ñòð. 12 |

extension of K in L is nite Galois. Then, Aut(L=K) is called the Galois

group of L=K, and denoted by G(L=K) again.

This innite Galois group is equipped with a natural topology, called pro-

nite topology, as will be stated later. For L M K with M being Galois

over K, we have a short exact sequence

1 ! G(L=M) ! G(L=K) ! G(M=K) ! 1: (1.1)

p

Exercise 1.1. Let K = Q , L = Q (n ), with n = e2 1=n being an n-

th primitive root of unity. Show that : G(L=K) (Z=n), where 2

=

(n)

G(L=K) is mapped to the unique () 2 (Z=n) such that (n ) = n , by

using the irreducibility of the minimal polynomial of n .

If we put L0 := Q (n jn 2 N), what is G(L0 =K)?

360

2. A short way to arithmetic fundamental groups

Let X be an arcwise connected topological space, and let a be a point of

X. The fundamental group of X with base point a, denoted by 1(X; a), is

dened to be the set of homotopy equivalence classes of closed paths from

a to itself. By the usual composition of paths, taking the homotopy classes,

1(X; a) becomes a group. For the composition rule, we denote by

0

the

path rst going along

and then along

0 .

For another choice of base point b, we have 1 (X; a) 1 (X; b), where

=

the isomorphism is well-dened up to an inner automorphism.

More generally, let 1 (X; a; b) denote the homotopy classes of paths from

a to b. Two paths

2 1 (X; a; b) and

0 2 1 (X; b; c) can be composed

to get an element

0

2 1 (X; a; c), and the system f1 (X; a; b)j(a; b 2

X)g constitutes a groupoid (i.e. a category where every homomorphism is

invertible).

Assume that X is a smooth complex algebraic variety. Let a be a point

of X.

Denition 2.1. Let Ma be the eld of the germs of meromorphic algebraic

functions at a on X, such that the germ has analytic continuation to a

nitely multivalued meromorphic function along any path on X.

Algebraic means that every h 2 Ma is algebraic over the rational function

eld C (X) of algebraic variety X. Functions like exp(z) are not in C (A 1 ).

Clearly Ma is a eld, which contains the rational function eld C (X) as

the single-valued meromophic functions. By analytic continuation, we have

an action

1 (X; a; b) Ma ! Mb ;

with h 2 Ma 7!

h 2 Mb being obtained by analytic continuation of the

germ h along the path

. In particular, 1 (X; a) acts on Ma from the left,

so we have a group homomorphism

1 (X; a) ! Aut(Ma =C (X));

since C (X) is single-valued and is xed elementwise by analytic continuation

along a closed path. Because h 2 Ma is assumed to be nitely multival-

ued, the set of all the branches S := f

hj

2 1 (X; a)g is nite. Hence,

the fundamental symmetric polynomials of S =: ff1 ; f2 ; : : : ; fn g are single-

valued, i.e. in C (X), since 1 (X; a) acts on S by permutations. Thus,

Arithmetic fundamental groups 361

Q

F(T) := n (T fi ) 2 C (X)[T ], which is divisible by the minimal polyno-

i=1

mial of h over C (X). Thus, any algebraic conjugate of h over C (X) is one

of the fi , and hence Ma =C (X) is a Galois extension.

Now we have a homomorphism

1 (X; a) ! G(Ma =C (X)):

Denition 2.2. We dene the algebraic fundamental group of a connected

smooth algebraic variety X over C by

alg

1 (X; a) := G(Ma =C (X)):

We have a group homomorphism

alg

1 (X; a) ! 1 (X; a);

by analytic continuation.

It can be proved that this morphism is \completion," i.e., the right-hand

side is obtained by a purely group-theoretical operation called \pronite

completion" of the left-hand side, and thus depends only on the homotopy

type of X, as we shall see in the next section. The arithmetic part will come

into sight when we consider X over a non algebraically closed eld K in x2.2.

2.1. Pronite groups and algebraic fundamental groups. We want

to describe the (possibly innite) Galois group G(L=K) for L := Ma , K :=

C (X). The answer is that it is the pronite completion of 1 (X; a).

To see this, for a while, let L=K be a general innite Galois extension.

We look at all nite Galois subextensions M=K in L. The surjectivity of

G(L=K) ! G(M=K) in (1.1) says that 2 G(L=K) determines a family

M 2 G(M=K) for each nite Galois extension M, so that they are com-

patible in the sense that if M M 0 , then M 7! M via natural morphism

0

G(M=K) ! G(M 0 =K).

In general, let be a directed set (i.e. a partially ordered set such that

for any ; 0 2 there is a common upper element 00 ; 0 in ), and

assume that we are given a family of nite groups G ( 2 ) together with a

group homomorphism G ! G for , with the following compatibility

condition: the composition of G ! G with G ! G coincides with

G ! G . We call this family a projective system of nite groups. (In

the terminology of category theory, this is merely a functor from to the

category of nite groups.) We dene its projective limit proj lim2 G as

follows. An element of proj lim G is a family ( )2 , 2 G , with the

362

property 7! for . This is a subset of the direct product:

Y

proj lim G G:

2 2

We equip G with the discrete (but compact, being nite) topology. Then,

the product is compact by Tikhonov's Theorem, and so is the projective

limit, being a closed subgroup of the compact group.

If we apply this notion to the case: is the set of nite Galois subextension

M=K equipped with the inclusion ordering M M 0 , M M 0 , we obtain

a projective system of nite groups G(M=K). Our previous observation says

that there is a morphism

G(L=K) ! proj lim G(M=K);

M

2 G(L=K) 7! (jM 2 G(M=K)). This is injective, since L being algebraic,

L is the union of all nite Galois extensions M. The above morphism is

surjective, since if we have an element of (M ) 2 proj limM G(M=K), then

the compatibility assures that the action of M restricts to that of M 0

if M M 0 . Thus, the actions M patch together to give an element of

G(L=K). Thus we have

G(L=K) = proj lim G(M=K):

M

A topological group which can be written as the projective limit of nite

groups is called a pronite group. G(L=K) is an example. It is known that

G is a pronite group if and only if it is a compact totally disconnected

Hausdor topological group.

Let G be an abstract group. The set of nite-index normal subgroups

:= fN / G j G=N : nite groupg is a directed set by N N 0 if and only if

N N 0. Then, we have a projective system of nite groups G=N, N 2 .

Denition 2.3. For an abstract group G, we dene its pronite completion

b

G to be

b

G := proj lim(G=N);

N

where N runs over the nite index normal subgroups of G. We have a natural

group homomorphism

b

G!G

by patching G ! G=N together, and its image is dense.

Arithmetic fundamental groups 363

Fix a prime l. Let N run over the nite index normal subgroups of G

with the quotient G=N being an l-group, and take the similar projective limit.

Then we obtain the pro-l completion of G, denoted by Gl .

Theorem 2.1. (SGA1[7, Corollary 5.2, p.337]) Let X be a connected alge-

braic variety over C . Then

alg

1(X; a) ! 1 (X; a)

gives an isomorphism of the pronite completion of the topological funda-

mental group and the algebraic fundamental group,

d = alg

1(X; a) ! 1 (X; a) 1 (X; a):

A rough sketch of the proof is as follows. It is enough to prove that there

is a one-to-one correspondence between

alg alg

fN / 1 (X; a) j 1 (X; a)=N : nite groupg

and

fM Ma j M=K is nite Galois g

so that

1 (X; a)=N G(M=K)

alg (2.1)

=

holds through

alg

1 (X; a) ! G(Ma =K) ! G(M=K);

since then by taking the projective limit of (2.1) we have the desired isomor-

phism.

For N, we construct a nite topological unramied covering pN : YN !

˜

X with aN 2 pN1 (a) xed. This can be done as follows. Let X be the

˜˜

universal covering of X, and a 2 X be a point above a. (These topological

notions are recalled in x3.1 below.) It is well known that 1 (X; a) acts on

˜ ˜

the covering X=X from the right. Then, by taking the quotient of X by N,

we have a nite connected unramied covering pN : YN ! X, with a point

aN 2 YN being xed as the image of a. A \Riemann existence theorem" in

˜

SGA1[7, Theorem 5.1, p.332] asserts that YN is actually an algebraic variety

over C . We have a homomorphism C (Y ) ! Ma induced by aN , namely, a

rational function h 2 C (Y ) is a multivalued meromorphic function on X,

and it can be regarded as a germ at a by restricting h to a neighbourhood

of aN 2 Y . This gives a correspondence N 7! C (Y ) Ma , such that

1(X; a)=N G(C (Y )=C (X)), as desired. The converse correspondence is

=

M=K 7! Ker(1 (X; a) ! G(M=K)).

We shall again discuss this construction in a later section x3.

364

2.2. Arithmetic fundamental groups. So far, the algebraic fundamental

group is obtained from the topological one by pronite completion, and it

ignores the algebraic structure altogether. An interesting part comes from

the absolute Galois groups of the eld over which the algebraic variety is

dened.

Let K C be a subeld, K the algebraic closure of K in C . Assume that

X is a smooth algebraic variety over K, which is geometrically connected.

This can be paraphrased by saying that the dening polynomials of X have

coecients in K, and X(C ) is connected as a complex variety.

If X is dened over K, then K(X) denotes the rational function eld of

X over K. If X is dened by polynomials, K(X) is the set of the functions

which can be described as rational expressions of the coordinate variables

with coecients in K.

Theorem 2.2. (c.f. SGA1[7, Chapter XIII p.393]) If Y ! X is a nite

unramied covering of a complex algebraic variety, and X is dened over

K, then there is a model YK ! XK of varieties over K, giving Y ! X by

base extension

K C . If C (Y )=C (X) is Galois, then so is K(Y )=K(X) and

their Galois groups are isomorphic.

Corollary 2.1. Let Maalg be the subeld of Ma of the algebraic elements

over K(X). Then

alg

G(Ma =K(X)) = G(Ma=C (X)):

alg

One can show that if X is dened over K, then Ma =K(X) is a Galois

extension.

Denition 2.4. If X is a smooth algebraic variety over K, then we dene

its arithmetic fundamental group

alg alg

1 (X; a) := G(Ma =K(X)):

Corollary 2.2. We have a short exact sequence

alg alg

1 ! 1 (X

K K; a) ! 1 (X; a) ! G(K=K) ! 1: (2.2)

alg

This is the exact sequence coming from Galois extensions Ma K(X)

K(X). Note that G(K(X)=K(X)) = G(K=K) since K(X)

K K = K(X),

i.e., since X is geometrically connected.

Now the left term of the short exact sequence (2.2) is the pronite com-

pletion of the topological fundamental group 1 (X; a), which depends only

on the homotopy type of X, while the right term G(K=K) is the absolute

Arithmetic fundamental groups 365

Galois group of K, which controls the arithmetic of K and depends only on

the eld K.

But the middle term, or the extension, highly depends on the structure of

X as an algebraic variety, for example if K is a number eld. Grothendieck

conjectured [8] that X is recoverable from the exact sequence of pronite

group, if X is \anabelian." Recently much progress has been done in this

direction, by H. Nakamura, F. Pop, A. Tamagawa, and S. Mochizuki, and

others. For example, the conjecture is true for curves over p-adic elds

with nonabelian fundamental groups. An exposition on these researches in

Japanese is available, and its English translation is to appear [25].

2.3. Arithmetic monodromy. The exact sequence (2.2) can be considered

to be an algebraic version of the \ber exact sequence" for X ! SpecK.

Let us consider a topological bration F ! B which is locally trivial.

Take a point b 2 B, and let Fb be the ber at b. Fix x 2 Fb . Assume

2 (B) = f1g. Then, we have the so-called homotopy exact sequence

1 ! 1 (Fb ; x) ! 1 (F; x) ! 1 (B; b) ! 1:

In the arithmetic case, F ! B is X ! SpecK, b is a : SpecK ! SpecK,

and Fb is X K K, therefore (2.2) is an analogue of the above. Note that

alg

1 (SpecK; a) = G(K=K) holds, if we argue in the etale fundamental group,

see x3 below.

In topology theory, it is often more convenient to consider the monodromy

of the bration p : F ! B, as follows. The rough idea is as follows: take

a closed path

2 1 (B; b). Then, because of the local triviality of the

bration, we can consider the family Fc := p 1 (c) where c moves along

as

a deformation of Fb . Then,

induces a homeomorphism of Fb to itself. Such

a homeomorphism of Fb is not unique, but its isotopy class is well dened.

Thus, 1 (B; b) acts on various homotopy invariants of Fb , like cohomologies

and homotopy groups. Such a representation of 1 (B; b) is called monodromy

representation associated to F ! B. Let us concentrate on the monodromy

on the fundamental group of Fb . In this case, we take x 2 Fb , and take

any lift

2 1 (B; b) to F with starting point x, and call it

. Since Fb is

˜

connected (i.e. 0 (Fb ) = f1g), we may adjust

so that it lies in 1 (F; x).

˜

The ambiguity of

is up to a composition with an element of 1 (Fb ; x). Take

˜

an element 2 1 (Fb ; x). Then, its deformation along

can be considered

to be

˜ 1 . In this way 1 (B; b) acts on 1 (Fb ; x), but the action is well

˜

dened up to the choice of

, i.e., up to the inner automorphism of 1 (Fb ; x).

˜

366

Then we have an outer monodromy representation associated to F ! B,

: 1 (B; b) ! Aut(1 (Fb ; x))=Inn(1 (Fb ; x)) =: Out(1 (Fb ; x)):

This is easier to treat than the exact sequence itself, since often the funda-

mental groups of a ber and the base are well understood.

An analogue can be considered for the arithmetic version. Take 2

alg

G(K=K), then take any lift 2 1 (X; b), and let it act by conjugation

˜

alg

7! ()˜ 1 on 2 1 (X

K; b). This gives an arithmetic version of the

˜

monodromy representation

alg

X : G(K=K) ! Out(1 (X

K; b)): (2.3)

The left-hand side depends only on K, and the right-hand side depends

only on the homotopy type of X

C , and thus X connects arithmetic and

topology. Sometimes X is called the outer Galois representation associated

with X.

2.4. The projective line minus two points. As a simplest example, we

consider the case where X=Q is an ane line minus one point f0g, with

coordinate function z, and take a = 1 as the base point. The topological

alg

fundamental group of this space is Z. An element h of Ma is nitely

multivalued, say, N-valued. Then, h is a single-valued function of w, if we

take an N-th root w of z. Since we assumed that h is algebraic over Q (X), h

alg

is a rational function of w with coecients in Q . Thus, Ma is generated by

the functions z 1=N := exp(log z=N), with branch xed so as to have positive

p

real values around 1. Put N := exp(2 1=N). The eld extensions

alg

Ma = Q (z 1=N jN 2 N) Q (X) = Q (z) Q (X) = Q (z)

give

b

alg

1 (X

Q ; a) = proj lim G(Q (z1=N )=Q (z)) = proj lim(Z=N) =: Z:

N N

Here, the identication G(Q (z 1=N )=Q (z)) = Z=N is given by 2 G(Q (z 1=N )=

Q (z)) 7! b 2 Z=N , where b is the unique element such that (z 1=N ) 7!

b

N z1=N . Then, the generator

of 1 (X; a) which goes around 0 counter-

clockwise gives an element of the Galois group which maps

p

: z1=N 7! exp(2 1=N)z 1=N ;

b

so

is, through G(Q (z 1=N )=Q (z)) = Z=N , identied with 1 2 Z. To compute

b b

X : G(Q =Q) ! Out(Z) = Aut(Z);

Arithmetic fundamental groups 367

b

it suces to see the action on 1 2 Z, or equivalently on

2 G(Q (z 1=N )=Q (z)),

b

since it generates Z topologically. We take a lift of 2 G(Q =Q) = G(Q (z)=Q(z))

to 2 G(Q (z 1=N jN 2 N)=Q (z)). For this, for example, we may take acting

˜ ˜

trivially on z 1=N . Then,

˜ 1 maps

˜

p p

˜ ˜

z1=N 7! z1=N 7! exp(2 1=N)z1=N 7! exp(2 1=N)() z 1=N =

() (z1=N ):

Here,

b

() = (()N ) 2 proj lim(Z=N) = Z

N

b

is the cyclotomic character G(Q =Q) ! Z, that is, ()N 2 Z=N is uniquely

()

determined by : N 7! N . This shows (

) =

() , so X is nothing

N

but the cyclotomic character

b b

: G(Q =Q) ! Z = Aut(Z):

2.5. The projective line minus three points. A basic result by Belyi

[3] says

Theorem 2.3. Let X be the projective line minus three points 0; 1; 1 over

Q , and take an a 2 X. Then,

d

X : G(Q =Q) ! Out(1 (X; a))

is injective.

d

The right-hand side 1 (X; a) is the pronite completion of the free group

F2 with two generators, say, x; y. This implies that G(Q =Q) is contained in

a purely group-theoretic object. One can x a lift

b

X : G(Q =Q) ! Aut(F2 )

b b

from Out(F2 ), then two elements X ()(x), X ()(y) in F2 characterise . It

is convenient to x the lift so that X () : x 7! x() ; y 7! f (x; y)y() f (x; y) 1

bb

with f (x; y) 2 [F2 ; F2 ]. This is possible in a unique way [3] [12].

There are roughly two directions of research:

(i) Characterize the image of X in a group-theoretic way.

(ii) Use (), f (x; y) to describe Y () for other varieties Y .

For (i), the Grothendieck-Teichmller group was introduced by Drinfeld [6],

u

d d

and its pronite version GT was given by Ihara [12] [13]. GT is the subgroup

b d b

of Aut(F2 ) given by three conditions [12], with G(Q =Q) ,! GT Aut(F2 )

being Belyi's injection.

d

It is known that GT acts on the pronite completion of the braid groups

c d c

Bn ([6], for the pronite case, [15, Appendix]). G(Q =Q) ! GT ! Aut(Bn ) is

368

known to come from the representation on the algebraic fundamental group

of the conguration space of n points on the ane line [15].

Recently, Hatcher, Lochak, Nakamura, Schneps [10] [24] gave a subgroup

d d

of GT, G(Q =Q) GT (actually there are several variations of ,

d

but whether = GT or not is still open), so that acts on the pronite

completion of mapping class groups of type (g; n) systematically.

These researches are closely related to the program by Grothendieck [8],

which tries to study G(Q =Q) by towers of moduli spaces, but here we don't

pursue this direction.

These studies go with (ii): rst the Galois action is described in terms

d

of () and f (x; y), then GT-action is described so that it generalizes the

Galois action.

d

It is still open whether G(Q =Q) = GT or not.

3. Arithmetic fundamental groups by etale topology

The denition of arithmetic fundamental groups in the previous section is

concrete, but not intrinsic. For example, it cannot be applied for the positive

characteristic case. A more sophisticated general denition uses the notion

of Galois category [7, Chapter V].

3.1. Unramied coverings of a topological space and ber functors.

In this section, we forget about algebraic varieties and consider only topo-

logical spaces. Though our aim is the theory of fundamental groups without

paths, this section illustrates how the categorical machinary works.

Let X be an arcwise connected topological space. Let p : Y ! X be an

unramied covering of X. That is, p : Y ! X is a surjective continuous

map such that for any x 2 X there exists an open neighbourhood U of x

with each connected component of p 1 (U) being homeomorphic to U via p.

It is a nite unramied covering if the number of the connected components

is nite. The map p is called a covering map.

It is well known that the connected unramied coverings of X and the

subgroups of 1 (X; a) are in one-to-one correspondence, but we shall start

by recalling this fact.

Fix a point a on X, and consider the category of connected unramied

coverings with one point specied. Its objects are the pairs of an unramied

connected covering pY : Y ! X and a point b 2 pY 1 (a), and its morphisms

are the continuous maps f : Y 0 ! Y compatible with the covering maps:

pY f = pY and f(b0 ) = b. We denote this category by Con(X; a).

0

Arithmetic fundamental groups 369

Assume that X is arcwise connected, locally arcwise connected, and locally

simply connected. Then there is a category equivalence between Con(X; a)

and fN : subgroup of 1 (X; a)g: The correspondence is: for (Y; b) 2 Con(X; a),

we have 1 (Y; b) ! 1 (X; a), which is injective, so 1 (Y; b) considered as a

subgroup corresponds to (Y; b). For the converse, for N 1 (X; a), we

consider the set of classes of paths

YN := fa path starting from a with arbitrary end point in Xg= N ;

where the equivalence is given by

N

0 if and only if they have the

same end point and the closed path

1

0 lies in the homotopy class in

N 1 (X; a). YN has a specied point aN , which corresponds to the trivial

path at a. Taking the end point of the path gives a map pN : YN ! X,

which is locally a homeomorphism by the assumption on locally arcwise

simply connectedness. Thus, (YN ; aN ) ! (X; a) is an object of Con(X; a).

If N is a normal subgroup of 1 (X; a), then pN : YN ! X is called a

Galois covering. In this case, 1 (X; a) acts on YN =X from the right. Take

2 1(X; a), and let act by [

] 2 YN 7! [

] 2 YN . We need to check

the well-denedness; if

N

0 then

N

0 should hold, which requires

that, if

1

0 2 N, then 1

1

0 2 N, that is, N must be normal. In

this case, we have

1 (X; a)=N Aut(YN =X)o ;

=

where o denotes the opposite, i.e., the group obtained by reversing the order

of multiplication.

If N is f1g, then Yf1g is called the universal covering of X, and denoted

˜

by X. Then, we have

1(X; a) Aut(X=X)o :

˜

=

Thus, if we can dene a universal covering without using paths, then we

can dene the fundamental group. Essentially this is the case for schemes,

a projective system (or a pro-object) plays the role of the universal cover-

˜

ing. However, if we adopt Aut(X=X) as the denition of the fundamental

group, it is not clear where the dependence on the choice of a disappeared.

The functoriality of 1 is also not clear, and the denition of fundamental

groupoid is dicult.

A smart idea is to use all (possibly non connected) coverings of X.

We consider the category CX of unramied coverings of X, i.e., an object

is a covering f : Y ! X, and a morphism is Y 0 ! Y compatible with f 0 ; f.

Thus, dierently from Con(X; a), a point is not specied, and the covering

370

may not be connected. Let a; b 2 X be points. The discrete set f 1 (a) Y

is called the ber of f at a. Take an element a0 2 f 1 (a): By lifting of the

path

2 1 (X; a; b) to a path starting from a0 in Y we have

2 1 (Y ; a0 ; b0 ),

˜

and the end point b0 of

is uniquely determined. Thus, we obtain a bijection

˜

cf (

) : f 1(a) ! f 1 (b); a0 7! b0 ;

in other words, an action of groupoid

cf : 1 (a; b) f 1 (a) ! f 1(b)

with (

; a0 ) 7! cf (

)(a0 ) = b0 . cf (

) is a bijection from f 1 (a) to f 1 (b),

which is compatible with morphisms Y 0 ! Y . The assignment (f : Y !

X) 7! f 1(a) gives a covariant functor

CX ! Sets

from the category of unramied coverings to the category of sets, which we

denote by Fa . Each element of 1 (X; a; b) gives a natural transformation

Fa 7! Fb , which is a natural equivalence.

On the other hand, a natural transformation Fa 7! Fb always comes

from some element of 1 (X; a; b). To see this, note that CX has a spe-

˜

cial object, namely, the universal covering p : X ! X. Actually, the ob-

˜

ject X represents the ber functor Fa , i.e., we have a natural isomorphism

Fa (Y ) HomC (X; Y ). To x the isomorphism, it is enough to x a point

˜

=

˜˜

a 2 X above a. Then, for every point b 2 f 1 (a), there exists a unique ho-

X

˜

momorphism X ! Y with a 7! b, by the property of the universal covering.

˜

We shall see that a natural transformation Fa ! Fb comes from an element

˜

of 1 (X; a; b). A natural transformation

: Fa ! Fb gives a map

(X) :

˜ ˜

Fa (X) = p 1(a) ! Fb (X) = p 1 (b). Then, once we choose a a 2 p 1(a),

˜

1 (b), which we denote ˜. Now, by the naturality of

˜a

we have

(X)(˜) 2 p b

˜ ˜

, it commutes with any automorphism of X=X. Since X=X is a Galois

˜˜

cover, for any a0 2 p 1 (a), there is an automorphism : X ! X such that

˜

(˜) = a0 . This shows that the image of a0 by

(X) must be (˜). Thus,

˜

a˜ b

˜

(X) : p 1(a) ! p 1 (b) is bijective, and it is uniquely determined, once ˜ is

˜ b

chosen, by compatibility with . Let

be a path from a to ˜ in X. (˜ ) for

˜b˜

˜

˜ ˜ ˜

2 Aut(X=X) gives the bijection Fa (X) ! Fb (X).

Since connected components of other coverings are quotients of the uni-

versal covering, it is easy to see that any choice of ˜ uniquely gives a natural

b

transformation Fa ! Fb , and it comes from the path from a to b which is the

projection of

to X. Thus, we have Hom(Fa ; Fb ) 1 (X; a; b) canonically.

˜ =

Arithmetic fundamental groups 371

This identies the groupoid f1 (X; a; b)ja; b 2 Xg with the groupoid

whose objects are Fa : CX ! Sets, and the morphisms are the natural

transformations Fa ! Fb , which are automatically invertible.

˜

Since Fa is representable by X, we can identify 1 (X; a) = Aut(Fa ) with

˜

the opposite group of Aut(X=X), because of the Yoneda lemma

Aut(Fa ) Aut(HomC (X; )) = Aut(X)o ;

˜ ˜

=

where is determined once a is specied.

˜

=

Now, Fa : CX ! Sets induces a functor from CX to the category 1 (X; a)-

set, whose objects are the sets with an action of 1 (X; a) = Aut(Fa ), and

whose morphisms are the maps compatible with this action. This functor

gives a categorical equivalence. To establish the equivalence, let S be a

1 (X; a)-set. We decompose S to orbits, and construct a covering corre-

sponding to each orbit, then take the direct sum. For an orbit, take one

point and let H be the stabilizer of the point. The covering corresponding

to H gives the desired covering for that orbit.

Proposition 3.1. The category CX of unramied coverings of an arcwise

and locally arcwise simply connected topological space X is categorically

equivalent to 1 (X; a)-set. The equivalence is given by the ber functor

Fa : (f : Y ! X) 7! f 1(a):

The 1 (X; a) action on f 1(a) comes from 1 (X; a) = Aut(Fa ).

3.2. Finite coverings. Roughly speaking, the etale fundamental groupoid

of a connected scheme is dened by using the category of unramied covers

in an algebraic sense. We don't have an analogue of real-one dimensional

\path", say, in the positive characteristic world, but we have a good category

and functors which allow us a categorical formulation of the algebraic (or

etale) fundamental groupoid.

Before proceeding to etale fundamental groups, we note what will occur if

we restrict CX to the category CX;fin of nite unramied coverings and the

ber functor

Fa : CX;fin ! Finsets;

where Finsets is the category of nite sets. This modication is essential

when we work in the category of algebraic varieties.

A problem is that this functor Fa is not representable by an object in

CX;fin. So, instead of the universal covering, we use a projective system,

called a pro-object, which represents the functor Fa . Let (P )(2) be the

system of all connected nite Galois coverings of X with one point above a

372

specied. That is, an object of P is a pair (aY ; Y ) with a connected nite

unramied covering pY : Y ! X and aY 7! a, and a morphism is Y 7! Y 0

which maps aY to a0Y . We consider (P ) as a projective system in CX;fin

(thus, aY for each Y gives no restriction on morphisms in CX;fin ).

It can be shown that

Hompro C ((P ); Z) := lim HomC (P ; Z) pZ 1 (a):=

!

X;f in X;f in

Then, one can show

d

Aut(Fa ) opposite Aut((P )) = proj lim Aut(P ) opposite 1 (X; a):

= =

The rst identity comes from the Yoneda lemma, and the last equality is

because Aut(P )o is the nite quotient of 1 (X; a) correspondinng to P .

Both equalities are xed because a system (a ) is xed. In this case, the

category CX;fin is equivalent to the category of nite sets with continuous

d d

action by 1 (X; a), which we call the category of nite 1 (X; a)-sets.

Proposition 3.2. CX;fin is categorically equivalent to the category of nite

d

1(X; a)-sets. The equivalence is given by the ber functor Fa .

3.3. Etale fundamental groups. In the following, we work in the category

of schemes. We shall only sketch the story of the etale fundamental groups

here. For the precise notions, see SGA1 [7].

Denition 3.1. Let f : X ! Y be a morphism of nite type, x 2 X,

y := f(x) 2 Y . We say f is unramied at x, if OX;x =f(my )OX;x is a nite

direct sum of nite separable eld extensions of k(y). If moreover f is

at

at x, then f is said to be etale at x. If f is etale at every point x 2 X, then

f is called etale. If moreover f is nite and Y is connected, then f : X ! Y

is called an etale covering.

If X and Y are algebraic varieties over an algebraically closed eld K C ,

and if x is a closed point, then it is known that f is etale at x 2 X if and only

if f is nite and unramied as an analytic morphism [7, Chapter XII]. Thus,

the etale morphisms correctly generalize the notion of unramied coverings.

Denition 3.2. Let X be a locally noetherian connected scheme. Let CX be

the category of etale coverings of X, i.e., objects are nite etale f : Y ! X,

and morphisms are Y 0 ! Y compatible with f 0; f.

This category is an analogue of that of nite unramied coverings of a

connected topological space X.

Arithmetic fundamental groups 373

There is a notion of Galois category. It consists of a category C and a

functor called the ber functor, F : C ! Finsets, and satises six axioms

stated in [7, Chapter V-x4], which we shall omit here. Once we have a Galois

category, we can dene its fundamental group with base point F as Aut(F ),

i.e., the group of natural transformation from F to itself. This becomes a

pronite group. Two ber functors F; G : C ! Finsets are non-canonically

isomorphic, and the set of natural transformations from F to G is a groupoid,

with objects ber functors and morphisms natural transformations.

Similarly to the topological case, one can show the category equivalence

between CX and the category of the nite sets with Aut(F )-continuous ac-

tion.

Theorem 3.1. [7, Chapter V] Let X be a locally noetherian connected scheme.

Take a geometric point a : Spec

! X, where

is an algebraically closed

eld. Then, the category CX of nite etale covers of X, with ber functor

Fa : CX ! Finset, (f : Y ! X) 7! f 1 (a) = Y X Spec

, is a Galois

category.

Denition 3.3.

alg

1 (X; a) := Aut(Fa ):

Thus, the category of nite etale covers of X is equivalent to the cagegory

alg

of nite sets with 1 (X; a)-action. Similarly to the topological case, we

may use (P ), the projective system of connected nite Galois cover of X,

with a geometric point a above a compatibly specied. Then, by forgetting

a, we may regard (P ) as a projective system in CX , which pro-represents

Fa . Then, we have

alg

1 (X; a) := Aut(Fa ) = proj lim(P ):

alg

In this setting, the functoriality of 1 is easy. For f : X ! Y , we have

alg alg

1 (X; a) ! 1 (Y; f(a)), since ( ) Y X is a functor f : CY ! CX , and

Fa f = Ff(a) : CY ! Finsets holds, so an element of Aut(Fa ) gives an

element of Aut(Ff(a) ), inducing

alg alg

Aut(Fa ) = 1 (X; a) ! Aut(Ff(a) ) = 1 (Y; f(a)):

Let X be a geometrically connected scheme over a eld K. Let a :

SpecK ! X be a geometric point. The sequence

X

K ! X ! SpecK

374

gives a short exact sequence

alg alg alg

1 ! 1 (X

K K; a) ! 1 (X; a) ! 1 (SpecK; SpecK) ! 1;

which is nothing but (2.2) (for a proof, see [7, Chapter X, XIII]).

It is easy to show that an object of CSpecK is a direct sum of a nite number

of nite separable extensions of K, and morphisms are usual homomorphisms

of algebras over K. A connected object is the spectrum of a eld. Once we

x a geometric point a : Spec

! SpecK, the pro-object which represents

the ber functor Fa is the system of nite Galois extensions of K inside

(this inclusion into

corresponds to choosing a point aN in the ber

YN ! X above a). Thus, we have

1 (SpecK; Spec

) contra proj lim Aut(SpecL=SpecK) contra proj lim G(L=K);

alg = =

L L

where L runs through the nite Galois extensions of K in

, and it is nothing

but G(K sep=K) where K sep

is the separable closure of K in

.

Proposition 3.3. Let

be an algebraically closed eld, and let K

be

a subeld. Then

1(SpecK; Spec

) = G(K sep=K)

holds, where K sep is the separable closure of K in

.

The geometric part of the fundamental group can be obtained as follows.

A theorem called \Riemann's existence theorem" in SGA1[7, Theorem 5.1

P.332] assures that a nite unramied covering of an algebraic variety X

over C is algebraic and etale over X, i.e., CX and CX;fin are categorically

equivalent. An argument in SGA1[7, Chapter XIII] says that if X is an

algebraic variety over an algebraically closed eld K C , then the base

change ( )

K C gives a category equivalence between CX

C and CX . These

category equivalences are compatible with ber functors, so we have

d

alg alg

1 (X; a) = 1 (X

C ; a) = AutC (Fa ) = 1 (X; a):

X;f in

Exercise 3.1. Show that the universal covering of the complex plane C mi-

nus 0 is still an algebraic variety, but that of C minus 0 and 1 is not. Even

in the former case, the covering map is not algebraic.

Exercise 3.2. Describe the category CX of nite etale coverings where X

is the ane line minus one point 0 over Q .

Arithmetic fundamental groups 375

4. Arithmetic mapping class groups

4.1. The algebraic stack Mg;n over SpecZ. I do not give the denition

of an algebraic stack, the denition of the fundamental group of an algebraic

stack, and so on, in this note, simply because of my lack of ability to state

it concisely. I would just like to refer to [5] for the denition of an algebraic

stack, the moduli stack of genus g curves and the moduli stack of stable

curves and to [19] for the case of n pointed genus g curves. For the arithmetic

fundamental group of the moduli stack, see [26] (but this article requires

prerequisites on etale homotopy [1]).

We just sketch the picture. Let g, n be integers with 2g 2 + n > 0. We

want to introduce the universal property of the moduli stack of n pointed

genus g curves.

Denition 4.1. A family of n pointed genus g curves over a scheme S (a

family of (g; n)-curves in short), C ! S, is a proper smooth morphism

C ! S, whose bers are a proper smooth curves of genus g, with n sections

s1; s2; : : : ; sn : S ! C given, where the images of the si do not intersect

each other, and C ! S is the complement of the image of these sections in

C .

What we want is the universal family Cg;n ! Mg;n , which itself is a

family of (g; n)-curves, with the universal property that for any family of

(g; n)-curves C ! S, we have a unique morphism S ! Mg;n such that C is

isomorphic to the base change Cg;n M S. Unfortunately, we don't have

g;n

such a universal family in the category of schemes. So, we need to enlarge

the category to that of algebraic stacks.

I just describe some properties of algebraic stacks here. The category of

algebraic stacks contains the category of schemes as a full subcategory, and

algebraic stacks behave similarly to schemes. In the category of algebraic

stacks, we have the correct universal family Cg;n ! Mg;n.

The notion of nite morphisms, etale morphisms, connectedness, etc. can

be dened for algebraic stacks. In particular, for a connected algebraic stack,

we have the category of its nite etale covers. It becomes a Galois category,

and we have its etale fundamental group.

The algebraic stack Mg;n is dened over SpecZ. But from now on, we

consider Mg;n over SpecQ.

4.2. The arithmetic fundamental group of the moduli stack. Takayuki

Oda [26] showed that the etale homotopy type of the algebraic stack Mg;n

Q

376

is the same as that of the analytic stack Man , and using Teichmller space, u

g;n

showed that the latter object has the etale homotopy type K( \ in the g;n; 1)

sense of Artin-Mazur [1], where g;n is the Teichmller-modular group or

u

the mapping class group of n-punctured genus g Riemann surfaces.

This shows as a corollary

1 (Mg;n Q ; a) bg;n;

alg =

and gives a short exact sequence

alg alg

1 ! 1 (Mg;n Q ; a) ! 1 (Mg;n ; a) ! G(Q =Q) ! 1: (4.1)

Also, the vanishing of 2 of Mg;n gives a short exact sequence

alg alg alg

1 ! 1 (Cg;n ; b) ! 1 (Cg;n; b) ! 1 (Mg;n ; a) ! 1; (4.2)

where a is a geometric point of Mg;n, Cg;n is the ber on a, b a geometric

point of Cg;n. Hence, Cg;n is a (g; n)-curve over an algebraically closed eld,

alg

and 1 (Cg;n ; b) is isomorphic to the pronite completion of the orientable

surface of (g; n)-type, i.e., the pronite completion of

g;n :=< 1 ; 1 ; : : : ; g ; g ;

1 ;

2 ; : : : ;

n ;

[1 ; 1 ][2 ; 2 ] [g ; g ]

1

n = 1 >; (4.3)

where

i are paths around the punctures, i ; i are usual generators of 1

of an orientable surface.

Once we are given a short exact sequence (4.2), in the same way as x2.3,

we have the monodromy representation

b

g;n : 1 (Mg;n ; a) ! Out(1 (Cg;n; b)) Out(g;n);

alg alg =

which is called arithmetic universal monodromy representation. This con-

tains the usual representation of the mapping class group g;n in the fun-

damental group of the orientable surface g;n , since the restriction of g;n

to

alg alg

1 (Mg;n

Q ; b) 1 (Mg;n ; b)

coincides with

bg;n ! Out(g;n); d

which comes from the natural homomorphism

g;n ! Out(g;n ):

This latter may be called the topological universal monodromy. What do

we get if we consider the arithmetic universal monodromy instead of the

Arithmetic fundamental groups 377

topological one? There is an interesting phenomenon: \arithmetic action

gives an obstruction to topological action."

5. A conjecture of Takayuki Oda

5.1. Weight ltration on the fundamental group. Let C be a (g; n)-

curve, and g;n be its (classical) fundamental group. We dene its weight

ltration as follows.

Denition 5.1. (Weight ltration on g;n.)

We dene a ltration on g;n

g;n = W 1 g;n W 2 g;n W 3 g;n

By W 1 := g;n ,

W 2 :=< [g;n ; g;n ];

1 ;

2 ; : : : ;

n >norm;

where <>norm denotes the normal subgroup generated by elements inside <>

and [; ] denotes the commutator product,

i are elements in the presentation

(4.3), and then

W N :=< [W i; W j ]ji + j = N >norm

inductively for N 3.

Fix a prime l. We dene a similar ltration on the pro-l completion

of lg;n. There, <>norm and [; ] are the topological closure of the normal

subgroup generated by the elements inside <>, the commutators, respectively.

It is easy to check that grj (g;n ) := W j =W j 1 is abelian, and is central

in g;n =W j 1. In other words, W is the fastest decreasing central ltration

with W 2 containing

1 ; : : : ;

n . It is known that each grj is a free Z-module

(free Zl-module, respectively for pro-l case) of nite rank [2] [18].

This notion of weight ltration came from the study of the mixed Hodge

structure on the fundamental groups, by Morgan and Hain [9], but for the

particular case of P1 f0; 1; 1g, Ihara had worked on this [11] independently,

from an arithmetic motivation.

For x 2 W i ; y 2 W j , [x; y] 2 W i j holds, and this denes a Z-bilinear

product gr i

gr j ! gr i j . We dene

Grg;n := 1 gr i (g;n ):

i=1

With the product [x; y], Grg;n becomes a Lie algebra over Z. For lg;n , we

have a Lie algebra over Zl.

378

Denition 5.2. (Induced ltration) We equip := Aut(g;n), with the

following ltration = I0 I 1 I 2 , called induced ltration:

2 I j , for any k 2 N and x 2 W k (g;n ),

(x)x 1 2 W k j (g;n) holds.

˜

We pushout this ltration to Out(g;n ). For an outer representation

: G ! Out(g;n )

of any group G, we pullback the ltration to G, and call it induced ltration:

G = I 0(G) I 1 (G) I 2 (G) :

The same kind of ltration is dened for G ! Out(lg;n ).

In this case, we dene

Gr(G) := 1 gr i (G) = 1 I i (G)=I i 1 (G)

i=1 i=1

(note that i starts from 1, not 0), then Gr(G) becomes a Lie algebra. By

denition, if we induce ltrations by G ! G0 ! Out(g;n ), then Gr(G) ,!

Gr(G0). By [18] [2], GrOut(g;n ) injects to GrOut(lg;n ), and hence if

G ! Out(g;n ) ! Out(lg;n )

factors through G0 ! Out(lg;n ), then GrG ,! GrG0 holds.

The natural homomorphism

g;n ! Out(g;n )

gives a natural ltration to g;n , which seems to go back to D. Johnson [17].

By composing with the natural morphism

d

Out(g;n ) ! Out(lg;n );

we have

alg

1 (Mg;n ; a) ! Out(lg;n );

alg alg

hence 1 (Mg;n ; a), 1 (Mg;n

Q ; a), is equipped with an induced ltra-

tion, and we have a natural injection

alg alg

Gr1 (Mg;n; a) ,! Gr1 (Mg;n

Q ; a);

and the image is a Lie algebra ideal.

Conjecture 5.1. (Conjectured by Takayuki Oda) The quotient of

alg

Gr(1 (Mg;n ; a))

by the ideal

alg

Gr(1 (Mg;n

Q ; a))

is independent of g; n for 2g 2 + n 0.

Arithmetic fundamental groups 379

This conjecture is almost proved by a collection of works by Nakamura,

Ihara, Takao, myself, et. al. [21] [23] [16],

Theorem 5.1. The quotient of

alg

Gr(1 (Mg;n ; a))

Z Q l

l

by

alg

Gr(1 (Mg;n

Q ; a))

Z Q l l

is independent of g; n for 2g 2 + n 0.

The signicance of this result is that for M0;3 = SpecQ, the Lie algebra

is understood to some extent by deep results such as Anderson-Coleman-

Ihara's power series and Soul's non-vanishing of Galois cohomology, and it

e

implies a purely topological consequence: an obstruction to the surjectivity

of the Johnson homomorphisms.

5.2. Obstruction to the surjectivity of Johnson morphisms. For sim-

plicity, assume n = 0, and hence g denotes the mapping class group

of genus g Riemann surfaces. Take 2 I m g , and take a suitable lift

2 I m Aut(g ) as in Denition 5.2. Then, () 1 2 W m 1 g for any

˜ ˜

2 g . The map

g ! W m 1 g ; 7! () 1

˜

gives a linear map g =W 1 g ! gr m 1 g . We denote H := g =W 1 g

for homology, then we have Poincare duality H H, and dene

=

hg; (m) := Ker(Hom(H; gr m 1 g ) ! gr m 2 g );

where

Hom(H; gr m 1 g ) ! gr m 2 g

comes from

Hom(H; gr m 1 g ) H

gr m 1 g ! gr m 2 g :

[;]

=

The lift in Aut(g ) is mapped into hg; (m). The ambiguity of taking the

˜

lift in Aut is absorbed by taking the quotient by the action of gr m g by

7! [; x] for x 2 gr m g , and we have an injective morphism

gr m ( g ) ,! hg; =gr m g ( Hom(H; gr m 1 g )=gr m g ):

This is called the Johnson homomorphism [17] (see Morita [22]).

D. Johnson proved that this is an isomorphism for m = 1, but for general

m it is not necessarily surjective; actually S. Morita gave an obstruction

called Morita-trace [22] for m odd, m 3.

380

We can dene the same ltration for

alg

1 (Mg ) ! Out(lg );

and then we have an injection

alg

gr m (1 (Mg )) ,! (hg; =gr m g )

Zl Hom(H; gr m 1 lg )=gr m lg :

Theorem 5.1 asserts that

alg alg

gr m (1 (Mg;n

Q )) ,! gr m (1 (Mg;n ))

is not surjective for some m, it has cokernel of rank independent of g; n. As

I am going to explain in the next section, for (g; n) = (3; 0), it is known that

this cokernel is nontrivial at least for m = 4k + 2, k 1 (and the rank has

a lower bound which is a linear function of m). Thus,

alg

gr m (1 (Mg;n

Q )) ,! hg; =gr m g

Zl

has also cokernel of at least that rank. This homomorphism is given by

Zl

from the Johnson homomorphism, hence this gives an obstruction to the

surjectivity of Johnson homomorphisms, which is dierent from Morita's

trace. The existence of such an obstruction was conjectured by Takayuki

Oda, and proved by myself [21] and H. Nakamura [23] independently.

5.3. The projective line minus three points again. Let P1 denote 011

the projective line minus three points over Q . This curve does not deform

over Q , and hence the universal family is trivial,

C0;3 = P1 and M0;3 = SpecQ:

011

Geometrically, thus, there is no monodromy, but arithmetically this has huge

monodromy as proved by Belyi (see x2.3).

Fix a prime l. We shall consider pro-l completion F2l of the free group F2

in two generators, so we have

b

alg

1 (P1

Q ; a) = F2 ! F2l :

011

Then, we have a group homomorphism

lP1 : G(Q =Q) ! Out(F2l ):

011

The weight ltration for the (g; n) = (0; 3) curve essentially coincides with

the lower central series

F2l = F2l (1) F2l (2) F2l (3)

Arithmetic fundamental groups 381

dened inductively by F2l (1) = F2l , F2l (m) = [F2l (m 1); F2l ] (here [; ] denotes

the closure of the commutator); the correspondence is

W 2m+1 (F2l ) = W 2m (F2l ) = F2l (m) (m 1):

Ihara [11] started to study the ltration of G(Q =Q) induced by this ltration,

independently of the notion of weight etc. Note that, the case of (0; 3) in

Theorem 5.1, the geometric part vanishes, so the quotient in the theorem is

nothing but just GrG(Q=Q) in this case.

The following is a corollary of the theory of power-series by Anderson,

Coleman, Ihara, together with Soul's nonvanishing of Galois cohomology

e

(there is a list of references, see the references in [11]).

Theorem 5.2. In the Lie algebra Gr(G(Q =Q), each gr m (G(Q =Q)) does

not vanish for odd m 3.

Roughly speaking, by using Anderson-Coleman-Ihara's power-series, one

can construct a homomorphism

gr 2m (G(Q =Q)) ! HomG(Q=Q) (1 (SpecZ[1=l]); Zl(m)):

It can be described as a particular Kummer cocycle, and the morphism does

not vanish for odd m 3 by Soul's result. The right-hand side is rank 1

e

up to torsion. An element 2m 2 gr 2m (G(Q =Q)) which does not vanish in

the right-hand side is called a Soul element.

e

The following conjecture is often contributed to Deligne [4].

Conjecture 5.2. (i) Gr(G(Q =Q))

Q l is generated by 2m (m 3; odd).

(ii) Gr(G(Q =Q))

Q l is a free graded Lie algebra.

The rank of Grm (G(Q =Q)) as Zl-module has a lower bound which is a linear

function of m, and these conjectures are veried for m 11 [20] [27], but

both conjectures seem to be still open. Ihara [14] recently showed that (ii)

implies (i).

382

References

[1] M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Math. 100, Springer, 1969.

[2] M. Asada and M. Kaneko, On the automorphism groups of some pro-l fundamental

groups, Advanced studies in Pure Math. 12 1987, 137{159.

[3] G.V. Bely, On Galois extensions of a maximal cyclotomic eld, Math USSR Izv. 14

(1980), 247-256.

[4] P. Deligne, Le groupe fondamental de la droite projective moins trois points, in \Galois

groups over Q," Publ. MSRI 16 1989, 79{298.

[5] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus,

Publ. Math. I.H.E.S 36 (1969) 75-110.

[6] V.G. Drinfel'd, On quasitriangular quasi-Hopf algebras and a group closely connected

with Gal(Q=Q), Algebra i Analiz 2 (1990), 114{148; English transl. Leningrad Math.

J. 2 (1991), 829{860.

[7] A. Grothendieck and M. Raynaud, Rev^tement Etales et Groupe Fondamental (SGA

e

1), Lecture Notes in Math. 224, Springer-Verlag 1971.

[8] A. Grothendieck, Esquisse d'un programme, mimeographed note (1984), in \Geomet-

ric Galois Actions 1," London Math. Soc. Lect. Note Ser. 242 (1997) 7{48.

[9] R. M. Hain, The geometry of the mixed Hodge structure on the fundamental group, Al-

gebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 247{282, Proc. Sympos.

Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI, 1987.

[10] A. Hatcher, P. Lochak and L. Schneps, On the Teichmller tower of mapping class

u

groups, to appear in J. Reine Angew. Math.

[11] Y. Ihara, Pronite braid groups, Galois representations and Complex multiplications,

Ann. Math. 123 (1986), 43{106.

[12] Y. Ihara, Braids, Galois groups, and some arithmetic functions, Proceedings of the

d

ICM 90 (I), 1991 99-120.

[13] Y. Ihara, On the embedding of Gal(Q=Q) into GT, in \The Grothendieck Theory of

Dessins d'Enfants," London Math. Soc. Lecture Note Series 200, Cambridge Univ.

Press, 1994, pp. 289{305.

[14] Y. Ihara, Some arithmetic aspects of Galois actions on the pro-p fundamental group

of P1 f0; 1; 1g, RIMS preprint 1229, 1999.

[15] Y. Ihara and M. Matsumoto, On Galois Actions on Pronite Completions of Braid

Groups, in AMS Contemporary Math. 186 \Recent Developments in the Inverse

Galois Problem." 1994, 173{200.

[16] Y. Ihara and H. Nakamura, On deformation of maximally degenerate stable marked

curves and Oda's problem, J. Reine Angew. Math. 487 (1997), 125{151.

[17] D. Johnson, A survey of the Torelli group, Contemporary Math. 20 (1983), 165-179.

[18] M. Kaneko, Certain automorphism groups of pro-l fundamental groups of punctured

Riemann surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 36 (1989), 363{372.

[19] F.F. Knudsen, The projectivity of the moduli space of stable curves II: The stacks

M , Math. Scad. 52 (1983), 161-199.

g;n

[20] M. Matsumoto, On the Galois image in the derivation algebra of 1 of the projective

line minus three points, AMS Contemporary Math. 186 \Recent Developments in

the Inverse Galois Problem" 1994, 201{213.

[21] M. Matsumoto, Galois representations on pronite braid groups on curves, J. Reine.

Angew. Math. 474 (1996), 169{219.

Arithmetic fundamental groups 383

[22] S. Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke.

Math. J. (1993), 699{726.

[23] H. Nakamura, Coupling of universal monodromy representations of Galois-

Teichmller modular groups, Math. Ann. 304 (1996), 99-119.

u

[24] H. Nakamura and L. Schneps, On a subgroup of Grothendieck-Teichmller group act-

u

ing on the tower of pronite Teichmller modular groups, preprint, available from

u

http://www.comp.metro-u.ac.jp/˜h-naka/preprint.html.

[25] H. Nakamura, A. Tamagawa and S. Mochizuki, Grothendieck's conjectures concerning

fundamental groups of algebraic curves, (Japanese) Sugaku 50 (1998), no. 2, 113{129.,

English translation is to appear in Sugaku Exposition.

[26] T. Oda, Etale homotopy type of the moduli spaces of algebraic curves, in \Geometric

Galois Actions 1," London Math. Soc. Lect. Note Ser. 242 (1997) 85{95.

[27] H. Tsunogai, On ranks of the stable derivation algebra and Deligne's problem, Proc.

Japan Acad. Ser. A Math. Sci. 73 (1997), no. 2, 29{31.

ñòð. 12 |