Math. J. Okayama Univ. 60 (2018), 175–208

INDECOMPOSABILITY OF VARIOUS PROFINITEGROUPS ARISING FROM HYPERBOLIC CURVES
Arata Minamide

Abstract. In this paper, we prove that the étale fundamental group ofa hyperbolic curve over an arithmetic field [e.g., a finite extension field of
Q or Qp ] or an algebraically closed field is indecomposable [i.e., cannot
be decomposed into the direct product of nontrivial profinite groups].
Moreover, in the case of characteristic zero, we also prove that the étale
fundamental group of the configuration space of a curve of the above type
is indecomposable. Finally, we consider the topic of indecomposability
in the context of the comparison of the absolute Galois group of Q with
the Grothendieck-Teichmüller group GT and pose the question: Is GT
indecomposable? We give an aﬃrmative answer to a pro-l version ofthis question.

Contents
Introduction
175
0. Notations and Conventions
179
1. Indecomposability of Profinite Groups
181
2. Indecomposability of Various Absolute Galois Groups
186
3. Indecomposability of Geometric Fundamental Groups of Curves 187
4. Indecomposability of Various Fundamental Groups
190
5. Indecomposability of k-schemes
198
6. Indecomposability of the Pro-l Grothendieck-Teichmüller Group 202
Acknowledgement
206
References
207

Introduction
In Introduction, we shall write ΠS for the étale fundamental group of aconnected noetherian scheme S. Moreover, for a field K, we shall write K(respectively, GK ) for an algebraic closure of K (respectively, the absolute
Galois group of K). In [9], [10], Grothendieck introduced the notion ofan “anabelian variety”. He refers to a variety V over a finitely generated
Mathematics Subject Classification. Primary 14H30; Secondary 11R99.
Key words and phrases. indecomposability, étale fundamental group, hyperbolic curve,
configuration space, Grothendieck-Teichmüller group.
175

176

A. MINAMIDE

extension field F of Q which may be “reconstructed” from the natural [outer]surjectionΠV ↠ GF
as an “anabelian variety”. But we do not know any general [rigorous] conditions which characterize the “anabelianity”. On the other hand, it isconsidered that the notion of the slimness of profinite groups is deeply related to the “anabelianity” [cf. [16]]. [We shall say that a profinite group
G is slim if every open subgroup of G is center-free.] In fact, in the one
dimensional case, the following equivalences are known: For a geometricallyconnected, smooth curve C over F ,C is anabelian ⇔ C is hyperbolic ⇔ ΠC×F F is nontrivial and slim
[cf., e.g., [27]].
In this paper, as a notion which is deeply related to the “anabelianity”,
we adopt the strong indecomposability of profinite groups. The term strong
indecomposability is defined as follows [cf. Definition 1.1]:
We shall say that a profinite group G is indecomposable if, for
any isomorphism of profinite groups G ∼= G1 × G2 , where G1 ,G2 are profinite groups, it follows that either G1 or G2 is thetrivial group. We shall say that G is strongly indecomposable
if every open subgroup of G is indecomposable.
Indeed, there is a similarity between the “anabelianity” [of a variety V ] andthe strong indecomposability [of a profinite group G], as follows:(a) Let ρ : GF → Out(ΠV ×F F ) be the natural outer Galois representation associated to V . To test the “anabelianity” of V , we need toconsider the Galois centralizer
ZOut(ΠV × F ) (Im(ρ))
F[cf. [16]]. In other words, we need to consider a subgroup Im(ρ) ⊆Out(ΠV ×F F ) and a subgroup ZOut(ΠV × F ) (Im(ρ)) ⊆ Out(ΠV ×F F ),F
which commutes with Im(ρ).
(b) To test the strong indecomposability of G, we need to consider, forevery open subgroup H of G, whether or not H has a decomposition
H = H1 × H2
— where H1 , H2 are nontrivial. In other words, we need to consider
a subgroup H1 ⊆ H and a subgroup H2 ⊆ H, which commutes with
H1 .

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

177

In this paper, we prove that various profinite groups, which appear inanabelian geometry, are, in fact, strongly indecomposable. In the following,
for a prime number l, we shall write G(l) for the maximal pro-l quotient of a
profinite group G. First, in the one dimensional case, we prove the following[cf. Theorems 3.1, 3.6; Proposition 3.2]:
Theorem A. Let k be an algebraically closed field of characteristic p ≥ 0;l ̸= p a prime number; X a smooth curve of type (g, r) over k. Then thefollowing hold:(l)(i) Suppose that p = 0. If 2g − 2 + r > 0, then ΠX , ΠX are slim andstrongly indecomposable.
(ii) Suppose that p > 0. If (g, r) ̸= (0, 0), (1, 0) (respectively, 2g −2+r >(l)0), then ΠX (respectively, ΠX ) is slim and strongly indecomposable.
We note that the characteristic zero case and the pro-l case of Theorem A
are well-known [cf. [25], Proposition 3.2].
Next, we consider the case that the base field is non-algebraically closed.
Let k be a field; V a geometrically connected scheme of finite type over k.
In particular, the composite V ×k k → V → Spec(k) induces the following
exact sequence of profinite groups
1 −−−−→ ΠV ×k k −−−−→ ΠV −−−−→ Gk −−−−→ 1[cf. [8], EXPOSÉ IX, Théorème 6.1]. In the following, for any prime numberl, we shall write
def

(l)

ΠlV = ΠV /Ker(ΠV ×k k ↠ ΠV × k )k(l)— where ΠV ×k k ↠ ΠV × k is the natural surjection. Then we can prove thekfollowing theorem [cf. Theorem 4.3]:
Theorem B. Let k be a field of characteristic p ≥ 0 such that Gk is slim andstrongly indecomposable; X a smooth curve of type (g, r) over k. Supposethat there exists a prime number l ̸= p satisfying the following condition:(∗lk ) For any finite extension field k ′ of k, the l-adic cyclo′
tomic character χk′ : Gk′ → Z×l of k is nontrivial.
Then the following hold:(i) Suppose that p = 0. If 2g − 2 + r > 0, then ΠX , ΠlX are slim andstrongly indecomposable.

178

A. MINAMIDE(ii) Suppose that p > 0. If (g, r) ̸= (0, 0), (1, 0) (respectively, 2g −2+r >0), then ΠX (respectively, ΠlX ) is slim and strongly indecomposable.
Next, we consider the higher dimensional case. We note that, in the higher dimensional case, configuration spaces of hyperbolic curves are typicalexamples of anabelian varieties [cf. [16]]. In this case, we prove the following[cf. Theorem 4.4]:
Theorem C. Let n be a positive integer; k a field of characteristic p ≥ 0such that Gk is slim and strongly indecomposable; l ̸= p a prime number; Xa hyperbolic curve over k; Xn the n-th configuration space associated to X.
Then the following hold:(i) Suppose that k is algebraically closed. If p = 0 (respectively, p > 0),(l)(l)
then ΠXn , ΠXn are (respectively, ΠXn is) slim and strongly indecomposable.
(ii) Suppose that k satisfies condition (∗lk ) appearing in the statementof Theorem B. If p = 0 (respectively, p > 0), then ΠXn , ΠlXn are(respectively, ΠlXn is) slim and strongly indecomposable.
For instance, Theorems B and C imply the following [cf. Corollary 4.6]:
Corollary D. Let n be a positive integer; k a field of characteristic p ≥ 0;l ̸= p a prime number; X a smooth curve of type (g, r) over k; Xn the n-th
configuration space associated to X. Then the following hold:(i) Suppose that k is a finitely generated extension field of either a number field or a mixed characteristic local field. If 2g − 2 + r > 0, thenΠXn , ΠlXn are slim and strongly indecomposable.
(ii) Suppose that k is a finitely generated transcendental extension fieldof a finite field. If (g, r) ̸= (0, 0), (1, 0) (respectively, 2g − 2 + r > 0),then ΠX (respectively, ΠlXn ) is slim and strongly indecomposable.
Theorem C also implies the following geometric result [cf. Corollary 5.7]:
Theorem E. Let n be a positive integer; k a field; X a hyperbolic curve
over k; Xn the n-th configuration space associated to X. Suppose that there
exists an isomorphism of k-schemes∼

Xn → Y ×k Z
— where Y , Z are k-schemes. Then it follows that either
Y ∼= Spec(k) or Z ∼= Spec(k).

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

179

Finally, we consider the Grothendieck-Teichmüller group GT. One fundamental problem in the theory of GT is the issue of whether or not thewell-known injection
GQ ,→ GTis bijective. In this paper, in connection with this problem, we consider thefollowing problem [cf. [33], §1.4]:
Suppose that GQ satisfies a [profinite] group-theoretic property (P). Then does GT also satisfy the property (P)?
In particular, we pose the question:
Is GT strongly indecomposable?
Here, we note that GQ is strongly indecomposable [cf. Corollary 2.3]. In
this paper, we give an aﬃrmative answer to a pro-l version of this question.
More precisely, we prove the following result [cf. Corollary 6.3]:
Theorem F. For any prime number l, the pro-l Grothendieck-Teichmüllergroup GTl is strongly indecomposable.
We note that, in the proof of Theorem F, the above similarity between the“anabelianity” and the strong indecomposability is eﬀectively used.

0. Notations and Conventions
Fields: A finite extension field of Q will be referred to as a number field. A
finite extension field of Qp for some prime number p will be referred to as amixed characteristic local field.
Topological groups: Let G be a Hausdorﬀ topological group, and H ⊆ G
a closed subgroup. Let us write ZG (H) for the centralizer of H in G. We
defshall write Z(G) = ZG (G) for the center of G.
We shall say that a profinite group G is elastic if it holds that every
topologically finitely generated closed normal subgroup N ⊆ H of an open
subgroup H ⊆ G of G is either trivial or of finite index in G. If G is elastic,
but not topologically finitely generated, then we shall say that G is very
elastic.
We shall say that a profinite group G is slim if for every open subgroup
H ⊆ G, the centralizer ZG (H) is trivial. A profinite group G is slim if andonly if every open subgroup of G has trivial center [cf. [21], Remark 0.1.3].
Note that every open subgroup of a slim profinite group is also slim. It is

180

A. MINAMIDE

easily verified that every finite closed normal subgroup N ⊆ G of a slim
profinite group G is trivial.
Let p be a prime number. Then we shall write G(p) for the maximal pro-p
quotient of a profinite group G. If G admits an open subgroup which is
pro-p, then we shall say that G is almost pro-p.
We shall write Gab for the abelianization of a profinite group G, i.e., thequotient of G by the closure of the commutator subgroup of G.
If G is a topologically finitely generated profinite group, then one verifies
easily that the topology of G admits a basis of characteristic open subgroups, which thus induces a profinite topology on Aut(G) [cf. [31], Proposition
4.4.3], hence also on Out(G).
Let X be a connected noetherian scheme. Then we shall write ΠX for theétale fundamental group of X [for some choice of basepoint]. For any fielddefk, we shall write Gk = ΠSpec(k) for the absolute Galois group of k.
Curves: Let S be a scheme and X a scheme over S. If (g, r) is a pair ofnonnegative integers, then we shall say that X → S is a smooth curve oftype (g, r) over S if there exist an S-scheme X which is smooth, proper,of relative dimension 1 with geometrically connected fibers of genus g, anda closed subscheme D ⊆ X which is finite étale of degree r over S suchthat the complement of D in X is isomorphic to X over S. By abuse ofterminology, we shall refer to g as the genus of X.
We shall say that X is a hyperbolic curve over S if there exists a pair (g, r)of nonnegative integers with 2g − 2 + r > 0 such that X is a smooth curveof type (g, r) over S.
Let X → S be a smooth curve of type (g, r). For positive integers n, i, jsuch that i < j ≤ n, writedefpi,j : Pn = X ×S . . . ×S X → X ×S X
for the projection of the product Pn of n copies of X → S to the i-th andj-th factors. Then we shall refer to as the n-th configuration space associatedto X → S the S-scheme
Xn → S
which is the open subscheme determined by the complement in Pn of theunion of the various inverse images via the pi,j [as (i, j) ranges over thepairs of positive integers ≤ n such that i < j] of the image of the diagonalembedding X ,→ X ×S X.

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

181

1. Indecomposability of Profinite Groups
In this section, we introduce the notion of the indecomposability of profinite groups, and prove [profinite] group-theoretic results which are neededin §4, §5.
Definition 1.1. (cf. [25], Definition 3.1) We shall say that a profinite group
G is indecomposable if, for any isomorphism of profinite groups G ∼= G1 ×G2 ,where G1 , G2 are profinite groups, it follows that either G1 or G2 is thetrivial group. We shall say that G is strongly indecomposable if every open
subgroup of G is indecomposable.

Lemma 1.2. Let G be a profinite group. If G is elastic, slim, and topologically finitely generated, then G is strongly indecomposable.
Proof. First, we note that any open subgroup of G is also elastic, slim, andtopologically finitely generated. Thus, to verify the assertion, it suﬃces toshow that G is indecomposable. Suppose that we have an isomorphism of
profinite groups G ∼= G1 × G2 such that G1 ̸= {1}. Then since G1 is a
nontrivial topologically finitely generated closed normal subgroup of G, [bythe elasticity of G] G1 is of finite index in G. In particular, G1 is an open
subgroup of G. Thus, by the slimness of G, we have G2 ⊆ ZG (G1 ) = {1}. □Remark 1.3. Lemma 1.2 is implicitly used in [25], Remark 3.2.1.

Proposition 1.4. Letp1 −−−−→ ∆ −−−−→ Π −−−−→ G −−−−→ 1
be an exact sequence of profinite groups. Then the following hold:(i) Suppose that ∆ is indecomposable, and G is center-free and indecomposable. Then if the natural outer Galois representation
G → Out(∆)
associated to the above exact sequence is nontrivial, then Π is also
indecomposable.

182

A. MINAMIDE(ii) Suppose that ∆ is nontrivial and center-free, and that G is nontrivial. Then if Π is indecomposable, then the natural outer Galois
representation
G → Out(∆)
associated to the above exact sequence is nontrivial.
Proof. (i) Suppose that Π = Π1 × Π2 , where Π1 , Π2 are nontrivial closed
normal subgroups of Π. Then since G is center-free, it follows from [25],Proposition 3.3 that there exist normal closed subgroups Hi ⊆ Πi [for i = 1,∼
2] such that Π1 /H1 × Π2 /H2 → G. In particular, since G is indecomposable,
we obtain that either Π1 /H1 = {1} or Π2 /H2 = {1}. Without loss of∼generality, we may assume that Π1 /H1 = {1}, so Π1 = H1 , Π2 /H2 → G.
∼
Thus, we have Π1 × H2 → ∆.
∼Now I claim that H2 ̸= {1}. Indeed, suppose that H2 = {1}, so Π1 → ∆,∼
Π2 → G. Then the extension determined by the exact sequence that appears
in the statement of Proposition 1.4 is isomorphic to the trivial extension of
G by ∆
1 −−−−→ ∆ −−−−→ ∆ × G −−−−→ G −−−−→ 1.
Thus, the natural outer Galois representation G → Out(∆) induced by the
conjugation action of G on ∆ is trivial. But this contradicts the assumptionthat the outer representation G → Out(∆) is nontrivial. This completes theproof of the claim.
Thus, ∆ ∼= Π1 × H2 gives a nontrivial decomposition, which contradicts
the indecomposability of ∆. This completes the proof that Π is indecomposable.
(ii) Suppose that the representation G → Out(∆) is trivial. Here, notethat both ∆ and ZΠ (∆) are normal closed subgroups of Π. Now I claim that
Π is generated by ∆ and ZΠ (∆). Indeed, let π ∈ Π. Then, by the trivialityof G → Out(∆), there exists an element δ ∈ ∆ such that
π · x · π −1 = δ · x · δ −1
for any x ∈ ∆. In particular, we have δ −1 · π ∈ ZΠ (∆), so π = δ · (δ −1 · π) ∈∆ · ZΠ (∆). This completes the proof of the claim. Thus, since ∆ is centerfree, i.e., ∆ ∩ ZΠ (∆) = Z(∆) = {1}, we obtain that Π ∼= ∆ × ZΠ (∆).
Here, we note that since p(ZΠ (∆)) = G is nontrivial, we have ZΠ (∆) ̸= {1}.
Therefore, since ∆ is nontrivial, we conclude that Π is not indecomposable,
a contradiction.
□

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

183

Remark 1.5.
(i) One cannot drop the center-freeness assumption in
the statement of Proposition 1.4, (i). Indeed, let S3 be the symmetric group on 3 letters; sgn : S3 → {±1} the homomorphism obtainedby taking signatures; Φ : S3 × S3 ↠ {±1} the homomorphism given
by assigning (σ1 , σ2 ) 7→ sgn(σ1 ) · sgn(σ2 ). In particular, we have thefollowing exact sequenceΦ

1 −−−−→ Ker(Φ) −−−−→ S3 × S3 −−−−→ {±1} −−−−→ 1.
Here, note that {±1} is not center-free. Then although Ker(Φ) and{±1} are indecomposable, and, moreover, the natural outer representation associated to this exact sequence is nontrivial, S3 × S3 is
not indecomposable.
(ii) One cannot drop the center-freeness assumption in the statement of
Proposition 1.4, (ii). Indeed, let us consider the exact sequence×l

1 −−−−→ Zl −−−−→ Zl −−−−→ Zl /lZl −−−−→ 1.
Here, note that Zl is not center-free. Then although Zl is indecomposable, the natural outer representation associated to this exact
sequence is trivial.
The following Lemma 1.6 (respectively, Lemma 1.7) is a variant of [13],Lemma 5 (respectively, [13], Lemma 23).
Lemma 1.6. Let G be a slim profinite group; H an open subgroup of G; αan automorphism of G. Suppose that it holds that α(h) = h for any h ∈ H.
Then α = idG .
def ∩Proof. Write N =g∈G (g · H · g−1 ). Here, we observe the following facts:(1) N is a normal open subgroup of G;(2) It holds that α(n) = n for any n ∈ N .
Now I claim that, for any g ∈ G, it holds that α(g) · g −1 ∈ ZG (N ). Indeed,
for any n ∈ N , we haveα(g) · g −1 · n · (α(g) · g −1 )−1 = α(g) · g −1 · n · g · α(g)−1= α(g) · α(g −1 · n · g) · α(g)−1= α(g) · α(g)−1 · α(n) · α(g) · α(g)−1= α(n)= n

184

A. MINAMIDE— where the second (respectively, fifth) equality follows from the facts (1),(2) (respectively, fact (2)). In light of the claim, it follows from the slimnessof G that α(g) · g −1 = 1, hence that α(g) = g. Therefore, we conclude thatα = idG .
□Lemma 1.7. Let
1 −−−−→ ∆′ −−−−→ Π′ −−−−→ G′ −−−−→ 1






γy
αy
βy
1 −−−−→ ∆ −−−−→ Π −−−−→ G −−−−→ 1
be a commutative diagram of profinite groups, where the horizontal sequences
are exact. Writeρ : G → Out(∆)(respectively, ρ′ : G′ → Out(∆′ ))for the natural outer representation associated to the lower (respectively,
upper) horizontal sequence. Then the following hold:(i) Suppose that ∆ is slim, and that the three vertical arrows are open
injections. Then Ker(ρ′ ) is an open subgroup of Ker(ρ). In particular, it holds that Im(ρ) is infinite if and only if Im(ρ′ ) is infinite.
(ii) Suppose that α is a surjection. Then it holds thatγ(Ker(ρ′ )) ⊆ Ker(ρ).
In particular, if γ is an open injection, and Im(ρ) is infinite, thenIm(ρ′ ) is infinite.
Proof. First, we consider assertion (i). We begin by observing that it follows
from Lemma 1.6, thatZΠ′ (∆′ ) ⊆ ZΠ (∆).
In particular, we haveKer(ρ′ ) = p(ZΠ′ (∆′ )) ⊆ p(ZΠ (∆)) = Ker(ρ).
Thus, it suﬃces to verify that Ker(ρ′ ) is open in Ker(ρ). Let
ϕ : N∆ (∆′ )/∆′ → Out(∆′ )be the outer representation induced by conjugation. Here, note that sinceN∆ (∆′ )/∆′ is finite [cf. the fact that ∆′ is open in ∆], Im(ϕ) is also finite.
Now we observe that it holds that ρ′ (G′ ∩ Ker(ρ)) ⊆ Im(ϕ). In particular,
we obtain the following exact sequence of profinite groupsρ′

1 −−−−→ Ker(ρ′ ) −−−−→ G′ ∩ Ker(ρ) −−−−→ Im(ϕ).

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

185

Thus, since Im(ϕ) is finite, it follows that Ker(ρ′ ) is open in G′ ∩ Ker(ρ).
On the other hand, since G′ ∩ Ker(ρ) is open in Ker(ρ) [cf. the fact that G′
is open in G], we conclude that Ker(ρ′ ) is open in Ker(ρ). This completes
the proof of assertion (i). Next, we consider assertion (ii). Let us verify
the inclusion γ(Ker(ρ′ )) ⊆ Ker(ρ). Let g ′ ∈ Ker(ρ′ ); π ′ ∈ Π′ an element
which mapped to g ′ via the surjection Π′ ↠ G′ . In particular, there exists
an element δ ′ ∈ ∆′ such that it holds that
π ′ · x · (π ′ )−1 = δ ′ · x · (δ ′ )−1 .
for any x ∈ ∆′ . Thus, since α is surjective, it holds thatβ(π ′ ) · y · (β(π ′ ))−1 = α(δ ′ ) · y · (α(δ ′ ))−1 .
for any y ∈ ∆. Therefore, since β(π ′ ) is mapped to γ(g ′ ) via the surjectionΠ ↠ G, we conclude that γ(g ′ ) ∈ Ker(ρ), hence that γ(Ker(ρ′ )) ⊆ Ker(ρ).
This completes the proof of assertion (ii).
□

Proposition 1.8. Letp1 −−−−→ ∆ −−−−→ Π −−−−→ G −−−−→ 1
be an exact sequence of profinite groups. Then the following hold:(i) If ∆, G are center-free (respectively, slim), then Π is also center-free(respectively, slim).
(ii) Suppose that ∆ is slim and strongly indecomposable, and that G isslim and strongly indecomposable. Then if the image of the natural
outer representation
G → Out(∆)
associated to the above exact sequence is infinite, then Π is also slim
and strongly indecomposable.
Proof. First, let us consider assertion (i). To verify assertion (i), it suﬃces
to verify the center-freeness portion of assertion (i). Suppose that ∆, G arecenter-free. Let π ∈ Z(Π). Then it follows that p(π) ∈ Z(G) = {1}, hence
that π ∈ ∆ ∩ Z(Π) ⊆ Z(∆) = {1}. Thus, we conclude that Π is also centerfree. This completes the proof of assertion (i). Next, we consider assertion(ii). The slimness portion of assertion (ii) follows from assertion (i). The
strong indecomposability portion of assertion (ii) follows from Proposition
1.4, (i); Lemma 1.7, (i).
□186

A. MINAMIDE

2. Indecomposability of Various Absolute Galois Groups
In this section, we review [profinite] group-theoretic properties of various
absolute Galois groups.
Theorem 2.1. Let k be a Hilbertian field [cf. [5], Chapter 12]. Then Gk isvery elastic, slim, and strongly indecomposable.
Proof. The very elasticity portion of Theorem 2.1 follows from [5], Lemma
16.11.5; [5], Proposition 16.11.6. Note that for any open subgroup H of Gk ,∼there exists a finite separable extension kH of k such that GkH → H. Here,by [5], Corollary 12.2.3, kH is also a Hilbertian field. Thus, to verify the slimness and the strong indecomposability portions of Theorem 2.1, it suﬃces
to show that Gk is center-free and indecomposable. But this center-freeness(respectively, indecomposability) follows from [5], Proposition 16.11.6 (respectively, a theorem of Haran-Jarden [cf. [11], Corollary 2.5]).
□
Remark 2.2. Let k be either a finite field or a mixed characteristic local
field. Then k is always non-Hilbertian. Indeed, Gk is topologically finitely
generated [cf. Proposition 2.4, below; [5], Lemma 16.11.5].

Corollary 2.3. The following types of fields are Hilbertian:(i) finitely generated extension fields of Q,(ii) finitely generated transcendental extension fields of an arbitrary field.
In particular, their absolute Galois groups are very elastic, slim, and strongly
indecomposable.
Proof. The first statement follows from [5], Theorem 13.4.2. The second
statement follows from the first, together with Theorem 2.1.
□
Proposition 2.4. Let k be a mixed characteristic local field. Then Gk iselastic, slim, topologically finitely generated, and strongly indecomposable.
Proof. The assertions follow from Lemma 1.2; [24], Theorem 1.7, (ii); [28],Theorem 7.4.1.
□

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

187

Remark 2.5. Let k be a positive characteristic local field, i.e., a field isomorphic to the field of formal Laurent series F((t)) over a finite field F.
Then, by applying the same argument as the argument applied in the proofof [21], Theorem 1.1.1, (ii), one may verify easily that Gk is slim. On theother hand, it is not clear to the author at the time of writing whether or not
the elasticity, as well as the [strong] indecomposability of Gk holds. [Notethat, in this case, Gk is not topologically finitely generated [cf., e.g., [28],Theorem 7.5.10].]

3. Indecomposability of Geometric Fundamental Groups ofCurvesIn this section, we discuss the indecomposability of the geometric fundamental group of a smooth [hyperbolic] curve.
First, let us recall the following well-known fact.
Theorem 3.1. Let k be an algebraically closed field of characteristic zero;(l)l a prime number; X a hyperbolic curve over k. Then ΠX , ΠX are elastic,(l)slim, and topologically finitely generated. In particular, ΠX , ΠX are strongly
indecomposable.
(l)

Proof. The fact that ΠX , ΠX are elastic (respectively, slim; topologicallyfinitely generated) follows from [25], Theorem 1.5 (respectively, [25], Proposition 1.4; [8], EXPOSÉ XIII, Corollaire 2.12). In particular, the strong
indecomposability portion of Theorem 3.1 follows from Lemma 1.2 [cf. also[25], Proposition 3.2; [25], Remark 3.2.1].
□
Thus, for the rest of this section, we consider the case of positive characteristic. The following Proposition is an immediate consequence of Theorem
3.1.
Proposition 3.2. Let k be an algebraically closed field of characteristic(l)p > 0; l ̸= p a prime number; X a hyperbolic curve over k. Then ΠX iselastic, slim, topologically finitely generated, and strongly indecomposable.
Proof. This follows from Theorem 3.1; [8], EXPOSÉ XIII, Corollaire 2.12.
□

188

A. MINAMIDE

The following Lemmas 3.3, 3.4 are well-known, but we review them briefly
for the sake of completeness.
Lemma 3.3. Let k be an algebraically closed field of characteristic p > 0;
X a smooth curve of type (g, r) over k such that the pair (g, r) satisfies(g, r) ̸= (0, 0), (1, 0). Then there exists a normal open subgroup N of ΠX
such that the Galois finite étale covering XN → X corresponding to N has
genus ≥ 2.
Proof. If g ≥ 2, then there is nothing to prove. Thus, we may assume thatg ≤ 1. First, we consider the case where g = 0, i.e., the unique smoothcompactification of X is isomorphic to P1k . Here, note that if we identify
the function field of P1k with k(t), where t is an indeterminate, then for any
Artin-Schreier equation
xp − x = t m(m ∈ Z>0 , p ∤ m),

one computes easily that the normalization of P1k in the extension fieldk(t)[x]/(xp − x − tm ) of k(t) determines a Galois finite ramified covering
ϕm : Cm → P1k of P1k branched only at ∞, where Cm is a smooth, proper
curve of genus (m−1)(p−1)[cf., e.g., [35], Example 8.16]. Thus, for any curve
2
X of type (0, r), where r > 0, by taking m to be suﬃciently large and pullingback ϕm via an embedding X ,→ A1k ⊂ P1k , we obtain a Galois finite étale
covering X ′ → X of X such that the genus of X ′ is ≥ 2. Next, we consider
the case where g = 1, i.e., the unique smooth compactification of X is anelliptic curve E. Note that by applying the Riemann-Roch Theorem to E,def
we obtain a finite morphism E1 = E \ {p} → A1k over k, where p ∈ E \ X
is a closed point of E. Next, let us observe that it follows from the genus 0case, which has already been verified, that there exists a Galois finite étale
covering C → A1k of A1k such that the genus of C is ≥ 2. Then any connectedcomponent of E1 ×A1 C determines a Galois finite étale covering C ′ → E1
kof E1 . Moreover, by applying the Hurwitz formula to the compactificationof the finite morphism C ′ ,→ E1 ×A1 C → C, it follows that the genus of C ′kis also ≥ 2. Thus, for any curve X of type (1, r), where r > 0, by pullingback C ′ → E1 via the natural open immersion X ,→ E1 , we obtain a Galois
finite étale covering X ′ → X of X such that the genus of X ′ is ≥ 2.
□
Lemma 3.4. In the notation of Lemma 3.3, let l ̸= p be a prime number.
Then for any normal open subgroup N of ΠX such that the connected finiteétale covering XN → X corresponding to N has genus ≥ 2, the conjugation
action of ΠX /N on N ab ⊗Zb Zl is faithful.

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

189

Proof. The assertion follows immediately from the argument given in [3],Lemma 1.14.
□
Remark 3.5. It is well-known that, if we replace “(g, r) ̸= (0, 0), (1, 0)” by“2g − 2 + r > 0”, then the “characteristic zero versions” of Lemmas 3.3, 3.4
also hold. In fact, such “characteristic zero versions” are used in the proofof [25], Proposition 3.2.
The following Theorem 3.6 is the main result of this section. We note thatthe slimness portion of Theorem 3.6 is well-known [cf. [34], Proposition 1.11].
But for the convenience of the reader, we also give a proof of the slimness
portion of Theorem 3.6, as an application of Lemmas 3.3, 3.4.
def

Theorem 3.6. In the notation of Lemma 3.3, G = ΠX is slim and strongly
indecomposable.
Proof. First, I claim that(∗1 ) Let H be an open subgroup of G; XH → X the connected finite étale covering corresponding to H. Then XH isa smooth curve of type ̸= (0, 0), (1, 0) over k.
Indeed, this follows immediately from the Hurwitz formula. In light of theclaim (∗1 ), to verify the slimness (respectively, strong indecomposability) of
G, it suﬃces to show that G is center-free (respectively, indecomposable).
Thus, let us first verify the center-freeness of G. Here, observe that, for
any open subgroups J ⊆ G, there exists a normal open subgroup J ′ of G
such that J ′ ⊆ J, and, moreover, the Galois finite étale covering XJ ′ → X
corresponding to J ′ has genus ≥ 2. Indeed, it follows from the claim (∗1 )and Lemma 3.3 that there exists an open subgroup J ′′ ⊆ J such that theconnected finite étale covering XJ ′′ → X corresponding to J ′′ [⊆ G] has
def ∩genus ≥ 2. Then J ′ = g∈G (g · J ′′ · g −1 ) is a normal open subgroup of G
such that J ′ ⊆ J, and, moreover, the Galois finite étale covering XJ ′ → X
corresponding to J ′ has genus ≥ 2 [cf. the Hurwitz formula]. In particular, the center-freeness of G follows from this observation, together with
Lemma 3.4. This completes the proof the center-freeness of G, hence also
the slimness of G.
Thus, it remains to show that G is indecomposable. Suppose that we have
an isomorphism of profinite groups G ∼= G1 × G2 such that G1 ̸= {1}, G2 ̸={1}. In particular, by the slimness of G, it follows that G1 , G2 are infinite[cf. §0]. Then, by applying Lemma 3.3, we obtain a normal open subgroup

190

A. MINAMIDEK ⊆ G such the Galois finite étale covering XK → X corresponding to K
has genus ≥ 2. Take an open subgroup K ′ ⊆ K, which may be identified
withG′1 × G′2
— where, for i = 1, 2, G′i is an open subgroup of Gi . In particular, G′i isnontrivial [cf. the infiniteness of Gi ]. Moreover, it follows from the Hurwitz
formula that the connected finite étale covering XK ′ → XK corresponding
to K ′ [⊆ K] has genus ≥ 2 [cf. the fact that the genus of XK is ≥ 2]. Thus,by replacing G (respectively, Gi ) by K ′ (respectively, G′i ), we may assume,
without loss of generality, that the genus of X is ≥ 2. Now I claim that(∗2 ) for every prime number l ̸= p, there exist finite quotients
G1 ↠ Q1 , G2 ↠ Q2 such that l divides the orders of Q1 , Q2 .
Indeed, suppose that l does not divide the order of any finite quotient of
G1 . Now let N1 ⊊ G1 be a proper normal open subgroup of G1 . Note thatdefby assumption, we have N1ab ⊗Zb Zl = {1}. Write N = N1 × G2 . Then since
the conjugation action of G/N ∼= G1 /N1 × {1} onab∼
N ab ⊗Zb Zl ∼= (N1ab ⊗Zb Zl ) × (Gabb Zl ) = {1} × (G2 ⊗Z
b Zl )
2 ⊗Z

is trivial, we conclude from Lemma 3.4 that G/N = {1}, a contradiction.
This completes the proof of the claim (∗2 ). In light of the claim (∗2 ), we
obtain an open subgroup Ui ⊆ Gi such that there exists a surjection
Ui ↠ Z/lZ.
(l)for i = 1, 2. In particular, it follows that Ui is nontrivial. Let U be theopen subgroup of G that may be identified with U1 × U2 [⊆ G1 × G2 ]. Thus,
we have(l)(l)
U (l) ∼
= U1 × U2 .
On the other hand, since the connected finite étale covering XU → X corresponding to U has genus ≥ 2 [cf. the Hurwitz formula], it follows from
Proposition 3.2, that U (l) is indecomposable — a contradiction. This completes the proof of the indecomposability of G, hence also the strong indecomposability of G.
□

4. Indecomposability of Various Fundamental Groups
In this section, by applying the results of §1, §2 and §3, we prove theindecomposability of various fundamental groups.

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS191

Let k be a field; k an algebraic closure of k; V a geometrically connected
scheme of finite type over k. In particular, the composite V ×k k → V →Spec(k) induces the following exact sequence of profinite groups
1 −−−−→ ΠV ×k k −−−−→ ΠV −−−−→ Gk −−−−→ 1[cf. [8], EXPOSÉ IX, Théorème 6.1]. In the following, for any prime numberl, we shall write
def

(l)

ΠlV = ΠV /Ker(ΠV ×k k ↠ ΠV × k )k(l)— where ΠV ×k k ↠ ΠV × k is the natural surjection.
kDefinition 4.1. Let k be a field of characteristic p ≥ 0; l ̸= p a prime
number. Then for the pair (k, l), we consider the following condition:(∗lk ) For any finite extension field k ′ of k, the l-adic cyclotomic character′χk′ : Gk′ → Z×l of k is nontrivial.
We shall say that k is l-cyclotomically full if the pair (k, l) satisfies condition(∗lk ).
Lemma 4.2. In the notation of Definition 4.1, the following hold:(i) k is l-cyclotomically full if and only if for any finite extension fieldk ′ of k, there exists a positive integer n such that k ′ does not containa primitive ln -th root of unity.
(ii) Let K be an extension field of k. Then if K is l-cyclotomically
full, then the same is true of k. Suppose further that K is a finitely
generated extension field of k. Then if k is l-cyclotomically full, thenthe same is true of K.
(iii) k is l-cyclotomically full if and only if the image of the l-adic cyclotomic character χk : Gk → Z×l of k is infinite.
(iv) Let X be a smooth curve of type (g, r) over k such that the pair(g, r) satisfies (g, r) ̸= (0, 0), (0, 1) (respectively, (g, r) ̸= (0, 0)) if
p = 0 (respectively, p > 0); k an algebraic closure of k. Writedef

Xk = X ×k k. Suppose, moreover, that k is l-cyclotomically full.
Then the image of the natural outer Galois representation
ρk : Gk → Out(ΠXk )associated to the exact sequence
1 −−−−→ ΠXk −−−−→ ΠX −−−−→ Gk −−−−→ 1

192

A. MINAMIDE

is infinite. If, moreover, (g, r) ̸= (0, 1), then the image of the naturally induced pro-l outer Galois representation
(l)

(l)

ρk : Gk → Out(ΠX )kis infinite.
(v) Let k be either a number field, or a mixed characteristic local field,
or a finite field. Suppose that K is a finitely generated extension
field of k. Then K is l-cyclotomically full.

Proof. Assertion (i) follows immediately from the definitions.
The first statement of assertion (ii) follows from assertion (i). The second
statement of assertion (ii) follows from assertion (i) and the following wellknown fact:
If K is a finitely generated extension field of k, then thealgebraic closure of k in K is a finite extension of k.
[In fact, let E ⊆ K be the algebraic closure of k in K; {x1 , . . . , xn } ⊆ K atranscendence basis of K/k. Then we obtain that [E : k] = [E(x1 , . . . , xn ) :k(x1 , . . . , xn )] ≤ [K : k(x1 , . . . , xn )] < +∞.]
We consider assertion (iii). First, let us prove necessity. Suppose thatthe image of χk is finite. Then the kernel H of χk is an open subgroup of∼
Gk . Thus, there exists a finite extension k ′ of k such that Gk′ → H. In∼
particular, the l-adic cyclotomic character χk′ : Gk′ → H ,→ Gk → Z×l of′k is trivial — a contradiction. Next, we prove suﬃciency. To this end,
let k ′ be a finite extension field of k. Write χk′ : Gk′ → Z×l for the l-adic
cyclotomic character of k ′ , H for the kernel of χk . Then if we identify Gk′∼
with an open subgroup of Gk , then Gk′ /Gk′ ∩ H [→ Im(χk′ )] corresponds to∼an open subgroup of Gk /H [→ Im(χk )]. On the other hand, since Im(χk ) is
infinite, we thus conclude that Im(χk′ ) is also infinite, hence, in particular,
nontrivial. This completes the proof of assertion (iii).
Next, we consider assertion (iv). First, suppose that (g, r) = (0, 1) [sop > 0]. Then note that there exists an open subgroup N of ΠXk such thatthe connected finite étale covering Z → Xk corresponding to N has genus≥ 2 [cf. Lemma 3.3]. Now, by applying [26], Proposition (1.4.1), (i), we
have an open subgroup H of ΠX such that H ∩ ΠXk = N . Let Y → X
be the connected finite étale covering corresponding to H [⊆ ΠX ]; k ′ thefinite extension field of k corresponding to the image of H via the surjection
ΠX ↠ Gk . Here, observe that there exists a finite extension field k ′′ of k ′defsuch that Y ′ = Y ×k′ k ′′ is a hyperbolic curve over k ′′ . Thus, we have theINDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

193

following commutative diagram of profinite groups
1 −−−−→ ΠY ′ × ′′ k −−−−→ ΠY ′ −−−−→ Gk′′ −−−−→ 1
k


≀


y
y
y
1 −−−−→ ΠY × ′ k −−−−→ ΠY −−−−→ Gk′ −−−−→ 1
k





y
y
y
1 −−−−→ ΠX×k k −−−−→ ΠX −−−−→ Gk −−−−→ 1— where the vertical arrows are open injections; the horizontal sequences
are exact. Here, note that, by applying the Hurwitz formura to∼[Y ′ ×k′′ k →] Y ×k′ k → Z,
it follows that the genus of Y ′ is ≥ 2. Thus, since ΠX×k k is slim [cf. Lemma
3.6], it follows from Lemma 1.7, (i), that the infiniteness of the image of thenatural outer representation
Gk′′ → Out(ΠY ′ × ′′ k )kimplies the infiniteness of Im(ρk ). Therefore, the case where (g, r) = (0, 1)
follows from the case where g ≥ 2. Thus, in the remainder of the proof ofassertion (iv), we may assume without loss of generality that (g, r) ̸= (0, 1).
Next, observe that to verify the infiniteness of Im(ρk ), it suﬃces to verify(l)the infiniteness of Im(ρk ). Moreover, by replacing k by a suitable finite(l)extension of k, it suﬃces to verify that ρk is nontrivial. Suppose that(l)
ρk is trivial. First, we assume that g ≥ 1. We write X for the smoothcompactification of X; J(X) for the Jacobian variety of X;
def

(l)

∆ = ΠX× k ;
k

def

(l)

∆ = ΠX× k .
kIn particular, we have the following commutative diagram of profinite groups
1 −−−−→ ∆ −−−−→ ΠlX −−−−→ Gk −−−−→ 1




y
y
1 −−−−→ ∆ −−−−→ ΠlX −−−−→ Gk −−−−→ 1— where the horizontal sequences are exact; the left-hand and central vertical arrows are the surjections induced by the open immersion X ,→ X
[cf. the definition of Πl(−) ]. Thus, it follows from Lemma 1.7, (ii), that thenatural outer Galois representation
ρ : Gk → Out(∆)

194

A. MINAMIDE

associated to X is trivial, hence that the composite
ρ

∼

ab

Gk → Out(∆) → Aut(∆ ) → Aut(Tl (J(X)))— where we write Tl (J(X)) for the l-adic Tate module of J(X); the second
arrow is the homomorphism induced by the abelianization; the third arrowab ∼is the isomorphism induced by the natural isomorphism ∆ → Tl (J(X))— i.e, the natural l-adic Galois representation associated to J(X) is trivial.
Then since, as is well-known [cf. the natural isomorphisms
2g
∧

∼

∼

2g
1
Hét
(J(X) ×k k, Zl ) → Hét
(J(X) ×k k, Zl ) → Zl (−g)

of Zl [Gk ]-modules discussed in [18], Remark 15.5; [17], Chapter VI, Theorem
11.1, (a)], the determinant of the l-adic Galois representation associated toJ(X) is a positive power of the l-adic cyclotomic character of k, we conclude
that some positive power of the l-adic cyclotomic character of k is trivial.
But this contradicts (iii). Next, we assume that g = 0 and r ≥ 2. Let us
first observe that, by replacing k by a suitable finite extension field of k, we
may assume without loss of generality that X is obtained by removing r − 2
k-rational point(s) from A1k \ {0}. Then we note that, in the above notation,
we have the following commutative diagram of profinite groups
1 −−−−→

∆


y

−−−−→

−−−−→ Gk −−−−→ 1

ΠlX


y

(l)

1 −−−−→ ΠA1 \{0} −−−−→ ΠlA1 \{0} −−−−→ Gk −−−−→ 1kk— where the horizontal sequences are exact; the left-hand and central vertical arrows are the surjections induced by the open immersion X ,→ A1k \ {0}[cf. the definition of Πl(−) ]. Thus, it follows from Lemma 1.7, (ii), that thenatural outer Galois representation(l)

Gk → Out(ΠA \{0} )kassociated to A1k \{0} is trivial, hence that the l-adic cyclotomic character of
1 (A1 \{0}, Z ) ∼ Z (−1).]k is trivial, a contradiction. [Here, we recall that Hétl = lkFinally, we consider assertion (v). To verify the assertion, it suﬃces toshow that k is l-cyclotomically full [cf. (ii)]. Thus, to verify the assertion,
it suﬃces to show that, for any finite extension field k ′ of k, there exists
a positive integer n such that k ′ does not contain a primitive ln -th root ofunity [cf. (i)]. But this follows from the well-known fact that for any finite
extension field k ′ of k, the group of roots of unity in k ′ is finite [cf. [19],Chapter 5; [30], Chapter 2, §4.3, §4.4].
□

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

195

Theorem 4.3. Let k be a field of characteristic p ≥ 0 such that Gk is slim
and strongly indecomposable; X a smooth curve of type (g, r) over k. Thenif k is l-cyclotomically full for a prime number l ̸= p, then the followinghold:(i) Suppose that p = 0. If 2g − 2 + r > 0, then ΠX , ΠlX are slim andstrongly indecomposable.
(ii) Suppose that p > 0. If (g, r) ̸= (0, 0), (1, 0) (respectively, 2g −2+r >0), then ΠX (respectively, ΠlX ) is slim and strongly indecomposable.
Proof. Let k be an algebraic closure of k. Then we have the following commutative diagram of profinite groups
1 −−−−→ ΠX×k k −−−−→ ΠX −−−−→ Gk −−−−→ 1




y
y
(l)

1 −−−−→ ΠX× k −−−−→ ΠlX −−−−→ Gk −−−−→ 1k— where the horizontal sequences are exact [cf. the definition of ΠlX ]. Therefore, Theorem 4.3 follows from Proposition 1.8, (ii); Theorem 3.1; Proposition 3.2; Theorem 3.6; Lemma 4.2, (iv).
□
Theorem 4.4. Let n be a positive integer; k a field of characteristic p ≥ 0such that Gk is slim and strongly indecomposable; l ̸= p a prime number; Xa hyperbolic curve over k; Xn the n-th configuration space associated to X.
Then the following hold:(i) Suppose that k is algebraically closed. If p = 0 (respectively, p > 0),(l)(l)
then ΠXn , ΠXn are (respectively, ΠXn is) slim and strongly indecomposable.
(ii) Suppose that k is l-cyclotomically full. If p = 0 (respectively, p > 0),then ΠXn , ΠlXn are (respectively, ΠlXn is) slim and strongly indecomposable.
Proof. First, let us consider assertion (i). For n ≥ 2, let Xn → Xn−1 bethe projection morphism obtained by forgetting the factor labeled n; x ageometric point of Xn−1 ; (Xn )x the fiber of Xn → Xn−1 over x. Then if
p = 0 (respectively, p ≥ 0), then we have the following exact sequence ofprofinite groups
1 −−−−→ Π(Xn )x −−−−→ ΠXn −−−−→ ΠXn−1 −−−−→ 1

196

A. MINAMIDE(respectively,
(l)

(l)

(l)

1 −−−−→ Π(Xn )x −−−−→ ΠXn −−−−→ ΠXn−1 −−−−→ 1 )
(l)

(l)

[cf. [25], Proposition 2.2, (i)]. We note that Π(Xn )x , ΠX1 , Π(Xn )x , and ΠX1
are slim and strongly indecomposable [cf. Theorem 3.1; Proposition 3.2].
Thus, since the natural outer representation
(l)

(l)

ΠXn−1 → Out(Π(Xn )x ) (respectively, ΠXn−1 → Out(Π(Xn )x ))associated to the above exact sequence is injective [cf. [2], Theorem 1; [2],the Remark following the proof of Theorem 1], by applying induction on n,(l)it follows from Proposition 1.8, (ii), that ΠXn , ΠXn are slim and strongly
indecomposable. This completes the proof of assertion (i). Next, we consider
assertion (ii). Let k be an algebraic closure of k. We note that we have thefollowing commutative diagram of profinite groups
1 −−−−→ ΠXn ×k k −−−−→ ΠXn −−−−→ Gk −−−−→ 1




y
y
(l)

1 −−−−→ ΠX × k −−−−→ ΠlXn −−−−→ Gk −−−−→ 1nk— where the horizontal sequences are exact [cf. the definition of ΠlXn ]. Here, let us observe that Xn ×k k may be naturally identified with the n-th
def
configuration space of X ×k k. Write ∆n = Π(X×k k)n . Now I claim that, if
p = 0 (respectively, p ≥ 0), then the image of the outer representation
ρn : Gk → Out(∆n )(l)(respectively, ρ(l)n : Gk → Out(∆n ))is infinite. Indeed, since k is l-cyclotomically full, it follows from Lemma
4.2, (iv), that the image of the outer representaion
ρ1 : Gk → Out(∆1 )

(l)

(l)

(respectively, ρ1 : Gk → Out(∆1 ))is infinite. Here, note that we have the following commutative diagram ofprofinite groups
1 −−−−→ ∆n −−−−→ ΠXn −−−−→ Gk −−−−→ 1




y
y
1 −−−−→ ∆1 −−−−→ ΠX −−−−→ Gk −−−−→ 1

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

197(respectively,(l)1 −−−−→ ∆n −−−−→ ΠlXn −−−−→ Gk −−−−→ 1




y
y
(l)

1 −−−−→ ∆1 −−−−→ ΠlX −−−−→ Gk −−−−→ 1 )— where the horizontal sequences are exact; the left-hand and central vertical arrows are the surjections induced by the projection Xn → X obtainedby forgetting the factors with labels ̸= n. Thus, we conclude from Lemma(l)1.7, (ii), that Im(ρn ) (respectively, Im(ρn )) is infinite. This completes theproof of the claim. In light of the claim, assertion (ii) follows from assertion(i) and Proposition 1.8, (ii).
□Corollary 4.5. Let n be a positive integer; k a Hilbertian field of characteristic p ≥ 0; X a smooth curve of type (g, r) over k; Xn the n-th configuration
space associated to X. Suppose that there exists a prime number l ̸= p suchthat k is l-cyclotomically full. Then the following hold:(i) Suppose that p = 0. If 2g − 2 + r > 0, then ΠXn , ΠlXn are slim andstrongly indecomposable.
(ii) Suppose that p > 0. If (g, r) ̸= (0, 0), (1, 0) (respectively, 2g −2+r >0), then ΠX (respectively, ΠlXn ) is slim and strongly indecomposable.
Proof. These assertions follow immediately from Theorems 2.1, 4.3, 4.4.

□

Corollary 4.6. Let n be a positive integer; k a field of characteristic p ≥ 0;l ̸= p a prime number; X a smooth curve of type (g, r) over k; Xn the n-th
configuration space associated to X. Then the following hold:(i) Suppose that k is a finitely generated extension field of either a number field or a mixed characteristic local field. If 2g − 2 + r > 0, thenΠXn , ΠlXn are slim and strongly indecomposable.
(ii) Suppose that k is a finitely generated transcendental extension fieldof a finite field. If (g, r) ̸= (0, 0), (1, 0) (respectively, 2g − 2 + r > 0),then ΠX (respectively, ΠlXn ) is slim and strongly indecomposable.
Proof. First, we note that every field k which appears in Corollary 4.6 isl-cyclotomically full [cf. Lemma 4.2, (v)]. Thus, in the case that k is Hilbertian [cf. Corollary 2.3] (respectively, non-Hilbertian, i.e., mixed characteristic local), the assertions follow from Corollary 4.5 (respectively, Proposition
2.4 and Theorem 4.4).
□

198

A. MINAMIDE

Remark 4.7. Let k0 be a positive characteristic local field, i.e., a field
isomorphic to the field of formal Laurent series F((t)) over a finite field F. In
the notation of Corollary 4.6, suppose that k is a finitely generated extension
field of k0 . Here, we note that k is l-cyclotomically full. [Indeed, to verify
that k is l-cyclotomically full, we may assume, without loss of generality, thatk = F((t)) [cf. Lemma 4.2, (ii)]. Then since the l-adic cyclotomic characterχk : Gk → Z×l of k factors as the composite of the natural [outer] surjection
Gk ↠ GF — which is induced by the natural injection F ,→ k — and thel-adic cyclotomic character GF → Z×l of F, we conclude that the image ofχk is infinite [cf. Lemma 4.2, (iii), (v)], hence that k is l-cyclotomically full[cf. Lemma 4.2, (iii)].] In particular:• Suppose that [k : k0 ] = +∞. In this case, since k is Hilbertian [cf.
Corollary 2.3], we can prove the following [cf. Corollary 4.5]:
If (g, r) ̸= (0, 0), (1, 0) (respectively, 2g − 2 + r > 0), then ΠX(respectively, ΠlXn ) is slim and strongly indecomposable.
• Suppose that [k : k0 ] < +∞. In this case, note that Gk is slim [cf.
Remark 2.5]. Thus, by applying the same arguments as the arguments applied in the proofs of Theorems 4.3, 4.4 [cf. also Proposition
1.8, (i)], we can prove the following:
If (g, r) ̸= (0, 0), (1, 0) (respectively, 2g − 2 + r > 0), then ΠX(respectively, ΠlXn ) is slim.
On the other hand, it is not clear to the author at the time of writing
whether or not ΠX (respectively, ΠlXn ) is [strongly] indecomposable.

5. Indecomposability of k-schemes
In this section, we introduce the notion of the indecomposability of kschemes, and give a criterion so that a k-scheme is indecomposable. As anapplication, we prove that the configuration space of a hyperbolic curve overa field of characteristic zero is indecomposable.
Definition 5.1. Let k be a field. We shall say that a k-scheme V is indecomposable if, for any k-isomorphism of k-schemes V ∼= Y ×k Z, where Y ,Z are k-schemes, it follows that either∼ Spec(k) or Z =∼ Spec(k).
Y =INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

199

Definition 5.2. (cf. [14], Definition 2.5; [32], Definition 2.25) Let k bea field; k an algebraic closure of k; l a prime number; X a geometricallyconnected, separated scheme of finite type over k. Then we shall say that
X is of LFG-type (respectively, l-LFG-type) if, for any connected, normal,
separated scheme Y of finite type over k and any nonconstant morphism
Y → X ×k k over k, the image of the outer homomorphismΠY → ΠX×k k

(l)

(l)

(respectively, ΠY → ΠX× k )kis infinite.

Remark 5.3. In the notation of Definition 5.2, the implication
X is of l-LFG-type ⇒ X is of LFG-type
holds.
The following Proposition is essentially proved in [32], Proposition 2.28
[cf. also [14], Proposition 2.7]. [Note that, in [32], the characteristic of thebase field is assumed to be zero.]
Proposition 5.4. Let n be a positive integer; k a field of characteristicp ≥ 0; X a hyperbolic curve over k; Xn the n-th configuration space of X.
Then, for any prime number l ̸= p, Xn is of l-LFG-type. In particular, Xn
is of LFG-type.
Proof. We may assume that k is algebraically closed. First, suppose thatn = 1. Let Y be a connected, normal, separated scheme of finite type overk. Then since any nonconstant k-morphism Y → X is dominant, it follows(l)(l)
from [14], Lemma 1.3, that ΠY → ΠX is open. In particular, ΠY → ΠX is(l)also open. Thus, we conclude from the well-known fact that ΠX is infinite(l)(l)
that the image of ΠY → ΠX is infinite.
Next, suppose that n > 1, and that the induction hypothesis is in force.
Let f : Y → Xn be a nonconstant k-morphism. We consider the compositefg : Y → Xn → Xn−1 ,where the second arrow is the projection morphism obtained by forgetting
the factor labeled n. If g is nonconstant, then it follows from the induction(l)(l)
hypothesis that the image of ΠY → ΠXn−1 is infinite, hence that the image
(l)

(l)

of ΠY → ΠXn is infinite. If g is constant, then we write x → Xn−1 for the

200

A. MINAMIDE

geometric point of Xn−1 through which g factors. We write (Xn )x for thefiber of Xn → Xn−1 over x. In particular, f factors as the composite
Y → (Xn )x → Xn ,where the first arrow is a nonconstant k-morphism; the second arrow is thenatural projection. We note that it follows from the case where n = 1 that
(l)
(l)
(l)
(l)
the image of ΠY → Π(Xn )x is infinite. Then since Π(Xn )x → ΠXn is injective
(l)

(l)

[cf. [25], Proposition 2.2, (i)], we conclude that the image of ΠY → ΠXn isalso infinite.
□

Lemma 5.5. Let k be a field; V a geometrically integral, separated schemeof finite type over k. Suppose that there exists an isomorphism of k-schemes∼

V → Y ×k Z
— where Y , Z are k-schemes. Then Y , Z are geometrically integral, separated schemes of finite type over k.
Proof. In the following, we shall identify V and Y ×k Z via the above isomorphism. First, note that since V is separated and of finite type over k,
it follows from [6], Proposition (5.5.1), (v); [6], Corollaire (6.3.9), that thenatural projections V → Y , V → Z are separated and of finite type. Now I
claim that Y , Z are quasi-compact. Indeed, let us first observe that [since
V is integral] V is nonempty, hence that Y , Z are nonempty. In particular,
it follows that the structure morphisms Y → Spec(k), Z → Spec(k) are surjective, hence that the natural projections V → Y , V → Z are surjective.
Thus, since V is quasi-compact, we conclude that Y , Z are quasi-compact.
This completes the proof of the claim. In light of the claim, it follows immediately that the structure morphisms Y → Spec(k), Z → Spec(k) arefaithfully flat and quasi-compact. Thus, we conclude from [7], Proposition(2.7.1), that Y → Spec(k), Z → Spec(k) are separated and of finite type.
To verify that Y , Z are geometrically integral over k, it suﬃces to show
that, for an algebraic closure k of k,
Y ×k k,Z ×k kare integral. But since the natural projections V ×k k → Y ×k k, V ×k k →Z ×k k are faithfully flat, the integrality of Y ×k k, Z ×k k follows from theintegrality of V ×k k.
□

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

201

Theorem 5.6. Let k be a field of characteristic p ≥ 0; l ̸= p a prime
number; k an algebraic closure of k; V a geometrically integral, separated
scheme of finite type over k. Suppose that one of the following conditions is
satisfied:(1) The following conditions are satisfied:(1-i) V is proper over k.
(1-ii) V is of LFG-type.
(1-iii) ΠV ×k k is indecomposable.
(2) The following conditions are satisfied:(2-i) p = 0.
(2-ii) V is of LFG-type.
(2-iii) ΠV ×k k is indecomposable.
(3) The following conditions are satisfied:(3-i) V is of l-LFG-type.
(l)(3-ii) ΠV × k is indecomposable.
kThen V is indecomposable.
Proof. To verify Theorem 5.6, it suﬃces to show the following claim(∗) Suppose that there exists an isomorphism of k-schemes∼

V → Y ×k Z
— where Y , Z are k-schemes. Then it follows that either
Y ∼= Spec(k) or Z ∼= Spec(k).
First, note that Y , Z are geometrically integral, separated schemes of finite
type over k [cf. Lemma 5.5]. Thus, to verify the claim (∗), we may assume
that k is algebraically closed. Then to verify the claim (∗), it suﬃces toshow that either
dim(Y ) = 0 or dim(Z) = 0.
Now we observe that, if either condition (1) or condition (2) (respectively,
condition (3)) is satisfied, then by the Künneth formula [cf. [8], EXPOSÉ
X, Corollaire 1.7; [8], EXPOSÉ XIII, Proposition 4.6; [29], Proposition 4.7],
there exists an isomorphism of profinite groups∼

ΠV → ΠY × ΠZ

(l) ∼

(l)

(l)

(respectively, ΠV → ΠY × ΠZ ).
(l)

Then since ΠV (respectively, ΠV ) is indecomposable, we may assume with(l)out loss of generality that ΠY = {1} (respectively, ΠY = {1}). Now we
fix a k-rational point z ∈ Z(k) of Z. Then we obtain a closed immersion

202

A. MINAMIDE∼∼

Y → Y ×k {z} ,→ Y ×k Z → V . Write Y ′ → Y for the [surjective] morphism
obtained by normalizing Y . Here, if we assume that dim(Y ) ≥ 1, then thecomposite Y ′ → Y ,→ V is nonconstant. Thus, since V is of LFG-type(respectively, l-LFG-type), the image of the outer homomorphism
ΠY ′ → ΠV

(l)

(l)

(respectively, ΠY ′ → ΠV )is infinite — a contradiction. Therefore, we conclude that dim(Y ) = 0. This
completes the proof of the claim (∗), hence also of Theorem 5.6.
□
Corollary 5.7. In the notation of Proposition 5.4, Xn is indecomposable.
Proof. This follows from Theorem 4.4, (i); Proposition 5.4; Theorem 5.6. □Remark 5.8. It is not clear to the author at the time of writing whether
or not there exists a geometric proof of Corollary 5.7.

6. Indecomposability of the Pro-l Grothendieck-Teichmüller
Group
In this section, we verify the indecomposability of the pro-l GrothendieckTeichmüller group GTl [cf. Corollary 6.3] as a consequence of a certain
anabelian result over finite fields [cf. [12], Remark 6, (iv)].
Let l be a prime number; k an algebraically closed field of characteristic
zero; F a field of characteristic char(F) ̸= l; F an algebraic closure of F;
defΠ = ΠP1 \{0,1,∞} ; S3 the symmetric group on 3 letters. Here, we observekthat the natural action of S3 on P1k \ {0, 1, ∞} induces injections
S3 ,→ Out(Π(l) ).

S3 ,→ Out(Π);

Thus, in the following, via these injections, we regard S3 as a subgroup ofOut(Π) and Out(Π(l) ). Moreover, if char(F) = 0 (respectively, char(F) ≥ 0),then let∼ρF : GF → Out(ΠP1 \{0,1,∞} ) → Out(Π)F(l)(respectively, ρF : GF∼(l)

→ Out(ΠP1 \{0,1,∞} ) → Out(Π(l) ))F

be the composite of the natural [pro-l] outer Galois representation with theisomorphism induced by the natural [outer] isomorphism
∼

(l)

∼

ΠP1 \{0,1,∞} → Π (respectively, ΠP1 \{0,1,∞} → Π(l) ).
FFINDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

203

Theorem 6.1. Let N be a closed subgroup of Out(Π) (respectively,
Out(Π(l) )) satisfying the following conditions:(l)(a) N contains an open subgroup of Im(ρQ ) (respectively, Im(ρQ )).
(b) It holds that N ⊆ ZOut(Π) (S3 ) (respectively, N ⊆ ZOut(Π(l) ) (S3 )).
Then N is slim (respectively, slim and strongly indecomposable).
Proof. First, let us consider the slimness portion of Theorem 6.1. To verify
the slimness of N , it suﬃces to show that for any open subgroup U of N ,U is center-free. Let σ ∈ Z(U ). We note that, by condition (a), there
exists a finite extension field E of Q such that Im(ρE ) ⊆ U (respectively,(l)Im(ρE ) ⊆ U ). Thus, it follows from [20], Theorem A, that∼∼σ ∈ ZOut(Π) (Im(ρE )) ← AutE (P1E \ {0, 1, ∞}) ← S3
∼

(l)

∼

(respectively, σ ∈ ZOut(Π(l) ) (Im(ρE )) ← AutE (P1E \ {0, 1, ∞}) ← S3 ),hence that, σ ∈ Z(S3 ) = {1} [cf. condition (b)]. Therefore, we conclude
that U is center-free, hence also that N is slim.
Next, let us verify the strong indecomposability of N [⊆ Out(Π(l) )]. To
verify the strong indecomposability of N , it suﬃces to show that for any
open subgroup U of N , U is indecomposable. Let p ̸= l be a prime number.
Here, we note that since the restriction(l)ρQ |GQp : GQp → Out(Π(l) )factors as the composite
(l)

ρ Fp

GQp ↠ GFp → Out(Π(l) )— where the first arrow is the natural [outer] surjection — it follows from(l)condition (a) that N contains an open subgroup of Im(ρFp ). Thus, there
exists a finite extension field F of Fp such that
def

(l)

G = Im(ρF ) ⊆ U.
Moreover, since Out(Π(l) ) is almost pro-l [cf. [1], Corollary 7], by replacing
F by a suitable finite extension field of F , we may assume without loss of(l)(l)
generality that ρF factors through the maximal pro-l quotient GF ↠ GF
of GF . Here, we note that since G is infinite [cf. Lemma 4.2, (iv), (v)], wehave G ∼= Zl .

204

A. MINAMIDE

Now suppose that we have an isomorphism of profinite groups U ∼= H1 ×
H2 . In the following, we shall identify U and H1 × H2 via this isomorphism.
Then I claim that it holds thateither G ∩ H1 ̸= {1}or G ∩ H2 ̸= {1}.

Indeed, suppose that G ∩ H1 = {1} and G ∩ H2 = {1}. In particular, it
follows that, for i = 1, 2, the composite
pri

G ,→ U = H1 × H2 ↠ Hi
— where pri is i-th projection — is injective. Thus, if we write Ki ⊆ Hi∼
for the image of the above composite, we obtain that G → Ki [∼= Zl ]. Here,note that we have inclusionsdef

G ⊆ K = K1 × K2 ⊆ H1 × H2 .
Now let us observe that it follows from Lemma 4.2, (iv), (v); [22], Corollary
2.7, (i); condition (b); [15], Lemma 3.5, that
ZU (G) ⊆ Out|C| (Π(l) )— where we write Out|C| (Π(l) ) for the group of outer automorphisms of Π(l)which fixes the conjugacy class of each cuspidal inertia subgroup of Π(l) .
Thus, since K [∼= Zl × Zl ] is abelian, we obtain thatK ⊆ ZU (G) ,→ Z×l— where “,→” is induced by the morphism “degP ” of [12], Definition 3.1,
which is injective by [12], Remark 6, (iv). Take an open subgroup J of Z×lsuch that J ∼= Zl . In particular, if we identify K with Zl × Zl , then there
exists a positive integer m such that lm Zl × lm Zl ⊆ J ∩ K ⊆ K. On theother hand, since lm Zl × lm Zl ̸= {0} is a closed subgroup of J [∼= Zl ], we
obtain that lm Zl × lm Zl ∼= Zl , a contradiction. This completes the proof ofthe claim.
In light of the claim, we may assume without loss of generality thatG ∩ H1 ̸= {1}.
Then since G ∩ H1 ⊆ G is a nontrivial closed subgroup of G ∼= Zl , it follows
that G∩H1 is open in G. Thus, by replacing F by a suitable finite extension,
we may assume without loss of generality that G ⊆ H1 . In particular, we
obtain thatH2 ⊆ ZU (G) ,→ Z×l— where “,→” denotes the arrow “,→” in the final display of the proof of theabove claim. Thus, it follows that H2 is abelian. On the other hand, sinceH2 is center-free [cf. the fact that N , hence also U is slim], we obtain thatH2 = {1}. Therefore, we conclude that U is indecomposable, as desired. □INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

205

In the following, we shall denote bypri : (P1k \ {0, 1, ∞})2 → P1k \ {0, 1, ∞}— where we denote by (P1k \ {0, 1, ∞})2 the second configuration space of
P1k \ {0, 1, ∞} — the i-th projection for i ∈ {1, 2}. We shall writedefΠ2 = Π(P1 \{0,1,∞})2 .
k

In particular, pri induces a(n) [outer] surjection(l)Π2 ↠ Π (respectively, Π2↠ Π(l) )(l)

which we denote by pi (respectively, pi ). Then we shall write
OutF (Π2 )(l)(respectively, OutF (Π2 ))(l)for the group of outer automorphisms σ of Π2 (respectively, Π2 ) such that(l)(l)σ(Ker(pi )) = Ker(pi ) (respectively, σ(Ker(pi )) = Ker(pi )) for i ∈ {1, 2}[cf. [23], Definition 1.1, (ii)].
Now let us recall the Grothendieck-Teichmüller groupGT
which was introduced by Drinfeld [cf. [4]]. This group is a subgroup of theautomorphism group of the free profinite group of rank 2 and important,e.g., from the point of view of the comparison with GQ [cf. [33], §1.4]. Let
us, moreover, recall that GT may be identified with
Im(OutF (Π2 ) → Out(Π)) ∩ ZOut(Π) (S3 ) [⊆ Out(Π)]— where OutF (Π2 ) → Out(Π) is the injection induced by p1 [cf. [23], Remark 1.11.1; [23], Corollary 1.12, (ii); [15], Theorem 2.3, (ii)]. Now we define
a pro-l analogue of GT as follows:
Definition 6.2. We shall write
def

(l)

GTl = Im(OutF (Π2 ) → Out(Π(l) )) ∩ ZOut(Π(l) ) (S3 ) [⊆ Out(Π(l) )]and refer to GTl as the pro-l Grothendieck-Teichmüller group [cf. [15], Definition 3.4, (i)].
Corollary 6.3. GT (respectively, GTl ) is slim (respectively, slim and strongly indecomposable).

206

A. MINAMIDE

Proof. To verify Corollary 6.3, it suﬃces to show that GT and GTl satisfy
conditions (a), (b) of Theorem 6.1. It is immediate that GT and GTl satisfy
condition (b). Now I claim that the following inclusions hold:Im(ρQ ) ⊆ GT,(l)

Im(ρQ ) ⊆ GTl .

Let us verify the first inclusion. [In a similar vein, one can verify the second
inclusion.] To verify the inclusion, we may assume that k = Q, and that ρQ
is the natural outer Galois representaion associated to P1Q \ {0, 1, ∞}. First,
let us observe that the image of the natural outer Galois representation
GQ → Out(Π2 )associated to (P1Q \ {0, 1, ∞})2 is contained in OutF (Π2 ) ⊆ Out(Π2 ), hence
that the homomorphism GQ → Out(Π2 ) factors as the composite of a homomorphism αQ : GQ → OutF (Π2 ) and the natural injection OutF (Π2 ) ,→Out(Π2 ). [Indeed, this observation follows from the fact that the naturalaction of GQ on (P1Q \ {0, 1, ∞})2 is compatible with pri for i ∈ {1, 2}.] In
particular, we obtain a commutative diagramαQ

GQ −−−−→ OutF (Π2 )


y
ρQ

GQ −−−−→ Out(Π)— where the right-hand vertical arrow is the injection induced by pr1 . In
light of the commutativity of this diagram, we conclude that
Im(ρQ ) ⊆ Im(OutF (Π2 ) → Out(Π)).
Thus, to verify the inclusion Im(ρQ ) ⊆ GT, it suﬃces to verify the inclusion
Im(ρQ ) ⊆ ZOut(Π) (S3 ). On the other hand, this follows immediately from
1 \ {0, 1, ∞}the fact that, for any σ ∈ S3 , g ∈ GQ , the automorphism of PQ

induced by σ commutes with the automorphism of P1Q \ {0, 1, ∞} induced

1 \ {0, 1, ∞}.] Thisby g. [Here, we note that S3 and GQ act naturally on PQ
completes the proof of the claim. Therefore, we conclude that GT and GTlsatisfy condition (a).
□Acknowledgement
I would like to thank Professors Shinichi Mochizuki and Yuichiro Hoshifor their suggestions, many helpful discussions, and warm encouragement. I
also would like to thank the referee for numerous fruitful comments. This
research was partially supported by Grant-in-Aid for JSPS Fellows (KAKENHI No. 16J02375).

INDECOMPOSABILITY OF VARIOUS PROFINITE GROUPS

207

References[1] M. P. Anderson, Exactness properties of profinite completion functors, Topology 13(1974), pp. 229-239.
[2] M. Asada, The Faithfulness of the Monodromy Representations Associated with Certain Families of Algebraic Curves, J. Pure Appl. Algebra 159 (2001), pp. 123-147.
[3] P. Deligne and D. Mumford, The Irreducibility of the Moduli Space of Curves ofGiven Genus, IHES Publ. Math. 36 (1969), pp. 75-109.
[4] V.G. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connectedwith Gal(Q/Q), Algebra i Analiz 2 (1990), pp. 149-181.
[5] M. Fried and M. Jarden, Field Arithmetic (Second Edition), Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, A Series of Modern Surveys in Mathematics
11, Springer-Verlag (2005).
[6] A. Grothendieck, Eléments de géométrie algébrique. I. Le langage des schémas, Inst.
Hautes Études Sci. Publ. Math. No. 4 (1960).
[7] A. Grothendieck, Eléments de géométrie algébrique. IV. Étude locale des schémas etdes morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. No. 24 (1965).
[8] A. Grothendieck, Revêtements Étales et Groupe Fondamental, Lecture Notes in Mathematics (SGA1) 224, Springer-Verlag (1971).
[9] A. Grothendieck, Sketch of a program, Geometric Galois Actions; 1. Around
Grothendieck’s Esquisse d’un Programme, London Math. Soc. Lect. Note Ser. 242,
Cambridge Univ. Press (1997), pp. 243-283.
[10] A. Grothendieck, Letter to G. Faltings, Geometric Galois Actions; 1. Around
Grothendieck’s Esquisse d’un Programme, London Math. Soc. Lect. Note Ser. 242,
Cambridge Univ. Press (1997), pp. 285-293.
[11] D. Haran and M. Jarden, Compositum of Galois extensions of Hilbertian fields, Ann.
Sci. École Norm. Sup. 24 (1991), pp. 739-748.
[12] Y. Hoshi, Absolute Anabelian Cuspidalization of Configuration Spaces of Proper
Hyperbolic Curves over Finite Fields, Publ. Res. Inst. Math. Sci. 45 (2009), pp.
661-744.
[13] Y. Hoshi, On Monodromically Full Points of Configuration Spaces of Hyperbolic
Curves, The Arithmetic of Fundamental Groups — PIA 2010, Contributions in Mathematical and Computational Sciences 2, Springer-Verlag (2012), pp. 167-207.
[14] Y. Hoshi, The Grothendieck Conjecture for Hyperbolic Polycurves of Lower Dimension, J. Math. Sci. Univ. Tokyo 21 (2014), pp. 153-219.
[15] Y. Hoshi and S. Mochizuki, Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves II, RIMS Preprint 1762 (November 2012).
[16] Y. Ihara and H. Nakamura, Some illustrative examples for anabelian geometry inhigh dimensions, Geometric Galois Actions; 1. Around Grothendieck’s Esquisse d’un
Programme, London Math. Soc. Lect. Note Ser. 242, Cambridge Univ. Press (1997),
pp. 127-138.
[17] J. S. Milne, Étale Cohomology, Princeton Mathematical Series 33, Princeton University Press (1980).
[18] J. S. Milne, Abelian Varieties, Arithmetic Geometry, Springer-Verlag (1986), pp. 103150.

208

A. MINAMIDE[19] J. S. Milne, Algebraic Number Theory,
http://www.jmilne.org/math/CourseNotes/ant.html.
[20] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math. 138(1999), pp. 319-423.
[21] S. Mochizuki, The Absolute Anabelian Geometry of Hyperbolic Curves, Galois Theory and Modular Forms, Kluwer Academic Publishers (2004), pp. 77-122.
[22] S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku Math.
J. 59 (2007), pp. 455-479.
[23] S. Mochizuki, On the Combinatorial Cuspidalization of Hyperbolic Curves, Osaka J.
Math. 47 (2010), pp. 651-715.
[24] S. Mochizuki, Topics in Absolute Anabelian Geometry I: Generalities, J. Math. Sci.
Univ. Tokyo 19 (2012), pp. 139-242.
[25] S. Mochizuki and A. Tamagawa, The Algebraic and Anabelian Geometry of Configuration Spaces, Hokkaido Math. J. 37 (2008), pp. 75-131.
[26] H. Nakamura, Galois Rigidity of Pure Sphere Braid Groups and Profinite Calculus,
J. Math. Sci. Univ. Tokyo 1 (1994), pp. 71-136.
[27] H. Nakamura, A. Tamagawa, S. Mochizuki, The Grothendieck conjecture on thefundamental groups of algebraic curves, Sugaku Expositions 14 (2001), pp. 31-53.
[28] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields (Second Edition), Grundlehren der mathematischen Wissenschaften 323, Springer-Verlag (2008).
[29] F. Orgogozo, Altérations et groupe fondamental premier à p, Bull. Soc. Math. France.
131 (2003), pp. 123-147.
[30] A. Robert, A Course in p-adic Analysis, Graduate Texts in Mathematics 198,
Springer-Verlag (2000).
[31] L. Ribes and P. Zalesskii, Profinite Groups (Second Edition), Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, A Series of Modern Surveys in Mathematics
40, Springer-Verlag (2009).
[32] K. Sawada, Pro-p Grothendieck conjecture for hyperbolic polycurves, RIMS Preprint
1853 (June 2016).
d : a survey, Geometric Galois[33] L. Schneps, The Grothendieck-Teichmüller group GT
Actions; 1. Around Grothendieck’s Esquisse d’un Programme, London Math. Soc.
Lect. Note Ser. 242, Cambridge Univ. Press (1997), pp. 183-203.
[34] A. Tamagawa, The Grothendieck Conjecture for Aﬃne Curves, Compositio Math.
109 (1997), pp. 135-194.
[35] K. Ueno, Algebraic Geometry 3, Translation of Mathematical Monographs 218,
American Mathematical Society (2003).
Arata Minamide
Research Institute for Mathematical Sciences
Kyoto University
Kyoto, 606-8502 Japane-mail address: minamide@kurims.kyoto-u.ac.jp(Received April 21, 2015 )(Accepted June 12, 2017 )