The m-step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields

概要

A summary of “The m-step solvable Grothendieck conjecture
for affine hyperbolic curves over finitely generated fields”
Naganori Yamaguchi
n anabelian geometry, the following conjecture, called the Grothendieck conjecture
for hyperbolic curves, is a central problem: If the (tame) arithmetic fundamental groups
of two hyperbolic curves over a field k are isomorphic as profinite groups (over the absolute Galois group Gk ), are these hyperbolic curves are isomorphic (over k)? For this
conjecture, the case where k is finitely generated over the rational number field Q and
the hyperbolic curves have genus 0 was settled by Hiroaki Nakamura, the case where k is
either a finite field or finitely generated over Q and the hyperbolic curves are affine was
settled by Akio Tamagawa, and finally, this conjecture was completely proved by Shinichi
Mochizuki when k is a sub-ℓ-adic field (i.e., a subfield of a field finitely generated over
the ℓ-adic local field Qℓ for a fixed prime ℓ). Thus, the Grothendieck conjecture has been
proved. However, we can consider the following extension of this conjecture:
(Q) If certain quotients of the (tame) arithmetic fundamental groups of two hyperbolic
curves over k are isomorphic as profinite groups (over Gk ), are these hyperbolic
curves isomorphic (over k)?
In particular, when the quotients are the maximal geometrically m-step solvable quotients
of the (tame) arithmetic fundamental group, we call (Q) the m-step solvable Grothendieck
conjecture, where m is a positive integer. This conjecture was proved in the case where
“k is an algebraic number field satisfying certain conditions, m ≥ 2, and the hyperbolic
curves are 4-punctured projective lines over k” by Hiroaki Nakamura, where “k is a subℓ-adic field and m ≥ 5” by Shinichi Mochizuki, and where “k is finitely generated over
the prime field, m ≥ 3, and the hyperbolic curves have genus 0 and satisfy a certain
condition” by the author.
In this paper, we prove the m-step solvable Grothendieck conjecture for affine hyperbolic curves in most cases, as follows.

(Notation) Let m be a positive integer. For i = 1, 2, let ki be a field of characteristic
pi ≥ 0 and Gki the absolute Galois group of ki . Let Xi be a proper, smooth curve
over ki and Ei a closed subscheme of Xi which is finite, ´etale over ki . Let gi be the
genus of Xi and ri the degree of Ei over ki . Set Ui := Xi − Ei . If pi > 0, then,
for n ∈ Z≥0 , we write Ui (n) for the n-th Frobenius twist of Ui over ki . For any
extension l over ki , we write Ui,l := Ui ×ki l. We write ΠUi for the tame arithmetic
fundamental group π1tame (Ui , ∗) of Ui and ΠUi for the tame geometric fundamental
[m]
group π1tame (Ui,ksep , ∗) of Ui . We define ΠUi as the m-step derived subgroup of ΠUi
m

[m]

[m]

(m)

and set ΠUi := ΠUi /ΠUi and ΠUi := ΠUi /ΠUi .
1

Finite field case
When ki is finite, the m-step solvable Grothendieck conjecture has not been proved
in any single case. Thus, the following theorem is a completely new result and even the
first result on the conjecture over finite fields.
Theorem A (Theorem 2.16, Corollary 2.22). Assume that k1 , k2 are finite and that U1
is affine hyperbolic.
(1) Assume that m satisfies
{
m≥2
m≥3
(m)

(if r1 ≥ 3 and (g1 , r1 ) ̸= (0, 3), (0, 4))
(if r1 < 3 or (g1 , r1 ) = (0, 3), (0, 4))

(m)

Then ΠU1 and ΠU2 are isomorphic as profinite groups if and only if U1 and U2 are
isomorphic as schemes.
(2) Assume that m ≥ 3. Let n be an integer satisfying m > n ≥ 2. Let Φ be an iso(m−n) ∼
(m−n)
morphism ΠU1
−
→ ΠU2 of profinite groups which is induced by an isomorphism
(m) ∼
(m)
ΠU1 −
→ ΠU2 of profinite groups. Then Φ is induced (up to inner automorphisms of
∼
(m−n)
ΠU2 ) by a unique isomorphism U1 −
→ U2 of schemes.
(1)

(m)

(Sketch of Proof) First, we reconstruct the decomposition groups of ΠUi from ΠUi by
(m)
using the (quasi-)sections of the natural projection Π Ui ↠ Gki . In this step, we always
(m)
face the difficulty that comes from the fact that we can only use data from Π Ui , not
the whole ΠUi . (Just for example, we face this difficulty when proving the separatedness
(1)
(m)
of decomposition groups of ΠUi .) Next, we reconstruct the curve Ui from ΠUi and its
decomposition groups. Then we reconstruct the multiplicative group and the addition
of the function field of Ui by using class field theory and a lemma of Tamagawa. This
settles (1), and, by applying (1) to coverings of Ui , we prove (2).
□

Finitely generated field case
Next, we consider the case that ki is a field finitely generated over the prime field. In
this case, we assume that k1 = k2 and write k and p instead of ki and pi , respectively.
The following theorem is the main result in this case.
Theorem B (Theorem 4.12, Corollary 4.18). Assume that k is a field finitely generated
over the prime field and that U1 is affine hyperbolic. Assume that U1,k does not descend
to a curve over Fp when p > 0.
(1) Assume that m satisfies
{
m≥4
m≥5

(if r1 ≥ 3 and (g1 , r1 ) ̸= (0, 3), (0, 4))
(if r1 < 3 or (g1 , r1 ) = (0, 3), (0, 4)).
(m)

(m)

Then, when p = 0 (resp. p > 0), ΠU1 and ΠU2 are isomorphic over Gk as profinite
groups if and only if U1 and U2 are isomorphic as k-schemes (resp. U1 (n1 ) and U2 (n2 )
are isomorphic as k-schemes for some pair n1 , n2 of non-negative integers).
2

(2) Assume that m ≥ 5. Let n be an integer satisfying m > n ≥ 4. Let Φ be an
(m−n) ∼
(m−n)
isomorphism ΠU1
−
→ ΠU 2
of profinite groups over Gk which is induced by an
(m) ∼
(m)
isomorphism ΠU1 −
→ ΠU2 of profinite groups over Gk . Then, when p = 0 (resp.
m−n
p > 0), Φ is induced (up to inner automorphisms of ΠU2 ) by a unique isomorphism
∼
∼
U1 −
→ U2 of k-schemes (resp. a unique isomorphism U1 (n1 ) −
→ U2 (n2 ) of k-schemes
for a unique pair n1 , n2 of non-negative integers satisfying n1 n2 = 0).
When p = 0, Theorem B(1) for m = 4, g1 ≥ 1 is a new result which is not covered
by the three previous results (by Nakamura, Mochizuki, and the author). When p > 0,
Theorem B for g1 ≥ 1 is a completely new result.

(Sketch of Proof) To show Theorem B, we need Theorem A(2) and the following theorem
on the m-step solvable version of the Oda-Tamagawa good reduction criterion for affine
hyperbolic curves.
Theorem C (Theorem 3.8) Assume that m ≥ 2. Let R be a discrete valuation ring, s
the closed point of Spec(R), ps the characteristic of the residue field of s, and (X, E) a
hyperbolic curve over the field of fractions K(R). Set U := X − E. Then (X, E) has
good reduction at s if and only if the image of the inertia group of GK in the outer
m
automorphism group of the maximal prime-to-p′s quotient of ΠU is trivial.
By using Galois descent theory, we can reduce the proof of Theorem B to the case that
the Jacobian variety of X1 has a level N structure for an integer N ≥ 3 which is not
divisible by p and Ei consists of k-rational points. We take an integral regular scheme
S of finite type over Spec(Z) with function field k. By replacing S with a suitable open
subscheme if necessary, we may assume that N is invertible on S and that there exists
a smooth curve (Xi , Ei ) over S whose generic fiber is isomorphic to (and identified with)
(Xi , Ei ). The main step of the proof is to show that the morphism ζi : S → Mg,r [N ]
classifying (Xi , Ei ) (with a suitable ordering of the sections in Ei and a suitable level N
structure) for i = 1, 2 coincide set-theoretically. By using this, Theorem A, and Theorem
C, (1) follows. By applying (1) to coverings of Ui , we prove (2). ...