arX

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0203

181v

2 [m

ath.

NT

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Oct

200

2

ON A GEOMETRIC DESCRIPTION OF Gal(Qp/Qp),

AND A p-ADIC AVATAR OF GT

YVES ANDRE

CONTENTS

1. Introduction 22. Geometric fundamental groups and Galois actions 53. GQ, GR and geometric fundamental groups 74. The tempered fundamental group of ap-adic manifold 125. A fundamental exact sequence 176. Out πtemp

1 for a curve 197. GQ, GQp

and geometric fundamental groups 228. The local Grothendieck-Teichmuller group atp 269. GQp

is the automorphism group of the temperedπ1-functor 32References 34

ABSTRACT. We develop a p-adic version of the so-called Grothendieck-

Teichmuller theory (which studies Gal(Q/Q) by means of its action on

profinite braid groups or mapping class groups). For every place v of Q, we

give some geometrico-combinatorial descriptions of the local Galois group

Gal(Qv/Qv) inside Gal(Q/Q). We also show that Gal(Qp/Qp) is the

automorphism group of an appropriate π1-functor in p-adic geometry.

Classification: 11R32, 14H30, 14G22, 14G20, 20F28, 20F36.

1

2 YVES ANDRE

1. INTRODUCTION

1.1. Profinite groups which are the absolute Galois groupGk = Gal(k/k)of some fieldk have been the object of extensive study, especially in the casewherek is a number field. In that case,Gk has more structure: it comesequipped with a constellation of closed subgroupsGkv = Gal(kv/kv) at-tached to the placesv of k, and fitting together in an arithmetically relevantway.

The problem then arises todescribe these local Galois groupsGkv in Gk.

1.2. A purely group-theoretic approach to this problem has been found byArtin (for archimedeanv) and Neukirch (for non-archimedeanv): namely,the subgroupsGkv ⊂ Gk for v|k real are exactly the subgroups of ordertwo; the subgroupsGkv for v non-archimedean are exactly the closed sub-groups which have, abstractly, the algebro-topological structure of the abso-lute Galois group of a local field1, and are maximal for this property,cf. [27,§XII.1]. This is however more a characterization than a description of thelocal Galois groups - rather, of the set of local Galois groupsGkv attachedto a fixed placev|k of k.

1.3. In this paper, we examine the problem from a completely differentviewpoint, aiming at ageometricsolution (with combinatorial flavour). Ourapproach is inspired by Grothendieck’s leitmotiv of studying Gk via itsouter action on the “geometric” algebraic fundamental group πalg

1 (Xk) ofsmooth geometrically connected algebraic varietiesX defined overk . Infact, the problem of describing the local Galois groups inGk has been raisedexplicitly by Grothendieck in the context of geometric actions on funda-mental groups and his dream of anabelian geometry ([11], note4)2.

1.4. In the simplest non-trivial case (k = Q andX = P1Q \ {0, 1,∞} ),

it turns out that the outer action ofGQ on the profinite groupπalg1 (XQ) is

faithful (Belyi). An embeddingQ → C being fixed,πalg1 (XQ) is the profi-

nite completion of the usual ‘transcendental’ fundamentalgroup πtop1 (X

an

)of the complex-analytic manifoldX

an

(a discrete free group of rank two).

1which is “known”,cf. [27, §VII.5].2we became aware of these lines on having another look at [11] just before completing

this work: “Parmi les points cruciaux de ce dictionnaire [anabelien], je prevois [...] unedescription des sous-groupes d’inertie deΓ [ = GQ], par ou s’amorce le passage de lacaracteristique zero a la caracteristiquep > 0, et a l’anneau absoluZ.”

3

One can then recoverGR inside GQ as the intersection ofGQ andOut πtop

1 (Xan

) in Out πalg1 (XQ) , (cf. 3.3.1, and 3.3.2 for a more general

statement).

1.5. We shall give a similar description of the local Galois groupGQp, em-

bedded intoGQ via a fixed embeddingQ → Cp =ˆQp. The required ingre-

dient is a fundamental group for rigid-analytic “p-adic manifolds” playingthe role ofπtop

1 . In particular, the profinite completion of such a fundamen-tal group should coincide with the algebraic fundamental groupπalg

1 , in thecase of an algebraicp-adic manifold.

Such a theory has been set up in [1] (for completely differentpurposes),and will be outlined below (§4). Let us just say here that its corner-stone isthe notion of tempered3 etale covering, which generalizes in a minimal wayboth finite etale coverings and infinite topological coverings.

The correspondingtempered fundamental groupsπtemp1 encapsulate com-

binatorial information about the bad reduction of all finiteetale coverings ofthe base. Their topology is a little complicated (for instance, the temperedfundamental group ofP1

Cp\{0, 1,∞} is complete but not locally compact,

hence is neither discrete nor profinite); but in dimension one, they have asuggestive combinatorial description as inverse limit of asequence of fun-damental groups of certain finite graphs of groups, which could be consid-ered asnon-archimedean analogues of Grothendieck’s ‘dessins d’enfants’.

1.6. For any smooth geometrically connected algebraic variety X overQp, there is a canonical outer action ofGQp

on πtemp1 (X

an

), whereXan

denotes the rigid-analytic manifold attached toXCp, cf. prop. 5.1.1.

Coming back to the case ofX = P1Q \ {0, 1,∞} , one version of our

main result is thatone can recoverGQpinside GQ as the intersection of

GQ and Out πtemp1 (X

an

) in Out πalg1 (XQ) , cf. thm. 7.2.1 (a more

general statement is given in 7.2.3).We also study the structure ofOut πtemp

1 (Xan

) for algebraic curvesX.We show that this group is, likeπtemp

1 (Xan

) itself, the inverse limit of asequence of finitely generated discrete groups ( 6.1.4).

1.7. According to the philosophy of the Grothendieck-Teichmuller theory,P1

Q\{0, 1,∞} should be seen as the special casen = 4 of the moduli spaceM0,r of curves of genus0 with r ordered marked points, and one shouldstudy as well the Galois action on the geometric fundamentalgroup of thesemoduli spaces (and also in higher genus). The main player here is Drinfeld’s

3following the referee’s advice, we have changed our previous terminology ‘temperate’to ‘tempered’

4 YVES ANDRE

Grothendieck-Teichmuller groupGT , a kind of “geometric upper bound”for GQ [8][14].

We introduce a closed subgroupGT p of the profinite groupGT , thelocalGrothendieck-Teichmuller group atp, defined in terms of the rigid-analyticmanifolds attached to the moduli spacesM0,r overCp (8.6.3). This groupis a kind of “ geometric upper bound” forGQp

: we prove thatGQpis the

intersection ofGQ and GT p in GT (thm. 8.7.1).This sheds some new light on the longstanding problem of how close is

GT to GQ (cf. 8.7.2).

1.8. Another attempt to describe the absolute Galois group of a numberfield k as the full automorphism group of some geometric structure consistsin looking at sufficiently manyk-varieties at a time and at the geometricfundamental group as a functor. Geometric algebraic fundamental groupsgive rise to a functorπalg

kfrom the categoryVk of smooth geometrically

connected varieties over a number fieldk to the categoryT of topologicalgroups up to inner automorphisms. Pop has shown thatGk = Aut πalg

k([29], unpublished).

On the other hand, for anyp-adic placev of k, there is a functorπtempCp

:Vk → T given by the tempered fundamental groups of associatedp-adicmanifolds.Our final result is thatGkv = Aut πtemp

Cp(thm. 9.2.2).

Thus in some sense, the arithmetic of finite extensions ofQp is embodiedin analytic geometry overCp.

5

2. GEOMETRIC FUNDAMENTAL GROUPS ANDGALOIS ACTIONS

2.1. LetX be a smooth geometrically connected algebraic variety overover a fieldk, endowed with a geometric pointx. Grothendieck’s alge-braic fundamental groupπalg

1 (X, x) is the profinite group which classifiesall finite etale (pointed) coverings of(X, x):

{finite etale coverings ofX} ∼ {finite πalg1 (X, x)-sets}.

This group depends onx only up to inner automorphism.Let k be a separable closure ofk, and letGk = Gal(k/k) stand for

the absolute Galois group. The groupπalg1 (Xk, x) is sometimes called the

geometric fundamental group ofX (pointed atx). There is a functorialexact sequence

{1} → πalg1 (Xk, x)→ πalg

1 (X, x)→ Gk → {1},

which splits canonically ifx comes from ak-rational point ofX. Whencea Galois action

Gkρ→ Out πalg

1 (Xk),

which lifts to an action

Gkρx→ Aut πalg

1 (Xk, x)

if x comes from ak-rational point ofX (hereOut = Aut/Inn denotes asusual the group of outer automorphisms (inρ, we drop the base point fromthe notation since it is irrelevant).

2.2. We assume henceforth thatchar k = 0. Under this assumption, itis known that the profinite groupsπalg

1 (Xk, x) are finitely generated, whichimplies thatAut πalg

1 (Xk, x) andOut πalg1 (Xk) are also finitely generated

(hence metrizable4) profinite groups. More precisely, there are only finitelymany open normal subgroups ofπalg

1 (Xk, x) of index dividing5 n, hencetheir intersectionUn is a characteristic6 open subgroup ofπalg

1 (Xk, x) (henceis preserved byGk). It follows thatUn = πalg

1 (Xn, xn) for a well-definedfinite Galois etale (pointed) covering(Xn, xn) → (Xk, x) with Xn geo-metrically connected; in fact, the tower of characteristicsubgroupsUn is

4recall that a profinite group is metrizable if and only if it has countably many opensubgroups, or, equivalently, if it is a countable inverse limit of finite groups, [32,§ 4.1.3][6, § IX.2.8], or else, if and only if it is a closed subgroup of a quotient of the profinite freegroup on two generatorsF2 [32, § 4.1.6] (a profinite variant of the fact that any countablediscrete group is a subgroup of a quotient ofF2 ); because of these equivalences, onesometimes says ‘separable’ instead of ‘metrizable’

5another possible choice: replace “dividingn” by “less or equal ton”; our preferenceis justifiede.g.by 4.7 below

6i.e. stable under every automorphism of the profinite groupπalg1

(Xk, x)

6 YVES ANDRE

the tower of geometric fundamental groups of a ‘tower’ of (pointed) finiteGalois etale coverings

. . . → Xn′ → Xn → . . . → Xk , n|n′.xn′ 7→ xn 7→ . . . 7→ x

We denote byΓn = πalg1 (Xk, x)/Un the Galois group of the coveringXn/Xk.

Notice that the above ‘tower’ is canonical and functorial inX, and the sameis true for the corresponding ‘tower’ of finite groups

. . .→ Γn′ → Γn → . . .→ {1} , n|n′.

(Moreover, its formation is compatible with extension of algebraically closedfieldsk → k′). One then has (cf. [32, § 3.4. ex.6] for the case ofAut)

πalg1 (Xk, x) = lim

←−Γn,

Aut πalg1 (Xk, x) = lim

←−AutΓn,

Out πalg1 (Xk) = lim

←−OutΓn.

The topology of the group of (outer) automorphisms defined bythe inverselimit is called thetopology of congruence subgroups; thus, the group of(outer) automorphisms ofπalg

1 (Xk, x) is closed for the (profinite) topologyof congruence subgroups.

2.2.1. Observation.7 For any normal closed subgroupU of πalg1 (Xk) con-

tained inUn, the quotientUn/U is a characteristic subgroup ofπalg1 (Xk)/U .

Indeed, by definition, the homomorphismsπalg1 (Xk) → πalg

1 (Xk)/U →Γn induce bijections between the sets of finite quotients of order dividingnof each of these groups.

In particular, the kernel ofΓn′ → Γn is characteristic, for any multiplen′

of n.

2.3. In this paper, ‘curve’ is an abbreviation for ‘smooth geometricallyconnected algebraic variety of dimension one’. A curveX overk is calledhyperbolicor anabelianif its geometric fundamental group is non-abelian.If X is affine, this just means thatXk is not isomorphic to the projectiveline minus one or two points.

The following result will be of constant use throughout the present paper:

2.3.1. Theorem (Belyi, Matsumoto[21]). Assume thatk is a number field.Then the Galois action

Gkρ→ Out πalg

1 (Xk)

is faithful for any hyperbolic affine curveX. �

7I am endebted to one of the referees for this observation

7

This relies in turn on Belyi’s theorem (used several times inthe sequel)according to which any curve defined overk admits a rational functionwhich is ramified only above0, 1,∞.

3. GQ, GR AND GEOMETRIC FUNDAMENTAL GROUPS

3.1. Nowk is a number field, and we fix an embeddingι : k → C. Forany smooth geometrically connected algebraick-varietyX with a geome-tric pointx, one has canonical isomorphisms

πalg1 (Xk, x) = πalg

1 (XC, x) =πtop

1 (Xan

C , x),

the latter group being the profinite completion of the usual topological fun-damental groupπtop

1 (Xan

, x) which classifies topological (= etale) cove-rings8 of the complex-analytic manifold attached toXC.

If ι(k) ⊂ R, there is a Galois action

GR = Z/2Zρ∞−→ Out πtop

1 (Xan

),

which lifts to an action

GR

ρ∞,x

−→ Aut πtop1 (X

an

, x)

if x comes from a real point ofX. This action is compatible with the (outer)action ofGk on the profinite completionπalg

1 (Xk, x).

3.2. Assume thatX is anaffine curve. Thenπtop1 (X

an

, x) is a free groupof finite rank, which implies that it is residually finite,i.e. embeds into itsprofinite completion. It follows that the following naturalhomomorphismsare injective:

πtop1 (X

an

, x) → πalg1 (Xk, x),

Aut πtop1 (X

an

, x) → Aut πalg1 (Xk, x),

Out πtop1 (X

an

) → Out πalg1 (Xk).

For the injectivity of the third homomorphism, one uses in addition thefollowing

3.2.1. Lemma. A free groupF of finite rank> 1 is its own normalizer inits profinite completionF .

For lack of reference9, we indicate a proof. Leta belong to the normalizerof F in F , and letx1, . . . , xr be a basis ofF . A classical result of Stebesays thatF is conjugacy-separated: ifx, y ∈ F are conjugate in every finite

8not necessarily finite9after submission of this paper, the reference [9, thm. 2] waspointed out to the author

by A. Tamagawa and one of the referees

8 YVES ANDRE

quotient ofF , they are conjugate inF [20, prop. 4.9]. In particular, thereare elementsai ∈ F such thataxia

−1 = aixia−1i . Hencea−1

i a belongsto the centralizer ofxi in F , which isxZ

i . Therefore, there are elementszi ∈ Z such thata = aix

zii . For i = 1, 2, this impliesxz1

1 x−z22 ∈ F , whence

z1, z2 ∈ Z, anda ∈ F . �

We denote byOutπtop1 (X

an

) the completion ofOutπtop1 (X

an

) with respectto the topology of congruence subgroups,i.e. the closure ofOut πtop

1 (Xan

)

in Out πalg1 (Xk):

Out πtop1 (X

an

) = lim←−

Im [Out πtop1 (X

an

)→ OutΓn]

(this limit is clearly a quotient of the profinite completionof Out πtop1 (X

an

)

and embeds intolim←−

OutΓn = Out πalg1 (Xk) ).

Since we shall have to compare repeatedly the (outer) automorphismgroup of a topological group with the (outer) automorphism group of itsprofinite completion10, we include here the

3.2.2. Sorite. a) Let φ : G → H be a surjective homomorphism of topo-logical groups, and letφ : G → H be the induced homomorphism of theirprofinite completions.Let τ be an automorphism11 of G, and letτ be the corresponding automor-phism ofG. If τ induces an automorphism ofH (via φ), thenτ induces anautomorphism ofH/ ker(H → H).b) LetF be an open subgroup of finite index of a topological groupG. We

assume thatG embeds into its profinite completionG. Letτ be as before anautomorphism ofG. If τ preserves the imageF ofF in G, thenτ preservesF .

Proof. a) ReplacingG andH by G/ ker(G → G) andH/ ker(H → H)respectively, one may assume that they embed into their profinite comple-tion. LetAut(G, kerφ) (resp.Aut(G, ker φ)) denote the group of automor-phisms ofG (resp.G) which preserve the kernel ofφ (resp. φ). The soritefollows from the equality

Aut(G, kerφ) = Aut(G, ker φ) ∩ AutG,

b) One hasF = F ∩G, whenceAut(G,F ) = Aut(G, F ) ∩ AutG.

�

10the profinite completion of a topological group is the inverse limit of its finite quotients(i.e. quotients byopensubgroups of finite index)

11in the category of topological groups, of course

9

3.3. For any hyperbolic affine curveX over a number fieldk (and a fixedcomplex embeddingι of k), we have encountered two closed subgroups ofOut πalg

1 (Xk): Out πtop1 (X

an

) andGk. We shall study their intersection,starting with the case ofP1 \ {0, 1,∞}.

3.3.1. Theorem. If X = P1k \ {0, 1,∞} , then

Out πtop1 (X

an

) ∩ Gk = Out πtop1 (X

an

) ∩ Gk =

{GR if ι(k) ⊂ R

{1} otherwise.

Proof. We already know thatGR ⊂ Out πtop1 (X

an

) ∩ Gk if ι(k) ⊂ R . Itremains to show thatOut πtop

1 (Xan

) ∩ Gk ⊂ GR (from which it followsthat the intersection is trivial ifι(k) 6⊂ R ). We may and shall assume thatk = Q.

Sinceπtop1 (X

an

) is a free group of rank two,Out πtop1 (X

an

) is faithfullyrepresented on the abelianization(πtop

1 (Xan

))ab ∼= Z2 according to a classi-cal theorem of Nielsen (cf. [20, § I.4.5]). HenceOut πtop

1 (Xan

) ∼= GL2(Z)

and Outπtop1 (X

an

) is a quotient of the profinite completionGL2(Z) andcommutes with the subgroup{±id} ⊂ GL2(Z) ∼= Out πtop

1 (Xan

). Thissubgroup lies inOut πtop

1 (Xan

) ∩ GQ, and coincides withGR. It followsthat Outπtop

1 (Xan

) ∩GQ commutes withGR. But it is well-known thatGR is its own centralizer inGQ (cf. [27, § 12.1.4]). �

For more general affine hyperbolick-curvesX, we have:

3.3.2. Theorem. Assume that some finite etale covering ofXk admits anon-constant rational function which omits at least three values. Assumealso that ι(k) ⊂ R . Then

Out πtop1 (X

an

) ∩ Gk = GR.

A fortiori, if the geometric pointx is k-rational, then the subgroup ofGk

which stabilizesπtop1 (X

an

, x) ⊂ πalg1 (Xk, x) is GR.

(Note that the assertion about ‘Aut’ is less precise that thestatement about‘Out’.)

10 YVES ANDRE

Proof. 12 From the assumption, there is a finite extensionk′ of k in C, afinite etale coveringY → Xk′ and a dominant morphism

Yϕ→ Z = P1

k′ \ {0, 1,∞}.

(We may also assume thatX, Y ,Z are endowed with compatiblek′-rationalgeometric pointsx, y, z respectively). The result will follow from 3.3.1 bythe following devissage:

3.3.3. Lemma. The statement

(∗)X,k Outπtop1 (X

an

) ∩ Gk ⊂ GR

satisfiesa) (∗)X,k′ ⇒ (∗)X,k if k′ is a finite extension ofk in C,b) (∗)Z,k ⇒ (∗)Y,k if there is a non-constant morphismf : Y → Z,c) (∗)Y,k ⇒ (∗)X,k if Y is a finiteetale covering ofX,

providedι(k) ⊂ R.

Proof. a)Out πtop1 (X

an

)∩ Gk is finite by(∗)X,k′, and containsGR becauseι(k) ⊂ R). We conclude by the following

3.3.4. Sublemma. (Artin, cf. [27, § 12.1.7]) If H is a closed subgroup ofGk containingGR as a subgroup of finite index, thenH = GR. �

For b), it is more convenient to deal withAut instead ofOut ; we passfrom the latter to the former using the following consequence of 2.3.1 (re-placing the assumption thatι(k) ⊂ R by the assumption thatx isk-rational,as we may usinga) and its trivial converse):

3.3.5. Sublemma. The natural homomorphism

HX := (Inn πalg1 (Xk, x).Aut π

top1 (X

an

, x)) ∩Gk → Out πtop1 (X

an

) ∩Gk

is an isomorphism (the first intersection is taken inAut πalg1 (Xk, x) , the

second one inOut πalg1 (Xk) ). �

12after submission of this paper, A. Tamagawa (and one of the referees) pointed tome that this result can also be deduced from [23, thm. 1.1, rmk. 2.1], even without theassumption that some finite etale covering ofXk admits a non-constant rational functionwhich omits at least three values: indeed, letΓg,r be the relevant mapping class group,viewed as a subgroup of index two in the groupOut∗πtop

1(X

an

) of outer automorphismspreserving the conjugacy class of the local monodromy at each puncture, and letΓg,r. Theassertion follows from two facts: a)Gk \ {1} does not intersect (the closure of) the imageof Γg,r in Out πalg

1(Xk) [23, thm. 1.1]; b)Out πtop

1(X

an

) ∩ Gk ⊂ Out∗πtop1

(Xan

)

11

b) The dominant morphismf induces a homomorphism

πtop1 (Y

an

, y)f∗→ πtop

1 (Zan

, z)

whose image is of finite index. We set, for short,

G = πtop1 (Z

an

, z), F = f∗πtop1 (Y

an

, y),

and denote byAut(G, F ) the group of automorphisms ofG preservingF .This homomorphismf∗ induces in turn a homomorphism

HY → (Inn G.Aut(G, F )) ∩Gk

which is injective (because of the intersection withGk in both terms). By3.3.4, it thus suffices to show that(Inn G.Aut(G, F )) ∩Gk is contained inHZ = (Inn G.AutG) ∩Gk. We shall prove the stronger statement

Aut(G, F ) ⊂ AutG.

To this aim, it will be convenient to replaceF by a subgroupF ′ which isstill stable underAut(G, F ) but has the advantage of beingnormal(and offinite index) inG. We take

F ′ =⋂

g,γ

Inn(γ(g))(F ) =⋂

γ

⋂

g

γ(Inn(g)F ) =⋂

γ

γ(⋂

g

Inn(g)F ),

where the intersection runs overg ∈ G, γ ∈ Aut(G, F ). This is clearly anormal subgroup ofG, stable underAut(G, F ). To see that it is of finiteindex, note that eachγ(

⋂g Inn(g)F ) is a subgroup of the same (finite)

index inG, and there are only finitely many such subgroups sinceG is offinitely type (cf. [32, § 4.1.2]).

It thus suffices to show thatAut(G, F ′) ⊂ AutG, or equivalently, thatthe subgroupF ′′ of G generated by elements of the formγ(g), g ∈ G, γ ∈

Aut(G, F ′) is nothing butG. Note thatF ′ is normal inF ′′. There are exactsequences{1} → F ′ → G → Γ → {1}, {1} → F ′ → G → Γ → {1}with Γ finite, whence an exact sequence

{1} → F ′′ ∩ F ′ → F ′′ → Γ→ {1}.

But F ′ is normal in the subgroupF ′′ ∩ F ′ of its profinite completionF ′.SinceF ′ is free of rank> 1, we conclude from 3.2.1 thatF ′′∩F ′ = F ′ ⊂ G,henceG = F ′′ sinceG surjects ontoG/F ′ = Γ.

c) By the previous step, we may replaceY/X by a bigger finite etalecovering, and thus assume thatπalg

1 (Yk, y) is a characteristic subgroup of

12 YVES ANDRE

πalg1 (Xk, x) with finite quotient groupΓ (the same property then holds for

the topological fundamental groups). The canonical homomorphism

Aut πalg1 (Xk, x)→ Aut πalg

1 (Yk, y)

has finite kernel, and induces a homomorphism

Out πalg1 (Xk)→ (Out πalg

1 (Yk))/Γ

which also has finite kernel. HenceOut πtop1 (Xk)∩Gk is finite ifOut πtop

1 (Yk)∩Gk is, and one concludes by 3.3.4. �

Remarks.1) Some heuristic arguments of Baire type on moduli spaces forcurves with marked points seem to indicate that not every affine hyperboliccurve satisfies the assumption of 3.3.2 (it would be nice to have a completeproof)13.

2) The assumption of 3.3.2 is fulfilled forX = an elliptic curveE minusone point. Indeed, by translation, one may assume that this point is theorigin O. ThenE minusE[2] (the 2-torsion points) is an abelian etalecovering ofE\{O}, and the quotient ofE\E[2] by the involutionP 7→ −Pis isomorphic toP1 \ {0, 1,∞, λ}.On the other hand, the argument of 3.3.1 applies directly toX = E \ {O}sinceπtop

1 (Xan

) ∼= F2. Note however that whereasOut πtop1 (X

an

) is faith-fully represented on the abelianization(πtop

1 (Xan

))ab = H1(Xan

,Z), itdoes not amount to the same to take the intersection with the image ofGk

in Out πalg1 (Xk) or in Out (πalg

1 (Xk))ab = GL (Het

1 (Xk, Z)) (the latter ismuch bigger in general).

3) The statements of 3.3.1 and 3.3.2 do not change if one replacesι byits complex conjugate; hence it is enough to fix an archimedean place ofk(instead ofι).

4. THE TEMPERED FUNDAMENTAL GROUP OF Ap-ADIC MANIFOLD

4.1. We fix a prime numberp and denote byCp as usual the completionof a fixed algebraic closureQp of Qp. LetK be a complete subfield ofCp,and letK be its algebraic closure inCp. We notice thatGK = Gal(K/K)is a metrizable profinite group (and even a finitely generatedprofinite groupif K = Cp or a finite extension ofQp).

Among the several approaches to analytic geometry overK, we havefound Berkovich’s one [2] most convenient for a discussion of unramifiedcoverings and fundamental groups; indeed, Berkovich’s spaces are locallyringed spaces in the usual sense, as opposed to Tate’s rigid spaces which areGrothendieck’s sites (Berkovich’s spaces contain ‘more points’ than rigid

13I am grateful to L. Ramero and P. Colmez for interesting discussions on this topic

13

spaces, but unlike the passage from classical algebraic varieties to schemes,all Berkovich non-classical points are closed).

In the sequel, by(analytic)K-manifold - orp-adic manifoldif K = Cp -we shall mean asmooth paracompact strictlyK-analytic spacein the senseof Berkovich [3]14. Any smooth algebraicK-variety X gives rise to ananalyticK-manifoldXan, its analytification.

According to Berkovich [4],K-manifolds are locally compact, locallyarcwise connected, andlocally contractible, hence subject to the usual the-ory of universal coverings and topological fundamental groups.

By geometric point or base point of aK-manifold, we mean a point de-fined over some algebraically closed complete extension ofK.

4.2. In the sequel, we shall have to consider etale coveringsof possiblyinfinite degree. Let us recall a definition [15][1] which applies both to thecomplex and to the non-archimedean situations:Y → X is anetale cover-ing (resp. a topological covering) if, locally on the baseX , it is a disjointunion of finite etale covering maps (resp.isomorphisms).

For complex manifolds, any etale covering gives rise to a topologicalcovering, and conversely.

On the other hand, for non-archimedeanK-manifolds, topological cove-rings give rise to etale coverings [15,§2.6], but not conversely: for instance,the ‘Kummer covering’

P1K \ {0,∞} → P1

K \ {0,∞}, z 7→ zn, n > 1

is a finite etale covering, but not a local homeomorphism of analytic K-manifolds (in fact the cardinality of the fiber, drops to1 at some non-classical points; in the more traditional rigid viewpoint,this is because thereis noadmissibleopen cover over which the Kummer covering splits).

Besides the topological fundamental groupπtop1 (X , x), which classifies

as usual the topological coverings of the pointed connectedK-manifold(X , x), there is the etale fundamental groupπet

1 (X , x), introduced by deJong [15], which classifies the (possibly infinite) etale coverings ofX . Infactπtop

1 (X , x) is a discrete quotient of the topological groupπet1 (X , x).

But neither group is a close analogue of the fundamental group of a com-plex manifold:• πtop

1 (X , x) is “too small”: for instance,πtop1 (P1

Cp\{0, 1,∞}, x) = {1}.

• πet1 (X , x) is “too big”: for instance,πet

1 (P1Cp, x) is non-trivial (in fact,

it is a huge non-abelian group [15,§ 7]).

14in [1], we used this term in a less restrictive sense, allowing a non-empty boundary

14 YVES ANDRE

4.3. In order to remedy this, we have introduced in [1] an intermediatefundamental groupπtemp

1 (X , x), the tempered fundamental group, whichclassifies all tempered (pointed) coverings of(X , x):

{∐

tempered coverings ofX} ∼ {discreteπtemp1 (X , x)-sets}.

By definition, an etale coveringY → X is said to betemperedif there is acommutative diagram of etale coverings

Zւ ց

T Yց ւX

whereZ → T is a (possibly infinite) topological covering, andT → X isa finite etale covering.

In some sense, this is the ‘minimal’ theory of etale coverings whichtakesinto account both the topological coverings and the finite etale coverings.

We refer to [1,§§ 1, 2] for a discussion of these coverings and the precisedefinition ofπtemp

1 (X , x) as a separated prodiscrete topological group. Thistopological group depends on the base pointx only up to inner automor-phism [1,§ 1.4.4].

Here, we shall content ourselves with the following useful ‘criterion’:

4.3.1. Proposition. [1, § 2.1.9]Let g : Y → X be a Galois etale coveringwith (discrete) groupG. If G is torsion-free (resp. virtually torsion-free15),theng is a topological covering (resp. a tempered covering). �

Remarks.1) Because of this proposition, infinite covering maps are ofteneasier to handle than finite ones, when the point is to check whether theyare local homeomorphisms; this may justify the detour to infinite coveringseven if one is primarily interested in finite ones.

2) We mention in passing that there is a theory of tempered fundamentalgroups based at tangential base points, in dimension one [1,§ 2.2].

4.4. Let us consider the case whenX = Xan

is the analytification of ageometrically connected smooth algebraicK-variety.

4.4.1. Proposition. [1, § 2.1.7] πtemp1 (X

an

, x) is a countable inverse limitof discrete finitely generated groups; it particular, it is apolish group.Its profinite completion is canonically isomorphic toπalg

1 (X, x). �

15recall that a discrete group is said to be virtually torsion-free is it admits a torsion-freesubgroup of finite index

15

(the proof of the second assertion relies on the Gabber-Lutkebohmertversion of Riemann’s existence theorem [19]).

Let us recall that, according to Bourbaki [6,§ IX.6.1], a topological groupis polishif it is metrizable, complete, and countable at infinity (it might notbe locally compact). These form a nice category of topological groups towork with: let {1} → N → G → H → {1} be a sequence of homomor-phisms of topological groups which is exact in the abstract sense. AssumethatN is closed inG. If G is polish, then so areN (and any closed subgroupof G) andG/N [6, § IX.2.8 prop. 12;§ IX.3.1 prop.4 ]), and ifH is alsopolish, then the bijective homomorphismG/N → H is an isomorphism [6,§ IX.5. ex. 28].

4.5. In the caseK = Cp, πtemp1 (X

an

, x) can be described as follows. Letus consider the ‘tower’ of finite Galois etale coverings

. . .→ Xn′ → Xn → . . .→ X , n|n′ , Gal(Xn/X) = Γn,

introduced in 2.2. Let us denote byXn the universal topological coveringof X

an

n . ThenXn is an etale Galois covering ofXan

[1, 2.1.2] with Galoisgroup∆n sitting in a extension

{1} → πtop1 (X

an

n , xn)→ ∆n → Γn → {1},

where the discrete groupπtop1 (X

an

n , xn) is finitely generated [1,§ 1.1.3].Whenn increases (by divisibility), these exact sequences form aninversesystem, which is canonical and functorial inX.

One hasπtemp1 (X

an

, x) = lim←−

∆n

(surjective transition maps), and there is a canonical morphism commutativediagram

{1} → πtemp1 (X

an

n ) → πtemp1 (X

an

) → Γn → {1}↓ ↓ ↓ ↓ ↓{1} → πtop

1 (Xan

n ) → ∆n → Γn → {1}

with surjective vertical maps,cf. [1, § 2.1.5., 2.1.8].On the other hand,in dimension one, πtop

1 (Xan

n ) has a simple combina-torial description as the fundamental group of the (dual) graph of incidenceof the semistable reduction ofXn in characteristicp, cf. [15, § 5.3.], in par-ticular it is a free group (notably,X

an

n is simply connected if it has goodor tree-like reduction). It follows that∆n is virtually free (i.e. admits afree subgroup of finite index), hence residually finite, which implies thatπtemp1 (X

an

, x) itself is residually finite, so that the following natural homo-morphism is injective [1,§ 2.1.6]:

πtemp1 (X

an

, x) → πalg1 (Xk, x).

16 YVES ANDRE

Remark.By the theorem of Karass-Pietrovsky-Solitar [16], each∆n, be-ing virtually free and finitely generated, is the fundamental group of a finitegraph of finite groups. It would be very interesting to exhibit a canonicalgeometric construction of such graphs of groups, which could be consideredasp-adic analogues of Grothendieck’s “dessins d’enfants”. Atpresent, suchconstruction have been proposed only in the special case where the projec-tive completion ofXn is a so-called Mumford curve,i.e. has a maximallydegenerate reduction [13][17], using the tree ofSL2 over a local field16.

4.6. From this description,πtemp1 (X

an

) can be easily computed whenXis a non-hyperbolic curve. For instance,• πtemp

1 (P1Cp\ {0,∞}) ∼= Z,

and ifX = Ej is an elliptic curve with invariantj,• πtemp

1 (Ean

j ) = πalg1 (Ej) ∼= Z× Z if |j|p ≤ 1 (and∆n

∼= (Z/nZ)2),

• πtemp1 (E

an

j ) ∼= Z× Z if |j|p > 1, i.e. if Ej has bad reduction(in this case, one has a commutative diagram

πtop1 (X

an

n ) → ∆n → Γn

↓∼= ↓∼= ↓∼=nZ → Z× (Z/nZ) → (Z/nZ)2 ).

WhenX is hyperbolic, there is no such simple explicit description; ingeneral,πtemp

1 (Xan

, x) is ‘lacunary’ in the sense that its open subgroups offinite index have many infinite discrete quotients. In fact,πtemp

1 (Xan

, x)encapsulates the combinatorial information about the reduction of all finiteetale coverings ofX.• For X = the projective line minusn ≥ 3 points,πtemp

1 (Xan

) is notlocally compact, and depends on the position of the missing points if n ≥ 4[1, § 2.3.12, 4.5.5]).

4.7. The subtle lacunary properties ofπtemp1 , on which the sequel depends,

really belong to the “profinite theory” and are lost if one passes to the max-imal prime-to-p quotient or to the pro-p completion; indeed:

4.7.1. Proposition. If X is a curve with good reduction (ifX is affine, wealso require that there is no confluence of the points at infinity by reduction),the natural homomorphism

πtemp1 (X

an

, x)→ πalg1 (X, x)(p

′) × πalg1 (X, x)(p)

to the product of the maximal prime-to-p quotient and the maximal pro-pquotient ofπalg

1 (X, x), is surjective.

16after submission of this paper and discussion with F. Kato, it seems that he is nowable to construct suchp-adic dessins d’enfants in the general case

17

Proof. Recall thatΓn is the quotient ofπalg1 (X) by the intersection of its

(finitely many) open subgroups of index dividingn. It follows that |Γn|divides a power ofn, and that there are sujective homomorphisms

Γ(p)n →→ Γpm1 , Γpm2 →→ Γ(p)

n ,

wherepm1 (resp. pm2) is the greatest power ofp dividing n (resp. |Γn|).From the equalityπalg

1 (X, x) = lim←−

Γn, we derive that

πalg1 (X, x)(p

′) = lim←−

(p,n)=1

Γn, πalg1 (X, x)(p) = lim

←−m

Γpm.

Sinceπtemp1 (X

an

, x) = lim←−

∆n, it thus suffices to show that ifn is primeto p (resp. is a power ofp), then∆n = Γn, which means thatX

an

n issimply-connected. By construction,Xn is a finite etale Galois coveringwith groupΓn of the curve with good reductionX, and|Γn| is prime top(resp. is a power ofp). A standard descent argument shows thatXn →X is actually defined over a complete discretely valued subfield K ⊂ Cp

with algebraically closed residue field. By Grothendieck’sspecializationtheorem [10] (resp.Raynaud’s specialization theorem [30, thm.1’]),Xn hasgood reduction (resp. tree-like reduction), henceX

an

n is simply-connected.�

Remark. It follows from this thatπtemp1 (P1

Cp

an

\ {0, 1,∞}) maps sur-jectively onto the free pro-nilpotent group of rank two (in contrast, it iswell-known thatπtop

1 (P1C

an

\ {0, 1,∞}) maps injectively into the free pro-nilpotent group of rank two).

5. A FUNDAMENTAL EXACT SEQUENCE

5.1. In this section, we establish an exact sequence relating the ‘arith-metic’ and ‘geometric’ tempered fundamental groups.

As above,K is a complete subfield ofCp. For any analyticK-manifoldX ,XCp

has finitely many connected components [15,§ 2.14], andX is saidto be geometrically connected if there is just one; it amounts to the same torequire that for every finite extensionL/K, XL is connected.

Let (X , x) be a geometrically connectedK-manifold endowed with ageometric point. LetL/K be a finite Galois extension contained inCp.There is a functorial exact sequence of separated prodiscrete groups [1,§2.1.8])

{1} → πtemp1 (XL, x)→ πtemp

1 (X , x)→ Gal(L/K)→ {1},

which splits canonically ifx comes from aK-rational point ofX. Whenthe Galois extensionL/K varies, these exact sequences are compatible in

18 YVES ANDRE

an obvious sense, and on passing to the limit, they provide another exactsequence of separated prodiscrete groups

{1} → lim←−L

πtemp1 (XL, x)→ πtemp

1 (X , x)→ GK → {1}

(the surjectivity ofπtemp1 (X , x) → GK can be proved exactly as in [15,

§2.12]).

5.1.1. Proposition. The canonical homomorphism

φ : πtemp1 (XCp

, x)→ lim←−L

πtemp1 (XL, x)

is an isomorphism. Therefore there is a functorial exact sequence

{1} → πtemp1 (XCp

, x)→ πtemp1 (X , x)→ GK → {1},

whence a Galois action

GKρ→ Out πtemp

1 (XCp),

which lifts to an action

GKρx→ Aut πtemp

1 (XCp, x)

if x comes from aK-rational point ofX.

Proof. We shall use the following

5.1.2. Lemma. Letφ : G→ H be a (continuous) homomorphism of sepa-rated prodiscrete groups. Letφ∗ be the induced functor (in the oppositedirection) between discreteH- andG-sets. Ifφ∗ is essentially surjective(resp. an equivalence), thenφ is injective (resp. an isomorphism).

cf. [1, § 1.4.9]. �

In order to prove the proposition, it thus suffices to show that φ∗ isan equivalence. It is clearly fully faithful, and the essential surjectivityamounts to saying that any tempered covering ofXCp

is defined over somefinite extension ofK. This is clear for finite etale coverings, and it sufficesto check that the universal topological coveringYCp

of any finite coveringYCp

of XCpis defined over some finite extension ofK.

Let us first record the following easy

5.1.3. Sublemma. LetY be a geometrically connectedK-manifold.i) For any finite extensionL/K, YL is a component of(Y)L.ii) The connected components of(Y)Cp

are defined over some finite exten-

sionK ′/K ([15, § 2.14]), and then(YK ′)Cpis connected.

iii) For any finite extensionL/K ′, YL = (YK ′)L: topological coverings ofYL come from topological coverings ofYK ′. �

19

The fact that the universal topological covering ofYCpis defined over

some finite extension ofK (in fact over the extensionK ′/K just introduced)follows from the following lemma (applied toT = YK ′):

5.1.4. Lemma. Let T be a geometrically connected and simply connectedK-manifold. ThenTCp

is simply connected (in the sense of coverings).

Proof. Let h : Z → TCpbe a topological covering: every pointt ∈ TCp

(classical or not) has an open neighborhoodV (t) such thath−1V (t) is iso-morphic to a disjoint sum of copies ofV (t) via h. By the argument of [3,p. 103], there is a finite Galois extensionL/K contained inCp and an opendomainUL(t) ⊂ TL such thatt ∈ (UL(t))Cp

⊂ V (t). The restriction ofh above(UL(t))Cp

, being a split covering, admits a canonicalL-structure.Saturating with respect toGal(L/K), one finds by descent an open domainU(t) ⊂ T such thatU(t)L = ∪σ∈Gal(L/K) UL(t)

σ and such that the restric-tion of h aboveU(t)Cp

admits a canonicalK-structure. By 5.1.3.iii), thisactually defines a topological coveringZ(t) → U(t). These topologicalcoverings glue together (using [3,§ 1.3.3.a]) and define a topological cov-ering ofT which is aK-structure forh, and which splits sinceT is simplyconnected (in the sense of coverings). �

Remark.It does happen that(Y)Cphas several connected components (for

a geometrically connectedK-manifoldY). The simplest example is givenby a twisted Tate elliptic curveY , cf. [2, § 4].

5.2. Let us assume thatX = Xan

for some geometrically connectedsmooth algebraicK-varietyX. Then, passing to the profinite completion,one gets a commutative diagram

{1} → πtemp1 (X

an

Cp, x) → πtemp

1 (Xan

, x) → GK → {1}↓ ↓ ↓

πtemp1 (X

an

Cp, x) → πtemp

1 (Xan, x) → GK

↓∼= ↓∼= ↓

{1} → πalg1 (XCp

, x) → πalg1 (X, x) → GK → {1}

and the last row is the canonical exact sequence. In particular, the actionsρ(resp.ρx) of 6.1.1 are compatible with the homonymous actions definedin2.1.

6. Out πtemp1 FOR A CURVE

6.1. In this section, we examine the groupOut πtemp1 (X

an

) whenX is asmooth algebraic curve overCp. We fix a base pointx.

20 YVES ANDRE

By characteristic quotientof a given topological groupG, we mean thequotient ofG by some characteristic open subgroup.

6.1.1. Lemma. πtop1 (X

an

, x) is a characteristic quotient ofπtemp1 (X

an

, x).

Remark.We do not know whether this holds for higher dimensionalX, ifπtop1 (X

an

, x) has torsion.

Proof. For a curve,πtop1 (X

an

, x) is a free, hence torsion-free, discrete group.By proposition 4.3.1, it can actually be described as the biggest torsion-freediscrete quotient ofπtemp

1 (Xan

, x). This group-theoretic characterizationmakes it clear that the kernel ofπtemp

1 (Xan

, x) → πtop1 (X

an

, x) is charac-teristic (and of course open). �

Let us recall from 4.5. that

πtemp1 (X

an

, x) = lim←−

∆n

where∆n sits in an exact sequence

{1} → πtop1 (X

an

n )→ ∆n → Γn → {1}.

6.1.2. Lemma. i) ∆n is a characteristic quotient ofπtemp1 (X

an

, x).ii) If n|n′, bothΓn and∆n are characteristic quotients of∆n′ . In particular(for n = n′), the free groupπtop

1 (Xan

n ) is a characteristic subgroup of∆n.

Proof. i) Let us consider again the commutative diagram from 4.5

{1} → πtemp1 (X

an

n ) → πtemp1 (X

an

) → Γn → {1}↓ ↓ ↓ ↓ ↓{1} → πtop

1 (Xan

n ) → ∆n → Γn → {1} .

By definition, the kernel of πtemp1 (Xan) = πalg

1 (X) → Γn is characteris-tic. This implies that the kernelπtemp

1 (Xan

n ) of πtemp1 (X

an

) → Γn is char-acteristic. On the other hand, the kernel ofπtemp

1 (Xan

n ) → πtop1 (X

an

n ) ischaracteristic, by 6.1.1. It follows that the kernel ofπtemp

1 (Xan

) → ∆n ischaracteristic.ii): the profinite completion∆n′ of ∆n′ is a quotient ofπalg

1 (X) whichmaps toΓn. By observation 2.2.1,Γn is a characteristic quotient of∆n′,hence of∆n′; in other words, the kernelKn′ of ∆n′ → Γn is characteristic.

The kernel ofKn′ → πtop1 (X

an

n ) is the image in∆n′ of the kernel ofπtemp1 (X

an

n ) → πtop1 (X

an

n ). By the same argument as in 6.1.1 (using 4.3.1),it can be characterized as the smallest normal subgroupN of Kn′ such thatKn′/N is torsion-free; this shows that it is characteristic inKn′, hence in∆n′. It follows that∆n is a characteristic quotient of∆n′ . �

21

6.1.3. Corollary.

Aut πtemp1 (X

an

, x) = lim←−

Aut∆n, Out πtemp1 (X

an

) = lim←−

Out∆n.

Indeed, the previous lemma makes sense of the inverse limitlim←−

Aut∆n

and provides a natural homomorphismAut πtemp1 (X

an

, x) → lim←−

Aut∆n,which has an obvious inverse (note however that the transition maps neednot be surjective). �

We endowAut πtemp1 (X

an

, x) andOut πtemp1 (X

an

, x) with the inverselimit topology. With this topology, it is clear thatAut πtemp

1 (Xan

, x) actscontinuously onπtemp

1 (Xan

, x) = lim←−

∆n.

6.1.4. Proposition. Out πtemp1 (X

an

) is a countable inverse limit of discretefinitely generated groups; in particular it is a polish group(cf. 4.4).

Proof. Indeed, finitely generated virtually free groups (such as∆n) havefinitely generated17 automorphism group (Krstic, McCool [24]). �

6.2. For a curve, we have seen thatπtemp1 (X

an

, x) = lim←−

∆n embeds into

its profinite completionπalg1 (X, x) = lim

←−Γn. A fortiori, Aut πtemp

1 (Xan

, x) =

lim←−

Aut∆n embeds intoAut πalg1 (X, x) = lim

←−AutΓn. For outer automor-

phisms, we also have:

6.2.1. Proposition. Out πtemp1 (X

an

) embeds intoOut πalg1 (X) ; in particu-

lar, it is residually finite.

Proof. If X is not hyperbolic,πtemp1 (X

an

) is abelian and coincides withπalg1 (X) except in the case of an elliptic curve with bad reduction, for which

πtemp1 (X

an

) ∼= Z× Z; in either case, the statement is trivial.We now assume thatX is hyperbolic. The assumption of hyperbolicity en-sures that the center ofπtemp

1 (Xan

, x) is trivial (an immediate consequenceof the fact that the center of its profinite completionπalg

1 (X, x) is trivial).Let us assume that ∈ πalg

1 (X, x) induces an automorphism ofπtemp1 (X

an

, x)by conjugation. We have to show ∈ πtemp

1 (Xan

, x).Let us fixn, and consider the composed epimorphism

πalg1 (X, x)→→ ∆n →→ Γn

(where∆n denotes the profinite completion of∆n, which sits in an exactsequence

{1} → πtop1 (Xan

n )→ ∆n → Γn → {1} ).

17even finitely presented

22 YVES ANDRE

Let πn be any element ofπtemp1 (X

an

, x) which has the same image as

in Γn. The image n of .π−1n in ∆n actually belongs to πtop

1 (Xan

n ), andinduces by conjugation an automorphism of the free groupπtop

1 (Xan

n ) (cf.6.1.1, 6.1.2). By 3.2.1, it follows that n ∈ πtop

1 (Xan

n ), hence the image of in ∆n actually belongs to∆n. Therefore ∈ πtemp

1 (Xan

, x) = lim←−

∆n.�

6.2.2. Corollary. If X is hyperbolic,πtemp1 (X

an

, x) is its own normalizer inits profinite completion. �

We denote byOut πtemp1 (X

an

) the closure ofOut πtemp1 (X

an

) inOut πalg1 (X):

Outπtemp1 (X

an

) = lim←−

Im [Out πtemp1 (X

an

)→ OutΓn]

(this limit is clearly a quotient of the profinite completionofOut πtemp1 (X

an

)

and embeds intolim←−

OutΓn = Out πalg1 (X) ).

Remark. For the problems studied in the sequel (cf. 8.7.2), it would beinteresting to know whetherOut πtemp

1 (Xan

) = lim←−

Out∆n is a profinitegroup or not, especially in the caseX = P1

Cp\ {0, 1,∞}. This involves the

question of finiteness of the outer automorphism groups of the finitely gen-erated virtually free groups∆n. In the abstract setting, a complete anwserto this kind of question has been given by M. Pettet in terms ofgraphs ofgroups [28]. His criteria apply for instance to Kato’s presentation ofp-adictriangle groups as amalgams [17], and could apply in the sameway to the‘p-adic dessins d’enfants’ invoked at the end of 4.5.

Another interesting open problem is whether the natural homomorphismOut πtemp

1 (X) → GL2(Z) is surjective forX = P1Cp

an

\ {0, 1,∞} (oneshows easily that the natural composed map

πtemp1 (X)→ πalg

1 (X)→ Het1 (X, Z) ∼= Z2

is surjective).

7. GQ, GQpAND GEOMETRIC FUNDAMENTAL GROUPS

7.1. Nowk is a number field, and we fix an embeddingιv : k → Cp. Wedenote bykv the topological closure ofk in Cp.

For any smooth geometrically connected algebraick-varietyX, we de-note byX

an

thep-adic analytic manifold attached toXCp.

Let us fix aCp-pointx of X. One has canonical isomorphisms

πalg1 (Xk, x) = πalg

1 (XCp, x) = πtemp

1 (Xan, x),

23

the latter group being the profinite completion of the tempered fundamentalgroupπtemp

1 (Xan

, x) which classifies tempered etale coverings ofXan

.There is an outer actionρ of Gk on πalg

1 (Xk, x) and a compatible outeraction ofGkv onπtemp

1 (Xan

, x) (6.1, 6.2).

7.2. Let us now assume thatX is a hyperbolic affine curve. We haveencountered two closed subgroups ofOut πalg

1 (Xk): Out πtemp1 (X

an

) andGk. As in 3.3, we shall study their intersection, starting withthe case ofP1 \ {0, 1,∞}. The following is a non-archimedean analogue of 3.3.1.

7.2.1. Theorem. If X = P1k \ {0, 1,∞} , then

Out πtemp1 (X

an

) ∩ Gk = Out πtemp1 (X

an

) ∩ Gk = Gkv .

Proof. We already know thatGkv ⊂ Out πtemp1 (X

an

, x)∩ Gk by 5.1.1 and5.2. It remains to show thatOut πtemp

1 (Xan

) ∩ Gk ⊂ Gkv .We choose a rational base pointx of P1

k \ {0, 1,∞}. Letσ ∈ Gk be suchthatρ(σ) ∈ Out πtemp

1 (Xan

). We have to show that

(∗) ∀j ∈ k \ {0, 1}, |j|p ≤ 1⇒ |jσ|p ≤ 1.

LetEj be an elliptic curve with invariantj ∈ k. According to Belyi, there isa surjective morphismϕ : Ej → P1 defined overQ(j) which is unramifiedoutsideΣ = ϕ−1({0, 1,∞}).

The idea of the proof is that if the image of someτ ∈ Aut πtemp1 (X

an

, x)in Outπtemp

1 (Xan

) is ‘sufficiently close’ toρ(σ), thenτ induces an isomor-phismπtemp

1 (Ean

j )→ πtemp1 (E

an

jσ ), whence(∗), since the algebraic structureof these fundamental groups is determined by the alternative ‘good/bad’reduction,cf. 4.6.

Let us explain the details. We fix an integerm so that the characteris-tic finite etale coveringXm of Xk defined in 2.2 dominatesEj \ Σ. Thisprovides a sequence of finite etale coverings with base points

(Xm, xm)→ (Ej \ Σ, y)→ (Xk, x).

One has a commutative diagram

πalg1 (Xm, xm)

ρxm,xσm(σ)

→ πalg1 (Xσ

m, xσm)

↓ ↓

πalg1 (Ej \ Σ, y)

ρy,yσ (σ)→ πalg

1 (Ejσ \ Σσ, yσ)

↓ ↓

πalg1 (Xk, x)

ρx(σ)→ πalg

1 (Xk, x).

By assumption,

ρ(σ) ∈ Outπtemp1 (X

an

) = lim←−

Im[Out πtemp1 (X

an

)→ OutΓn]

24 YVES ANDRE

This implies the existence ofτ ∈ Aut πtemp1 (X

an

, x), ∈ πalg1 (Xk, x)

such thatρx(σ) andτ◦ad() have the same image inΓm. Sinceπalg1 (Xm, xm)

is open in the profinite completionπalg1 (Xk, x) of πtemp

1 (Xan

, x), one has

πalg1 (Xk, x) = πtemp

1 (Xan

, x).πalg1 (Xm, xm),

which allows to take ∈ πalg1 (Xm, xm). Then there is a similar commuta-

tive diagram built onτ

πalg1 (Xm, xm) → πalg

1 (Xσm, x

σm)

↓ ↓

πalg1 (Ej \ Σ, y)

τ ′→ πalg

1 (Ejσ \ Σσ, yσ)

↓ ↓

πalg1 (Xk, x)

τ→ πalg

1 (Xk, x),

which comes, by profinite completion, from a commutative diagram

πtemp1 (X

an

m , xm) → πtemp1 (Xσ

m

an

, xσm)

↓ ↓

πtemp1 (E

an

j \ Σ, y)τ ′→ πtemp

1 (Ean

jσ \ Σσ, yσ)

↓ ↓

πtemp1 (X

an

, x)τ→ πtemp

1 (Xan

, x)

where horizontal maps are isomorphisms (τ ′ stands for the induced isomor-phism).

On the other hand, the open immersionEan

j \Σ → Ean

j induces a (strict)surjective homomorphismπtemp

1 (Ean

j \ Σ, y) → πtemp1 (E

an

j ) [1, § 4.5.5.b],which is nothing but the abelianization homomorphism (thisbecomes clearon the profinite completions, or using 3.2.2.a). One thus hasanother com-mutative diagram

πtemp1 (E

an

j \ Σ, y)τ ′→ πtemp

1 (Ean

jσ \ Σσ, yσ)

↓ ↓

πtemp1 (E

an

j )τ ′ab

→ πtemp1 (E

an

jσ )

where horizontal maps are isomorphisms.If |j|p ≤ 1, Ej has good reduction and it follows thatπtemp

1 (Ean

j ) =

πalg1 (Ej), henceπtemp

1 (Ean

jσ ) = πalg1 (Ejσ), which implies in turn thatEjσ

has also good reduction atp, whence|jσ|p ≤ 1. �

7.2.2. Corollary. If X = P1k \ {0, 1,∞} , the image ofOut πtemp

1 (Xan

) inOut πalg

1 (Xk) is not dense in any open subgroup. �

25

For more general affine hyperbolick-curvesX, we have the followingnon-archimedean analogue of 3.3.2:

7.2.3. Theorem. Assume that some etale covering ofXk admits a non-constant rational function which omits at least three values. Then

Out πtemp1 (X

an

) ∩ Gk = Gkv .

A fortiori, if the geometric pointx is k-rational, then the subgroup ofGk

which stabilizesπtemp1 (X

an

, x) ⊂ πalg1 (Xk, x) is Gkv .

Proof. From the assumption, there is a finite extensionk′ of k in Cp, a finiteetale coveringY → Xk′ and a dominant morphism

Yϕ→ Z = P1

k′ \ {0, 1,∞}.

(We may also assume thatX, Y ,Z are endowed with compatiblek′-rationalgeometric pointsx, y, z respectively). The result will follow from 7.2.1 bythe following devissage:

7.2.4. Lemma. The statement

(∗)X,k Out πtemp1 (X

an

) ∩ Gk ⊂ Gkv

satisfiesa) (∗)X,k′ ⇒ (∗)X,k if k′ is a finite extension ofk in Cp ,b) (∗)Z,k ⇒ (∗)Y,k if there is a non-constant morphismf : Y → Z.c) (∗)Y,k ⇒ (∗)X,k if Y is a finiteetale covering ofX,

Proof. b) and c) are completely parallel to the proof of 3.3.2, replacingC

by Cp, πtop1 by πtemp

1 , the reference to 3.2.1 by a reference to 6.2.2, andsublemma 3.3.4 by

7.2.5. Sublemma. (Neukirch, cf.[27, § 12.1.10]) If H is a closed subgroupofGk containingGkv as a subgroup of finite index, thenH = Gkv . �

As for a): letH stand forOut πtemp1 (X

an

) ∩ Gk. It containsGkv . Onthe other hand,Out πtemp

1 (Xan

) ∩ Gk′, which is nothing butGk′v due to(∗)X,k′, is a subgroup of finite index ofH. ThereforeH is closed inGk andcontainsGkv as a subgroup of finite index. One concludes by 7.2.4.�

Remarks.1) One version of a celebrated theorem of Mochizuki [25] saysthat for a hyperbolic curveXkv overkv, the centralizer ofGkv inOut πalg

1 (XCp)

is isomorphic toAutXkv . Denoting byXan

thep-adic manifold attachedto XCp

, it follows that the center ofOut πtemp1 (X

an

) is isomorphic to thecenter ofAutXkv . In particular, it is trivial ifXkv = P1

kv\ {0, 1,∞}.

2) FromGkv , one can recover the inertia group and the Frobenius ele-ments by a purely group-theoretic recipe [27,§ 12.1.8].

26 YVES ANDRE

3) Theorems 7.2.1 and 7.2.3 do not change if one composesιv with anyelement ofGkv . Thus it is enough to fix ap-adic placev of k (instead ofιv).

8. THE LOCAL GROTHENDIECK-TEICHMULLER GROUP ATp

8.1. In this section, we define a “p-local” version of the Grothendieck-Teichmuller groupGT , which plays with respect toGQp

the role thatGTplays with respect toGQ.

For r ≥ 4, letM0,r be the moduli space for curves of genus0 withr ordered marked points. This smooth geometrically connected Q-varietycarries an action of the symmetric groupSr (permutation of the markedpoints). It admits a canonical smooth compactificationM0,r (Deligne-Mumford-Knudsen), such that the boundary∂M0,r is a divisor with normalcrossings; overC, this compactification is simply-connected [5,§ 1.1], anda fortiori, πalg

1 (M0,r,Q) = {1} (from now on, we drop the implicit basepoint from the notation).

For r = 4,M0,4∼= P1

Q \ {0, 1,∞}, M0,4∼= P1

Q, and the action ofS4

factors through the standard action ofS3. In general,M0,r is isomorphicto the complement in(P1

Q \ {0, 1,∞})r−3 of the partial diagonals defined

by the equality of two coordinates.Forgetting the last marked point provides anSr-equivariant surjective

morphismpr : M0,r+1 → M0,r with fibers isomorphic toP1 minus rpoints.

8.2. The fundamental groupπtop1 (M

an

0,r,C) (amapping class group in genuszero) is generally denoted byΓ0,r, but we add a superscripttop in order todistinguish it from subsequent variants.

The fibrationpr gives rise to an exact sequence of fundamental groups

{1} → Fr−1 → Γtop0,r+1 → Γtop

0,r → {1}

whereFr−1∼= πtop

1 (P1C

an

\ {x1, . . . , xr}) is the discrete free group onr− 1generators. On the other hand, according to a classical result of Nielsen-Zieschang (cf. [33]), the corresponding homomorphism

Γtop0,r → OutFr−1

induces an isomorphism

Γtop0,r∼= Out♮ πtop

1 (P1C

an

\ {x1, . . . , xr}) ∼= ker[OutFr−1→ GLr−1(Z)]

(the sign♮ indicates the subgroup of (outer) automorphisms which fix theconjugacy classes of the local monodromies at the pointsxi). [From there,using the surjectivity ofAutFn → GLn(Z) [20, § I.4.4] , one derives

27

the structure ofAutFn (resp. OutFn) for any n ≥ 1: it is an exten-sion ofGLn(Z) by an iterated extension of free groups of successive ranks2, 3, . . . , n (resp.2, 3, . . . , n− 1)].

8.3. TheGrothendieck-Teichmuller group GT introduced by Drinfel’d18

in connection with the theory of quantum groups is a closed subgroup ofthe profinite groupAut F2. If x, y denote fixed topological generators ofF2, it consists of automorphisms of the formx 7→ xλ, y 7→ f−1yλf forλ ∈ Z∗ and appropriatef ∈ F ′

2 satisfying three well-known equations I, II,III, which we do not recall here (cf. [8] [14]).

The bigger group obtained by discarding equation III (the pentagonalequation) is sometimes denoted byGT 0.

8.4. For anyr ≥ 4, we setΓalg0,r := πalg

1 (M0,r,Q). ThusΓalg0,r is identified

with the profinite completion ofΓtop0,r if one fixes an embeddingι : Q → C.

Following Nakamura [26], we denote byOut♭ Γalg0,r the subgroup of outer

automorphismsσ of Γalg0,r which sends the conjugacy class of a generator of

the inertia group at each component of∂M0,r to the conjugacy class of thesame generator raised to some powerλ(σ) ∈ Z∗ (the sameλ(σ) for eachcomponent). There is an obvious homomorphism

λ : Out♭ Γalg0,r → Z∗, σ 7→ λ(σ).

We denote by GT(r)

:= Out♭Sr

Γalg0,r the subgroup of elements which

commute with the action ofSr. This notation is motivated by:

8.4.1. Theorem. (Harbater, Schneps[12])1) TheSr-equivariant surjective morphismM0,r+1

pr−→ M0,r induces

canonical homomorphisms

Out♭ Γalg0,r+1 → Out♭ Γalg

0,r , GT(r+1) qr

→ GT(r)

compatible with theλ-map.

2) For all r ≥ 5, qr : GT(r+1)

→ GT(r)

is an isomorphism,3) An embeddingι : Q → C being fixed, there is a natural commutativediagram

GT(5) q4→ GT

(4)

↓∼= ↓∼=

GT → GT 0

18unlike the usage in the present paper,GT is just a (traditional) notation and does notindicate the profinite completion of some groupGT ; this notation helps distinguishing theprofinite Grothendieck-Teichmuller group from other variants (notably the pro-unipotentone related to mixed Tate motives)

28 YVES ANDRE

where the vertical maps are isomorphisms. �

(ι is used to identifyF2 with Γalg0,4 based at some tangential base point).

This result (partly based on results in [26]) provides a ‘purely algebraic’description ofGT (independent of the topology ofC), and at the same time,a natural interpretation of the Drinfel’d-Ihara embeddingGQ → GT (as anembedding of typeρ, with the notation of 2.1). The question of how closeis GT toGQ is a famous open problem.

8.5. In view of the results of§7, it is natural to try to replace the algebraicfundamental groups which occur in the previous theorem by tempered fun-damental groups in thep-adic context.

We fix a prime numberp, and consider the polish groupsπtemp1 (M

an

0,r,Cp).

It is likely that these are residually finite, but having no proof, we are led tointroduce the followingp-adic avatar of the mapping class group in genuszero:

Γtemp0,r := πtemp

1 (Man

0,r,Cp)/ ker[πtemp

1 (Man

0,r,Cp)→ πtemp

1 (Man

0,r,Cp)].

ThusΓtemp0,r is, like πtemp

1 (Man

0,r,Cp), a polish group, and embeds into its

profinite completion, which can be identified with theΓalg0,r if one fixes an

embeddingιp : Q → Cp.Note thatΓtemp

0,4 is our familiar groupπtemp1 (P1

Cp

an

\ {0, 1,∞}).

8.5.1. Proposition. The morphismpr : M0,r+1 → M0,r induces strictsurjective homomorphisms

πtemp1 (M

an

0,r+1,Cp)→→ πtemp

1 (Man

0,r,Cp), Γtemp

0,r+1 →→ Γtemp0,r .

Proof. From the descriptionM0,r,Cp= (P1

Cp\{0, 1,∞})r−3\∆,we deduce

a factorization ofpr into a closed embeddingj : M0,r+1,Cp→ A1

Cp×

M0,r,Cpfollowed by the second projection. This second projection induces

an isomorphism onπtemp1 : this follows from the fact that any finite etale

covering ofA1Cp×M0,r,Cp

is of the formA1Cp×M1 for some finite etale

coveringM1 ofM0,r,Cp(in the algebraic or analytic category, this does not

matter thanks to [19]), and from the fact that any topological covering ofA1

an

Cp×M

an

1 is of the formA1an

Cp×M

an

2 for some topological coveringMan

2

ofMan

1 (sinceπtop1 (A1

an

Cp) = {1}, andA1

an

Cp×M

an

1 is homotopy equivalent

to the naive cartesian product ofA1an

CpandM

an

1 [4, prf. of 10.1]).On the other hand,j is the inclusion of the complement ofr smooth

hyperplanes, and it follows from [1,§ 4.5.5.b)] that it induces a strict sur-jection onπtemp

1 (this result uses heavily the fact that one is dealing withpolish groups). �

29

Remark.One hasπtemp1 (M

an

0,r,Cp) = {1}: indeed, we have seen that any fi-

nite etale covering ofM0,r,Cpis trivial (in the algebraic or analytic category

[19]), and on the other hand, sinceM0,r is Zariski open in the affine space

Ar−3, πtop1 (M

an

0,r,Cp) = πtop

1 (Ar−3Cp

) = {1} [1, § 1.1.4].

This suggests thatπtemp1 (M

an

0,r,Cp)might ‘come’ in some sense from neigh-

borhoods of points of maximal degeneration in∂Man

0,r,Cp.

8.6. For anyr ≥ 4, we denote byOut♭ Γtemp0,r the subgroup of outer au-

tomorphismsσ of Γtemp0,r which send the conjugacy class of the local mon-

odromy at each component of∂M0,r to the conjugacy class of the samelocal monodromy raised to some powerλ(σ) ∈ Z∗ (the sameλ(σ) for eachcomponent); we refer to [1,§ 4.5.4] for the precise definition of this conju-gacy class.

8.6.1. Notation. • We denote byGT(r)p := Out♭

SrΓtemp0,r the subgroup of

Out♭ Γtemp0,r of elements which commute with the action of the symmetric

groupSr.

•We denote byGT(r)

p the closure of the image ofGT(r)p in GT

(r) ∼= GT .

(This notation is consistent with the notationGT of the Grothendiek-Teich-muller group, but it should not be confused with the profinite completionGT

(r)p of GT

(r)p ; we do not know whether the natural epimorphismGT

(r)p →

GT(r)

p is an isomorphism, butcf. 8.7.2.).

Note thatGT(4)

p is a closed subgroup ofOutπtemp1 (P1

Cp

an\ {0, 1,∞}).

8.6.2. Proposition. 1) TheSr-equivariant surjective morphismpr inducescanonicalinjectivehomomorphisms

GT (r+1)p

qr→ GT (r)

p , GT(r+1)

p

qr→ GT

(r)

p .

2) The canonical homomorphismGT (r)p → GT

(r)

p is injective; in particular,

GT (r)p is residually finite.

Proof. The part of 1) concerningGT(r)

p follows directly from 8.4.1. From

the embeddingΓtemp0,r → Γalg

0,r , we derive an embeddingAut♭ Γtemp0,r →

Aut♭ Γalg0,r . The “Aut” version of 8.4.1 (cf. [12, 3.1]) says thatpr induces

an injective homomorphismAut♭ Γalg0,r+1 → Aut♭ Γalg

0,r . By application ofthe sorite 3.2.2.a) toG = Γtemp

0,r+1, H = Γtemp0,r andφ = pr,∗, we derive that

pr induces an injective homomorphismAut♭ Γtemp0,r+1 → Aut♭ Γtemp

0,r . Finally,

30 YVES ANDRE

using the surjectivity ofΓtemp0,r+1 →→ Γtemp

0,r (8.5.1), we see thatpr induces inturn injective homomorphisms

Out♭ Γtemp0,r+1 → Out♭ Γtemp

0,r , GT (r+1)p →GT (r)

p ,

which proves 1).For 2), we compose the mapsqr up tor = 4 and get an injective homo-

morphismGT (r)p →GT (4)

p ⊂ OutΓtemp0,4 . But we know (6.2.1) thatOutΓtemp

0,4

embeds intoOutΓalg0,4. Since the composed mapGT (r)

p →OutΓalg0,4 factors

throughGT(r)

p , assertion 2) follows. �

8.6.3. Definition. • We defineGTp to be the intersection of the images oftheGT (r)

p in GT (4)p (for all r ≥ 4).

•We define thelocal Grothendieck-Teichmuller group atp to be the closure

GT p of GTp in GT(4)

p .

From 8.6.2.1), we get

8.6.4. Corollary. The abstract groupGTp acts on the ‘tower’ of temperedmapping class groups(Γtemp

0,r )r≥4 . �

On the other hand,GT p can be viewed as a closed subgroup of theGrothendieck-Teichmuller groupGT via the Harbater-Schneps isomorphism

GT(4) ∼= GT 0.

8.7. We fix ap-adic placev of Q. Then the mapsρ from §2 and§5 giverise to commutative squares of injective homomorphisms

GQp→ GT p GQp

→ GT (r)p

↓ ↓ ↓ ↓

GQ → GT 0 GQ → GT(r)

which are compatible via the embeddingsqr. It turns out that these squaresare cartesian:

8.7.1. Theorem. GQp= GQ ∩GT p = GQ ∩ GT p in GT 0 . Moreover, for

anyr ≥ 4, GQp= GQ ∩GT (r)

p = GQ ∩ GT(r)

p in GT(r)

.

Proof. This follows immediately from 7.2.1, since

GT(4)

p ⊂ Outπtemp1 (P1

Cp

an\ {0, 1,∞}). �

In the following corollary, we identify allGT(r)

with GT for r ≥ 5(according to 8.4.1).

31

8.7.2. Corollary. If GQ = GT , then all these groupsGTp = GT (r)p =

GT p = GT(r)

p would coincide withGQpin GT , for r ≥ 5.

In particular GT (5)p = Out♭

S5Γtemp0,5 would be a finitely generated

profinite group19. �

It is tempting to try to put this property in the wrong (cf. the remark after6.2.2)...

Remarks.1) The analogue forGT p of the inertia subgroup ofGQpis the

kernel of the composed mapGT pλ→ Z∗ = (Z(p′))∗ × Z∗

p → (Z(p′))∗.

2) The image ofGQpby λ is pZ × Z∗

p. A natural question is whether the

same holds for the image ofGT p or evenGT(4)

p . After submission of thispaper, A. Tamagawa has indicated to the author an argument according towhich this should follow from the main results of [31], notably theorem 5.2and proposition 5.3 ofloc. cit.

3) Sinceλ takes its values inZ∗, GT(r)p acts on the quotient ofΓtemp

0,r

by the closed normal subgroup generated by some fixed integral powers ofthe local monodromies at the components of the boundary. This quotientis an orbifold fundamental group in the sense of [1], and its discrete repre-sentations correspond to certain algebraic vector bundlesonM0,r,Cp

withfuchsian integrable connection and prescribed exponents [1, §4.5.9], whichform a p-adic analytic subvariety of the corresponding algebraic modulispace of connections. In particular,GQp

acts through a finite group on eachof thesep-adic analytic subvarieties.

4) As in the remark at the end of§7, one derives from Mochizuki’s theo-

rem (and 8.6.2) that the center of any of the groupsGTp, GT (r)p , GT p, GT

(r)

p

is trivial.5) In complete analogy with the above, one can introduce the archimedean

avatarsGT (r)∞ , GT

(r)

∞ , GT∞, GT∞ (replacingCp by C andπtemp1 by πtop

1 ).It turns out that allthese groups coincide withGR for any r ≥ 4. This isproved by the same argument as in parti) of the proof of 3.3.1, replacingthe final reference to [27,§ 12.1.4] by a reference to [18, prop. 4.ii]:GR isits own centralizer inGT .

6) Drinfel’d’s presentation ofGT uses topological generatorsx, y ofF2 which are the standard local monodromies at0 and1 in πtop

1 (P1C

an\

{0, 1,∞}) (based at a suitable tangential base point). One could ask whetherthere is an analogous presentation ofGT of p-adic flavour, using local mon-odromies at0 and1 in πtemp

1 (P1Cp

an\{0, 1,∞}). It turns out that the answer

19topologically generated by three elements,cf. [27, §7.4.1])

32 YVES ANDRE

is negative, as a consequence of the following observation:using techniquesof graphs of groups, we have shown in [1,§ 6.4.6] that forp = 2, one cannotfind local monodromiesx, y, z at0, 1,∞ respectively such thatxyz = 1.

9. GQpIS THE AUTOMORPHISM GROUP OF THE TEMPERED

π1-FUNCTOR

9.1. In this section, we deal with automorphisms of fundamental groupsconsidered as functors.Let Vk be the category of all smooth geometrically connected algebraic va-rieties defined overk.Let T be the category of separated topological groups up to inner automor-phism: a morphism in this category is a continuous homomorphismG→ Hgiven up to conjugation by an element ofH (in particular,AutT (G) =OutG).

In this article, we have been mainly concerned with the following func-tors

πalg

k, πtemp

Cp: Vk → T

πalg

k(X) := πalg

1 (Xk, x), πtempCp

(X) := πtemp1 (X

an

Cp, x),

given an embeddingk → Cp (note that in the categoryT , these fundamen-tal groups do not depend on the choice ofx up tocanonicalisomorphism;hence the above functors are uniquely defined up to unique isomorphism;similarly, πtemp

Cpdepends only on thep-adic placev of k induced by the

embedding, up to unique isomorphism).Let kv be the topological closure ofk in Cp. Functoriality of the mapsρ

considered in§2 and§5 may be expressed as follows: there are canonicalhomomorphisms

Gkρ→ Aut πalg

k, Gkv

ρ→ Aut πtemp

Cp

which build a commutative square ifk is theclosure ofk in Cp:

Gkv

ρ→ Aut πtemp

Cp

↓ ↓

Gkρ→ Aut πalg

k.

9.2. The following result had been conjectured by Oda and Matsumoto[22]:

9.2.1. Theorem. (Pop [29], unpublished) Gkρ→ Aut πalg

kis an isomor-

phism. �

33

(In fact, Pop proves more:Gk coincides withAut πalg

k |Wfor any big enough

full subcategoryW of Vk; we do not know any minimal choice forW, butit suffices thatW contains the open subsets of the projective spaces).

As a consequence of Pop’s theorem and theorem 7.2.1, we get

9.2.2. Theorem. Gkv

ρ→ Aut πtemp

Cpis an isomorphism.

Proof. One has a sequence of homomorphisms

Gkv

ρ→ Aut πtemp

Cp

Aut πalg

k

9.2.1= Gk

ր ցց ր

Out πtemp1 (P1

Cp

an\ {0, 1,∞})

Outπalg1 (P1

k\{0, 1,∞})

where the two maps on the right are injective. By 7.2.1, the intersection ofGk andOutπtemp

1 (P1Cp

an\ {0, 1,∞}) in Out πalg

1 (P1k\ {0, 1,∞}) is Gkv .

It follows thatρ : Gkv→Aut πtempCp

admits a left inverseν. It remains toshow thatker ν is trivial.

Note thatker ν = ker[Aut πtempCp→ Aut πalg

k]. It follows, by 6.2.1, that

the evaluation of any element ofker ν on any curveX is the identity (asouter automorphism of the fundamental group). In order to reduce the caseof an arbitrary smooth geometrically connectedk-varietyV to the case ofa curveX, we choose a quasi-projective dense open subsetU ⊂ V and anembedding ofU in some projective spacePN

k . According to a theorem ofBertini-Deligne [7, 1.4], there is a dense open subsetU of the grassmannianof linear varieties of codimensiondimU − 1 in PN

k such that for allL ∈U(k), the curveX = L ∩ U is geometrically connected and smooth, andthe homomorphism

πtop1 (X

an

C )→ πtop1 (U

an

C )

is surjective (for any fixed complex embeddingk → C). It follows that

πalg1 (Xk)→ πalg

1 (Uk)

is also surjective; on the other hand

πalg1 (Uk)→ πalg

1 (Vk)

is also surjective. Since the evaluation of any element ofker ν onX is theidentity, the evaluation of any element ofker ν on V is also the identity,henceker ν = {1}. �

Acknowledgements.A suggestion by Leila Schneps is at the origin of thispaper: when I asked her, a few years ago, about potential usesof temperedfundamental groups in Grothendieck-Teichmuller theory,she immediately

34 YVES ANDRE

pointed out the problem of characterizing geometrically the local Galoisgroups. I am very grateful for this pertinent suggestion; I also thank her,as well as Pierre Lochak, for many useful discussions about Grothendieck-Teichmuller theory. Finally, I am much indebted to A. Tamagawa for hiscomments on a first version of this text.

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