<< стр. 12(всего 12)СОДЕРЖАНИЕ
allowed to be in nite. It is called an (in nite) Galois extension, if any nite
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 in nite 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
de ned 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-de ned 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 f1 (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.
De nition 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)):
De nition 2.2. We de ne 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 \pro nite
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. Pro nite groups and algebraic fundamental groups. We want
to describe the (possibly in nite) Galois group G(L=K) for L := Ma , K :=
C (X). The answer is that it is the pro nite completion of 1 (X; a).
To see this, for a while, let L=K be a general in nite 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 de ne its projective limit proj lim2 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 pro nite group. G(L=K) is an example. It is known that
G is a pro nite 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 .
De nition 2.3. For an abstract group G, we de ne its pro nite 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 pro nite 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 unrami ed 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 unrami ed 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 pro nite 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
de ned.

Let K  C be a sub eld, 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 de ning polynomials of X have
coecients in K, and X(C ) is connected as a complex variety.
If X is de ned over K, then K(X) denotes the rational function eld of
X over K. If X is de ned by polynomials, K(X) is the set of the functions
which can be described as rational expressions of the coordinate variables
with coecients in K.
Theorem 2.2. (c.f. SGA1[7, Chapter XIII p.393]) If Y ! X is a nite
unrami ed covering of a complex algebraic variety, and X is de ned 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 sub eld 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 de ned over K, then Ma =K(X) is a Galois
extension.
De nition 2.4. If X is a smooth algebraic variety over K, then we de ne
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 pro nite 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  that X is recoverable from the exact sequence of pro nite
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 .
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 de ned.
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
˜

de ned 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 ane 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 coecients 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 identi cation 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 , identi ed with 1 2 Z. To compute
b b
X : G(Q =Q) ! Out(Z) = Aut(Z);
Arithmetic fundamental groups 367

b
it suces 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
 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 pro nite 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  .
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-Teichmller group was introduced by Drinfeld ,
u
d d
and its pro nite version GT was given by Ihara  . GT is the subgroup
b d b
of Aut(F2 ) given by three conditions , with G(Q =Q) ,! GT  Aut(F2 )
being Belyi's injection.
d
It is known that GT acts on the pro nite completion of the braid groups
c d c
Bn (, for the pro nite case, [15, Appendix]). G(Q =Q) ! GT ! Aut(Bn ) is
368

known to come from the representation on the algebraic fundamental group
of the con guration space of n points on the ane line .
Recently, Hatcher, Lochak, Nakamura, Schneps   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 pro nite
completion of mapping class groups of type (g; n) systematically.
These researches are closely related to the program by Grothendieck ,
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 de nition 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 de nition uses the notion
of Galois category [7, Chapter V].
3.1. Unrami ed 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
unrami ed 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 unrami ed 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 unrami ed 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 unrami ed
coverings with one point speci ed. Its objects are the pairs of an unrami ed
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 speci ed 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-de nedness; 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 de ne a universal covering without using paths, then we
can de ne 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 de nition 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 de nition of fundamental
groupoid is dicult.
A smart idea is to use all (possibly non connected) coverings of X.
We consider the category CX of unrami ed 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, di erently from Con(X; a), a point is not speci ed, 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 unrami ed 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 identi es the groupoid f1 (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 speci ed.
˜
=
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 unrami ed 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 de ned by using the category of unrami ed 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 unrami ed coverings and the
ber functor
Fa : CX;fin ! Finsets;
where Finsets is the category of nite sets. This modi cation 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

speci ed. That is, an object of P is a pair (aY ; Y ) with a connected nite
unrami ed 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 .
De nition 3.1. Let f : X ! Y be a morphism of nite type, x 2 X,
y := f(x) 2 Y . We say f is unrami ed 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 unrami ed as an analytic morphism [7, Chapter XII]. Thus,
the etale morphisms correctly generalize the notion of unrami ed coverings.
De nition 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 unrami ed 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 satis es six axioms
stated in [7, Chapter V-x4], which we shall omit here. Once we have a Galois
category, we can de ne its fundamental group with base point F as Aut(F ),
i.e., the group of natural transformation from F to itself. This becomes a
pro nite 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.
De nition 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 speci ed. 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 sub eld. 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 unrami ed 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 ane 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 de nition
of an algebraic stack, the de nition 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  for the de nition of an algebraic
stack, the moduli stack of genus g curves and the moduli stack of stable
curves and to  for the case of n pointed genus g curves. For the arithmetic
fundamental group of the moduli stack, see  (but this article requires
prerequisites on etale homotopy ).
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.
De nition 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 de ned 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 de ned over SpecZ. But from now on, we
consider Mg;n over SpecQ.
4.2. The arithmetic fundamental group of the moduli stack. Takayuki
Oda  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 Teichmller space, u
g;n
showed that the latter object has the etale homotopy type K( \ in the g;n; 1)
sense of Artin-Mazur , where g;n is the Teichmller-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 pro nite completion of the orientable
surface of (g; n)-type, i.e., the pro nite 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 de ne its weight
ltration as follows.
De nition 5.1. (Weight ltration on g;n.)
We de ne 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 de ne 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  .
This notion of weight ltration came from the study of the mixed Hodge
structure on the fundamental groups, by Morgan and Hain , but for the
particular case of P1 f0; 1; 1g, Ihara had worked on this  independently,
from an arithmetic motivation.
For x 2 W i ; y 2 W j , [x; y] 2 W i j holds, and this de nes a Z-bilinear
product gr i
gr j ! gr i j . We de ne
Grg;n := 1 gr i (g;n ):
i=1
With the product [x; y], Grg;n becomes a Lie algebra over Z. For lg;n , we
have a Lie algebra over Zl.
378

De nition 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 de ned for G ! Out(lg;n ).
In this case, we de ne
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
de nition, if we induce ltrations by G ! G0 ! Out(g;n ), then Gr(G) ,!
Gr(G0). By  , 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 .
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
Gr1 (Mg;n; a) ,! Gr1 (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.   ,
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 signi cance 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 De nition 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 de ne
=
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  (see Morita ).
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  for m odd, m  3.
380

We can de ne 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 di erent from Morita's
trace. The existence of such an obstruction was conjectured by Takayuki
Oda, and proved by myself  and H. Nakamura  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

de ned 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  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 ).
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 .
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 veri ed for m  11  , but
both conjectures seem to be still open. Ihara  recently showed that (ii)
implies (i).
382

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