Math. J. Okayama Univ. 48 (2006), 57–71

GLOBAL SOLVABLY CLOSED ANABELIAN GEOMETRY
Shinichi MOCHIZUKI
Abstract. In this paper, we study the pro-Σ anabelian geometry of hyperbolic curves, where Σ is a nonempty set of prime numbers, over Galois
groups of “solvably closed extensions” of number ﬁelds — i.e., inﬁnite
extensions of number ﬁelds which have no nontrivial abelian extensions.
The main results of this paper are, in essence, immediate corollariesof the following three ingredients: (a) classical results concerning thestructure of Galois groups of number ﬁelds; (b) an anabelian result of
Uchida concerning Galois groups of solvably closed extensions of number
ﬁelds; (c) a previous result of the author concerning the pro-Σ anabeliangeometry of hyperbolic curves over nonarchimedean local ﬁelds.

1. Introduction
In this paper, we study various properties of solvably closed Galois groupsof number ﬁelds, i.e., Galois groups of ﬁeld extensions of number ﬁelds that
admit no nontrivial abelian ﬁeld extensions [cf. Deﬁnition 1, (i)]. In §1,
we show that such Galois groups satisfy many of the properties of absolute
Galois groups of number ﬁelds that are of importance in the context of anabelian geometry. In particular, this includes properties concerning Galois
cohomology, center-free-ness, decomposition groups of valuations, and topologically ﬁnitely generated closed normal subgroups. In §2, after reviewing
a fundamental result of Uchida [cf. [11]] to the eﬀect that solvably closed
Galois groups of number ﬁelds are anabelian, we apply the various results
obtained in §1 to give a new version of the main result of [6] concerning thepro-Σ anabelian geometry of hyperbolic curves, where Σ is a nonempty setof prime numbers, in the context of solvably closed Galois groups of number
ﬁelds. Finally, in §3, we observe that “relatively small” solvably closed Galois groups of number ﬁelds exist in “substantial abundance”. For instance,in the case of punctured elliptic curves, it is possible in many instances toobtain solvably closed Galois groups of number ﬁelds that are, on the one
hand, “large enough” to be compatible with the outer Galois action on thepro-Σ geometric fundamental group of the punctured elliptic curve [i.e., in
the sense that this outer Galois action of the Galois group of the number
ﬁeld factors through the quotient determined by the solvably closed extension], but, on the other hand, “small enough” to be linearly disjoint fromMathematics Subject Classiﬁcation. Primary 14H30; Secondary 11R99.
Key words and phrases. solvably closed, number ﬁeld, Galois group, anabelian geometry, hyperbolic curve.
57

58

S. MOCHIZUKI

various ﬁeld extensions arising from the l-torsion points of the elliptic curve,
for a prime number l ∈/ Σ.
Acknowledgment. The author wishes like to thank Akio Tamagawa for bringing the results exposed in Theorems 2.4, 3.1 of the text to his attention.
2. Basic Properties
We begin by deﬁning the notion of a solvably closed Galois group of anumber ﬁeld and showing that such Galois groups satisfy many properties
that are well-known in the case of absolute Galois groups of number ﬁelds.
Let F be a number ﬁeld [i.e., a ﬁnite extension of the ﬁeld of rational
numbers], F an algebraic closure of F , and Fe ⊆ F a [not necessarily ﬁnite!]def
def
Galois extension of F . Write GF = Gal(F /F ), QF = Gal(Fe/F ). Thus,
one may think of QF as a quotient GF ³ QF of GF .
Deﬁnition 1.
(i) We shall say that a ﬁeld is solvably closed if it has no nontrivial abelian
extensions. If Fe is solvably closed, then we shall say that Fe/F is a solvably
closed extension and refer to QF as a solvably closed Galois group of thenumber ﬁeld F .
(ii) If G is any proﬁnite group, and p is a prime number, then we shallwrite
cdp (G)
for the smallest positive integer i such that H j (G, A) = 0 for all continuous
p-torsion G-modules A and all j > i, if such an integer i exists; if such andefinteger i does not exist, then we set cdp (G) = ∞ [cf. [8], Deﬁnition 3.3.1].
Remark 1. Observe that the Galois group QF is solvably closed if and only
if, for any open subgroup HQ ⊆ QF , whose inverse image in GF we denote
by HG ⊆ GF , the surjection induced on maximal pro-solvable quotients
sol
sol
HG
³ HQ

by the quotient morphism HG ³ HQ is an isomorphism.
Remark 2. Thus, if we denote by Fesol ⊆ F the maximal solvable [Galois]
extension of Fe, then one veriﬁes immediately that Gal(Fesol /F ) is a solvably
closed Galois group of the number ﬁeld F . In particular, [by taking Fe = F ,it follows that] the maximal pro-solvable quotient Gsol
F of GF is a solvably
closed Galois group of the number ﬁeld F .

GLOBAL SOLVABLY CLOSED ANABELIAN GEOMETRY

59

Remark 3. One veriﬁes immediately that any open subgroup of a solvably
closed Galois group of a number ﬁeld is again a solvably closed Galois groupof a number ﬁeld.
Proposition 2.1 (Galois Cohomology of Solvably Closed Galois Groups).
Suppose that QF is a solvably closed Galois group of the number ﬁeld
F . Then:(i) The natural surjection GF ³ QF induces an isomorphism∼

H i (QF , A) → H i (GF , A)
for all continuous torsion QF -modules A and all integers i ≥ 0. In particular, if F contains a square root of −1, then cdp (QF ) = 2 for all prime
numbers p.
(ii) Let p be a prime number; suppose that F contains a primitive p-th
root of unity. Then for any automorphism σ of the ﬁeld Fe that preserves
and acts nontrivially on F ⊆ Fe, the automorphism induced by σ of the setof one-dimensional Fp -subspaces of the Fp -vector space
H 2 (QF , Fp )is nontrivial.
def

Proof. First, we consider assertion (i). Write JF = ker(GF ³ QF ). To
show the desired isomorphism, it follows immediately from the Leray-Serre
spectral sequence associated to the extension 1 → JF → GF → QF → 1
that it suﬃces to show that H i (JF , A) = 0 for all i ≥ 1. Since
H i (H, A)
H i (JF , A) ∼lim=−→
JF ⊆H⊆GF[where H ranges over the open subgroups of GF containing JF ], we thus
conclude the desired vanishing as follows: If i ≥ 3, then the fact that
H i (H, A) = 0 follows from the fact that cdp (H) ≤ 2, for H suﬃciently
small [i.e., H that correspond to totally imaginary extensions of F — cf.
[8], Proposition 8.3.17]. If i = 2, then we recall that by the well-known“Hasse Principle for central simple algebras” [cf., e.g., [8], Corollary 8.1.16;
the discussion of [8], §7.1], it follows that we have an exact sequenceM
H 2 (Gv , Fp (1)) → Fp → 0
0 → H 2 (GF , Fp (1)) →v

where the “(1)” denotes a “Tate twist”; v ranges over the valuations of F ;Gv denotes the decomposition group of v in GF , which is well-deﬁned up to
conjugation; and we recall in passing that the restriction to the various direct
summands of the map to Fp induces an isomorphism H 2 (Gv , Fp (1)) ∼= Fp for

60

S. MOCHIZUKI

all nonarchimedean v. Thus, by applying the analogue for H of this exact
sequence for GF , together with the Grunwald-Wang Theorem [which assuresthe existence of global abelian ﬁeld extensions with prescribed behavior at
a ﬁnite number of valuations — cf., e.g., [8], Corollary 9.2.3], we conclude
immediately thatlim
H 2 (H, A) = 0−→
H[where H ranges over the open subgroups of GF containing JF ]. When i = 1,
the fact thatlim
H 1 (H, A) = 0−→
H

follows formally from the deﬁnition of a “solvably closed” Galois group [cf.
Deﬁnition 1, (i)]. Now the statement concerning cdp (QF ) follows immediately from the isomorphism just veriﬁed, together with the fact that, if F
contains a square root of −1 [hence is totally imaginary], then cdp (GF ) = 2[cf. [8], Proposition 8.3.17; the exact sequence just discussed concerning
H 2 (GF , Fp (1))]. This completes the proof of assertion (i).
Finally, we observe that assertion (ii) follows immediately from the exact
sequence just discussed concerning
H 2 (GF , Fp (1)) ∼= H 2 (QF , Fp (1)) ∼= H 2 (QF , Fp )[cf. assertion (i); our assumption that F contains a primitive p-th rootof unity], together with Tchebotarev’s density theorem [cf., e.g., [3], Chapter
VIII, §4, Theorem 10], which implies that if we write F0 ⊆ F for the subﬁeld
ﬁxed by σ, then there exist two distinct nonarchimedean valuations v1 , v2 of
F0 that split completely in F . That is to say, if w1 , w2 are valuations of F
lying over v1 , v2 , respectively, then there exists an element h ∈ H 2 (QF , Fp ) ∼=H 2 (GF , Fp (1)) [where we note that this isomorphism is compatible with thenatural actions by σ, up to multiplication by an element of F×p ] which maps
to a nonzero element of the direct sum in the above sequence whose uniquenonzero components are the components labeled by v1 , v2 ; thus, σ(Fp · h) 6=
Fp · h, as desired.
¤
Remark 4. As was pointed out to the author by the referee, one may generalize Proposition 2.1, (i), substantially if one assumes the Bloch-Kato conjecture — i.e., the assertion that the cup product∪ : H 1 (GK , Fp (1))⊗i → H i (GK , Fp (i))
induces a surjection for every integer i ≥ 1, every prime number p, andevery ﬁeld K of characteristic zero. Indeed, if GK ³ QK is a quotient by ae ofclosed normal subgroup JK ⊆ GK corresponding to a ﬁeld extension K

GLOBAL SOLVABLY CLOSED ANABELIAN GEOMETRY

61

K which has no nontrivial abelian extensions, then to show that the naturalmorphism
H i (QK , A) → H i (GK , A)
is an isomorphism for all integers i ≥ 0 and continuous torsion QK -modules
A, it suﬃces to verify [cf. the proof of Proposition 2.1, (i)], in the case
A = Fp , that for all open subgroups H ⊆ GK containing JK , an arbitraryclass ∈ H i (H, A) vanishes upon restriction to a suﬃciently small open sube has nogroup H1 ⊆ H containing JK ; but this follows from the fact that K
nontrivial abelian extensions if i = 1, hence by the Bloch-Kato conjecture if
i ≥ 2.
Before proceeding, we recall that a proﬁnite group ∆ is slim if every open
subgroup of ∆ has trivial centralizer in ∆ [cf. [5], Deﬁnition 0.1, (i)].
Corollary 2.2 (Slimness). Every solvably closed Galois group of anumber ﬁeld is slim.
Proof. Suppose that QF is solvably closed. Let HQ ⊆ QF be an open subgroup, σ ∈ QF an element of the centralizer of HQ . Write FH ⊆ Fe for theextension of F deﬁned by HQ . Since QF is solvably closed, by taking HQ to
be suﬃciently small, we may assume that FH contains a p-th root of unity,
for some prime number p. Note that since σ commutes with HQ , it follows
that σ acts trivially on H 2 (HQ , Fp ). Thus, by applying Proposition 2.1, (ii),to the action of σ on Fe/FH , we conclude that σ acts trivially on FH , i.e.,that σ ∈ HQ . On the other hand, since HQ may be taken to be arbitrarily
small, it thus follows that σ = 1, as desired.
¤
The next two results, concerning decomposition groups and topologicallyﬁnitely generated closed normal subgroups, respectively, are well-known inthe case of absolute Galois groups [cf., e.g., [8], Corollary 12.1.3; [2], Proposition 16.11.6].
Proposition 2.3 (Decomposition Groups). Suppose that QF is a solvably
closed Galois group of the number ﬁeld F . Let v, w be valuations of F
such that v 6= w; write Gv , Gw ⊆ QF for the corresponding decompositiongroups [which are well-deﬁned up to conjugation] in QF and Fv , Fw for thecorresponding completions of F . Then:(i) Suppose that F contains a square root of −1, and that v, w are
nonarchimedean; let K be a ﬁnite extension of Fv . Then there exists a
ﬁnite Galois extension of F contained in Fe whose restriction to Fv contains
K and whose restriction to Fw is the trivial extension.
(ii) Suppose that v, w are archimedean; let K be a nontrivial ﬁnite
extension of Fv . Then there exists a quadratic extension of F contained in

62

S. MOCHIZUKI

Fe whose restriction to Fv contains K and whose restriction to Fw is thetrivial extension.
(iii) The surjection GF ³ QF induces an isomorphism of Gv with thedecomposition group of v in GF . In particular, if v is nonarchimedean,
then Gv is slim and torsion-free.
T(iv) Gv Gw = {1}.
(v) Suppose that v is archimedean (respectively, nonarchimedean).
Then the normalizer (respectively, commensurator) of Gv in QF is equal
to Gv .
Proof. First, we consider assertion (i). Since the absolute Galois group of
Fv is pro-solvable [cf., e.g., [8], Chapter VII, §5], we may assume, by recursion, that K is an abelian extension of Fv . Since, moreover, F contains asquare root of −1, it follows that we may apply the Grunwald-Wang Theorem[cf., e.g., [8], Corollary 9.2.3] to F . Now assertion (i) follows immediately
by applying the Grunwald-Wang Theorem to F . Assertion (ii) follows by
considering the quadratic extension of F determined by taking the squareroot of an element f ∈ F which is < 0 at v and either > 0 or nonreal
at w [where we note that the existence of such an f follows immediatelyfrom the fact that the valuations v, w are distinct]. In the nonarchimedeancase, assertion (iii) follows formally from assertion (i), together with thewell-known facts that the absolute Galois group of a nonarchimedean local
ﬁeld is slim [cf., e.g., [5], Theorem 1.1.1, (ii)] and [of ﬁnite cohomologicaldimension — cf., e.g., [8], Corollary 7.2.5 — hence] torsion-free. In thearchimedean case, assertion (iii) follows, for instance, by considering the extension of F obtained by adjoining a square root of −1. To verify assertion(iv), let us ﬁrst observe that if at least one of v, w is nonarchimedean, thenit follows from the torsion-free-ness portion of assertion (iii) that both v, w
are nonarchimedean [cf. also the well-known fact that the absolute Galois
group of an archimedean local ﬁeld is ﬁnite, of order ≤ 2!], and, moreover,that [from the point of view of verifying assertion (iv)] one may replace F
by a ﬁnite abelian extension of F that satisﬁes the hypothesis of assertion(i). Now assertion (iv) follows immediately from assertions (i), (ii), (iii).
Finally, assertion (v) follows formally from assertion (iv) [together with thetorsion-free-ness portion of assertion (iii) in the nonarchimedean case]. ¤Theorem 2.4 (Topologically Finitely Generated Closed Normal Subgroups).
Suppose that Fe is a Galois extension of the number ﬁeld F such that for someprime number p, Fe has no cyclic extensions of degree p [e.g., a solvably
closed extension of F ]. Then every topologically ﬁnitely generated closed
normal subgroup N ⊆ QF is trivial.

GLOBAL SOLVABLY CLOSED ANABELIAN GEOMETRY

63

Proof. Although this fact only follows formally from the statement of [2],Proposition 16.11.6, in the case where Fe is algebraically closed, as was explained to the author by A. Tamagawa, the proof given in [2] generalizes
immediately to the case of arbitrary Fe [i.e., as in the statement of Theorem2.4]: Indeed, if we write L ⊆ Fe for the Galois [since N is normal] ﬁeld extension of F determined by N , and assume that N is nontrivial, then it follows
that there exists a proper normal open subgroup N1 ⊆ N of N . Thus, N1
determines a ﬁnite Galois extension L1 /L of degree > 1. Now let us recall
that number ﬁelds [such as F ] are Hilbertian [cf., e.g., [2], Theorem 13.4.2].
Thus, by [2], Theorem 13.9.1, (b) [i.e., “Weissauer’s extension theorem for
Hilbertian ﬁelds”], we conclude that L1 is Hilbertian, hence, by [repeatedapplication of] [2], Theorem 16.11.2, that L1 admits Galois extensions with
Galois group isomorphic to a product of an arbitrary ﬁnite number of copies
of Z/pZ. By our assumption on Fe, it follows that such Galois extensions of
L1 are contained in Fe, hence that N1 admits ﬁnite quotients isomorphic to a
product of an arbitrary ﬁnite number of copies of Z/pZ. But this contradicts
the assumption that N is topologically ﬁnitely generated.
¤
3. Anabelian Results
Next, we consider the anabelian geometry of hyperbolic curves, in thecontext of solvably closed Galois groups of number ﬁelds.
The following result is due to K. Uchida [cf. the main theorem of [11]]:
Theorem 3.1 (Solvably Closed Galois Groups are Anabelian). For i =1, 2, let Fei /Fi be a solvably closed extension of a number ﬁeld Fi ; writedef
Qi = Gal(Fei /Fi ). Then passing to the induced morphism on Galois groups
determines a bijection between the set of isomorphisms of topologicalgroups∼Q1 → Q2∼
and the set of isomorphisms of ﬁelds Fe1 → Fe2 that map F1 onto F2 .
Next, let us assume that we have been given a hyperbolic curve [cf., e.g.,[5], §0, for a discussion of hyperbolic curves] over F . Let Σ be a nonempty
set of prime numbers. Write∆X
for the maximal pro-Σ quotient of the geometric fundamental group π1 (X ×FF ) of X [relative to some basepoint]. Here, we note in passing that Σ may
be recovered from ∆X as the set of prime numbers that occur as factors oforders of ﬁnite quotients of ∆X . Thus, one has a natural outer action
GF → Out(∆X )64

S. MOCHIZUKI

of GF on ∆X .
Lemma 3.2 (Slimness). ∆X is slim.
Proof. This follows immediately by considering Galois actions on abelianizations of open subgroups of ∆X — cf. the proof of [5], Lemma 1.3.1, in
the case where Σ is the set of all prime numbers. Another [earlier] approachto proving the slimness of ∆X is given in [7], Corollary 1.3.4.
¤
Deﬁnition 2. We shall say that the [not necessarily solvably closed!] extension Fe/F , or, alternatively, the Galois group QF , is Σ-compatible with X
if the natural outer action
GF → Out(∆X )factors through the quotient GF ³ QF . Thus, if QF is Σ-compatible withX, then one obtains an exact sequence of proﬁnite groups
1 → ∆X → ΠX → QF → 1
by pulling back the natural exact sequence
1 → ∆X → Aut(∆X ) → Out(∆X ) → 1[which is exact by Lemma 3.2!] via the resulting homomorphism QF →Out(∆X ). Here, we note that since [it is an easily veriﬁed tautology that] theétale fundamental group π1 (X) of X may be recovered as the result of pullingback this natural exact sequence via the homomorphism GF → Out(∆X ),it thus follows that ΠX may be thought of as a quotient of π1 (X).
Proposition 3.3 (Geometric Subgroups are Characteristic). For i = 1, 2,
def
let Fei /Fi be a solvably closed extension of a number ﬁeld Fi ; Qi =Gal(Fei /Fi ); Σi a nonempty set of prime numbers; Xi a hyperbolic curve
over Fi with which Qi is Σi -compatible; 1 → ∆Xi → ΠXi → Qi → 1
the resulting exact sequence of proﬁnite groups [cf. Deﬁnition 2]. Then any
isomorphism of topological groups∼

ΠX1 → ΠX2
maps ∆X1 isomorphically onto ∆X2 . In particular, Σ1 = Σ2 .
Proof. We give two proofs of Proposition 3.3. The ﬁrst proof consists of simply observing [cf. the proof of [5], Lemma 1.1.4, (i), via [5], Theorem 1.1.2]∼
that the image of ∆X1 under the composite of the isomorphism ΠX1 → ΠX2
with the surjection ΠX2 ³ Q2 forms a topologically ﬁnitely generated closed
normal subgroup of Q2 , hence is trivial, by Theorem 2.4.
The second proof of Proposition 3.3 only uses Theorem 2.4 in the wellknown case of an absolute Galois group of a number ﬁeld. Moreover, when

GLOBAL SOLVABLY CLOSED ANABELIAN GEOMETRY

65

either Σ1 or Σ2 is not equal to the set of all prime numbers, then this second
proof does not use Theorem 2.4 at all.
For i = 1, 2, let Hi ⊆ ΠXi be corresponding [i.e., relative to the given∼
isomorphism ΠX1 → ΠX2 ] normal open subgroups; write Hi ³ Ji for thequotients determined by the quotients ΠXi ³ Qi . By taking the Hi to be
suﬃciently small, we may also assume that the number ﬁelds determined by
the Ji contain square roots of −1. Thus, by Proposition 2.1, (i), it follows
that
cdp (Hi ) = 2 + d(p, i)
where d(p, i) is equal to 1 or 2 [depending on whether Xi is aﬃne or proper]∼
if p ∈ Σi and d(p, i) = 0 if p ∈/ Σi . Since H1 → H2 , we thus conclude that
Σ1 = Σ2 , and that X1 is aﬃne if and only if X2 is. Now if Σ1 = Σ2 isthe set of all prime numbers, and X1 , X2 are aﬃne, then it follows from
Matsumoto’s injectivity theorem [cf. [4], Theorem 2.1] that the ﬁeld Fei is analgebraic closure of Fi ; thus, in this case, Proposition 3.3 follows from [5],Lemma 1.1.4, (i) [i.e., Theorem 2.4 for absolute Galois groups of number
ﬁelds].
Next, let us suppose that there exists a prime number p such that p ∈/ Σ1 ,defp∈/ Σ2 . This implies that every ﬁnite quotient group of Di = ker(Hi ³ Ji )has order prime to p, hence [by consideration of the Leray-Serre spectral
sequence associated to the surjection Hi ³ Ji ] that, for i = 1, 2, the natural
homomorphism
H 2 (Ji , Fp ) → H 2 (Hi , Fp )is an isomorphism. In particular, it follows that ∆Xi acts trivially on
H 2 (Hi , Fp ). Thus, the natural action of ΠXi on H 2 (Hi , Fp ) factors throughthe quotient ΠXi ³ Qi /Ji . Now, by taking Hi to be suﬃciently small,
we may assume [since Qi is solvably closed!] that the extension ﬁeld of Fi
determined by Ji contains a primitive p-th root of unity. Thus, by Proposition 2.1, (ii), we conclude that the action of Qi /Ji on H 2 (Hi , Fp ) is faithful.
∼∼
Since the isomorphism ΠX1 → ΠX2 induces an isomorphism H1 → H2 ,∼hence an isomorphism H 2 (H1 , Fp ) → H 2 (H2 , Fp ) which is compatible withthe respective actions of ΠX1 , ΠX2 , we thus conclude that the isomorphism∼
ΠX1 → ΠX2 preserves the kernels of the surjections ΠXi ³ Qi /Ji , hence thatthe subgroup ∆Xi = ker(ΠXi ³ Qi ) may be recovered as the intersection of
the kernels of the surjections ΠXi ³ Qi /Ji , by letting the Hi range over all
suﬃciently small normal open subgroups of ΠXi . This completes the proofof Proposition 3.3 in the case where there exists a prime number p such thatp∈/ Σ1 , p ∈/ Σ2 .
Finally, we consider the case where X1 , X2 are proper. Let p be a prime
number; suppose that the Hi have been taken to be suﬃciently small so

66

S. MOCHIZUKI

that the number ﬁelds determined by the Ji contain a primitive p-th root ofunity and a square root of −1 [which, by Proposition 2.1, (i), implies thatdef

cdp (Ji ) = 2]. Since Di = ker(Hi ³ Ji ) also satisﬁes cdp (Di ) ≤ 2, it thus
follows from the Leray-Serre spectral sequence associated to the extension
1 → Di → Hi → Ji → 1 that there is a natural isomorphism
H 4 (Hi , Fp ) ∼= H 2 (Ji , Fp ) ⊗ H 2 (Di , Fp )which is compatible with the natural action of ΠXi on the various cohomology modules involved. Here, we note that [by the well-known structure of
the cohomology of the geometric fundamental group of an algebraic curve]∆Xi ⊆ ΠXi acts trivially on H 2 (Di , Fp ). Thus, Proposition 3.3 follows in
the present case by applying Proposition 2.1, (ii), as in the argument givenin the preceding paragraph.
¤
Theorem 3.4 (The Anabelian Geometry of Hyperbolic Curves over Solvably Closed Galois Groups). For i = 1, 2, let Fei /Fi be a solvably closed exdef
tension of a number ﬁeld Fi ; Qi = Gal(Fei /Fi ); Σi a nonempty set of prime
numbers; Xi a hyperbolic curve over Fi with which Qi is Σi -compatible;
1 → ∆Xi → ΠXi → Qi → 1 the resulting exact sequence of proﬁnite groups
ei → Xi the pro-ﬁnite étale covering of Xi deter[cf. Deﬁnition 2]; X
mined by ΠXi [regarded as a quotient of the étale fundamental group of Xi ].
Then passing to the induced morphism on fundamental groups determines abijection between the set of isomorphisms of topological groups∼

ΠX1 → ΠX2∼

e2 ,e1 → Xand the set of compatible pairs of isomorphisms of schemes X∼
X1 → X2 .
∼Proof. By Proposition 3.3, any isomorphism ΠX1 → ΠX2 induces an isomor∼
phism Q1 → Q2 , hence, by Theorem 3.1, a compatible pair of isomorphisms∼∼
of ﬁelds Fe1 → Fe2 , F1 → F2 . Thus, we may apply “Theorem A” of [6] to the∼isomorphism ΠX1 → ΠX2 to conclude that this isomorphism arises from a
∼ e
∼
e1 →
unique compatible pair of isomorphisms of schemes X
X2 , X1 → X2 , asdesired.
¤4. Some Examples
Finally, we conclude by observing that in various situations, Σ-compatible
solvably closed extensions which are, moreover, “relatively small” [e.g., bycomparison to the entire absolute Galois group of a number ﬁeld] exist in
substantial abundance.

GLOBAL SOLVABLY CLOSED ANABELIAN GEOMETRY

67
def

Proposition 4.1 (The Case of a Single Prime Number). Let Σ = {r},where r is a prime number.
(i) Let ∆ be a topologically ﬁnitely generated pro-r group. [Thus,
since ∆ is topologically ﬁnitely generated, its topology admits a base of characteristic open subgroups, which determine a natural proﬁnite topology onOut(∆).] Write ∆ ³ ∆ab for the abelianization of ∆. Then the kernel ofthe natural morphism of proﬁnite groupsOut(∆) → Aut(∆ab ⊗ Fr )is a pro-r [hence, in particular, pro-solvable!] group.
(ii) Let X be a hyperbolic curve over F . Then there exists a ﬁnite Galois extension F1 over F such that the maximal solvable extension [whichdef
is solvably closed — cf. Remark 2] Fe = F1sol of F1 is Σ-compatible withX.
Proof. First, we consider assertion (i). Since ∆ admits a base of characteristic open subgroups, it suﬃces to verify assertion (i) when ∆ is a ﬁnite
group of order a power of r. But then consideration of the [manifestly characteristic!] lower central series of ∆ reveals that any automorphism α of ∆that induces the identity on ∆ab ⊗ Fr is “unipotent upper triangular” withrespect to the ﬁltration given by the lower central series; thus, the order of
α is a power of r. This completes the proof of assertion (i). Assertion (ii)follows formally from assertion (i) and the deﬁnitions.
¤
Proposition 4.2 (Basic Properties of Special Linear Groups). Let l be aprime number. Write SL2 (Fl ) for the special linear group of 2 by 2
defmatrices with coeﬃcients in Fl , P SL2 (Fl ) = SL2 (Fl )/{±1}.
(i) Suppose that l ≥ 5. Then P SL2 (Fl ) is a simple ﬁnite group.
(ii) No proper subgroup of SL2 (Fl ) surjects onto P SL2 (Fl ).
(iii) P SL2 (F2 ), P SL2 (F3 ), as well as every proper subgroup of P SL2 (Fl )[for arbitrary l], is either solvable or isomorphic to P SL2 (F5 ).
Proof. Assertions (i), (ii), (iii) are well-known — cf., e.g., [10], Chapter IV,§3.4, Lemmas 1, 2; [1], §1.2.
¤
Remark 5. The proper subgroups H of SL2 (Fl ) may be analyzed as follows:
If H is of order divisible by l, then H contains a subgroup U of order l.
×
Since F×l , Fl2 are of order prime to l, such a subgroup U is generated bya unipotent matrix; thus, [by possibly replacing H¡ with¢ a conjugate of H]
we may assume that U is generated by a matrix 10 11 . In particular, [as is
well-known or easily computed] the normalizer of U is given by the solvable

68

S. MOCHIZUKI

subgroup of upper triangular matrices of SL2 (Fl ). ¡ Thus,¢ ¡ if ¢U fails to benormal in H, the fact that SL2 (Fl ) is generated by 10 11 , 11 01 implies that
H = SL2 (Fl ), in contradiction to our assumption that H is proper. That
is to say, since H is proper, we conclude that H is solvable, as desired. On
the other hand, if the order of H is prime to l, then H may be classiﬁed
by applying the Hurwitz formula to the tamely ramiﬁed Galois covering
P1F → P1F /H [arising from the natural action of SL2 on P1F , where Fl is anlll
algebraic closure of Fl ], which gives rise to fairly restrictive conditions on theramiﬁcation indices of this covering. In particular, if H is non-abelian, then,by taking an appropriate isomorphism of P1F /H with P1F , one concludesllthat this covering is ramiﬁed over the three points “0”, “1”, and “∞” ofP1F , with ramiﬁcation index 2 at “0”, ramiﬁcation index ∈ {2, 3} at “1”,land ramiﬁcation index ∈ {3, 4, 5} (respectively, arbitrary, ≥ 2) at “∞” if
the ramiﬁcation index at “1” is equal to 3 (respectively, 2). Now it is anelementary exercise to classify the possible groups H that may occur. For
instance, by considering modular curves, it follows immediately that the case
H = P SL2 (F5 ) corresponds to the case where the ramiﬁcation indices are(2, 3, 5).
Proposition 4.3 (Linear Disjointness I). Let l > 5 be a prime number; r adefprime number 6= l; Σ = {r}; X a once-punctured elliptic curve over anumber ﬁeld F . Suppose further that F contains an l-th root of unity,
and that the resulting homomorphism
GF → SL2 (Fl )determined by the action of the absolute Galois group GF of F on the ltorsion points of the elliptic curve E compactifying X is surjective. Then
there exists a solvably closed extension Fe/F which is Σ-compatible
with X, and, moreover, linearly disjoint [over F ] from the extension K of
F determined by the kernel of the homomorphism GF → SL2 (Fl ).
Proof. Write L ⊆ K for the extension of F determined by the kernel ofthe homomorphism GF → P SL2 (Fl ) [obtained by composing the homomorphism GF → SL2 (Fl ) with the natural surjection SL2 (Fl ) ³ P SL2 (Fl )].
Then it follows immediately from Proposition 4.2, (ii), that any Galois extension of F is linearly disjoint from K if and only if it is linearly disjoint
from L. Now observe that Gal(L/F ) ∼= P SL2 (Fl ) is simple [cf. Proposition
4.2, (i)] and non-abelian. Thus, by Proposition 4.1, (i), it suﬃces to showthat the ﬁnite Galois extension R of F determined by the kernel of the homomorphism GF → GL2 (Fr ) arising from the Galois action on the r-torsion
points of E is linearly disjoint from L. On the other hand, again sinceGLOBAL SOLVABLY CLOSED ANABELIAN GEOMETRY69

Gal(L/F ) is simple and non-abelian, this linear disjointness property follows from the fact [cf. Proposition 4.2, (iii); our assumption that r 6= l > 5]that no subquotient of GL2 (Fr ) [or, equivalently, P SL2 (Fr ), since P SL2 (Fl )is simple and nonabelian] is isomorphic to P SL2 (Fl ). This completes theproof of Proposition 4.3.
¤
Proposition 4.4 (Linear Disjointness II). Let l > 5 be a prime number; Σ anonempty set of prime numbers such that l ∈/ Σ; X a once-punctured
elliptic curve over a number ﬁeld F with stable reduction over the
ring of integers OF of F ; Fµ the extension of F obtained by adjoining anl-th root of unity. Suppose further that l ≥ [F : Q] + 2; that [Fµ : F ]divides (l − 1)/2 [which implies that the homomorphism
GF → P GL2 (Fl ) = GL2 (Fl )/F×ldefdetermined by the action of the absolute Galois group GF of F on the ltorsion points of the elliptic curve E compactifying X factors through theimage of P SL2 (Fl ) in P GL2 (Fl )]; that the resulting homomorphism GF →P SL2 (Fl ) is surjective; and that, for each prime l of F lying over l at
which E has bad reduction, the following condition is satisﬁed:
Write Fl for the completion of F at l. Thus, the elliptic
curve E ×F Fl is a Tate curve, hence has a well-deﬁned“q-parameter” ql in the ring of integers OFl . Then thevaluation of ql is prime to l.
Then:(i) There exists an extension Fe/F which is Σ-compatible with X, and,moreover, linearly disjoint [over F ] from the extension K of F determinedby the kernel of the homomorphism GF → P SL2 (Fl ).
(ii) Write Kµ for the extension of F determined by the kernel of thehomomorphism GF → GL2 (Fl ) [arising from the Galois action on the ldef
torsion points of E]. Thus, Fµ ⊆ Kµ ; write Feµ = Fµ · Fe for the composite
extension [over F ]. Then the maximal solvable extension Feµsol of Feµ forms
a solvably closed extension of Fµ which is Σ-compatible with X and,moreover, linearly disjoint over Fµ from the extension Kµ of Fµ .
Proof. First, we consider assertion (i). Let Fe/F be the extension determinedby the kernel of the homomorphism GF → Out(∆X ) [cf. Deﬁnition 2]. Letl be a prime of F lying over l. Since P SL2 (Fl ) is simple [cf. Proposition
4.2, (i)], to complete the proof of assertion (i), it suﬃces to show that thecomposite [i.e., over F ] ﬁeld extension K · Fe is not equal to Fe. Thus, supposethat K · Fe = Fe. Since l ∈/ Σ, if E has good reduction at l, then it follows

70

S. MOCHIZUKI

that Fe/F is unramiﬁed at l; similarly, if E has bad reduction at l, then thefact that l 6∈ Σ implies that Fe/F is tamely ramiﬁed at l. On the other
hand, if E has good reduction at l, then the fact that K ⊆ K · Fe = Fe isunramiﬁed at l implies, by applying, for instance, results of Raynaud on
the “fully faithfulness of restriction to the generic ﬁber” for ﬁnite ﬂat groupschemes over moderately ramiﬁed discrete valuation rings [cf. [9], Corollaire
3.3.6, (1); our assumption that l ≥ [F : Q] + 2, which implies that thering of integers OFl is indeed “moderately ramiﬁed”], that, if we write E for
the stable model of the elliptic curve E over OFl and E[l] for the kernel ofmultiplication by l on E, then E[l] may be written as a direct productE[l] ∼=G×G
of two copies of some ﬁnite ﬂat group scheme G over OFl [which implies, forinstance, that the tangent space of E[l], hence also of E, is even-dimensional!]— a contradiction. Finally, if E has bad reduction at l, then the fact thatK ⊆ K · Fe = Fe is tamely ramiﬁed at l contradicts our assumption concerning
the “valuation of the q-parameter” [which implies that K is wildly ramiﬁed
at l]. This completes the proof of assertion (i).
To verify assertion (ii), let us ﬁrst observe that by Proposition 4.2, (i) [cf.
our assumption that l > 5], (ii), and the surjectivity assumption in the statement of the present Proposition 4.4, we have Gal(Kµ /Fµ ) ∼= SL2 (Fl ). Now,
by applying Proposition 4.2, (ii), as in the proof of Proposition 4.3, assertion(ii) follows immediately from assertion (i), together with the simplicity [andnon-solvability] of P SL2 (Fl ).
¤References[1] R. W. Carter, Simple groups of Lie type, Pure and Applied Mathematics 28, John
Wiley & Sons (1972).
[2] M. Fried and M. Jarden, Field Arithmetic (second edition), Springer-Verlag (2005).
[3] S. Lang, Algebraic number theory, Addison-Wesley Publishing Co. (1970).
[4] M. Matsumoto, Galois representations on proﬁnite braid groups on curves, J. Reine
Angew. Math. 474 (1996), pp. 169-219.
[5] S. Mochizuki, The Absolute Anabelian Geometry of Hyperbolic Curves, Galois Theory
and Modular Forms, Kluwer Academic Publishers (2003), pp. 77-122.
[6] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math. 138(1999), pp. 319-423.
[7] H. Nakamura, Galois rigidity of pure sphere braid groups and proﬁnite calculus, J.
Math. Sci. Univ. Tokyo 1 (1994), pp. 71-136.
[8] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number ﬁelds, Grundlehren derMathematischen Wissenschaften 323, Springer-Verlag (2000).
[9] M. Raynaud, Schémas en groupes de type (p, . . . , p), Bull. Soc. Math. France 102(1974), pp. 241-280.
[10] J.-P. Serre, Abelian l-adic Representations and Elliptic Curves, Benjamin (1968).

GLOBAL SOLVABLY CLOSED ANABELIAN GEOMETRY71[11] K. Uchida, Isomorphisms of Galois groups of solvably closed Galois extensions,Tôhoku Math. J. 31 (1979), pp. 359-362.
Shinichi Mochizuki
Research Institute for Mathematical Sciences
Kyoto University
Kyoto, 606-8502 Japane-mail address: motizuki@kurims.kyoto-u.ac.jp(Received April 6, 2006 )