there exists an open subgroup U of GS such that the mor-phisms φi,U are isomorphisms, where

φi,U : H i(U, Fp) → H iet(SpecOKU

\ SKU, Fp).

Then, by using the long exact sequence of Proposition 4.10, one deducesthe triviallity of H2

et(X(KS), Fp)U, i.e. H2

et(X(KS), Fp) = 1. It su�ces totake the same exact sequence with GS instead of U.(ii) - clear. ♦

Let U be an open subgroup of GS. Denote by χn(U) the Euler-Poincarécharacteristic truncated at the order 3, associated to the Galois cohomol-ogy groups H i(U, Fp) of U and denote by χet(U) the Euler-Poincaré char-acteristic associated to the étale cohomology groups H i

et(Spec OU\S, Fp):

χn(U) =n∑

i=0

(−1)idpHi(U, Fp),

χet(U) =∑i≥0

(−1)idpHiet(Spec OU \ S, Fp).

By Class Field Theory, one knows an upper bound for the p-rank ofH2(GS,Fp). When S contains no places above p, an exact calculationof this rank is at the heart of the construction of asymptotically exactextensions. In all cases, one has:

χ2(GS) ≤ (−δS + r1 + r2) + δp,S,

where δS =∑

v∈S∩Sp[Kv : Qp], and δp,S is equals to 1 when K contains

the p-roots of the unity and S is empty, 0 otherwise.Recall a result that appears in [21]:

15

Proposition 4.12 (Schmidt). � Suppose p > 2 or K be a totallyimaginary �eld . One has:

χet(GS) = −δS + r1 + r2.

In particular, for U ⊂O GS, the following holds:

χet(U) = (GS : U)χet(GS).

Now it is possible to compare χ3(GS) with χet(GS) :

Proposition 4.13. � One has (for GS 6= 1):

χ3(GS) ≤ χet(GS).

When cd(GS) ≤ 2 with S non-empty, the equality holds if and only if GS

is cohomologically étale.

Proof. � One knows:

χ3(GS) = χ2(GS)− dpH3(GS,Fp)

≤ (−δS + r1 + r2) + δp,S − dpH3(GS,Fp)

= χet(GS) + δp,S − dpH3(GS,Fp)

Then χ3(GS) ≤ χet(GS)+1 and more χ3(GS) ≤ χet(GS) with the eventualexception: H3(GS, Fp) = 1 and δp,S = 1. In this last case, the coho-mological dimension of GS is at most 2 and this one of Spec OK \ S is3.Suppose: χ2(GS) = χet(GS)+ 1. Then for all open subgroup U of GS, onehas:

χ3(U) = χ2(U)= [GS : U] χ2(GS)= [GS : U] (χet(GS) + 1)= χet(U) + [GS : U]

which contradicts the inequality χ3(U) ≤ χet(U) + 1.When the cohomological dimension of GS is at most 2 (with S noneempty), the morphism φ2 is an isomorphism and then H2

et(X(KS), Fp) =1. ♦

4.3. The mixed case. � Let S ⊂ Sp be a none empty subset of Sp.Denote by ϕS the morphism:

ϕS : Zp ⊗ EK →∏v∈S

U1v ,

16 CHRISTIAN MAIRE

U1v being the principal local units at v. When S = Sp, Leopoldt's con-

jecture predicts the injectivity of the map ϕS.If L/K is a �nite extension, we will denote by ϕL,S the morhism

Zp ⊗ EL →∏

v∈S(L)

Uv.

If we assume Leopoldt's conjecture, it is possible to give a quick proof ofthe fact that the cohomological dimension of GSp is at most 2. By usingthis remark, one can prove the following:

Theorem 4.14 ([16]). � Suppose that the kernel of ϕL,S is boundedalong the extension KS/K. The, GS is �nite or the cohomological di-mension of GS is at most 2.

The problem of the condition of the previous Theorem is that it is notexplicit. Suppose now that S is su�cently �large� i.e. such that KS/Kcontains a Zp-extension K∞/K. Put HS := Gal(KS/K∞) and XS :=Hab

S . The pro-p-group XS is a Zp[[T ]]-module. Let (ρS, µS, λS) be theirinvariants. It is possible to give explicit conditions which imply that thecohomological dimension of GS is at most 2. See for example, [18], [16]...:

Proposition 4.15. � If ϕS is injective and if µS = 0, then the coho-mological dimension of GS is at most 2.

Now, the �rst invariant ρS is related to χ2(GS).

Proposition 4.16. � Let Kn be the steps of K∞/K.(i) If the sequence (rkZp(kerϕKn,S))n of the Zp-rank of the kernels of themorphisms ϕKn,S is bounded, then

ρS = δS − (r1 + r2).

(ii) If the morphism ϕS is injective, then the sequence (rkZp(kerϕnS))n is

bounded and moreover,ρS = −χ2(GS).

Let us give some examples.

Corollary 4.17. � (i) Let K = Q(√−d) be an iamginary quadratic

�eld, where d is square free. Let p be a splitting prime in K/Q and putS = {p}, where p is a place of K above p. Then GS is cohomologicallyétale.(ii) The examples of [16]. Let K = Q(θ), where θ is a root of x4 + x3 +8x2−4x+2 = 0. The signature of this �eld is (0, 2). Put S = {(θ)}. Here

17

(θ) is a prime ideal above 2 with index of rami�cation over Q equals to 3.Note that K has two primes above 2. Then the cohomological dimensionof GS is 2, χ2(GS) = −1, and GS is cohomologically étale.

Proof. � In these two cases, µS = 0. Moreover, conditions (i) and (ii)of Proposition 4.16 are satis�ed. Hence χ2(GS) = r1 + r2− δS = χet(GS).The groups GS being of cohomological dimension at most 2, one concludeswith Proposition 4.13. ♦

5. The group GS and wild rami�cation

For all this section, we assume that GS is not trivial. All results of thispart are inspired by [18] and [19].

5.1. The setup. � Let Sp be the set of all places of K above p. PutS ′ = S ∪ Sp. (A priori, S can contain some places above p.) Let KS′

be the maximal pro-p-extension of K unrami�ed outside S ′. Put GS′ =Gal(KS′/K). For the next, we will consider only the situation wherep > 2 or where K is totally imaginary. In this case, the cohomologicaldimension of GS′ is 1 or 2.The group GS is a quotient of GS′ . Let H be the closed and normalsubgroup of GS′ generated by the inertia groups of all places v ∈ Sp inKS′/K. Then, GS′/H ' G. Put X = Hab. Then, it is well-known thatH is a Zp[[GS]]-module, where Zp[[GS]] := lim

U←Zp[GS/U].

The ring Zp[[GS]] is local and its projective dimension is 1 + cd(GS).As an application of Nakayama's lemma, one obtains:

Proposition 5.1. � The Zp[[GS]-module X is �nitely generated.

Proof. � The Hochschild-Serre spectrale sequence applied to the shortexact sequence

1 −→ H −→ GS′ −→ GS −→ 1,

shows

· · · −→ H1(GS′ , Fp) −→ H1(H, Fp)GS −→ H2(GS, Fp) −→ ...

As H1(GS′ , Fp) and H2(GS, Fp) are �nite, one deduces that XGS/p =

(H1(H, Fp)GS)∗ is �nite. Then by Nakayama's lemma, one has the result.

♦

Next recall the following:

18 CHRISTIAN MAIRE

Proposition 5.2. � Leopoldt's conjecture is equivalent to the trivialityof H2(GS′ , Qp/Zp). Hence, if one assumes Leopoldt's conjecture for allnumber �elds and the prime number p,

H2(H, Qp/Zp) = lim→

K⊂F⊂L

H2(Gal(Kp/F , Qp/Zp) = 0.

5.2. The cohomological dimension of GS and the structure ofX . � From now on, one assumes Leopoldt's conjecture for all number�elds and the prime number p.First, we have:

Proposition 5.3. � 1) If cd(GS) ≤ 2, then the projective dimension ofthe Zp[[GS]]-module X is at most 1.2) If X is Zp[[GS]]-free, then cd(GS) ≤ 2.

Proof. � Let

1 −→ W −→ F −→ GS′ −→ 1

and

1 −→ R −→ F −→ GS −→ 1

be two presentations of the groups GS′ and GS where F is the free pro-p-groups on dp(GS′) generators.As GS is a quotient of GS′ , one gets the exact sequence:

1 −→ W −→ R −→ H −→ 1.

From the homology version of the Hochschild-Serre spectral sequence,one gets:

1 −→ H2(H, Qp/Zp)∗ −→ (W ab)H −→ Rab −→ X −→ 1.

As one assumes Leopoldt's conjecture, H2(H, Qp/Zp) = 0. Recall a resultof Brumer:

Lemma 3 (Brumer,[3], theorem 5.2). � Let G be a �nitely gener-ated pro-p-group. Let

1 −→ R −→ F −→ G −→ 1,

be a presentation of G, where F is a free pro-p-group on d generators.Then, the group G is of cohomological dimension at most 2 if and only ifRab is free as Zp[[G]]-module.

19

As the cohomological dimension of GS′ is at most 2, the module W ab isfree over Zp[[GS′ ]] and so (W ab)H is free as Zp[[GS]]-module.One �nally obtains the exact sequence (Nguyen, [19], theorem 1.4):

0 −→ Zp[[GS]]s −→ Rab −→ X −→ 0.

Suppose �rst that cd(GS) ≤ 2. Then by Brumer's lemma, 1) holds.If now X is Zp[[GS]]-free, thenRab is Zp[[GS]]-free and by Brumer's lemma,one has 2). ♦

Lemma 5.4. � If the pro-p-group GS is of cohomological dimension atmost 2, then H1(GS,X ∗) = 1.

Proof. � As one assumes Leopoldt's conjecture, thanks to Proposi-tion 5.2 and the fact that GS′ is of dimension at most 2, it comesH i(H, Qp/Zp) = 0, for i > 1. Then the Hochschild-Serre spectralesequence H i(GS′/H, Hj(H, Qp/Zp)) =⇒ H i+j(GS′ , Qp/Zp) allow us toobtain the following long exact sequence

H1(GS, Qp/Zp)� � // H1(GS′ , Qp/Zp) // H1(H, Qp/Zp)

GS

��H1(GS, H1(H, Qp/Zp))

����

H2(GS′ , Qp/Zp)oo H2(GS, Qp/Zp)oo

H3(GS, Qp/Zp) 1

and the result follows. ♦

One obtains the main result of this section.

Theorem 5.5. � The Zp[[GS]]-module X is free if and only if:

(i) cd(GS) ≤ 2 and(ii) the natural morphism Tor(GS′

ab) → GSab, where Tor(GS′

ab) is thetorsion part of GS′

ab, is injective.

Moreover, when the Zp[[GS]]-module X is free, its dimension is

r2 + χ2(GS).

Proof. � Since X is �nitely generated as Zp[[GS]]-module, let

1 −→ N −→ F −→ X −→ 1,

20 CHRISTIAN MAIRE

be a minimal presentation of X , where F = Zp[[GS]]r. Hence, X is free ifand only if, N = 1, i.e. if and only if NGS

= 1 (by Nakayama's lemma).Passing to the GS-homology of the previous sequence, one obtains:

1 // H1(GS,X ) // NGS// Zr

p// XGS

// 1

So �rst suppose that X is a free Zp[[GS]]-module. Thanks to Proposition5.3, one has only to prove (ii). By the previous exact sequence, XGS

' Zrp.

The homology version of the spectral sequence H i(GS′/H, Hj(H, Qp/Zp)) =⇒H i+j(GS′ , Qp/Zp) gives

H2(GS, Zp)� � // XGS

// GabS′

// // GabS ,(1)

and then the natural morphism GS′ab → GS

ab restricted to the torsionpart is injective.

Now suppose �rst that cd(GS) ≤ 2. The group H2(GS, Qp/Zp) is p-divisible of �nite corank and then H2(GS, Zp) is Zp-free. If moreover thenatural morphism GS′

ab → GSab restricted to the torsion part is onto,

then, thanks to (1), XGSis Zp-free. Then, the map ϕ : Zr

p → XGSis an

isomorphism. Moreover, by lemma 5.4, H1(GS,X ) = 1, NGS= 1 and

then X is Zp[[GS]]-free.

When X is free of rank r, then H1(H, Qp/Zp)GS' Zr

p. Hence taking theZp-rank of the exact sequence of lemma 5.4, one obtains

rkZpGabS − rkZpGab

S′ + r − rkZpH2(GS, Zp) = 0

As cd(GS) ≤ 2, then H2(GS, Qp/Zp) is Zp-free and thus one has

rkZpH2(GS, Zp)− rkZpGabS = χ2(GS)− 1.

Then, by assuming Leopoldt's conjecture (rkZpGabS′ = r2 + 1), one �nally

obtains:

r = r2 + χ2(GS).

♦

5.3. Examples. � Before giving some examples, one has to computethe kernel of the map: Tor(Gab

S′ ) → GabS . To do this, we use a result of

Gras [7], Theorem 2.6, Chapter III.

21

Proposition 5.6. � Assume S ∩ Sp = ∅. The map Tor(GabS′ ) → Gab

S isinjective if and only if

#Tor(⊕v|pU

1v ⊕v∈S k×v /ϕS′(Zp ⊗ EK)

)= #

(⊕v∈Sk×v /ϕS(Zp ⊗ EK)

).

Proof. � Denote by I the subgroup generated by the inertia groups ofall the places in Sp. Then: 1 −→ I −→ Gab

S′ −→ GabS −→ 1. To show

that the map Tor(GabS′ ) → Gab

S is injective is equivalent to showing thatI ∩ Tor(Gab

S′ ) = 1. As S ∩ Sp = ∅, the maximal p-extension KabS of K

unrami�ed outside p is �nite, contains the p-Hilbert Class �eld KH of Kand moreover

KabS ∩ K̃ = KH ∩ K̃,

where K̃ is the compositum of all Zp-extension of K.

K̃

Tor(GabS′ )

KabS′

I

KH KabS

K

Here KabS′ is the maximal pro-p-extension of K, unrami�ed outside S ′. Of

course: GabS = Gal(Kab

S /K) and GabS′ = Gal(Kab

S′/K).

Hence:I ∩ Tor(Gab

S′ ) = Tor(I) ∩ Tor(GabS′ )

= [KabS′ : Kab

S K̃]

= [KabS′ : KHK̃]/[Kab

S : KH ]

By Theorem 2.6, chapter of [7], the �rst index [KabS′ : KHK̃] is exactly

#Tor(⊕v|pU

1v ⊕v∈S k×v /ϕS′(Zp ⊗ EK)

), and the second index is well-

konwn: it compares the order of a ray class group to the order of thep-class group. Here it is: # (⊕v∈Sk×v /ϕS(Zp ⊗ EK)). ♦

Corollary 5.7. � Let us conserve the notations of section 5. Let S bea �nite set of places of K, S ∩ Sp = ∅, and such that the cohomologicaldimension of GS is 2. Put X = Hab, where H = Gal(KS′/KS). Then, inthe two following situations X is Zp[[GS]]-free.

22 CHRISTIAN MAIRE

(i) K = Q.(ii) K/Q is an imaginary quadratic �eld, di�erent from Q(j) for p = 3.

Proof. � In these situations the group Zp⊗EK is trivial. Hence, thanksto Proposition 5.6 the result is clear if we note that for all v|p, U1

v iswithout torsion. ♦

Example 5.8. � We recall that we assume Lepoldt's conjecture.(i) The examples of Labute (and of Schmidt) allow us to illustrate theprevious corollary. Typically, let's take p = 3 and S = {7, 19, 61, 163}.Then Labute proved that for this case, cd(GS) = 2. Put S ′ = S ∪ {3},H = Gal(KS′/KS) and X := Hab. Then X is a Zp[[GS]]-free module ofrank 1.(ii) Thanks to the work of Vögel [27], one can give some examples withbase �eld K an imaginary quadratic �eld. Let K = Q(i). Take p = 3and let S be the set of all places above 229 and 241. Then Vögel provedin this case that cd(GS) = 2. Put S ′ = S ∪ {3}, H = Gal(KS′/KS) andX := Hab. Then X is a Zp[[GS]]-free module of rank 2.

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24 CHRISTIAN MAIRE

March 23, 2007

Christian Maire, Équipe Émile Picard, Institut de Mathématiques de Toulouse,Université de Toulouse • E-mail : maire@univ-tlse2.fr