The pro-C anabelian geometry of number fields

概要

A summary of
“The pro-C anabelian geometry of number fields”
Ryoji Shimizu
This is a summary of [Shimizu3]. Let K be a number field and C a full class of finite
groups. We write K C /K for the maximal pro-C Galois extension of K, and GCK for its Galois
group, which is the maximal pro-C quotient of the absolute Galois group of K.
The Neukirch-Uchida theorem, which is one of the most important results in anabelian
geometry, states that the isomorphisms of the absolute Galois groups of number fields arise
functorially from unique isomorphisms of fields (cf. [Uchida]). Moreover, various generalizations of the Neukirch-Uchida theorem, where one replaces the absolute Galois groups by
their various quotients, have been studied by many mathematicians (e.g. [Ivanov], [Ivanov2],
[Sa¨ıdi-Tamagawa], [Shimizu] and [Shimizu2]). These results prompt the following natural
question:
For i = 1, 2, let Ki be a number field, Ci a nontrivial full class of finite groups, and
∼
σ : GCK11 → GCK22 an isomorphism. Is σ induced by a unique isomorphism between K2C2 /K2
and K1C1 /K1 ?
In this paper, we answer this question under some assumptions. Write Σ(C) for the set
of prime numbers p with Z/pZ ∈ C. For a set S of nonarchimedean primes of K, set
∑
∑
−s
−s
def
def
p∈S N(p)
p∈S N(p)
,
δ
(S)
=
lim
inf
δsup (S) = lim sup
inf
1
1
s→1+0
log s−1
log s−1
s→1+0
and if δsup (S) = δinf (S), then write δ(S) (the Dirichlet density of S) for them. The most
important result is the following:
Theorem A ([Shimizu3, Theorem 7.1]). Assume that δsup (Σ(Ci )) > 0 for at least one i ∈
{1, 2}. Then σ is induced by a unique isomorphism between K2C2 /K2 and K1C1 /K1 .
The goal of this paper is to prove this theorem. As in proofs of the Neukirch-Uchida
theorem (cf. [NSW]) and other previous results, we first characterize group-theoretically
some data in GCK (e.g. the decomposition groups), and then obtain an isomorphism of fields
using them. In the following, we describe the structure of this paper in more detail.
In §1, we begin by collecting results on the decomposition groups of GCK . For example, we
prove that the decomposition groups of GCK are canonically isomorphic to the maximal pro-C
quotients of the absolute Galois groups of p-adic fields, and that for two distinct primes of
1

K C , the intersection of their decomposition groups is trivial. The uniqueness in the question
follows immediately from this.
In §2, we recover group-theoretically a certain subset of the set of decomposition groups
of GCK . More precisely:
C,l
Theorem B ([Shimizu3, Theorem 2.11]). Let l ∈ Σ(C). Set PK,f
= {p ∈ PK,f \ PK,l |
C
C
µl ⊂ Kp }, where PK,f (resp. PK,l ) and Kp stand for the set of nonarchimedean primes
(resp. nonarchimedean primes above l) of K and the maximal pro-C Galois extension of Kp ,
C,l
can be recovered
respectively. Then the set of decomposition groups in GCK at primes in PK,f
C
group-theoretically from GK .
def

Further, by using this result, the pro-C factor of the l-adic cyclotomic character GCK →
Aut(µl∞ )C = Zl ∗,C can be recovered group-theoretically from GCK .
In §3, we study fundamental properties of the “local correspondence”:
Definition C ([Shimizu3, Definition 3.1]). For i = 1, 2, let Si ⊂ PKi ,f . We say that
∼
a bijection ϕ : S1 (K1C1 ) → S2 (K2C2 ) is a local correspondence between S1 and S2 for σ if
∼
σ(Dp1 (K1C1 /K1 )) = Dϕ(p1 ) (K2C2 /K2 ) for any p1 ∈ S1 (K1C1 ). Let ϕ : S1 (K1C1 ) → S2 (K2C2 ) be a
local correspondence between S1 and S2 for σ. We say that ϕ satisfies condition (Char) (resp.
condition (Deg)) if for any p1 ∈ S1 (K1C1 ), the residue characteristics (resp. the local degrees)
of p1 |K1 and ϕ(p1 )|K2 coincide.
By Theorem B, we obtain a local correspondence between PKC11 ,f and PKC22 ,f for σ, where
C,l
C
= ∪l∈Σ(C) PK,f
PK,f
.
In §4, we study separatedness of the decomposition groups in G(Q(∪l∈Σ(C) µl )/Q)C =
G(QC ∩ Q(∪l∈Σ(C) µl )/Q). Together with a certain part of the cyclotomic character recovered
in §2, we prove that if δsup (Σ(Ci )) > 0 for at least one i ∈ {1, 2}, then the local correspondence
between PKC11 ,f and PKC22 ,f (resp. (PKC11 ,f ∩ cs(K1 /Q))(K1 ) and (PKC22 ,f ∩ cs(K2 /Q))(K2 )) for σ
def

def

C
satisfies condition (Char) (resp. conditions (Char) and (Deg)), where PK,f
= {p ∈ PQ,f |
C
}.
PK,p ⊂ PK,f
In §5, we develop two ways to show that isomorphisms of Galois groups of number fields
are induced by field isomorphisms under some assumptions. The two results ([Shimizu3,
Proposition 5.3 and Proposition 5.8] are based on [Uchida2] and [Shimizu2], respectively.
However, these previous works cannot be applied to the proof of our question as they are,
so that the goal of §5 is to modify them by inventing some new methods. In the proof of
Theorem A, we use the former:

Proposition D ([Shimizu3, Proposition 5.3]). Let S0 ⊂ PQ,f . Assume that the following
conditions hold:
(a) #Σ(C1 ) = ∞.
(b) δinf (cs(K1 /Q) \ S0 ) = 0.
2

(c) There exists a local correspondence between S0 (K1 ) and S0 (K2 ) for σ satisfying conditions (Char) and (Deg).
Then σ is induced by a unique isomorphism between K2C2 /K2 and K1C1 /K1 .
In the latter result, which will be used in the proof of Theorem E, we need to assume the
existence of a complex prime for the number field, however, the assumption on the Dirichlet
densities of the sets between which the local correspondence exists is weaker than the former.
C
) = 1.
In §6, as a preparation for Theorem A, we prove that if δsup (Σ(C)) > 0, then δ(PK,f
In §7, using the results obtained so far, we prove Theorem A and its corollaries, for example about the group of outer isomorphisms of GCK (cf. [Shimizu3, Corollary 7.4]). Further,
even when δ(Σ(Ci )) = 0 for each i ∈ {1, 2}, most of the results in this paper are still valid,
so that we can answer the question under some technical conditions:
def

Theorem E ([Shimizu3, Theorem 7.5]). For i = 1, 2, let Si ⊂ PKi ,f , and write Si = {p ∈
PQ,f | PKi ,p ⊂ Si }. Assume that the following conditions hold:
(a) For one i, 2 ∈ Σ(Ci ) or Ki has a complex prime.
(b) There exists a local correspondence between S1 and S2 for σ, satisfying condition
(Char).
(c) For any finite Galois subextension L1 of K1C1 /K1 , δsup (S1 ∩ cs(L1 /Q)) > 0.
(d) S2 ∩ cs(K2 /Q) ̸= ∅ .
Then σ is induced by a unique isomorphism between K2C2 /K2 and K1C1 /K1 .
By using this result, we can prove the following analog of “the relative Grothendieck Conjecture”:
Corollary F ([Shimizu3, Corollary 7.9]). Assume that the following conditions hold:
(a) For one i, 2 ∈ Σ(Ci ) or Ki has a complex prime.
(b) The following diagram commutes:
GCK11

FF
FF
FF
FF
"

≃
σ

/ GC2
K2
yy
y
y
yy
|y
y

1
Gab,C
,
Q

where the diagonal arrows are the restrictions.
Then σ is induced by a unique isomorphism between K2C2 /K2 and K1C1 /K1 .
3

Moreover, by Theorem E, Conjecture G is reduced to Conjecture H (cf. [Shimizu3, Proposition 7.12]).
Conjecture G ([Shimizu3, Conjecture 7.10]). Assume that the following condition holds:
(a) For one i, 2 ∈ Σ(Ci ) or Ki has a complex prime.
Then σ is induced by a unique isomorphism between K2C2 /K2 and K1C1 /K1 .
Conjecture H ([Shimizu3, Conjecture 7.11]). As we have seen after Definition C, there
exists a local correspondence ϕ between PKC11 ,f and PKC22 ,f for σ. Then ϕ satisfies condition
(Char).

References
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