# A refined version of the geometrically $m$-step solvable Grothendieck
conjecture for genus $0$ curves
over finitely generated fields

Naganori Yamaguchi Affiliation: Tokyo Institute of Technology, 2-12-1
Ookayama, Meguro-ku, Tokyo 152-8550, JAPAN,
e-mail<EMAIL_ADDRESS>
###### Abstract

In the present paper, we show a new result on the geometrically $2$-step
solvable Grothendieck conjecture for genus $0$ curves over finitely generated
fields. More precisely, we show that two genus $0$ hyperbolic curves over a
finitely generated field $k$ are isomorphic as $k$-schemes (up to Frobenius
twists) if and only if the geometrically maximal $2$-step solvable quotients
of their étale fundamental groups are isomorphic as topological groups over
the absolute Galois group of $k$.

Keywords: anabelian geometry, étale fundamental group, Grothendieck conjecture

2020 Mathematics Subject Classification. Primary: 14H30, Secondary: 14E20.

###### Contents

  1. 1 Group-theoretical reconstructions of various invariant
  2. 2 Separatedness property for decomposition groups over finite fields
  3. 3 A group-theoretical reconstruction of decomposition groups over finite fields
  4. 4 The weak bi-anabelian results over finite fields
  5. 5 Specialization
  6. 6 The weak bi-anabelian results over finitely generated fields

## Introduction

Let $k$ be a field of characteristic $p\geq 0$. In the letter to G. Faltings
(cf. [5]), A. Grothendieck gave the following conjecture:

######  Conjecture (The Grothendieck conjecture).

Assume that $k$ is a field finitely generated over the prime field and let
$G_{k}$ be the absolute Galois group of $k$. Then an “anabelian” variety $U$
over $k$ may be reconstructed group-theoretically from the étale fundamental
group $\pi^{\acute{e}\mathrm{t}}_{1}(U,\ast)$ of $U$ with a geometric point
$\ast$ and its natural projection $\pi^{\acute{e}\mathrm{t}}_{1}(U,\ast)\to
G_{k}$.

In the conjecture, we consider an “anabelian” variety, whose definition has
not yet been established perfectly. However, in the letter, A. Grothendieck
conjectured that hyperbolic curves are “anabelian”, where a smooth curve of
type $(g,r)$ is hyperbolic if $2-2g-r<0$, in other words,
$(g,r)\neq(0,0),(0,1),(0,2),(1,0)$ (cf. Notation and conventions Curves). The
Grothendieck conjecture for hyperbolic curves was proved by H. Nakamura [14],
A. Tamagawa [20], and S. Mochizuki [10]. Let $m$ be a non-negative integer. In
the present paper, we consider the following variant of the Grothendieck
conjecture by replacing $\pi^{\acute{e}\mathrm{t}}_{1}(X,\ast)$ with the
geometrically maximal $m$-step solvable quotient

$\pi_{1}^{\acute{e}\mathrm{t}}(U,\ast)^{(m)}\overset{\mathrm{def}}{=}\pi_{1}^{\acute{e}\mathrm{t}}(U,\ast)/\pi_{1}^{\acute{e}\mathrm{t}}(U_{k^{\mathrm{sep}}},\ast)^{[m]}$

of $\pi_{1}^{\acute{e}\mathrm{t}}(U,\ast)$, where $k^{\mathrm{sep}}$ stands
for a separable closure of $k$ and
$\pi_{1}^{\acute{e}\mathrm{t}}(U_{k^{\mathrm{sep}}},\ast)^{[m]}$ stands for
the $m$-th commutator subgroup of
$\pi_{1}^{\acute{e}\mathrm{t}}(U_{k^{\mathrm{sep}}},\ast)$ (cf. Notation and
conventions Groups).

######  Conjecture (The (geometrically) $m$-step solvable Grothendieck
conjecture).

An ”anabelian” variety $U$ over $k$ may be reconstructed group-theoretically
from $\pi^{\acute{e}\mathrm{t}}_{1}(U,\ast)^{(m)}$ and its natural projection
$\pi^{\acute{e}\mathrm{t}}_{1}(U,\ast)^{(m)}\to G_{k}$.

This conjecture is stronger than the Grothendieck conjecture as
$\pi_{1}^{\acute{e}\mathrm{t}}(U,\ast)^{(m)}$ is smaller than
$\pi_{1}^{\acute{e}\mathrm{t}}(U,\ast)$, and was studied by H. Nakamura [13],
S. Mochizuki [10], and the author [23] [24] when $U$ is a hyperbolic curve.
(In recent years, the $m$-step solvable type conjecture has new results that
are not limited to the Grothendieck conjecture. For example, M. Saïdi and A.
Tamagawa obtained results for the $m$-step solvable Neukirch-Uchida theorem
(cf. [17]), and S.-H. Hyeon obtained results for the local case (cf. [7]).) In
the present paper, we give two new results for the $m$-step solvable
Grothendieck conjecture for genus $0$ hyperbolic curves. For the sake of
readability, we explain the main theorems of the present paper in two cases
where $p=0$ and where $p>0$.

The characteristic $0$ case

First, we consider the case where $p=0$. Let $U_{i}$ a hyperbolic curve of
type $(g_{i},r_{i})$ over $k$ for $i=1,2$. The following conjecture is
referred to as the $weak$ $bi$-$anabelian$ $form$ $of$ $the$ $m$-$step$
$solvable$ $Grothendieck$ $conjecture$ for $p=0$:

######  Conjecture A (for ($m$, $k$, $U_{1}$, $U_{2}$)).

$U_{1}\underset{k}{\simeq}U_{2}\Longleftrightarrow\pi_{1}^{\acute{e}\mathrm{t}}(U_{1},\ast)^{(m)}\underset{G_{k}}{\simeq}\pi_{1}^{\acute{e}\mathrm{t}}(U_{2},\ast)^{(m)}$

Here, the right side is considered with one fixed $m$. The implication
$\Rightarrow$ is clear, and hence the difficulty of showing Conjecture A lies
in the implication $\Leftarrow$. Since
$\pi_{1}^{\acute{e}\mathrm{t}}(U,\ast)^{(2)}$ is smaller than
$\pi_{1}^{\acute{e}\mathrm{t}}(U,\ast)^{(m)}$ for $m\geq 3$, Conjecture A for
$m=2$ is stronger than that of $m\geq 3$, in other words, Conjecture A for
($2$, $k$, $U_{1}$, $U_{2}$) implies Conjecture A for ($m$, $k$, $U_{1}$,
$U_{2}$). Moreover, Conjecture A has a counterexample when $m=1$, and hence
the case where $m=2$ is the strongest result. The following are previous
results for Conjecture A:  Assume that $g_{1}=0$ and one of the following
conditions a-c:

  1. (a)

$m\geq 2$, $k$ is an algebraic number field satisfying one of the following
conditions aiaii:

    1. (i)

$k$ is a quadratic field $\neq\mathbb{Q}(\sqrt{2})$

    2. (ii)

There exists a prime ideal $\mathfrak{p}$ of $O_{k}$ unramified in
$k/\mathbb{Q}$ such that $|O_{k}/\mathfrak{p}|=2$ (e.g., $k=\mathbb{Q}$),

and $U_{1}=\mathbb{P}_{k}^{1}-\\{0,1,\infty,\lambda_{1}\\}$ with
$\lambda_{1}\in k-\\{0,1\\}$ (cf. [13, Theorem A]).

  2. (b)

$m\geq 5$ and $k$ is a sub-$\ell$-adic field (i.e., a subfield of a field
finitely generated over $\mathbb{Q}_{\ell}$) for some prime $\ell$ (cf. [10,
Theorem A′]).

  3. (c)

$m\geq 3$ and $k$ is a field finitely generated over $\mathbb{Q}$ (cf. [23,
Theorem 2.4.1]).

Then Conjecture A for ($m$, $k$, $U_{1}$, $U_{2}$) holds. Remark that b has a
result for the case where $g_{1}>0$. The first main theorem of the present
paper is as follows:

######  Theorem A (cf. Corollary 6.5).

Assume that $m\geq 2$, that $k$ is a field finitely generated over
$\mathbb{Q}$, that $g_{1}=0$, and that $r_{1}\neq 4$. Then the following
holds:

$U_{1}\underset{k}{\simeq}U_{2}\Longleftrightarrow\pi_{1}^{\acute{e}\mathrm{t}}(U_{1},\ast)^{(m)}\underset{G_{k}}{\simeq}\pi_{1}^{\acute{e}\mathrm{t}}(U_{2},\ast)^{(m)}$

In other words, Conjecture A for ($m$, $k$, $U_{1}$, $U_{2}$) holds.

One of the important points of Theorem A is that it contains the case where
$m=2$. As already mentioned, the case where $m=2$ is the strongest result for
the $m$-step solvable Grothendieck conjecture, and the previous result a by H.
Nakamura is the only previous result for $m=2$. (At the time of writing the
present paper, the author does not know of any examples where Conjecture A for
$m=2$ holds when $g_{1}>0$ and $(g_{1},r_{1})\neq(1,1$).) By using Theorem A
and the previous result a, we obtain the following corollary.

######  Corollary A.

Let $\Lambda_{1},\Lambda_{2}$ be subsets of $\mathbb{Q}\cup\\{\infty\\}$
satisfying $|\Lambda_{1}|\geq 3$. Then the following holds:

$\mathbb{P}_{\mathbb{Q}}^{1}-\Lambda_{1}\underset{\mathbb{Q}}{\simeq}\mathbb{P}_{\mathbb{Q}}^{1}-\Lambda_{2}\Longleftrightarrow\pi_{1}^{\acute{e}\mathrm{t}}(\mathbb{P}_{\mathbb{Q}}^{1}-\Lambda_{1},\ast)^{(2)}\underset{G_{\mathbb{Q}}}{\simeq}\pi_{1}^{\acute{e}\mathrm{t}}(\mathbb{P}_{\mathbb{Q}}^{1}-\Lambda_{1},\ast)^{(2)}$

In particular, Conjecture A for
($m,\mathbb{Q},\mathbb{P}_{\mathbb{Q}}^{1}-\Lambda_{1},\mathbb{P}_{\mathbb{Q}}^{1}-\Lambda_{2}$)
holds when $m\geq 2$.

###### Proof..

If $|\Lambda_{1}|\geq 5$, then the assertion follows from Theorem A. If
$|\Lambda_{1}|=4$, then the assertion follows from the previous result a. The
case where $m\geq 3$ follows from the case where $m=2$. ∎

Clearly, we may extend Corollary A to the case where the base field is a field
satisfying one of the conditions aiaii in the previous result a.

The positive characteristic case

Next, we consider the case where $p>0$. In this case, we need to consider a
(an $n$-th) $relative$ $Frobenius$ $twist$ $U_{i}(n)$ of $U_{i}$ over $k$ (cf.
Definition 4.4), since even though the morphism may not be an isomorphism of
schemes, it induces an isomorphism between étale fundamental groups. The weak
bi-anabelian form of the $m$-step solvable Grothendieck conjecture for the
case where $p>0$ is as follows:

######  Conjecture B (for ($m$, $k$, $U_{1}$, $U_{2}$)).

There exists a pair $(n_{1},n_{2})$ of non-negative integers with
$n_{1}n_{2}=0$ such that the following holds:

$U_{1}(n_{1})\underset{k}{\simeq}U_{2}(n_{2})\Longleftrightarrow\pi_{1}^{\mathrm{tame}}(U_{1},\ast)^{(m)}\underset{G_{k}}{\simeq}\pi_{1}^{\mathrm{tame}}(U_{2},\ast)^{(m)}$

Here, $\pi_{1}^{\mathrm{tame}}(U_{i},\ast)$ stands for the $tame$
$fundamental$ $group$ of $U_{i}$ with a geometric point $\ast$. If, moreover,
$U_{1,\overline{k}}$ does not descend to a curve over
$\overline{\mathbb{F}}_{p}$, then the pair $(n_{1},n_{2})$ is unique.

Here, the right side is considered with one fixed $m$. Similar to Conjecture
A, the difficulty of showing Conjecture B lies in the implication
$\Leftarrow$, and Conjecture B for the case where $m=2$ implies Conjecture B
for the case where $m\geq 3$. Conjecture B has the following previous results:
Assume that $g_{1}=0$ and one of the following conditions d-f:

  1. (d)

$m\geq 3$, $k$ is a field finitely generated over $\mathbb{F}_{p}$, and the
curve $U^{\text{cpt}}_{1,\overline{k}}-S$ does not descend to a curve over
$\overline{\mathbb{F}}_{p}$ for each $S\subset
U^{\text{cpt}}_{1,\overline{k}}-U_{1,\overline{k}}\ \text{with}\ |S|=4$, where
$U_{1}^{\text{cpt}}$ stands for a smooth compactification of $U_{1}$ (cf. [23,
Theorem 2.4.1]).

  2. (e)

$m\geq 2$ (resp. $m\geq 3$), $k$ is a finite field, and $r_{1}\neq 4$ (resp.
$r_{1}=4$) (cf. [24, Theorem 2.16]).

  3. (f)

$m\geq 4$ (resp. $m\geq 5$), $k$ is a field finitely generated over
$\mathbb{F}_{p}$, and $r_{1}\neq 4$ (resp. $r_{1}=4$) (cf. [24, Theorem
4.12]).

Then Conjecture B for ($m$, $k$, $U_{1}$, $U_{2}$) holds. Remark that ef have
results for $g_{1}>0$. Unlike in Conjecture A, if $k$ is a finite field, then
Conjecture B for $m=2$ is proved in general genus by the previous result e
except for the case where $r_{1}=4$. In the present paper, we extend this
result to the case of the pro-prime-to $p$ setting. The second main theorem of
the present paper is as follows:

######  Theorem B (cf. Corollaries 4.6, 6.5).

Assume that $m\geq 2$, that $k$ is a field finitely generated over
$\mathbb{F}_{p}$, that $g_{1}=0$, and that $r_{1}\neq 4$. Then there exists a
pair $(n_{1},n_{2})$ of non-negative integers with $n_{1}n_{2}=0$ such that
the following holds:

$U_{1}(n_{1})\underset{k}{\simeq}U_{2}(n_{2})\Longleftrightarrow\pi_{1}^{\mathrm{tame}}(U_{1},\ast)^{(m,p^{\prime})}\underset{G_{k}}{\simeq}\pi_{1}^{\mathrm{tame}}(U_{2},\ast)^{(m,p^{\prime})}$

Here, $\pi_{1}^{\mathrm{tame}}(U_{i},\ast)^{(m,p^{\prime})}$ stands for the
geometrically maximal $m$-step solvable pro-prime-to-$p$ quotient of
$\pi_{1}^{\mathrm{tame}}(U_{i},\ast)$ (cf. Notation and conventions Étale
fundamental groups). If, moreover, $U_{1,\overline{k}}$ does not descend to a
curve over $\overline{\mathbb{F}}_{p}$, then the pair $(n_{1},n_{2})$ is
unique.

The assumption “$r_{1}\neq 4$” in Theorems A and B comes from a reconstruction
of decomposition groups (cf. Section 3), and we cannot remove this if we use
the proof method of the present paper. The author does not know whether the
case where $m=2$ and $r_{1}=4$ is true or not at the time of writing the
present paper. The following is a corollary of Theorem B.

######  Corollary B.

Assume that $m\geq 2$, that $k$ is a field finitely generated over
$\mathbb{F}_{p}$, that $g_{1}=0$, and that $r_{1}\neq 4$. Then Conjecture B
for ($m$, $k$, $U_{1}$, $U_{2}$) holds. In particular, the previous results
def follow from Theorem B except for the case where $r_{1}=4$.

###### Proof..

Since we can reconstruct group-theoretically
$\pi_{1}^{\mathrm{tame}}(U_{i},\ast)^{(m,p^{\prime})}$ from
$\pi_{1}^{\mathrm{tame}}(U_{i},\ast)^{(m)}$ (cf. Lemma 1.1), the assertion
follows from Theorem B. ∎

As already mentioned, Conjectures A and B are referred to as the weak bi-
anabelian form of the $m$-step solvable Grothendieck conjecture. The word
“weak” comes from the fact that we do not consider the functoriality in these
conjectures. The weak bi-anabelian form can be rephrased as follows:

$\mathrm{Isom}_{k}(U_{1},U_{2})\neq\emptyset\iff\mathrm{Isom}_{G_{k}}(\pi_{1}^{\acute{e}\mathrm{t}}(U_{1},\ast)^{(m)},\pi_{1}^{\acute{e}\mathrm{t}}(U_{2},\ast)^{(m)})\neq\emptyset$

(Here, we only consider the case where $p=0$ for simplicity.) Considering the
functoriality of the fundamental groups, we can formulate the following
$strong$ $bi$-$anabelian$ $form$ $of$ $the$ $m$-$step$ $solvable$
$Grothendieck$ $conjecture$:

######  Conjecture C.

Fix a non-negative integer $n$. Then the image of the natural morphism

$\mathrm{Isom}_{k}(U_{1},U_{2})\to\mathrm{Isom}_{G_{k}}(\pi_{1}^{\acute{e}\mathrm{t}}(U_{1},\ast)^{(m)},\pi_{1}^{\acute{e}\mathrm{t}}(U_{2},\ast)^{(m)})/\mathrm{Inn}(\pi_{1}^{\acute{e}\mathrm{t}}(U_{2,k^{\mathrm{sep}}},\ast)^{m})$

coincides with the image of
$\mathrm{Isom}_{G_{k}}(\pi_{1}^{\acute{e}\mathrm{t}}(U_{1},\ast)^{(m+n)},\pi_{1}^{\acute{e}\mathrm{t}}(U_{2},\ast)^{(m+n)})/\mathrm{Inn}(\pi_{1}^{\acute{e}\mathrm{t}}(U_{2,k^{\mathrm{sep}}},\ast)^{m+n})$.

The previous results b, c, e, and f have results for Conjecture C. In the
present paper, we do not discuss Conjecture C, since results for Conjecture C
that is induced by Theorems A and B are essentially contained in [24,
Corollary 2.22, Theorem 4.16]. For more details, see Remark 6.7.

In Section 1, we show group-theoretical reconstructions of various invariant
and conditions (Lemma 1.1). In Section 2, we introduce (geometric) sections
and show a separatedness property for decomposition groups by using the Kummer
homomorphism (Proposition 2.3). In Section 3, we construct a group-theoretical
reconstruction of decomposition groups of
$\pi_{1}^{\mathrm{tame}}(U_{i},\ast)^{(\mathrm{ab},p^{\prime})}$ from
$\pi_{1}^{\mathrm{tame}}(U_{i},\ast)^{(2,p^{\prime})}$ if $(g_{i},r_{i},m)$
satisfies certain conditions (Proposition 3.6). In Section 4, we show the pro-
prime-to-$p$ version result of the $m$-step solvable Grothendieck conjecture
for affine hyperbolic curves of $p$-rank $0$ over finite fields (Theorem 4.5).
In Section 5, we introduce the specialization morphisms of smooth curves over
henselian regular local rings. In Section 6, we show the result of the pro-
prime-to-$p$ version of the $m$-step solvable Grothendieck conjecture for
genus $0$ hyperbolic curves over finitely generated fields of characteristic
$p\geq 0$ (Corollary 6.5).

## Notation and conventions

Integers and prime numbers We fix an integer $m$ that is greater than or equal
to ${\bf 0}$ in the present paper. Remark that this definition of $m$ is
different from the form described in [23] and [24], since we assumed $m\geq 1$
in [23] and [24]. We shall write $\mathbb{Z}$ for the set of all integers, and
for any $a\in\mathbb{Z}$, we shall write $\mathbb{Z}_{\geq a}$ for the set of
all integers that is greater than or equal to $a$. Let $\mathfrak{Primes}$ be
the set of all rational prime numbers and $\Sigma$ a non-empty subset of
$\mathfrak{Primes}$. If, for some rational prime number $\ell$,
$\Sigma=\\{\ell\\}$ (resp. $\Sigma=\mathfrak{Primes}-\\{\ell\\}$,
$\Sigma=\mathfrak{Primes}$), then we shall write “$\ell$”
(resp.“$\ell^{\prime}$”, “(no letter)”) instead of “$\Sigma$” in below
notation.

Groups We always assume that a homomorphism between profinite groups is
continuous. For any profinite group $P$, we shall write $\mathrm{Aut}(P)$ for
the group of automorphisms of $P$, $\mathrm{Inn}(P)$ for the group of inner
automorphisms of $P$, and
$\mathrm{Out}(P)\overset{\mathrm{def}}{=}\mathrm{Aut}(P)/\mathrm{Inn}(P)$ for
the group of outer automorphisms of $P$. We shall write $\overline{[P,P]}$ for
the closed subgroup of $P$ generated (topologically) by all commutators of
$P$, and refer to it as the commutator subgroup of $P$. We shall write
$P^{[0]}\overset{\mathrm{def}}{=}P$ and
$P^{[m]}\overset{\mathrm{def}}{=}\overline{[P^{[m-1]},P^{[m-1]}]}$ if $m\geq
1$. We shall write $P^{m}\overset{\mathrm{def}}{=}P/P^{[m]}$, and refer to it
as the maximal $m$-step solvable quotient of $P$. If there is no risk of
confusion, we shall write $P^{\mathrm{ab}}$ instead of $P^{1}$. We shall write
$P^{\Sigma}$ for the maximal pro-$\Sigma$ quotient of $P$. Note that the
quotients $(P^{m})^{\Sigma}$ and $(P^{\Sigma})^{m}$ of $P$ coincide with the
projective limit $\varprojlim_{H}P/H$, where $H$ runs over all open normal
subgroups $H$ of $P$ for which the quotient $P/H$ is a pro-$\Sigma$ group and
an $m$-step solvable group. Hence we may write
$P^{m,\Sigma}\overset{\mathrm{def}}{=}(P^{m})^{\Sigma}(=(P^{\Sigma})^{m})$.

Curves For any scheme $S$, any scheme $\mathcal{X}$ over $S$, and any
(possibly empty) closed subscheme $\mathcal{E}$ of $\mathcal{X}$, we shall
refer to the pair $(\mathcal{X},\mathcal{E})$ as a $smooth$ $curve$ $of$
$type$ $(g,r)$ $over$ $S$ if the following conditions (a)-(c) hold.

  1. (a)

$\mathcal{X}$ is flat, proper, of finite presentation, and of relative
dimension one over $S$.

  2. (b)

For any geometric point $\ast$ of $S$, the geometric fiber
$\mathcal{X}_{\ast}$ at $\ast$ is smooth, proper, connected, and of relative
dimension one over $\ast$ with genus $g$.

  3. (c)

The composition of $\mathcal{E}\hookrightarrow\mathcal{X}\rightarrow S$ is
finite, étale, and of degree $r$.

If there is no risk of confusion, we shall refer to the complement
$\mathcal{U}\overset{\mathrm{def}}{=}\mathcal{X}-\mathcal{E}$ as a smooth
curve of type $(g,r)$ over $S$. We shall refer to $\mathcal{U}$ as hyperbolic
if $2-2g-r<0$ (in other words, $(g,r)\neq(0,0),(0,1),(0,2),(1,0)$).

Étale fundamental groups Let $k$ be a field of characteristic $p\geq 0$,
$\overline{k}$ an algebraic closure of $k$, and $k^{\mathrm{sep}}$ the
separable closure of $k$ in $\overline{k}$. Let $(X,E)$ be a smooth curve of
type $(g,r)$ over $k$, and we set $U\overset{\mathrm{def}}{=}X-E$. Let
$\Omega$ be an algebraically closed field containing the field of fractions
$K(U_{k^{\mathrm{sep}}})$ of $U_{k^{\mathrm{sep}}}$. Then we shall define the
following notation:

$G_{k}\overset{\mathrm{def}}{=}\mathrm{Gal}(k^{\mathrm{sep}}/k),\ \ \ \ \ \
\Pi_{U}\overset{\mathrm{def}}{=}\pi_{1}^{\mathrm{tame}}(U,\ast),\ \ \
\mathrm{and}\ \ \
\overline{\Pi}_{U}\overset{\mathrm{def}}{=}\pi_{1}^{\mathrm{tame}}(U_{k^{\mathrm{sep}}},\ast).$

Here, $\ast:\mathrm{Spec}(\Omega)\rightarrow U_{k^{\mathrm{sep}}}(\rightarrow
U)$ stands for a geometric point, and
$\pi_{1}^{\mathrm{tame}}(\ast_{1},\ast_{2})$ stands for the $tame$
$fundamental$ $group$ of a (locally Noetherian) scheme $\ast_{1}$ and its
geometric point $\ast_{2}$. We shall refer to $\Pi_{U}$ (resp.
$\overline{\Pi}_{U}$) as the $arithmetic$ $fundamental$ $group$ (resp. the
$geometric$ $fundamental$ $group$) of $U$. These groups satisfy the basic
exact sequence that is called $homotopy$ $exact$ $sequence$ as follows:

$1\to\overline{\Pi}_{U}\to\Pi_{U}\to G_{k}\to 1$

Geometrically maximal $m$-step solvable quotients Let $\Sigma$ be a non-empty
subset of $\mathfrak{Primes}$, and we set
$\Sigma^{{\dagger}}\overset{\mathrm{def}}{=}\Sigma-\\{p\\}$. Then we shall
write $\tilde{\mathcal{K}}^{\Sigma}(U)\subset\Omega$ for the maximal tamely
ramified pro-$\Sigma$ Galois extension of $K(U_{k^{\mathrm{sep}}})$ in
$\Omega$ unramified on $U$, and $\tilde{\mathcal{K}}^{m,\Sigma}(U)$
$\subset\Omega$ for the maximal $m$-step solvable pro-$\Sigma$ Galois
extension of $K(U_{k^{\mathrm{sep}}})$ in $\tilde{\mathcal{K}}^{\Sigma}(U)$.
We shall write
$\tilde{U}^{\Sigma}\overset{\mathrm{def}}{=}\tilde{U}^{U,\Sigma}$ (resp.
$\tilde{X}^{\Sigma}\overset{\mathrm{def}}{=}\tilde{X}^{U,\Sigma}$) for the
integral closure of $U$ (resp. $X$) in $\tilde{\mathcal{K}}^{\Sigma}(U)$, and
set
$\tilde{E}^{\Sigma}\overset{\mathrm{def}}{=}\tilde{E}^{U,\Sigma}\overset{\mathrm{def}}{=}\tilde{X}^{\Sigma}-\tilde{U}^{\Sigma}$.
Similarly, we shall define the notation
$\tilde{U}^{m,\Sigma}\overset{\mathrm{def}}{=}\tilde{U}^{U,m,\Sigma}$,
$\tilde{X}^{m,\Sigma}\overset{\mathrm{def}}{=}\tilde{X}^{U,m,\Sigma}$, and
$\tilde{E}^{m,\Sigma}\overset{\mathrm{def}}{=}\tilde{E}^{U,m,\Sigma}$. We
shall define

$\Pi^{(\Sigma)}_{U}\overset{\mathrm{def}}{=}\Pi_{U}/\mathrm{Ker}(\overline{\Pi}_{U}\twoheadrightarrow\overline{\Pi}^{\Sigma}_{U})\
\text{ and }\
\Pi^{(m,\Sigma)}_{U}\overset{\mathrm{def}}{=}\Pi_{U}/\mathrm{Ker}(\overline{\Pi}_{U}\twoheadrightarrow\overline{\Pi}^{m,\Sigma}_{U}).$

Since
$\mathrm{Ker}(\overline{\Pi}_{U}\twoheadrightarrow\overline{\Pi}^{\Sigma}_{U})$
and
$\mathrm{Ker}(\overline{\Pi}_{U}\twoheadrightarrow\overline{\Pi}^{m,\Sigma}_{U})$
are characteristic subgroups of $\overline{\Pi}_{U}$, these quotient are well
defined and induce the exact sequences
$1\to\overline{\Pi}^{\Sigma}_{U}\to\Pi^{(\Sigma)}_{U}\to G_{k}\to 1$ and
$1\to\overline{\Pi}^{m,\Sigma}_{U}\to\Pi^{(m,\Sigma)}_{U}\to G_{k}\to 1.$
(Similarly, for any closed subgroup $H$ of $\Pi_{U}$, we shall write
$\overline{H}$ for the kernel of the composition of the natural homomorphisms
$H\hookrightarrow\Pi_{U}\twoheadrightarrow G_{k}$, and define $H^{(\Sigma)}$,
$H^{(m,\Sigma)}$, respectively.) Note that
$\Pi^{(\Sigma)}_{U}=\mathrm{Gal}(\tilde{\mathcal{K}}^{\Sigma}(U)/K(U))$ and
$\Pi^{(m,\Sigma)}_{U}=\mathrm{Gal}(\tilde{\mathcal{K}}^{m,\Sigma}(U)/K(U))$ by
definition, and, if $\Sigma$ contains a prime different from $p$, we have that

$\overline{\Pi}^{m,\Sigma}_{U}\text{ is trivial }\iff``m=0\text{ or
}(g,r)=(0,0),(0,1)^{\prime\prime},$

and that

$\overline{\Pi}^{m,\Sigma}_{U}\text{ is abelian }\iff``m=0,1\text{ or
}(g,r)=(0,0),(0,1),(0,2),(1,0)^{\prime\prime}.$

We shall define
$\overline{\Pi}^{m,0^{\prime}}_{U}\overset{\mathrm{def}}{=}\overline{\Pi}^{m}_{U}$
and $\Pi^{(m,0^{\prime})}_{U}\overset{\mathrm{def}}{=}\Pi^{(m)}_{U}$.

Tate Modules We shall write $J_{X}$ for the Jacobian variety of $X$, and
$T_{\Sigma}(J_{X})\overset{\mathrm{def}}{=}\varprojlim_{n}J_{X}(\overline{k})[n]$
for the pro-$\Sigma$ Tate module of $J_{X}$, where $n$ runs over all integers
whose prime factors are in $\Sigma$. We call
$\mathrm{rank}_{\mathbb{Z}_{\ell}}(T_{\Sigma}(J_{X}))$ the $\ell$-rank of
$J_{X}$ for any prime $\ell$.

Inertia groups and decomposition groups For any closed point $\tilde{v}$ of
$\tilde{X}^{\Sigma}$ (resp. $\tilde{X}^{m,\Sigma}$), we shall write

$D_{\tilde{v}}\overset{\mathrm{def}}{=}D_{\tilde{v},M}\overset{\mathrm{def}}{=}\\{\gamma\in
M\mid\gamma(\tilde{v})=\tilde{v}\\}$
$I_{\tilde{v}}\overset{\mathrm{def}}{=}I_{\tilde{v},M}\overset{\mathrm{def}}{=}\\{\gamma\in
M\mid\gamma(\tilde{v})=\tilde{v},\ \gamma\text{ acts trivially on
}\kappa(\tilde{v})\\}\\},$

and refer to these as the $decomposition$ $group$ $at$ $\tilde{v}$, the
$inertia$ $group$ $at$ $\tilde{v}$, respectively. Here, $M$ stands for
$\Pi^{(\Sigma)}_{U}$ (resp. $\Pi^{(m,\Sigma)}_{U}$). We shall write
$\mathrm{Dec}(\Pi^{(m,\Sigma)}_{U})$,
$\mathrm{Dec}^{\mathrm{cusp}}(\Pi^{(m,\Sigma)}_{U})$,
$\mathrm{Iner}(\Pi^{(m,\Sigma)}_{U})$ for the $\Pi^{(m,\Sigma)}_{U}$-set of
all decomposition groups of $\Pi^{(m,\Sigma)}_{U}$, $\Pi^{(m,\Sigma)}_{U}$-set
of all decomposition groups at cusps of $\Pi^{(m,\Sigma)}_{U}$, the
$\overline{\Pi}^{m,\Sigma}_{U}$-set of all inertia groups of
$\Pi^{(m,\Sigma)}_{U}$, respectively. It is clear that the inertia group is
trivial if $\tilde{v}$ is on $U$. We shall write
$\mathrm{Iner}^{\neq\\{1\\}}(\Pi^{(m,\Sigma)}_{U})$ for the
$\Pi^{(m,\Sigma)}_{U}$-set of all non-trivial inertia groups of
$\Pi^{(m,\Sigma)}_{U}$.

##  1 Group-theoretical reconstructions of various invariant

Let $k$ be a field of characteristic $p\geq 0$, $\Sigma$ a non-empty subset of
$\mathfrak{Primes}$, $(X,E)$ a smooth curve of type $(g,r)$ over $k$, and we
set $U\overset{\mathrm{def}}{=}X-E$. In this section, we reconstruct group-
theoretically various invariant and conditions from $\Pi_{U}^{(m,\Sigma)}$
(cf. Lemma 1.1 below). The sentence “reconstruct group-theoretically from
$\Pi_{U}^{(m,\Sigma)}$” means that there exists a functorial group-theoretical
algorithm for the invariant from $\Pi_{U}^{(m,\Sigma)}$. This is contained in
“mono-anabelian geometry” (cf. [11, Remark 1.9.8]). Some invariant in Lemma
1.1 below have functorial group-theoretical algorithms from
$\Pi_{U}^{(m,\Sigma)}$ (in other words, we can reconstruct group-theoretically
from $\Pi_{U}^{(m,\Sigma)}$). However, in the present paper, these are given
only under the letter in “bi-anabelian geometry” for simplicity.

First, we introduce the weight filtration of
$\overline{\Pi}^{\mathrm{ab},\Sigma}_{U}$. We obtain the outer Galois
representation $G_{k}\rightarrow\mathrm{Out}(\overline{\Pi}^{m,\Sigma}_{U})$
from the homotopy exact sequence (cf. Notation and conventions Geometrically
maxima $m$-step solvable quotients). Let us regard
$\overline{\Pi}^{\mathrm{ab},\Sigma}_{U}$ as a $G_{k}$-module by using the
Galois representation. Then we have the following isomorphism and exact
sequence of $G_{k}$-modules:

$\overline{\Pi}_{U}^{\mathrm{ab},\Sigma}\xrightarrow{\sim}T_{\Sigma}(J_{X})\ \
\ \ \ \text{when }r=0$
$0\rightarrow\hat{\mathbb{Z}}^{\Sigma^{{\dagger}}}(1)\rightarrow\mathbb{Z}[E(k^{\mathrm{sep}})]\bigotimes_{\mathbb{Z}}\hat{\mathbb{Z}}^{\Sigma^{{\dagger}}}(1)\xrightarrow{f}\overline{\Pi}_{U}^{\mathrm{ab},\Sigma}\rightarrow
T_{\Sigma}(J_{X})\rightarrow 0\ \ \ \ \text{when }r\neq 0$ (1.1)

Here, $\mathbb{Z}[E(k^{\mathrm{sep}})]$ is the free $\mathbb{Z}$-module with
the basis $E(k^{\mathrm{sep}})$ and is regarded as a $G_{k}$-module via the
natural $G_{k}$-action on $E(k^{\mathrm{sep}})$,
$\hat{\mathbb{Z}}^{\Sigma^{{\dagger}}}(1)$ is the $1\text{-}th$ Tate twist of
$\hat{\mathbb{Z}}^{\Sigma^{{\dagger}}}$, the second left-arrow of lower exact
sequence is defined by sending $1$ to the diagonal $\Sigma_{v\in
E(k^{\mathrm{sep}})}v\otimes 1$, and $f$ is defined as the morphism sending
$v\otimes 1$ to a (topological) generator of the inertia group
$I_{v,\overline{\Pi}_{U}^{\mathrm{ab},\Sigma}}$ at $v\in E(k^{\mathrm{sep}})$.
If $k$ is a field finitely generated over the prime field, then the
$G_{k}$-representations on
$\mathbb{Z}[E(k^{\mathrm{sep}})]\bigotimes_{\mathbb{Z}}\hat{\mathbb{Z}}^{\Sigma^{{\dagger}}}(1)$
and $T_{\Sigma}(J_{X})$ have (Frobenius) weights $-2$ and $-1$, respectively
(cf. [24, Lemma 1.2] (or [14, (2.5)])). Let us write
$W_{-2}(\overline{\Pi}_{U}^{\mathrm{ab},\Sigma})$ for the maximal weight $-2$
subgroup of $\overline{\Pi}_{U}^{\mathrm{ab},\Sigma}$. (Note that
$W_{-2}(\overline{\Pi}_{U}^{\mathrm{ab},\Sigma})=\mathrm{Im}(f)$ holds.)

Next, we show group-theoretical reconstructions of various invariant as
follows:

######  Lemma 1.1.

Let $i=1$, $2$. Let $k_{i}$ be a field finitely generated over the prime field
of characteristic $p_{i}\geq 0$. Let $(X_{i},E_{i})$ be a smooth curve of type
$(g_{i},r_{i})$ over $k_{i}$, and we set
$U_{i}\overset{\mathrm{def}}{=}X_{i}-E_{i}$. Let $\Sigma_{i}$ be a non-empty
subset of $\mathfrak{Primes}$. Let $\Phi$ be an isomorphism
$\Pi^{(m,\Sigma_{1})}_{U_{1}}\xrightarrow{\sim}\Pi^{(m,\Sigma_{2})}_{U_{2}}$.
Then the following hold:

  1. (1)

$k_{1}$ is finite if and only if $k_{2}$ is finite.

  2. (2)

Let $n$ be an integer with $m\geq n\geq 0$. Then $\Phi$ maps
$(\overline{\Pi}^{m,\Sigma_{1}}_{U_{1}})^{[n]}$ to
$(\overline{\Pi}^{m,\Sigma_{2}}_{U_{2}})^{[n]}$. In particular, $\Phi$ induces
a unique isomorphism
$\Phi^{n}:\Pi^{(n,\Sigma_{1})}_{U_{1}}\xrightarrow{\sim}\Pi^{(n,\Sigma_{2})}_{U_{2}}$
that satisfies the commutativity of the following diagram:

$\textstyle{\Pi^{(m,\Sigma_{1})}_{U_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\Pi^{(m,\Sigma_{2})}_{U_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi^{(n,\Sigma_{1})}_{U_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi^{n}}$$\textstyle{\Pi^{(n,\Sigma_{2})}_{U_{2}}}$

  3. (3)

If $k_{1}$ is infinite, then $p_{1}=p_{2}$.

If, moreover, we assume that $\overline{\Pi}_{U_{1}}^{m,\Sigma_{1}}$ is non-
trivial, and that $\Sigma_{i}^{{\dagger}}=\mathfrak{Primes}-\\{p_{i}\\}$ for
$i=1,2$ (resp. and that $\Sigma_{1}^{{\dagger}}\neq\emptyset$, $k_{1}=k_{2}$,
and $\Phi^{0}$ is identity morphism of $G_{k_{1}}$) when $k_{1}$ is finite
(resp. infinite). Then the following hold:

  1. (4)

If $k_{1}$ is finite, then $\Phi^{0}$ maps the Frobenius element
$\mathrm{Fr}_{k_{1}}$ of $G_{k_{1}}$ to $\mathrm{Fr}_{k_{2}}$ of $G_{k_{2}}$.

  2. (5)

$\Phi^{1}$ maps $W_{-2}(\overline{\Pi}_{U_{1}}^{\mathrm{ab},\Sigma_{1}})$ to
$W_{-2}(\overline{\Pi}_{U_{2}}^{\mathrm{ab},\Sigma_{2}})$.

  3. (6)

$g_{1}=g_{2}$.

  4. (7)

$r_{1}\geq 2$ holds if and only if $r_{2}\geq 2$ holds. If , moreover, either
“$m\geq 2$”, “$r_{1}\geq 2$”, or “$r_{1}\geq 1$ and $r_{2}\geq 1$”, then
$r_{1}=r_{2}$.

  5. (8)

Let $n$ be an integer with $m>n\geq 1$, then the isomorphism $\Phi^{n}$
preserves inertia groups. If, moreover, $(g_{1},r_{1})\neq(0,2)$,
$p_{1}\notin\Sigma_{1}$, and either “$m\geq 4$” or “$m\geq 3$ and $r_{1}\neq
1$”, then the isomorphism $\Phi^{n}$ preserves decomposition groups at cusps.

  6. (9)

If $k_{1}$ is finite, then the cardinalities $|k_{1}|$ and $|k_{2}|$ are the
same. In particular, $p_{1}=p_{2}$.

  7. (10)

The $\ell$-rank of $J_{X_{1}}$ is equal to the the $\ell$-rank of $J_{X_{2}}$
for $\ell$.

###### Proof..

1 Since $\overline{\Pi}_{U_{i}}$ is the tame fundamental group, we have a
surjection from a finitely generated free profinite group to
$\overline{\Pi}_{U_{i}}$ (cf. [6]). Hence
$\overline{\Pi}_{U_{i}}^{m,\Sigma_{i}}$ is also finitely generated. We know
that $G_{k_{i}}$ is finitely generated if and only if $k_{i}$ is finite. Thus,
$k_{i}$ is finite if and only if $\Pi_{U_{i}}^{(m,\Sigma_{i})}$ is finitely
generated. This completes the proof.

2 If $k_{i}$ is finite, then the assertion follows from [24, Lemma 1.8(1)]. If
$k_{i}$ is infinite, then $k_{i}$ is a Hilbertian field and $G_{k}$ does not
have a normal closed finitely generated subgroup (cf. [3, Theorem 13.4.2] and
[4, Theorem 15.10]). Hence $\overline{\Pi}^{m,\Sigma_{i}}_{U_{i}}$ is the
maximal normal closed finitely generated subgroup of
$\Pi^{(m,\Sigma_{i})}_{U_{i}}$. In particular, $\Phi$ maps
$\overline{\Pi}^{m,\Sigma_{1}}_{U_{1}}$ to
$\overline{\Pi}^{m,\Sigma_{2}}_{U_{2}}$. Therefore, the assertion follows from
the fact that $(\overline{\Pi}^{m,\Sigma_{i}}_{U_{i}})^{[n]}$ is reconstructed
group-theoretically from $\overline{\Pi}^{m,\Sigma_{i}}_{U_{i}}$.

3 By 2 , we get the isomorphism $\Phi^{0}$ between $G_{k_{1}}$ and
$G_{k_{2}}$. Hence the assertion follows from [15, Proposition 1.14].

4 See [24, Proposition 1.9(2)] or [16, Proposition 1.15(viii)].

5 The assertion follows from 4.

6 See [24, Proposition 1.7(1)].

7 See [24, Proposition 1.7(2)].

8 The first assertion follows from [24, Proposition 1.12(1)]. The second
assertion follows from the first assertion and the fact that a decomposition
group at cusps of $\Pi_{U_{i}}^{(m-1,\Sigma)}$ coincides with the normalizer
of an inertia group of $\Pi_{U_{i}}^{(m-1,\Sigma)}$ (cf. [24, Lemma 1.11(3)]).

9 See [24, Proposition 1.9(3)].

10 The assertion follows from (1.1), 2, 5, and the isomorphism
$T_{\Sigma}(J_{X})\simeq\overline{\Pi}_{X}^{\mathrm{ab},\Sigma}$. ∎

##  2 Separatedness property for decomposition groups over finite fields

Let $k$ be a field of characteristic $p\geq 0$, $\Sigma$ a non-empty subset of
$\mathfrak{Primes}$, $(X,E)$ a smooth curve of type $(g,r)$ over $k$, and we
set $U\overset{\mathrm{def}}{=}X-E$. In this section, we introduce (geometric)
sections and show a separatedness property for decomposition groups by using
Kummer homomorphism, where “separatedness property” means, for any closed
points $\tilde{v}_{1}$, $\tilde{v}_{2}\in\tilde{X}^{m,\Sigma}$,
$\tilde{v}_{1}=\tilde{v}_{2}$ holds if and only if
$D_{\tilde{v}_{1},\Pi^{(m,\Sigma)}_{U}}=D_{\tilde{v}_{2},\Pi^{(m,\Sigma)}_{U}}$
holds (cf. Proposition 2.3 below). When $\Sigma=\mathfrak{Primes}$ and $k$ is
finite, we have the result for the separatedness property for decomposition
groups of $\Pi^{(m,\Sigma)}_{U}$ (cf. [24, Proposition 2.6]). However, we can
extend easily [24, Proposition 2.6] not only to the case where $k$ is finite,
but also to the case where $k$ is a perfect field finitely generated over the
prime field, since the assumption that $k$ is finite is not used in the proof
of [24, Proposition 2.6]. Moreover, if the $p$-rank of $J_{X}$ is $0$, then we
can also obtain a result of the case where $\Sigma=\mathfrak{Primes}-\\{p\\}$.
We show the separatedness property when $\Sigma=\mathfrak{Primes}-\\{p\\}$ and
$k$ is a perfect field finitely generated over the prime field under the
assumption that the $p$-rank of $J_{X}$ is $0$ (cf. Proposition 2.3 below).
Let us first define the following notation.

######  Definition 2.1.

Let $G$ be an open subgroup of $G_{k}$ with natural inclusion
$\iota:G\hookrightarrow G_{k}$. We shall refer to a homomorphism
$s:G\to\Pi^{(m,p^{\prime})}_{U}$ as a section (of $\iota$) if
$\mathrm{pr}\circ s=\iota$, where $\mathrm{pr}$ stands for the natural
projection $\mathrm{pr}:\Pi^{(m,p^{\prime})}_{U}\to G_{k}$. If $u$ is a
$\overline{k}^{G}$-rational point of $U$ and
$\tilde{u}\in\tilde{X}^{m,p^{\prime}}$ is a closed point over $u$, then the
natural morphism $\tilde{u}\to\tilde{X}^{m,p^{\prime}}$ induces a section
$s_{\tilde{u}}:G\to
D_{\tilde{u},\Pi_{U}^{(m,p^{\prime})}}\subset\Pi^{(m,p^{\prime})}_{U}$ (of
$\iota$). We shall refer to a section $s$ as $geometric$ if there exists a
closed point $\tilde{v}$ of $\tilde{X}^{m,p^{\prime}}$ such that $s(G)\subset
D_{\tilde{v},\Pi_{U}^{(m,p^{\prime})}}$. In this case, we shall say that the
geometric section $s$ is at $\tilde{v}$ (or at $v$, where $v$ is the images of
$\tilde{v}$ by the natural map $\tilde{X}^{m,p^{\prime}}\twoheadrightarrow
X$). (Note that $\tilde{v}$ is uniquely determined by the section $s$ (cf.
Proposition 2.3(a)$\Leftrightarrow$(d) below).) We write
$\mathcal{S}_{G}\overset{\mathrm{def}}{=}\mathcal{S}_{G,\Pi^{(m,p^{\prime})}_{U}}$
for the set of all sections (of $\iota$) and
$\mathcal{S}^{\mathrm{geom}}_{G}\overset{\mathrm{def}}{=}\mathcal{S}^{\mathrm{geom}}_{G,\Pi^{(m,p^{\prime})}_{U}}$
for the set of all geometric sections (of $\iota$). Via the natural action of
$\Pi^{(m,p^{\prime})}_{U}$, we regard $\mathcal{S}_{G}$ and
$\mathcal{S}^{\mathrm{geom}}_{G}$ as $\Pi^{(m,p^{\prime})}_{U}$-sets.

Let $s,s^{\prime}:G\to\Pi^{(\mathrm{ab},p^{\prime})}_{U}$ be geometric
sections at closed points $\tilde{v}$, $\tilde{v}^{\prime}$ of
$\tilde{X}^{\mathrm{ab},p^{\prime}}$, respectively. Then the images $v$,
$v^{\prime}\in X^{\mathrm{cl}}$ of $\tilde{v}$, $\tilde{v}^{\prime}$,
respectively, are $\overline{k}^{G}$-rational points (cf. [20, Lemma 2.6]).
When $g\geq 1$ and $k$ is perfect, the Kummer exact sequence $1\to\mu_{N}\to
T(J_{X})\to T(J_{X})\to 1$ for $N\in\mathbb{Z}_{\geq 1}$ induces the Kummer
homomorphism:

$\textstyle{J_{X}(\overline{k}^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J_{X}(\overline{k}^{G})/NJ_{X}(\overline{k}^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(G,J_{X}(\overline{k})[N])}$
(2.1)

When
$U=\mathbb{G}_{m,k}\overset{\mathrm{def}}{=}\mathbb{P}_{k}^{1}-\\{0,\infty\\}$,
the (torally) Kummer exact sequence $1\to\mu_{N}\to T(\mathbb{G}_{m,k})\to
T(\mathbb{G}_{m,k})\to 1$ induces the (torally) Kummer homomorphism:

$\textstyle{\mathbb{G}_{m,k}(\overline{k}^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}_{m,k}(\overline{k}^{G})/\mathbb{G}_{m,k}(\overline{k}^{G})^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}(G,\mathbb{G}_{m,k}(\overline{k})[N])}$
(2.2)

If the perfect field $k$ satisfies the following two conditions:
$\cap_{N}\mathbb{G}_{m,k}(L)^{N}=\\{1\\}$ and $\cap_{N}(N\cdot A(L))=\\{0\\}$
for any finite extension $L$ of $k$ and any abelian variety $A$ over $L$, then
the projective limits of the Kummer homomorphisms:

$J_{X}(\overline{k}^{G})\to\varprojlim_{N\in\mathbb{Z}_{\geq
1}}J_{X}(\overline{k}^{G})/NJ_{X}(\overline{k}^{G})\to
H^{1}_{\mathrm{cont}}(G,T(J_{X}))$ (2.3)

and

$\mathbb{G}_{m,k}(\overline{k}^{G})\to\varprojlim_{p\nmid
N}\mathbb{G}_{m,k}(\overline{k}^{G})/\mathbb{G}_{m,k}(\overline{k}^{G})^{N}\to
H^{1}_{\mathrm{cont}}(G,T_{p^{\prime}}(\mathbb{G}_{m,k}))\ =\
H^{1}_{\mathrm{cont}}(G,T(\mathbb{G}_{m,k}))$ (2.4)

are injective (cf. [8, Lemma 8.1]). These properties are called
$Kummer$-$faithful$, see [12] for more detail. By the following lemma, we can
distinguish which geometric sections are above which closed points of $U$.
This method was used in [20, Lemma 2.6] and [24, Lemma 2.3(3)] when
$\Sigma=\mathfrak{Primes}$.

######  Lemma 2.2.

Assume that $k$ is a perfect field finitely generated over prime field and
that the $p$-rank of $J_{X}$ is zero. We set the map

$j_{G,\Pi_{U}^{(\mathrm{ab},p^{\prime})}}:\mathcal{S}_{G,\Pi_{U}^{(\mathrm{ab},p^{\prime})}}\times\mathcal{S}_{G,\Pi_{U}^{(\mathrm{ab},p^{\prime})}}\to
H^{1}_{\mathrm{cont}}(G,\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U})$

that sends a pair $(s_{1},s_{2})$ to the cohomology class of the (continuous)
1-cocycle $G\rightarrow\overline{\Pi}^{1,p^{\prime}}_{U}$, $\sigma\mapsto
s_{1}(\sigma)s_{2}(\sigma)^{-1}$. Let
$s,s^{\prime}:G\to\Pi^{(\mathrm{ab},p^{\prime})}_{U}$ be geometric sections at
closed points $\tilde{v}$, $\tilde{v}^{\prime}$ of
$\tilde{X}^{\mathrm{ab},p^{\prime}}$, respectively. Let $v$, $v^{\prime}\in
X^{\mathrm{cl}}$ be the images of $\tilde{v}$, $\tilde{v}^{\prime}$,
respectively. (Note that these are $\overline{k}^{G}$-rational points (cf.
[20, Lemma 2.6])). Then the following (1)-(2) hold:

  1. (1)

If $g\geq 1$ and $r=0$, then the image of the degree $0$ divisor
$v-v^{\prime}$ by the Kummer homomorphism (2.3) coincides with
$j_{G,\Pi_{U}^{(\mathrm{ab},p^{\prime})}}(s,s^{\prime})$.

  2. (2)

If $U=\mathbb{G}_{m,k}$ (in other words, $g=0$, $r=2$, $E(\overline{k})=E(k)$)
and $v$, $v^{\prime}\in U$, then the image of $v/v^{\prime}$ by the Kummer
homomorphism (2.4) coincides with
$j_{G,\Pi_{U}^{(\mathrm{ab},p^{\prime})}}(s,s^{\prime})$.

###### Proof..

(1) Let $N$ be an element of $\mathbb{Z}_{\geq 1}$. It is sufficient to show
that the image of $v/v^{\prime}$ by the Kummer homomorphism (2.1) coincides
with the cohomology class of the 1-cocycle

$G\rightarrow J_{X}(\overline{k})[N],\ \sigma\mapsto
s_{\tilde{v}}(\sigma)-s_{\tilde{v}^{\prime}}(\sigma).$

If $N$ is not divided by $p$, then the coincidence follows from [19, Corollary
71], since $N$-multiplicative morphism is finite, étale. If $N$ is divided by
$p$, then the coincidence is clear, since the $p$-rank of $J_{X}$ is zero.
This completes the proof.

(2) By Kummer theory, the image of $v-v^{\prime}$ by the Kummer homomorphism
(2.2) coincides with the cohomology class of the 1-cocycle

$G\rightarrow\mathbb{G}_{m,k}(\overline{k})[N],\ \sigma\mapsto
s_{\tilde{v}}(\sigma)s_{\tilde{v}^{\prime}}(\sigma)^{-1}$

if $v$, $v^{\prime}\in U$. It implies the assertion. ∎

######  Proposition 2.3.

Let $\tilde{v},\tilde{v}^{\prime}$ be closed point of
$\tilde{X}^{\mathrm{ab},p^{\prime}}$. Assume that $k$ is a perfect field
finitely generated over the prime field, that the $p$-rank of $J_{X}$ is $0$,
that $(g,r)\neq(0,0)$, $(0,1)$, and that
$\tilde{v},\tilde{v}^{\prime}\notin\tilde{E}^{\mathrm{ab},p^{\prime}}$ when
$(g,r)=(0,2)$. Then the following conditions (a)-(d) are equivalent:

  1. (a)

$\tilde{v}=\tilde{v}^{\prime}$.

  2. (b)

$D_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}=D_{\tilde{v}^{\prime},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$.

  3. (c)

$D_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$ and
$D_{\tilde{v}^{\prime},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$ are commensurable.

  4. (d)

The image of $D_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}\cap
D_{\tilde{v}^{\prime},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$ in $G_{k}$ is open
and the images of $\tilde{v}$, $\tilde{v}^{\prime}$ by
$\tilde{X}^{\mathrm{ab},p^{\prime}}\twoheadrightarrow X_{\overline{k}}$ are
the same.

If, moreover, $\tilde{v}\notin\tilde{E}^{\mathrm{ab},\Sigma}$ when
$(g,r)=(0,3)$, then the conditions (a)-(d) and the following condition (e) are
equivalent:

  1. (e)

The image of $D_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}\cap
D_{\tilde{v}^{\prime},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$ in $G_{k}$ is open.

###### Proof..

The implications (a)$\Rightarrow$(b)$\Rightarrow$(c)$\Rightarrow$(e) are
clear. Let us state the following three claims:

  1. (i)

(d)$\Rightarrow$(a)

  2. (ii)

(c)$\Rightarrow$(d) when $(g,r)=(0,3)$ and $\tilde{v},\
\tilde{v}^{\prime}\in\tilde{E}^{1,p^{\prime}}$.

  3. (iii)

(e)$\Rightarrow$(d) under the additional assumption
“$\tilde{v}\notin\tilde{E}^{\mathrm{ab},\Sigma}$ when $(g,r)=(0,3)$”.

The implications
(e)$\Rightarrow$(d)$\Rightarrow$(a)$\Rightarrow$(b)$\Rightarrow$(c)$\Rightarrow$(e)
follow from iiii under the additional assumption
“$\tilde{v}\notin\tilde{E}^{\mathrm{ab},\Sigma}$ when $(g,r)=(0,3)$”.
Moreover, the implications
(c)$\Rightarrow$(d)$\Rightarrow$(a)$\Rightarrow$(b)$\Rightarrow$(c) follow
from iiiiii in general cases. Therefore, the assertion holds if iiiiii hold.
Thus, it is sufficient to show the claims iiiiii.

First, we show (i). The second assumption of (d) implies that
$\tilde{v},\tilde{v}^{\prime}\notin\tilde{E}^{\mathrm{ab},p^{\prime}}$ holds
if and only if $\tilde{v}^{\prime}\notin\tilde{E}^{\mathrm{ab},p^{\prime}}$
holds. Moreover, the assumption “$(g,r)\neq(0,0)$, $(0,1)$, and
$\tilde{v}^{\prime}\notin\tilde{E}^{\mathrm{ab},p^{\prime}}$ when
$(g,r)=(0,2)$” implies that
$\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}/I_{\tilde{v}^{\prime}}\neq\\{1\\}$.
Since the image of $D_{\tilde{v}}\cap D_{\tilde{v}^{\prime}}$ in $G_{k}$ is
open, by taking finite extension of $k$ if necessary, we may assume that the
image of $D_{\tilde{v}}\cap D_{\tilde{v}^{\prime}}$ coincides with $G_{k}$.
Let $v$ be the image of $\tilde{v}$, $\tilde{v}^{\prime}$ by
$\tilde{X}^{\mathrm{ab},p^{\prime}}\twoheadrightarrow X_{\overline{k}}$. Then
$v$ is a $k$-rational point, and hence $G_{k}$ acts on
$I_{\tilde{v}^{\prime}}\subset\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}$.
The second assumption of (d) implies that there exists
$\gamma\in\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}$ such that
$\tilde{v}^{\prime}=\gamma\tilde{v}$. This implies that
$s_{\tilde{v}^{\prime}}=(s_{\tilde{v}})^{\gamma}$. Let $t\in G_{k}$ be an
element and $\tilde{t}\in D_{\tilde{v}}\cap D_{\tilde{v}^{\prime}}$ an inverse
image of $t$. Since $\gamma\tilde{t}\gamma^{-1}\in\gamma
D_{\tilde{v}}\gamma^{-1}=D_{\tilde{v}^{\prime}}$ and
$\tilde{t}^{-1}\gamma\tilde{t}\in\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}$,
we obtain that
$\tilde{t}^{-1}\gamma\tilde{t}\gamma^{-1}\in\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}\cap
D_{\tilde{v}^{\prime}}=I_{\tilde{v}^{\prime}}$. In other words,
$\tilde{t}^{-1}\overline{\gamma}\tilde{t}=\overline{\gamma}$ holds in
$\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}/I_{\tilde{v}^{\prime}}$. Since
$k$ is a field finitely generated over the prime field, the action
$G_{k}\curvearrowright\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}/I_{\tilde{v}^{\prime}}$
has weights $-1$ and $-2$. This implies that
$(\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}/I_{\tilde{v}^{\prime}})^{G_{k}}=\\{1\\}$.
Therefore, we obtain that
$\overline{\gamma}\in(\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}/I_{\tilde{v}^{\prime}})^{G_{k}}=\\{1\\}$,
and hence $\gamma\in I_{\tilde{v}^{\prime}}$ holds. Thus,
$\tilde{v}=\gamma^{-1}\tilde{v}^{\prime}=\tilde{v}^{\prime}$ holds. This
completes the proof of (i).

Next, we show (ii). It is clear that (c) implies that the image of
$D_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}\cap
D_{\tilde{v}^{\prime},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$ in $G_{k}$ is open.
Moreover, (c) implies that $I_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$
and $I_{\tilde{v}^{\prime},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$ are
commensurable. Therefore, the images of $\tilde{v}$, $\tilde{v}^{\prime}$ by
$\tilde{E}^{\mathrm{ab},p^{\prime}}\twoheadrightarrow E_{\overline{k}}$ are
the same by [24, Lemma 1.11(1)]. Hence (ii) holds.

Finally, we show (iii). Since the image of $D_{\tilde{v}}\cap
D_{\tilde{v}^{\prime}}$ in $G_{k}$ is open, by taking finite extension of $k$
if necessary, we may assume that the image of $D_{\tilde{v}}\cap
D_{\tilde{v}^{\prime}}$ coincides with $G_{k}$. Let $v,v^{\prime}\in X$ be the
images of $\tilde{v},\tilde{v}^{\prime}$, respectively. Then $v$, $v^{\prime}$
are $k$-rational points. Moreover, by taking finite extension of $k$ if
necessary, we may assume that $E(k)=E(\overline{k})$. The condition (e)
implies the condition (e) for the curve $U\cup\\{v,v^{\prime}\\}$, and the
condition (a) for $U\cup\\{v,v^{\prime}\\}$ clearly implies the condition (a).
Hence we may assume that $v,v^{\prime}\in U$. (Note that, if $g=0$, then
$r(U\cup\\{v,v^{\prime}\\})\geq 2$ by the assumptions “$(g,r)\neq(0,0)$,
$(0,1)$”,
“$\tilde{v},\tilde{v}^{\prime}\notin\tilde{E}^{\mathrm{ab},p^{\prime}}$ when
$(g,r)=(0,2)$”, and “$\tilde{v}\notin\tilde{E}^{\mathrm{ab},\Sigma}$ when
$(g,r)=(0,3)$”.) Hence the second assertion of (d) follows from Lemma 2.2.
Thus, (iii) holds. This completes the proof. ∎

######  Remark 2.4.

One of the important corollaries of the separatedness property is that it
implies the bijectivity of the natural map from $\tilde{X}^{m,p^{\prime}}$ to
$\mathrm{Dec}(\Pi^{(m,p^{\prime})}_{U})$. In particular, if we have a group-
theoretical reconstruction of $\mathrm{Dec}(\Pi^{(m,p^{\prime})}_{U})$, then
we can reconstruct $\tilde{X}^{m,p^{\prime}}$ from $\Pi^{(m,p^{\prime})}_{U}$
group-theoretically by using this bijectivity. This argument was used in [20]
under the classical setting (and in [24] under the $m$-step setting). We will
use this argument in Section 4.

######  Definition 2.5.

Let $P$ be a topological group and $H$ a closed subgroup of $P$. We define

$\mathrm{N}_{P}(H)\overset{\mathrm{def}}{=}\\{x\in P\mid xHx^{-1}=H\\},$
$\mathrm{Comm}_{P}(H)\overset{\mathrm{def}}{=}\\{x\in P\mid xHx^{-1}\text{ and
}H\text{ are commensurable.}\\}$

and refer to them as the $normalizer$, $commensurator$ of $H$ in $P$,
respectively.

######  Corollary 2.6.

Assume that $k$ is a perfect field finitely generated over the prime field,
that the $p$-rank of $J_{X}$ is $0$, that $(g,r)\neq(0,0)$, $(0,1)$, and that
$\tilde{v},\tilde{v}^{\prime}\notin\tilde{E}^{\mathrm{ab},p^{\prime}}$ when
$(g,r)=(0,2)$. Then, for any finite extension $L$ of $k$ and any closed point
$\tilde{v}$ of $\tilde{X}^{\mathrm{ab},p^{\prime}}$, the following hold:

$D_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}=\mathrm{N}_{\Pi_{U}^{(\mathrm{ab},p^{\prime})}}\left(D_{\tilde{v},\Pi_{U_{L}}^{(\mathrm{ab},p^{\prime})}}\right)=\mathrm{Comm}_{\Pi_{U}^{(\mathrm{ab},p^{\prime})}}\left(D_{\tilde{v},\Pi_{U_{L}}^{(\mathrm{ab},p^{\prime})}}\right)$

###### Proof..

Since $D_{\tilde{v},\Pi_{U_{L}}^{(\mathrm{ab},p^{\prime})}}$ is a normal
subgroup of $D_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$, it is clear
that $D_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$ is contained in
$\mathrm{N}_{\Pi_{U}^{(\mathrm{ab},p^{\prime})}}(D_{\tilde{v},\Pi_{U_{L}}^{(\mathrm{ab},p^{\prime})}})(\subset\mathrm{Comm}_{\Pi_{U}^{(\mathrm{ab},p^{\prime})}}(D_{\tilde{v},\Pi_{U_{L}}^{(\mathrm{ab},p^{\prime})}})$.
Let $x$ be an element of
$\mathrm{Comm}_{\Pi_{U}^{(\mathrm{ab},p^{\prime})}}(D_{\tilde{v},\Pi_{U_{L}}^{(\mathrm{ab},p^{\prime})}})$.
Then
$xD_{\tilde{v},\Pi_{U_{L}}^{(\mathrm{ab},p^{\prime})}}x^{-1}=D_{x\cdot\tilde{v},\Pi_{U_{L}}^{(\mathrm{ab},p^{\prime})}}$
and $D_{\tilde{v},\Pi_{U_{L}}^{(\mathrm{ab},p^{\prime})}}$ are commensurable.
Hence, by Proposition 2.3a$\Leftrightarrow$c, $x\cdot\tilde{v}=\tilde{v}$
holds. Therefore, $x\in D_{\tilde{v},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$
holds. This completes the proof. ∎

##  3 A group-theoretical reconstruction of decomposition groups over finite
fields

Let $k$ be a finite field of characteristic $p>0$, $(X,E)$ a smooth curve of
type $(g,r)$ over $k$, and we set $U\overset{\mathrm{def}}{=}X-E$. In this
section, we first give a group-theoretical reconstruction of geometric
sections of $\Pi^{(\mathrm{ab},p^{\prime})}_{U}\twoheadrightarrow G_{k}$ from
$\Pi^{(2,p^{\prime})}_{U}$ (cf. Lemma 3.1 below). Next, we construct a group-
theoretical reconstruction of decomposition groups of
$\Pi^{(\mathrm{ab},p^{\prime})}_{U}$ from $\Pi^{(2,p^{\prime})}_{U}$ if
$(g,r,m)$ satisfies certain conditions (cf. Proposition 3.6 below). This
construction is based on [20, Section 3] (and [24, Section 2]), however,
considering the case of pro-prime-to-$p$, all arguments are re-constructed in
the case of pro-prime-to-$p$ with the assumption “the $p$-rank of $J_{X}$ is
$0$” coming from Lemma 2.2. (At the time of writing the present paper, this
assumption is essential and the author cannot yet remove it.)

First of all, we will give a group-theoretical reconstruction of geometric
sections. One of the important observations is that we can describe whether a
section is geometric or not by using Lefschetz trace formula, since $k$ is a
finite field. This excellent observation was made in [20, Proposition (0.7)].
Via this observation, we obtain the following lemma (cf. [20, Section 3]):

######  Lemma 3.1.

Let $i=1$, $2$. Let $k_{i}$ be a finite field of characteristic $p_{i}$. Let
$(X_{i},E_{i})$ be a smooth curve of type $(g_{i},r_{i})$ over $k_{i}$, and we
set $U_{i}\overset{\mathrm{def}}{=}X_{i}-E_{i}$. Let $\Phi$ be an isomorphism
$\Pi^{(2,p_{1}^{\prime})}_{U_{1}}\xrightarrow{\sim}\Pi^{(2,p_{2}^{\prime})}_{U_{2}}$.
Let
$\Phi^{1}:\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\xrightarrow{\sim}\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}$
be the isomorphism induced by $\Phi$ (cf. Lemma 1.12). Then $\Phi$ naturally
induces a bijective map
$\mathcal{S}_{G,\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}}\xrightarrow{\sim}\mathcal{S}_{G,\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}}$
that is compatible with the actions of
$\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}$,
$\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}$ relative to $\Phi^{1}$, for any
open subgroup $G$ of $G_{k}$. Moreover, the bijective map preserves geometric
sections.

###### Proof..

As Lemma 1.19 implies that $p_{1}=p_{2}$ and that $k_{1}=k_{2}$, we may write
$p\overset{\mathrm{def}}{=}p_{1}(=p_{2})$ and
$k\overset{\mathrm{def}}{=}k_{1}(=k_{2})$. Let $G$ be an open subgroup of
$G_{k}$ with a natural inclusion $\iota:G\hookrightarrow G_{k}$. First, we
claim that a section
$s\in\mathcal{S}_{G,\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}}}$ is geometric if
and only if, for any open subgroup $H$ of
$\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}}$ containing $s(G)$,
$X_{i,H}(\overline{k}^{G})$ is non-empty, where $X_{i,H}$ stands for the
covering of $X_{i}$ that corresponds to $H$. Let $\tilde{v}$ be a closed point
of $\tilde{X}_{i}^{\mathrm{ab},p^{\prime}}$ with $s(G)\subset D_{\tilde{v}}$
and $v_{H}$ the image of $\tilde{v}$ in $X_{i,H}$. Then
$\overline{k}^{G}\supset\overline{k}^{\mathrm{pr}(D_{\tilde{v}}\cap
H)}=\kappa(v_{H})$, and hence $v_{H}\in X_{i,H}(\overline{k}^{G})$.
Conversely, since $X_{i,H}(\overline{k}^{G})$ is finite and non-empty, there
exists an element $v\in
X_{i,s(G)}(\overline{k}^{G})=\varprojlim_{H}X_{i,H}(\overline{k}^{G})$ by
Tychonoff’s theorem, where $H$ runs over all open subgroups of
$\Pi_{U_{i}}^{(\mathrm{ab},p^{\prime})}$ containing $s(G)$. Let
$\tilde{v}\in\tilde{X}_{i}^{\mathrm{ab},p^{\prime}}$ be a closed point above
$v$ and $\mathrm{pr}:\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}}\to G_{k}$ the
natural projection. Then $\mathrm{pr}|_{s(G)}(D_{\tilde{v}}\cap
s(G))=G=\mathrm{pr}|_{s(G)}(s(G))$, and hence $D_{\tilde{v}}\supset s(G)$,
since $\mathrm{pr}|_{s(G)}$ is injective. Therefore, the claim follows. The
assertion follows from Lemma 1.15, the claim, and the Lefschetz trace formula

$|X_{i,H}(\overline{k}^{G})|=1+|\overline{k}^{G}|-\mathrm{tr}_{\mathbb{Z}_{\ell}}(\mathrm{Fr}_{\overline{k}^{G}}\mid\overline{H}^{\mathrm{ab},\ell}/W_{-2}(\overline{H}^{\mathrm{ab},\ell})),$

where $\ell$ stands for a rational prime number different from $p$. ∎

Next, we consider a group-theoretical reconstruction of
$\mathrm{Dec}(\Pi_{U}^{(\mathrm{ab},p^{\prime})})$ from
$\Pi_{U}^{(2,p^{\prime})}$. By definition, for any geometric section
$s\in\mathcal{S}_{G}^{\mathrm{geom}}\overset{\mathrm{def}}{=}\mathcal{S}^{\mathrm{geom}}_{G,\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$,
there exists $D_{s}\in\mathrm{Dec}(\Pi_{U}^{(\mathrm{ab},p^{\prime})})$ such
that $s(G)\subset D_{s}$. If “the $p$-rank of $J_{X}$ is $0$” and “$r_{1}\geq
4$ when $g_{1}=0$”, then such an element of
$\mathrm{Dec}(\Pi_{U}^{(\mathrm{ab},p^{\prime})})$ is uniquely determined by
$s$ (cf. Proposition 2.3a$\Leftrightarrow$e). Hence we obtain the following
unique map

$D_{\ast}:\mathcal{S}_{G}^{\mathrm{geom}}\twoheadrightarrow\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U}),\
\ \ s\mapsto D_{s}$ (3.1)

that satisfies $s(G)\subset D_{s}$ for any
$s\in\mathcal{S}_{G}^{\mathrm{geom}}$. Note that the map $D_{\ast}$ is
$\Pi^{(\mathrm{ab},p^{\prime})}_{U}$-equivalence map, since the actions of
$\Pi^{(\mathrm{ab},p^{\prime})}_{U}$ on $\mathcal{S}_{G}^{\mathrm{geom}}$ and
$\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})$ are conjugacy actions.

######  Definition 3.2.

We shall define $\Pi_{U}^{(m,p^{\prime})}$-sets
$\mathcal{QS}\overset{\mathrm{def}}{=}\mathcal{QS}_{\Pi^{(m,p^{\prime})}_{U}}\overset{\mathrm{def}}{=}\varinjlim_{G}\mathcal{S}_{G,\Pi^{(m,p^{\prime})}_{U}}$
and
$\mathcal{QS}^{\mathrm{geom}}\overset{\mathrm{def}}{=}\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(m,p^{\prime})}_{U}}\overset{\mathrm{def}}{=}\varinjlim_{G}\mathcal{S}^{\mathrm{geom}}_{G,\Pi^{(m,p^{\prime})}_{U}}$,
where the inductive limits run over all open subgroup $G$ of $G_{k}$. We shall
refer to an element of $\mathcal{QS}$, $\mathcal{QS}^{\mathrm{geom}}$ as a
$quasi$-$section$, a $geometric$ $quasi$-$section$ (of
$\Pi^{(m,p^{\prime})}_{U}$), respectively. If there is no risk of confusion,
we shall use the notation $s\in\mathfrak{s}$ for any element
$s\in\mathcal{S}_{G}\overset{\mathrm{def}}{=}\mathcal{S}_{G,\Pi^{(m,p^{\prime})}_{U}}$
that is mapped to $\mathfrak{s}$ by the natural map
$\mathcal{S}_{G}\to\mathcal{QS}$. The map (3.1) induces the following
$\Pi_{U}^{(\mathrm{ab},p^{\prime})}$-equivalence maps:

$D_{\ast}:\mathcal{QS}^{\mathrm{geom}}\twoheadrightarrow\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U}),\
\ \ \mathfrak{s}\mapsto D_{\mathfrak{s}},$ (3.2)
$I_{\ast}:\mathcal{QS}^{\mathrm{geom}}\twoheadrightarrow\mathrm{Iner}(\Pi^{(\mathrm{ab},p^{\prime})}_{U}),\
\ \ \mathfrak{s}\mapsto I_{\mathfrak{s}}$ (3.3)

We write
$\mathcal{QS}^{\mathrm{geom,cusp}}\overset{\mathrm{def}}{=}\mathcal{QS}^{\mathrm{geom,cusp}}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$
for the set of geometric quasi-section $\mathfrak{s}$ that satisfies
$D_{\mathfrak{s}}\in\mathrm{Dec}^{\mathrm{cusp}}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})$.

These definition induce the following commutative diagram:

$\textstyle{\mathcal{QS}^{\mathrm{geom,cusp}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Dec}^{\mathrm{cusp}}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Iner}^{\neq\\{1\\}}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{QS}^{\mathrm{geom}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Iner}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})}$$\textstyle{\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}\backslash\mathcal{QS}^{\mathrm{geom}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})}$$\textstyle{(X_{\overline{k}})^{\mathrm{cl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$
(3.4)

Here, the injectivity of the natural map
$(\tilde{X}^{\mathrm{ab},p^{\prime}})^{\mathrm{cl}}\to\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})$
(and hence the injectivity of the natural map
$(X_{\overline{k}})^{\mathrm{cl}}\rightarrow\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U}$))
follows from Proposition 2.3a$\Leftrightarrow$c. Remark that
$\mathrm{Iner}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})$ and
$\mathcal{QS}^{\mathrm{geom}}$ are reconstructed group-theoretically from
$\Pi_{U}^{(2,p^{\prime})}$ by Lemma 1.1 and Lemma 3.1.

######  Lemma 3.3.

Let $i=1$, $2$. Let $k_{i}$ be a finite field of characteristic $p_{i}$. Let
$(X_{i},E_{i})$ be a smooth curve of type $(g_{i},r_{i})$ over $k_{i}$, and we
set $U_{i}\overset{\mathrm{def}}{=}X_{i}-E_{i}$. Let $\Phi^{1}$ be an
isomorphism
$\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\xrightarrow{\sim}\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}$.
Assume that the $p$-rank of $J_{X}$ is $0$, that $r_{1}\geq 5$ when $g_{1}=0$,
that the natural bijection
$\mathcal{QS}_{\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}}\xrightarrow{\sim}\mathcal{QS}_{\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}}$
induced by $\Phi^{1}$ preserves geometric quasi-sections, and that $\Phi^{1}$
preserves inertia groups. Then the following conditions (A)-(B) are
equivalent:

  1. (A)

There exists a bijection
$\overline{\Pi}_{U_{1}}^{\mathrm{ab},p_{1}^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}})\xrightarrow{\sim}\overline{\Pi}_{U_{2}}^{\mathrm{ab},p_{2}^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}})$
such that the following diagram commutes:

$\textstyle{\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\rho_{1}}$$\textstyle{\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{2}}$$\textstyle{\overline{\Pi}_{U_{1}}^{\mathrm{ab},p_{1}^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\overline{\Pi}_{U_{2}}^{\mathrm{ab},p_{2}^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}})}$
(3.5)

Here, the vertical arrows $\rho_{1}$ and $\rho_{2}$ stand for the surjection
in (3.4) and the upper horizontal arrow stands for the bijection induced by
the assumption of $\Phi^{1}$.

  2. (B)

$\Phi^{1}$ preserves decomposition groups.

###### Proof..

If $\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}$ is trivial, then the assertion
is clear. Hence we may assume that
$\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}$ is non-trivial. As Lemma 1.169
imply that $g_{1}=g_{2}$, that $p_{1}=p_{2}$, and that $k_{1}=k_{2}$, we may
write $g\overset{\mathrm{def}}{=}g_{1}(=g_{2})$,
$p\overset{\mathrm{def}}{=}p_{1}(=p_{2})$, and
$k\overset{\mathrm{def}}{=}k_{1}(=k_{2})$. We also have that the $p$-rank of
$J_{X_{1}}$ coincides with the $p$-rank of $J_{X_{2}}$ by Lemma 1.110. For
simplicity, we write
$\mathcal{QS}^{\mathrm{geom}}_{i}\overset{\mathrm{def}}{=}\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}}}$
and
$\mathcal{QS}^{\mathrm{geom,cusp}}_{i}\overset{\mathrm{def}}{=}\mathcal{QS}^{\mathrm{geom,cusp}}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}}}$.
Lemma 1.16 implies that $r_{1}\in\\{0,1\\}$ if and only if
$r_{2}\in\\{0,1\\}$. We know that the map
$D_{\ast}:\mathcal{QS}_{i}^{\mathrm{geom}}\twoheadrightarrow\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}})$
in (3.2) is injective on the set
$\mathcal{QS}^{\mathrm{geom}}_{i}-\mathcal{QS}^{\mathrm{geom,cusp}}_{i}$. If
$r_{i}\in\\{0,1\\}$, then $\mathcal{QS}^{\mathrm{geom,cusp}}_{i}=\emptyset$.
Hence $D_{\ast}$ is bijective when $r_{i}\in\\{0,1\\}$. Therefore, the
conditions AB are clearly true when $r_{1}\in\\{0,1\\}$. Thus, we may assume
that $r_{1}\geq 2$. In this case, the implication (B)$\Rightarrow$(A) is
clear. Hence it is sufficient to show the implication A$\Rightarrow$B. Assume
the condition A. As Lemma 1.17 implies that $r_{1}=r_{2}$, we may write
$r\overset{\mathrm{def}}{=}r_{1}(=r_{2})$. Since the map $D_{*}$ in (3.2) is
$\Pi_{U_{i}}^{(\mathrm{ab},p^{\prime})}$-equivalent, the map
$\rho_{i}:\mathcal{QS}^{\mathrm{geom}}_{i}\to\overline{\Pi}_{U_{i}}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}})$
in (3.5) factors through
$\overline{\Pi}_{U_{i}}^{\mathrm{ab},p^{\prime}}\backslash\mathcal{QS}^{\mathrm{geom}}_{i}$,
and then we write
$\alpha_{i}:\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U_{i}}\backslash\mathcal{QS}^{\mathrm{geom}}_{i}\to\overline{\Pi}_{U_{i}}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}})$
for the map induced by $D_{*}$. (Here, we use $r\geq 4$ when $g=0$ to define
the map (3.1).)

$\textstyle{\mathcal{QS}^{\mathrm{geom}}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{i}}$$\textstyle{\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U_{i}}\backslash\mathcal{QS}^{\mathrm{geom}}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{i}}$$\textstyle{\overline{\Pi}_{U_{i}}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}})}$$\textstyle{(X_{i,\overline{k}})^{\mathrm{cl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$

Let us state the following claims:

  1. (i)

$\mathcal{QS}^{\mathrm{geom,cusp}}_{i}=\\{\mathfrak{s}\in\mathcal{QS}^{\mathrm{geom}}_{i}\mid|\alpha_{i}^{-1}(\rho_{i}(\mathfrak{s}))|>1\\}$.
(Remark that the claim also holds when $(g,r)=(0,4)$.)

  2. (ii)

The following diagram commutes:

$\textstyle{\mathcal{QS}^{\mathrm{geom,cusp}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathcal{QS}^{\mathrm{geom,cusp}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Iner}^{\neq\\{1\\}}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{Iner}^{\neq\\{1\\}}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{2}})}$
(3.6)

Here, the vertical arrows stand for the surjection induced by (3.3), the upper
horizontal arrow stands for the bijection in i, and the lower horizontal arrow
stands for the bijection induced by the assumption of $\Phi^{1}$.

  3. (iii)

For any $\mathfrak{s}\in\mathcal{QS}^{\mathrm{geom,cusp}}_{i}$ (resp.
$\mathcal{QS}^{\mathrm{geom}}_{i}-\mathcal{QS}^{\mathrm{geom,cusp}}_{i}$),
$D_{\mathfrak{s}}$ coincides with the subgroup

$\langle\\{\mathrm{Im}(s)\cdot I_{\mathfrak{s}}\mid
G\overset{\mathrm{op}}{\subset}G_{k},\
s\in\mathcal{S}^{\mathrm{geom}}_{G,\Pi_{U_{i}}^{(\mathrm{ab},p^{\prime})}}\text{
satisfying }s\in\mathfrak{s}\\}\rangle$ (3.7) $(\text{resp.
}\langle\\{\mathrm{Im}(s)\mid G\overset{\mathrm{op}}{\subset}G_{k},\
s\in\mathcal{S}^{\mathrm{geom}}_{G,\Pi_{U_{i}}^{(\mathrm{ab},p^{\prime})}}\text{
satisfying }s\in\mathfrak{s}\\}\rangle)$

of $\Pi_{U_{i}}^{(\mathrm{ab},p^{\prime})}$.

The bijection
$\mathcal{QS}^{\mathrm{geom}}_{1}\xrightarrow{\sim}\mathcal{QS}^{\mathrm{geom}}_{2}$
maps $\mathcal{QS}^{\mathrm{geom,cusp}}_{1}$ to
$\mathcal{QS}^{\mathrm{geom,cusp}}_{2}$ by A and i. Moreover, the diagram
(3.6) is commutative by ii. These results imply that $\Phi^{1}$ preserves the
subgroup (3.7). Thus, $\Phi^{1}$ preserves decomposition groups by iii, and
hence the assertion holds if iiiiii hold. Therefore, it is sufficient to show
the claims iiiiii.

First, we show i. Since
$D_{\ast}:\mathcal{QS}^{\mathrm{geom}}_{i}\twoheadrightarrow\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}})$
in (3.2) is injective on
$\mathcal{QS}^{\mathrm{geom}}_{i}-\mathcal{QS}^{\mathrm{geom,cusp}}_{i}$, this
implies that $|\alpha_{i}^{-1}(\rho_{i}(\mathfrak{s}))|=1$ for
$\mathfrak{s}\in\mathcal{QS}^{\mathrm{geom}}_{i}-\mathcal{QS}^{\mathrm{geom,cusp}}_{i}$.
Hence we may assume that
$\mathfrak{s}\in\mathcal{QS}^{\mathrm{geom,cusp}}_{i}$. Let
$\mathcal{S}\subset\mathcal{QS}^{\mathrm{geom,cusp}}_{i}$ be the inverse image
of $D_{\mathfrak{s}}$ by $D_{\ast}$. For any
$\mathfrak{s}^{\prime}\in\mathcal{S}$ and any
$a\in\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U}$ such that
$a\cdot\mathfrak{s}=\mathfrak{s}^{\prime}$, we have that
$aI_{\mathfrak{s}}a^{-1}=I_{a\cdot\mathfrak{s}}=I_{\mathfrak{s}^{\prime}}(=I_{\mathfrak{s}})$,
and hence $a\in I_{\mathfrak{s}}$ holds. Thus,
$I_{\mathfrak{s}}\backslash\mathcal{S}=\overline{\Pi}_{U_{i}}^{\mathrm{ab},p^{\prime}}\backslash\mathcal{S}\
(\subset\alpha_{i}^{-1}(\rho_{i}(\mathfrak{s})))$ holds. By [24, Lemma 1.10]
and the assumption $r\neq 1$, $I_{\mathfrak{s}}$ is isomorphic to an inertia
group of $\Pi^{(p^{\prime})}_{U}$. In particular, $I_{\mathfrak{s}}\backslash
S\simeq\varprojlim_{k^{\prime}}\mathrm{H}^{1}(G_{k^{\prime}},\hat{\mathbb{Z}}^{p^{\prime}}(1))\xrightarrow{\sim}\overline{k}^{\times}$
holds. Hence i holds.

Next, we show ii. Since
$|\mathrm{Iner}^{\neq\\{1\\}}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}})|=1$
when $r=2$, we may assume that $r\geq 3$. Let $G$ be a open subgroup of
$G_{k}$ such that $(G_{k}:G)\geq r!$, where $r!$ stands for the factorial of
$r$. (Note that $G_{k}\simeq\hat{\mathbb{Z}}$). Then the natural action of $G$
on $E_{i}(\overline{k})$ is trivial, since the natural action of $G_{k}$ on
$E_{i}(\overline{k})$ factors through the permutation action of symmetric
group $S_{r}$ on $E_{i}(\overline{k})$. Thus, we may assume that the natural
action of $G_{k}$ on $E_{i}(\overline{k})$ is trivial by replacing $G_{k}$
with $G$ if necessary. (In other words, we may assume that
$E_{i}(\overline{k})=E_{i}(k)$.) By the assumption that $r\geq 5$ when $g=0$
and the remark that i also holds when .$(g,r)=(0,4)$, for any inertia subgroup
$I$ of $\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U_{i}}$, we can apply the
proof of i to the following diagram:

$\textstyle{\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}}/I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{i,I}}$$\textstyle{(\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U_{i}}/I)\backslash\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}}/I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{i,I}}$$\textstyle{(\overline{\Pi}_{U_{i}}^{\mathrm{ab},p^{\prime}}/I)\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}}/I)}$$\textstyle{(X_{i,\overline{k}})^{\mathrm{cl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$

Here, $\alpha_{i,I}$ and $\rho_{i,I}$ are defined in the same way as
$\alpha_{i}$ and $\rho_{i}$, respectively. Hence, for any
$\mathfrak{s}\in\mathcal{QS}^{\mathrm{geom,cusp}}_{i}$, $I_{\mathfrak{s}}$ is
the unique inertia subgroup $I$ of
$\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U_{i}}$ satisfying
$|\alpha_{i,I}^{-1}(\rho_{i,I}(\mathfrak{s}))|=1$. (The uniqueness property
follows from i and $r\geq 2$.) Thus, we get the map (3.3) group-theoretically,
and hence the diagram (3.6) is commutative by A. Therefore, ii holds

Finally, we show iii. The group (3.7) is clearly contained in
$D_{\mathfrak{s}}$. Since $G_{k}\simeq\hat{\mathbb{Z}}$, there exist an open
subgroup $G$ of $G_{k}$ and a section
$s\in\mathcal{S}^{\mathrm{geom}}_{G,\Pi_{U_{i}}^{(\mathrm{ab},p^{\prime})}}$
such that $\mathfrak{s}=[s]$ and that $\mathrm{Im}(s)\cdot
I_{\mathfrak{s}}=D_{\mathfrak{s}}$. Hence the group (3.7) coincide with
$D_{\mathfrak{s}}$. Therefore, iii holds. This completes the proof. ∎

######  Definition 3.4.

Let $G$, $G^{\prime}$ be open subgroups of $G_{k}$,
$a\in\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U}$,
$s\in\mathcal{S}_{G,\Pi^{(m,p^{\prime})}_{U}}$, and
$s^{\prime}\in\mathcal{S}_{G^{\prime},\Pi^{(m,p^{\prime})}_{U}}$. When $g>0$,
we define the following condition for the multiple
$(G,s,G^{\prime},s^{\prime},a)$:

  * (†)

$s(G)\cdot W_{-2}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})$ and
$a(s^{\prime}(G^{\prime}))a^{-1}\cdot
W_{-2}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})$ are commensurable.

When $g=0$, we define the following condition for the multiple
$(G,s,G^{\prime},s^{\prime},a)$:

  * (†)

There exists
$I\in\mathrm{Iner}^{\neq\\{1\\}}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})$ such
that, for any subset
$\mathcal{I}\subset\mathrm{Iner}^{\neq\\{1\\}}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}})-\\{I\\}$
whose cardinality is $2$, $s(G)\cdot\langle I^{\prime}\mid
I^{\prime}\notin\mathcal{I}\rangle$ and
$a(s^{\prime}(G^{\prime}))a^{-1}\cdot\langle I^{\prime}\mid
I^{\prime}\notin\mathcal{I}\rangle$ are commensurable. (Here, $\langle
I^{\prime}\mid I^{\prime}\notin\mathcal{I}\rangle$ stands for the subgroup of
$\Pi_{U_{i}}^{(\mathrm{ab},p^{\prime})}$ generated by all inertia groups
except for the elements of $\mathcal{I}$.)

If $\tilde{G}$ be an open subgroup of $G$ and $\tilde{s}$ be the restriction
of $s$ on $\tilde{G}$, then the condition (†) for
$(G,s,G^{\prime},s^{\prime},a)$ holds if and only if the condition (†) for
$(\tilde{G},\tilde{s},G^{\prime},s^{\prime},a)$ holds. Hence, for any
$\mathfrak{s}$,
$\mathfrak{s}^{\prime}\in\mathcal{QS}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$,
we define a relation $\mathfrak{s}\sim\mathfrak{s}^{\prime}$ as there exists a
multiple $(G,s,G^{\prime},s^{\prime},a)$ such that $s\in\mathfrak{s}$,
$s^{\prime}\in\mathfrak{s}^{\prime}$ and that $(G,s,G^{\prime},s^{\prime},a)$
satisfies the condition (†).

######  Lemma 3.5.

Assume that the $p$-rank of $J_{X}$ is zero, and that $r\geq 5$ when $g=0$.
Let $\mathfrak{s},\ \mathfrak{s}^{\prime}$ be elements of
$\mathcal{QS}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U}}^{\mathrm{geom}}$. We write
$\rho:\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U}}\twoheadrightarrow\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})$
for the surjection induced by $D_{\ast}$ (cf. (3.4)). Then

$\mathfrak{s}\sim\mathfrak{s}^{\prime}\iff\rho(\mathfrak{s})=\rho(\mathfrak{s}^{\prime})$

holds. In particular, the relation $\sim$ is an equivalent relation of
$\mathcal{QS}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U}}^{\mathrm{geom}}$ and the
following diagram is commutative:

$\textstyle{\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{\ast}}$$\textstyle{\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{X}^{\mathrm{ab},p^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{(\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U}}/\sim)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})}$$\textstyle{(X_{\overline{k}})^{\mathrm{cl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$
(3.8)

Here, the bijectivity of the right-side vertical arrows follows from
Proposition 2.3a$\Leftrightarrow$c.

###### Proof..

The commutativity of (3.8) follows from the first assertion. Hence it is
sufficient to show the first assertion. First, we assume that $g\geq 1$. Let
us assume that $\mathfrak{s}\sim\mathfrak{s}^{\prime}$ and let
$(G,s,G^{\prime},s^{\prime},a)$ be the multiple of definition. Let
$\overline{s}$, $\overline{s}^{\prime}$, $\overline{a}$ be the images of $s$,
$s^{\prime}$, $a$ in
$\mathcal{S}^{\mathrm{geom}}_{G,\Pi_{X}^{(\mathrm{ab},p^{\prime})}}$,
$\mathcal{S}^{\mathrm{geom}}_{G^{\prime},\Pi_{X}^{(\mathrm{ab},p^{\prime})}}$,
$\overline{\Pi}_{X}^{\mathrm{ab},p^{\prime}}$, respectively. Then
$\overline{s}(G)$ and
$\overline{a}(\overline{s}^{\prime}(G^{\prime}))\overline{a}^{\prime}$ are
commensurable in $\Pi_{X}^{(\mathrm{ab},p^{\prime})}$. Hence Proposition
2.3a$\Leftrightarrow$e implies that $\overline{s}(G)$ and
$\overline{a}(\overline{s}^{\prime}(G^{\prime}))\overline{a}^{\prime}$ are
contained in the same decomposition group of
$\Pi_{X}^{(\mathrm{ab},p^{\prime})}$. Since
$\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})\xrightarrow{\sim}\overline{\Pi}_{X}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{X})(\xleftarrow{\sim}(X_{\overline{k}})^{\mathrm{cl}}$),
we obtain that $\rho(\mathfrak{s})=\rho(\mathfrak{s}^{\prime})$. Next, we
assume that $\rho(\mathfrak{s})=\rho(\mathfrak{s}^{\prime})$. Then there
exists open subgroups $G$, $G^{\prime}$ of $G_{k}$, elements
$\overline{a}\in\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{X}$,
$\overline{s}\in\mathcal{S}_{G,\Pi^{(\mathrm{ab},p^{\prime})}_{X}}$, and
$\overline{s}^{\prime}\in\mathcal{S}^{\mathrm{geom}}_{G^{\prime},\Pi^{(\mathrm{ab},p^{\prime})}_{X}}$
such that $\overline{s}(G)$ and
$\overline{a}(\overline{s}^{\prime}(G^{\prime}))\overline{a}^{\prime}$ are
contained in the same decomposition groups of
$\Pi_{X}^{(\mathrm{ab},p^{\prime})}$. Since any decomposition groups of
$\Pi_{X}^{(\mathrm{ab},p^{\prime})}$ are isomorphic to
$G_{k}(\cong\hat{\mathbb{Z}})$), $\overline{s}(G)$ and
$\overline{a}(\overline{s}^{\prime}(G^{\prime}))\overline{a}^{\prime}$ are
commensurable in $\Pi_{X}^{(\mathrm{ab},p^{\prime})}$. Hence
$\mathfrak{s}\sim\mathfrak{s}^{\prime}$ holds.

Next, we assume that $g=0$. By replacing $G_{k}$ with an open subgroup
$\tilde{G}$ of $G_{k}$ satisfying $(G_{k}:\tilde{G})\geq r!$ if necessary, we
may assume that $E(\overline{k})=E(k)$. Let us assume that
$\mathfrak{s}\sim\mathfrak{s}^{\prime}$ and let $I$ and
$(G,s,G^{\prime},s^{\prime},a)$ be the inertia group and the multiple of
definition. Since $r\geq 5$, there exists an subset $\mathcal{I}$ of
$\mathrm{Iner}^{\neq\\{1\\}}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}})-\\{I,I_{\mathfrak{s}},I_{\mathfrak{s}^{\prime}}\\}$
whose cardinality is $2$. Let $\overline{s}$, $\overline{s}^{\prime}$,
$\overline{a}$ be the images of $s$, $s^{\prime}$, $a$ in
$\mathcal{S}^{\mathrm{geom}}_{G,\Pi_{U}^{(\mathrm{ab},p^{\prime})}/\langle
I^{\prime}\mid I^{\prime}\notin\mathcal{I}\rangle}$,
$\mathcal{S}^{\mathrm{geom}}_{G^{\prime},\Pi_{U}^{(\mathrm{ab},p^{\prime})}/\langle
I^{\prime}\mid I^{\prime}\notin\mathcal{I}\rangle}$,
$\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}/\langle I^{\prime}\mid
I^{\prime}\notin\mathcal{I}\rangle$, respectively. Then $\overline{s}(G)$ and
$\overline{a}(\overline{s}^{\prime}(G^{\prime}))\overline{a}^{\prime}$ are
commensurable in $\Pi_{U}^{(\mathrm{ab},p^{\prime})}/\langle I^{\prime}\mid
I^{\prime}\notin\mathcal{I}\rangle$. Hence Proposition 2.3a$\Leftrightarrow$e
implies that $\overline{s}(G)$ and
$\overline{a}(\overline{s}^{\prime}(G^{\prime}))\overline{a}^{\prime}$ are
contained in the same decomposition group of
$\Pi_{U}^{(\mathrm{ab},p^{\prime})}/\langle I^{\prime}\mid
I^{\prime}\notin\mathcal{I}\rangle$. (Here, we need to use the assumption that
the cardinality of $\mathcal{I}$ is $2$.) Since
$\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U})\xrightarrow{\sim}(\overline{\Pi}_{U}^{\mathrm{ab},p^{\prime}}/\langle
I^{\prime}\mid
I^{\prime}\notin\mathcal{I}\rangle)\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U}/\langle
I^{\prime}\mid
I^{\prime}\notin\mathcal{I}\rangle)(\xleftarrow{\sim}(X_{\overline{k}})^{\mathrm{cl}}$),
we obtain that $\rho(\mathfrak{s})=\rho(\mathfrak{s}^{\prime})$. Next, we
assume that $\rho(\mathfrak{s})=\rho(\mathfrak{s}^{\prime})$. This implies
that $I_{\mathfrak{s}}=I_{\mathfrak{s}^{\prime}}$. Let us take a subset
$\mathcal{I}\subset\mathrm{Iner}^{\neq\\{1\\}}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}})-\\{I_{\mathfrak{s}}\\}$
whose cardinality is $2$. Then there exists open subgroups $G$, $G^{\prime}$
of $G_{k}$, elements
$\overline{a}\in\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U}/\langle
I^{\prime}\mid I^{\prime}\notin\mathcal{I}\rangle)$,
$\overline{s}\in\mathcal{S}_{G,\Pi^{(\mathrm{ab},p^{\prime})}_{U}}/\langle
I^{\prime}\mid I^{\prime}\notin\mathcal{I}\rangle)$, and
$\overline{s}^{\prime}\in\mathcal{S}^{\mathrm{geom}}_{G^{\prime},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}/\langle
I^{\prime}\mid I^{\prime}\notin\mathcal{I}\rangle)$ such that
$\overline{s}(G)$ and
$\overline{a}(\overline{s}^{\prime}(G^{\prime}))\overline{a}^{\prime}$ are
contained in the same decomposition groups of
$\Pi_{U}^{(\mathrm{ab},p^{\prime})}/\langle I^{\prime}\mid
I^{\prime}\notin\mathcal{I}\rangle)$ that is not at cusps. Since any
decomposition groups of $\Pi_{U}^{(\mathrm{ab},p^{\prime})}/\langle
I^{\prime}\mid I^{\prime}\notin\mathcal{I}\rangle)$ that is not at cusps are
isomorphic to $G_{k}(\cong\hat{\mathbb{Z}})$), $\overline{s}(G)$ and
$\overline{a}(\overline{s}^{\prime}(G^{\prime}))\overline{a}^{\prime}$ are
commensurable in $\Pi_{U}^{(\mathrm{ab},p^{\prime})}/\langle I^{\prime}\mid
I^{\prime}\notin\mathcal{I}\rangle)$. Hence
$\mathfrak{s}\sim\mathfrak{s}^{\prime}$ holds. This completes the proof. ∎

######  Proposition 3.6.

Let $i=1$, $2$. Let $k_{i}$ be a finite field of characteristic $p_{i}$. Let
$(X_{i},E_{i})$ be a smooth curve of type $(g_{i},r_{i})$ over $k_{i}$, and we
set $U_{i}\overset{\mathrm{def}}{=}X_{i}-E_{i}$. Let $\Phi$ be an isomorphism
$\Pi^{(2,p_{1}^{\prime})}_{U_{1}}\xrightarrow{\sim}\Pi^{(2,p_{2}^{\prime})}_{U_{2}}$.
Let $\Phi^{1}$ be the isomorphism
$\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\xrightarrow{\sim}\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}$
induced by $\Phi$ (cf. Lemma 1.12). Assume that the $p_{1}$-rank of
$J_{X_{1}}$ is zero and that $r_{1}\geq 5$ when $g_{1}=0$. Then the following
hold:

  1. (1)

The isomorphism $\Phi^{1}$ preserves decomposition groups.

  2. (2)

If, moreover, $r\neq 1$, then $\Phi^{1}$ preserves decomposition groups at
cusps.

###### Proof..

(1) As Lemma 1.1679 imply that $g_{1}=g_{2}$, that $r_{1}=r_{2}$, that
$p_{1}=p_{2}$, and that $k_{1}=k_{2}$, we may write
$g\overset{\mathrm{def}}{=}g_{1}(=g_{2})$,
$r\overset{\mathrm{def}}{=}r_{1}(=r_{2})$,
$p\overset{\mathrm{def}}{=}p_{1}(=p_{2})$, and
$k\overset{\mathrm{def}}{=}k_{1}(=k_{2})$. We have that the $p$-rank of
$J_{X_{1}}$ coincides with the $p$-rank of $J_{X_{2}}$ by Lemma 1.110. By
Lemma 3.5, we have that the relation $\sim$ in Definition 3.4 induces an
isomorphism
$(\mathcal{QS}^{\mathrm{geom}}_{\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}}}/\sim)\xrightarrow{\sim}\overline{\Pi}_{U_{i}}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{i}})$.
By Lemma 1.158, Lemma 3.1, and Lemma 3.5, this isomorphism induces an
isomorphism
$\overline{\Pi}_{U_{1}}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{1}})\xrightarrow{\sim}\overline{\Pi}_{U_{2}}^{\mathrm{ab},p^{\prime}}\backslash\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime})}_{U_{2}})$
that makes the diagram (3.5) commutative. Hence, by Lemma 3.3A$\Rightarrow$B,
$\Phi^{1}$ preserves decomposition groups. This completes the proof.

(2) By Lemma 1.18, $\Phi^{1}$ preserves inertia groups. Since the intersection
of a decomposition group and $\overline{\Pi}_{U_{i}}^{\mathrm{ab},p^{\prime}}$
is an inertia group, we can characterize whether a decomposition group is at
cusps or not. (When $r=1$, all inertia groups are trivial!) Thus, the
assertion follows from 1. ∎

We define two non-standard letters for the Isom-set.

######  Definition 3.7.

For any scheme $S_{i}$ and any scheme $T_{i}$ over $S_{i}$ ($i=1$, $2$), we
write $\mathrm{Isom}(T_{1}/S_{1},T_{2}/S_{2})$ for the following set:

$\left\\{(\tilde{F},F)\in\mathrm{Isom}_{\mathrm{Sch}}(T_{1},T_{2})\times\mathrm{Isom}_{\mathrm{Sch}}(S_{1},S_{2})\middle|\vbox{
\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.01633pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.23854pt\raise 6.61111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\tilde{F}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.01633pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-28.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.01633pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
40.03265pt\raise-28.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.27953pt\raise-33.2461pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{F}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
32.2785pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.2785pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{S_{2}}$}}}}}}}\ignorespaces}}}}\ignorespaces}\text{ is
commutative.}\right\\}$

For any group $M_{i}$ and any set $C_{i}$ with a group action
$M_{i}\curvearrowright C_{i}$ ($i=1$, $2$), we write
$\mathrm{Isom}(M_{1}\curvearrowright C_{1},M_{2}\curvearrowright C_{2})$ for
the following set:

$\left\\{(f,g)\in\mathrm{Isom}_{\mathrm{Grp}}(M_{1},M_{2})\times\mathrm{Isom}_{\mathrm{Set}}(C_{1},C_{2})\middle|\vbox{
\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.69238pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.1021pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 76.58894pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-28.81888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
76.58894pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
86.38477pt\raise-28.81888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Aut}(C_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
27.5469pt\raise-32.81888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{g\circ(-)\circ g^{-1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
65.69238pt\raise-39.31888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
65.69238pt\raise-39.31888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Aut}(C_{2})}$}}}}}}}\ignorespaces}}}}\ignorespaces}\text{
is commutative.}\right\\}$

If $(f,g)\in\mathrm{Isom}(M_{1}\curvearrowright C_{1},M_{2}\curvearrowright
C_{2})$, then we say that the isomorphism $f$ is compatible with the actions
of $M_{1}$, $M_{2}$ relative to the isomorphism $g$.

######  Remark 3.8.

Let $i$, $k_{i}$, $p_{i}$, $X_{i}$, $E_{i}$, $g_{i}$, $r_{i}$, and $U_{i}$ be
the same letters as in Proposition 3.6. Assume that the $p_{1}$-rank of
$J_{X_{1}}$ is zero, and that $r_{1}\geq 5$ when $g_{1}=0$. The separatedness
property of the decomposition groups (cf. Proposition 2.3) implies the
bijectivity of the natural map from
$\tilde{X}_{i}^{\mathrm{ab},p^{\prime}_{i}}$ to
$\mathrm{Dec}(\Pi^{(\mathrm{ab},p^{\prime}_{i})}_{U_{i}})$. Hence there exists
a bijective map between
$\mathrm{Isom}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\curvearrowright\tilde{X}_{1}^{\mathrm{ab},p_{1}^{\prime}},\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}\curvearrowright\tilde{X}_{2}^{\mathrm{ab},p_{2}^{\prime}})$
and
$\mathrm{Isom}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\curvearrowright\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}),\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}\curvearrowright\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}))$.
Moreover, Proposition 3.61 , induces a bijective map from
$\mathrm{Isom}(\Pi^{(2,p_{1}^{\prime})}_{U_{1}},\Pi^{(2,p_{2}^{\prime})}_{U_{2}})$
to
$\mathrm{Isom}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\curvearrowright\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}),\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}\curvearrowright\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}))$.
To summarize, we have the following maps:

$\displaystyle\mathrm{Isom}(\Pi^{(2,p_{1}^{\prime})}_{U_{1}},\Pi^{(2,p_{2}^{\prime})}_{U_{2}})$
$\displaystyle\rightarrow$
$\displaystyle\mathrm{Isom}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\curvearrowright\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}),\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}\curvearrowright\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}))$
$\displaystyle\xrightarrow{\sim}$
$\displaystyle\mathrm{Isom}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\curvearrowright\tilde{X}_{1}^{\mathrm{ab},p_{1}^{\prime}},\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}\curvearrowright\tilde{X}_{2}^{\mathrm{ab},p_{2}^{\prime}})$
$\displaystyle\to$ $\displaystyle\mathrm{Isom}(G_{k_{1}}\curvearrowright
X_{1,\overline{k}_{1}},G_{k_{2}}\curvearrowright X_{2,\overline{k}_{2}})$

When $r_{1}\neq 1$, the images of the first vertical arrow preserve
decomposition groups at cusps by Proposition 3.62, and hence it induce
elements of
$\mathrm{Isom}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\curvearrowright\tilde{U}_{1}^{\mathrm{ab},p_{1}^{\prime}},\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}\curvearrowright\tilde{U}_{2}^{\mathrm{ab},p_{2}^{\prime}})$.
In the next section, we will prove that the images in
$\mathrm{Isom}(G_{k_{1}}\curvearrowright
U_{1,\overline{k}_{1}},G_{k_{2}}\curvearrowright U_{2,\overline{k}_{2}})$ of
$\mathrm{Isom}(\Pi^{(2,p_{1}^{\prime})}_{U_{1}},\Pi^{(2,p_{2}^{\prime})}_{U_{2}})$
are actually scheme-theoretic under suitable assumptions (cf. Theorem 4.5
below).

##  4 The weak bi-anabelian results over finite fields

Let $k$ be a finite field of characteristic $p>0$, $(X,E)$ a smooth curve of
type $(g,r)$ over $k$, and we set $U\overset{\mathrm{def}}{=}X-E$. In this
section, we show a result of the pro-prime-to-$p$ version of the $m$-step
solvable Grothendieck conjecture for affine hyperbolic curves of $p$-rank $0$
over finite fields (cf. Theorem 4.5 below). More precisely, we show that two
affine hyperbolic curves of $p$-rank $0$ are isomorphic as schemes if and only
if the geometrically maximal $2$-step solvable pro-prime-to-$p$ quotients of
their tame fundamental groups are isomorphic as profinite groups. If we
consider the geometrically maximal $2$-step solvable profinite quotients of
the tame fundamental groups, then the assertion was proved in [24, Theorem
2.16]. To show this, first we show a group-theoretical reconstruction of the
multiplicative group $K(U)^{\times}$ from $\Pi^{(2,p^{\prime})}_{U}$ if
$(g,r,m)$ satisfies certain conditions (cf. Lemma 4.3 below). The
reconstruction uses the decomposition groups of
$\Pi^{(\mathrm{ab},p^{\prime})}_{U}$ and contents of Section 3. Next, we deal
with the additivity of $K(U)$ by using [20, Lemma 4.7] (cf. the proof of
Theorem 4.5 below). Almost all objects in this section have functorial group-
theoretical algorithms from $\Pi^{(2,p^{\prime})}_{U}$. However, the author is
not aware that the additivity has a functorial group-theoretical algorithm
from $\Pi^{(2,p^{\prime})}_{U}$, and hence results in this section are given
only in “bi-anabelian geometry”. Finally, we obtain the result of the $m$-step
solvable pro-prime-to-$p$ version of the Grothendieck conjecture for affine
hyperbolic curves of $p$-rank $0$ over finite fields (cf. Theorem 4.5 below).
The key aspect of Theorem 4.5 below is that the pro-prime-to-$p$ version
result of the Grothendieck conjecture allows us to avoid the $m$-step solvable
good reduction criterion for hyperbolic curves (cf. [24, Theorem 3.2]) when we
prove results for fields finitely generated over prime fields (cf Remark 5.3
below.) This is an important difference between [24] and the present paper,
since we can improve [24, Theorems 2.16, 4.12] by this.

First of all, we show results in local theory. Let $v$ be a closed point of
$X$. Let$K(U)_{v}$ be the $v$-adic completion of $K(U)$,
$\mathrm{ord}_{v}:K(U)_{v}^{\times}\twoheadrightarrow\mathbb{Z}$ the
surjective valuation, $O_{v}\overset{\mathrm{def}}{=}\left\\{a\in
K(U)_{v}\mid\mathrm{ord}_{v}(a)\geq 0\right\\}$ the valuation ring of
$K(U)_{v}$, $m_{v}\overset{\mathrm{def}}{=}\left\\{a\in
K(U)_{v}\mid\mathrm{ord}_{v}(a)\geq 1\right\\}$ the maximal ideal of $O_{v}$,
$U_{v}^{(0)}\overset{\mathrm{def}}{=}O_{v}^{\times}(=\left\\{a\in
K(U)_{v}\mid\mathrm{ord}_{v}(a)=0\right\\})$ the multiplicative group of
$O_{v}$, and $U_{v}^{(n)}\overset{\mathrm{def}}{=}1+m_{v}^{n}$ the $n$-th
higher unit group of $K(U)_{v}$, where $n\in\mathbb{Z}_{\geq 1}$. We write
$F_{\tilde{v}^{1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$ for the inverse image
of the (discrete) subgroup $\langle\mathrm{Fr}_{\kappa(v)}\rangle$ in
$G_{\kappa(v)}$ by
$D^{\mathrm{ab}}_{\tilde{v}^{1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}\rightarrow
G_{\kappa(v)}$, where $\tilde{v}^{1}$ is a closed point of
$\tilde{X}^{\mathrm{ab},p^{\prime}}$ above $v$. The following lemma is well-
known, however, we will show it here for the sake of convenience.

######  Lemma 4.1.

Assume that $r\neq 1$, that the $p$-rank of $J_{X}$ is $0$, and that $U$ is
hyperbolic. Let $\tilde{v}$ be a closed point of $\tilde{X}$. Let
$\tilde{v}^{\prime}$, $\tilde{v}^{\prime 1}$, $v$ be the images of $\tilde{v}$
in $\tilde{X}^{p^{\prime}}$, $\tilde{X}^{\mathrm{ab},p^{\prime}}$, $X$,
respectively. Then the following hold:

  1. (1)

The following diagram is commutative:

$\textstyle{\widehat{K(U)_{v}^{\times}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ord}_{v}}$$\scriptstyle{\sim}$$\scriptstyle{\rho_{v}}$$\textstyle{G_{K(U)_{v}}^{\mathrm{ab}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{\mathrm{ab}}_{\tilde{v}^{\prime},{\Pi^{(p^{\prime})}_{U}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{D^{\mathrm{ab}}_{\tilde{v}^{\prime
1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hat{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{G_{\kappa(v)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{\kappa(v)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{\kappa(v)}}$
(4.1)

Here, the upper left-side horizontal arrow $\rho_{v}$ is the local reciprocity
isomorphism, and lower left-side horizontal arrow $\hat{\mathbb{Z}}\to
G_{\kappa(v)}$ is defined by the map that send $1$ to
$\mathrm{Fr}_{\kappa(v)}$. Moreover, the composite morphism
$\widehat{K(U)_{v}^{\times}}\twoheadrightarrow
D^{\mathrm{ab}}_{\tilde{v}^{\prime 1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$ of
the upper horizontal arrows induces an isomorphism
$K(U)_{v}^{\times}/U_{v}^{(0)}\xrightarrow{\sim}F_{\tilde{v}^{\prime
1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$ (resp.
$K(U)_{v}^{\times}/U_{v}^{(1)}\xrightarrow{\sim}F_{\tilde{v}^{1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$)
if $v\in U^{\mathrm{cl}}$ (resp. $v\in E$).

  2. (2)

Let $H^{{\dagger}}$, $H$ be open subgroups of
$\Pi^{(\mathrm{ab},p^{\prime})}_{U}$ with
$\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U}\subset H^{{\dagger}}\subset H$.
Let $\tilde{v}^{\prime 1}_{H}$, $\tilde{v}^{\prime 1}_{H^{{\dagger}}}$,
$v_{H}$, $v_{H^{{\dagger}}}$ are the images of $\tilde{v}$ in
$\tilde{X}^{\mathrm{ab},p^{\prime}}_{H}$,
$\tilde{X}^{\mathrm{ab},p^{\prime}}_{H^{{\dagger}}}$, $X_{H}$,
$X_{H^{{\dagger}}}$ respectively. Then the following diagram is commutative:

$\textstyle{K(U_{H^{{\dagger}}})_{v_{H^{{\dagger}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{\tilde{v}^{\prime
1}_{H^{{\dagger}}},(H^{{\dagger}})^{(\mathrm{ab},p^{\prime})}}}$$\textstyle{K(U_{H})_{v_{H}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{\tilde{v}^{\prime
1}_{H},H^{(\mathrm{ab},p^{\prime})}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here, the left-side vertical arrow is the natural inclusion and the right-side
vertical arrow is the restriction of the transfer morphism
$D^{\mathrm{ab}}_{\tilde{v}^{\prime 1}_{H},H^{(\mathrm{ab},p^{\prime})}}\to
D^{\mathrm{ab}}_{\tilde{v}^{\prime
1}_{H^{{\dagger}}},(H^{{\dagger}})^{(\mathrm{ab},p^{\prime})}}$ of
$D_{\tilde{v}^{\prime
1}_{H^{{\dagger}}},(H^{{\dagger}})^{(\mathrm{ab},p^{\prime})}}\to
D_{\tilde{v}^{\prime 1}_{H},H^{(\mathrm{ab},p^{\prime})}}$

###### Proof..

By the assumption of $r\neq 1$ and [24, Lemma 1.10], the natural projection
$D_{\tilde{v}^{\prime},\Pi^{(p^{\prime})}_{U}}\twoheadrightarrow
D_{\tilde{v}^{\prime 1},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$ is an
isomorphism, and hence the upper right-side horizontal arrow
$D^{\mathrm{ab}}_{\tilde{v}^{\prime},\Pi^{(p^{\prime})}_{U}}\twoheadrightarrow
D^{\mathrm{ab}}_{\tilde{v}^{\prime 1},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}$ in
(4.1) is also an isomorphism. Moreover, the natural surjection
$\Pi^{(\mathrm{ab})}_{U}\twoheadrightarrow\Pi^{(\mathrm{ab},p^{\prime})}_{U}$
is isomorphic, since the $p$-rank of $J_{X}$ is $0$, and
$D_{\tilde{v},\Pi_{U}}\twoheadrightarrow
D_{\tilde{v}^{1},\Pi_{U}^{(\mathrm{ab})}}$ is an isomorphism by the assumption
of $r\neq 1$ and [24, Lemma 1.10]. Therefore,
$D_{\tilde{v}^{\prime},\Pi^{(p^{\prime})}_{U}}^{\mathrm{ab}}\xrightarrow{\sim}D^{\mathrm{ab}}_{\tilde{v}^{\prime
1},\Pi_{U}^{(\mathrm{ab},p^{\prime})}}\xleftarrow{\sim}D^{\mathrm{ab}}_{\tilde{v}^{1},\Pi_{U}^{(\mathrm{ab})}}\xleftarrow{\sim}D^{\mathrm{ab}}_{\tilde{v},\Pi_{U}}$
hold.

(1) By the class field theory, the left-side square in (4.1) is commutative.
The middle and right-side squares are also commutative by definition. Hence
the commutativity of (4.1) holds. Let $\pi$ be an uniformizer of $K(U)_{v}$.
Then $\widehat{K(U)_{v}^{\times}}\simeq
U_{v}^{(0)}\times\pi^{\hat{\mathbb{Z}}}\simeq
U_{v}^{(1)}\times\mathbb{Z}/(p^{[\kappa(v):k]}-1)\mathbb{Z}\times\pi^{\hat{\mathbb{Z}}}$.
By the class field theory, the local reciprocity isomorphism $\rho_{v}$ maps
isomorphically $U^{(0)}_{v}$, $U_{v}^{(1)}$ to the inertia subgroup of
$G_{K(U)_{v}}^{\mathrm{ab}}$, wild inertia subgroup of
$G_{K(U)_{v}}^{\mathrm{ab}}$, respectively. Moreover, by the class field
theory, $\rho_{v}$ induces an isomorphism between $K(U)_{v}^{\times}$ and the
inverse image of $\langle\mathrm{Fr}_{\kappa(v)}\rangle$ by the map
$G_{K(U)_{v}}^{\mathrm{ab}}\to G_{\kappa(v)}$. Hence, by the commutativity of
(4.1) and the fact that $\Pi_{U}$ is the tame fundamental group of $U$, we
obtain that
$K(U)_{v}^{\times}/U_{v}^{(0)}\xrightarrow{\sim}F_{\tilde{v}^{\prime
1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$ (resp.
$K(U)_{v}^{\times}/U_{v}^{(1)}\xrightarrow{\sim}F_{\tilde{v}^{\prime
1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$) if $v\in U^{\mathrm{cl}}$ (resp.
$v\in E$).

(2) We have the following diagram:

$\textstyle{\widehat{K(U_{H^{{\dagger}}})^{\times}_{v_{H^{{\dagger}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\rho_{v_{H^{{\dagger}}}}}$$\textstyle{G_{K(U_{H^{{\dagger}}})_{v_{H^{{\dagger}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{\mathrm{ab}}_{\tilde{v}^{\prime
1}_{H^{{\dagger}}},(H^{{\dagger}})^{(\mathrm{ab},p^{\prime})}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{\kappa(v_{H^{{\dagger}}})}}$$\textstyle{\widehat{K(U_{H})^{\times}_{v_{H}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\rho_{v_{H}}}$$\textstyle{G_{K(U_{H})_{v_{H}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{transfer}}$$\textstyle{D^{\mathrm{ab}}_{\tilde{v}^{\prime
1}_{H},H^{(\mathrm{ab},p^{\prime})}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{transfer}}$$\textstyle{G_{\kappa(v_{H})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here, the right-side vertical arrow stands for the map that send
$\mathrm{Fr}_{\kappa(v_{H})}$ to
$\mathrm{Fr}_{\kappa(v_{H^{{\dagger}}})}^{e}$, where $e$ stands for the
ramification index of
$K(U_{H^{{\dagger}}})_{v_{H^{{\dagger}}}}/K(U_{H})_{v_{H}}$. Then the left-
side square is commutative by class field theory. The middle and right-side
square also commutative by definition. Hence the assertion follows from the
commutativity. ∎

We write $F_{v}$ instead of $F_{\tilde{v}^{\prime
1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$, since the group $F_{\tilde{v}^{\prime
1},\Pi^{(\mathrm{ab},p^{\prime})}_{U}}$ only depends on $v$ (up to
isomorphism) by Lemma 4.1(1). Next, we consider the global setting. Let
$\mathrm{ord}_{v}:K(U)^{\times}\twoheadrightarrow\mathbb{Z}$ be the unique
surjective valuation associated to $v$, $O_{X,v}:=\left\\{a\in
K(U)\mid\mathrm{ord}_{v}(a)\geq 0\right\\}$ the valuation ring of $K(U)$ at
$v$, and $m_{X,v}$ the maximal ideal of $O_{X,v}$. We have that
$\Gamma(U,O_{X})=\left\\{a\in K(U)\mid\mathrm{ord}_{v}(a)\geq 0\text{ for each
}v\in U^{\mathrm{cl}}\right\\}$.

######  Definition 4.2.

We define the group

$K(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\times}\overset{\mathrm{def}}{=}\mathrm{Ker}(\prod_{{v}\in
X^{\mathrm{cl}}}\
^{\prime}F_{v}\rightarrow(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\mathrm{ab}})$

and the composite morphism

$\widetilde{\mathrm{ord}}_{v}:K(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\times}\to
F_{v}\twoheadrightarrow F_{v}/\mathrm{Ker}(F_{v}\rightarrow G_{k})\
(=\langle\mathrm{Fr}_{\kappa(v)}\rangle),$

where $\prod_{v}^{\prime}F_{v}$ stands for the restricted direct product of
$F_{v}$ ($v\in X^{\mathrm{cl}}$) with respect to
$\mathrm{Ker}(F_{v}\rightarrow G_{k})$ that turns out to coincide with the
direct sum of $F_{v}$. We define the multiplicative monoid

$K(\Pi_{U}^{(\mathrm{ab},p^{\prime})})\overset{\mathrm{def}}{=}K(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\times}\cup\left\\{0\right\\}.$

######  Lemma 4.3.

Assume that $r\neq 1$, that the $p$-rank of $J_{X}$ is $0$, and that $U$ is
affine hyperbolic. Then the following hold:

  1. (1)

The morphism $K(U)^{\times}\to K(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\times}$
induced by the commutative diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(U)^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{A}_{K(U)}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{K(U)}^{\mathrm{ab}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{{v}\in
X^{\mathrm{cl}}}\
^{\prime}F_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\mathrm{ab}},}$
(4.2)

is an isomorphism, where $\mathbb{A}_{K(U)}^{\times}$ is the idele group of
$K(U)$ (i.e. the restricted direct product of $K(U)^{\times}_{v}$ ($v\in
X^{\mathrm{cl}}$) with respect to $U_{v}^{(0)}$).

  2. (2)

Let $v$ be a closed point of $X$. Then the following diagram is commutative:

$\textstyle{K(U)^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ord}_{v}}$

$\sim$

$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{K(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\mathrm{ord}}_{v}}$$\textstyle{F_{v}/\mathrm{Ker}(F_{v}\rightarrow
G_{k})}$ (4.3)

Here, the right-hand vertical arrow stands for the morphism
$\mathbb{Z}\rightarrow
G_{k};1\mapsto\mathrm{Fr}_{\kappa(v)}(=\mathrm{Fr}_{k}^{[G_{k}:G_{\kappa(v)}]})$.

  3. (3)

Let $v$ be a closed point of $U$ (resp. $E$). Then the isomorphism
$K(U)^{\times}\xrightarrow{\sim}K(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\times}$
in (1) maps $U_{v}^{(0)}$ (resp. $U_{v}^{(1)}$) to
$\mathrm{Ker}(K(\Pi_{U}^{(\mathrm{ab},p^{\prime})})^{\times}\rightarrow
F_{v})$.

  4. (4)

Let $H^{{\dagger}}$, $H$ be open subgroups of
$\Pi^{(\mathrm{ab},p^{\prime})}_{U}$ with
$\overline{\Pi}^{\mathrm{ab},p^{\prime}}_{U}\subset H^{{\dagger}}\subset H$.
Then the following diagram is commutative:

$\textstyle{K(U_{H^{{\dagger}}})^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{K((H^{{\dagger}})^{(\mathrm{ab},p^{\prime})})^{\times}}$$\textstyle{K(U_{H})^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{K(H^{(\mathrm{ab},p^{\prime})})^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(4.4)

Here, the horizontal arrows stands for the morphisms in (1) and the right-side
vertical arrow is defined as the restriction of the morphism $\prod_{{v}\in
X^{\mathrm{cl}}_{H}}\ ^{\prime}F_{v}\to\prod_{{v}\in
X^{\mathrm{cl}}_{H^{{\dagger}}}}\ ^{\prime}F_{v}$ induced by the transfer
morphism $F_{\tilde{v}^{\prime 1}_{H},H^{(\mathrm{ab},p^{\prime})}}.\to
F_{\tilde{v}^{\prime
1}_{H^{{\dagger}}},(H^{{\dagger}})^{(\mathrm{ab},p^{\prime})}}$. (Here,
$\tilde{v}^{\prime 1}_{H}$ is an inverse image of $v$ in
$\tilde{X}_{H}^{\mathrm{ab},p^{\prime}}$ and $\tilde{v}^{\prime
1}_{H^{{\dagger}}}$ is an inverse image of $\tilde{v}^{\prime 1}_{H}$ in
$\tilde{X}_{H^{{\dagger}}}^{\mathrm{ab},p^{\prime}}$.)

###### Proof..

(1) The assertion follows from global class field theory, Lemma 4.11, and
Definition 4.2. (Here, we use the assumption that $U$ is affine.)

(2) The assertion follows from 1 and Lemma 4.11.

(3) The assertion follows from 1 and Lemma 4.11.

(4) The assertion follows from 1 and Lemma 4.12 ∎

######  Definition 4.4.

For any scheme $S$ over $\mathbb{F}_{p}$. we define the absolute Frobenius
morphism $F_{S}:S\rightarrow S$ as the identity map on the underlying
topological space and the $p$-power endomorphism on the structure sheaf. For
any scheme $X$ over $S$, we have the following commutative diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{X}}$$\scriptstyle{F_{X/S}}$$\textstyle{X(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{S}}$$\textstyle{S}$

Here, $X(1)\overset{\mathrm{def}}{=}X\times_{S,F_{S}}S$ and we refer to it as
the Frobenius twist of $X$ over $S$. We shall write
$X(0)\overset{\mathrm{def}}{=}X$ and $X(n)\overset{\mathrm{def}}{=}X(n-1)(1)$
for any $n\in\mathbb{Z}_{\geq 0}$. The morphism $F_{X/S}:X\rightarrow X(1)$
induced by the universality of the fiber product is called the relative
Frobenius morphism of $X$ over $S$.

By the natural injection
$\mathrm{Isom}(U_{1,\overline{k}_{1}}/U_{1},U_{2,\overline{k}_{2}}/U_{2})\to\mathrm{Isom}(G_{k_{1}}\curvearrowright
X_{1,\overline{k}_{1}},G_{k_{2}}\curvearrowright X_{2,\overline{k}_{2}})$ (cf.
Definition 3.7), we identify
$\mathrm{Isom}(U_{1,\overline{k}_{1}}/U_{1},U_{2,\overline{k}_{2}}/U_{2})$ its
image. Then we obtain the main result of this section as follows:

######  Theorem 4.5 (Weak bi-anabelian result over finite fields).

Let $i=1$, $2$. Let $k_{i}$ be a finite field of characteristic $p_{i}$. Let
$(X_{i},E_{i})$ be a smooth curve of type $(g_{i},r_{i})$ over $k_{i}$, and we
set $U_{i}\overset{\mathrm{def}}{=}X_{i}-E_{i}$. Assume that the $p_{1}$-rank
of $J_{X_{1}}$ is zero, that $r_{1}\geq 3$, and that $r_{1}\geq 5$ when
$g_{1}=0$. Then the image of the composite map of

$\displaystyle\mathrm{Isom}(\Pi^{(2,p_{1}^{\prime})}_{U_{1}},\Pi^{(2,p_{2}^{\prime})}_{U_{2}})$
$\displaystyle\rightarrow$
$\displaystyle\mathrm{Isom}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\curvearrowright\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}),\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}\curvearrowright\mathrm{Dec}(\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}))$
$\displaystyle\xrightarrow{\sim}$
$\displaystyle\mathrm{Isom}(\Pi^{(\mathrm{ab},p_{1}^{\prime})}_{U_{1}}\curvearrowright\tilde{X}_{1}^{\mathrm{ab},p_{1}^{\prime}},\Pi^{(\mathrm{ab},p_{2}^{\prime})}_{U_{2}}\curvearrowright\tilde{X}_{2}^{\mathrm{ab},p_{2}^{\prime}})$
$\displaystyle\to$ $\displaystyle\mathrm{Isom}(G_{k_{1}}\curvearrowright
X_{1,\overline{k}_{1}},G_{k_{2}}\curvearrowright X_{2,\overline{k}_{2}})$

is contained in
$\mathrm{Isom}(U_{1,\overline{k}_{1}}/U_{1},U_{2,\overline{k}_{2}}/U_{2})$.
Here, the first arrow is the map induced by Proposition 3.6 and the second
arrow is the inverse map of the bijection induced by Proposition 2.3 (cf.
Remark 3.8).

###### Proof..

If
$\mathrm{Isom}(\Pi^{(2,p_{1}^{\prime})}_{U_{1}},\Pi^{(2,p_{2}^{\prime})}_{U_{2}})=\emptyset$,
then the assertion clearly holds. Hence we may assume that
$\mathrm{Isom}(\Pi^{(2,p_{1}^{\prime})}_{U_{1}},\Pi^{(2,p_{2}^{\prime})}_{U_{2}})\neq\emptyset$.
Let
$\Phi\in\mathrm{Isom}(\Pi^{(2,p_{1}^{\prime})}_{U_{1}},\Pi^{(2,p_{2}^{\prime})}_{U_{2}})$.
As Lemma 1.1679 imply that $g_{1}=g_{2}$, that $r_{1}=r_{2}$, that
$p_{1}=p_{2}$, and that $k_{1}=k_{2}$, we may write
$g\overset{\mathrm{def}}{=}g_{1}(=g_{2})$,
$r\overset{\mathrm{def}}{=}r_{1}(=r_{2})$,
$p\overset{\mathrm{def}}{=}p_{1}(=p_{2})$, and
$k\overset{\mathrm{def}}{=}k_{1}(=k_{2})$. (Note that $U_{1}$ and $U_{2}$ are
hyperbolic by the assumption of $(g,r)$.) We have that the $p$-rank of
$J_{X_{1}}$ coincides with the $p$-rank of $J_{X_{2}}$ by Lemma 1.110.

Since “the $p_{1}$-rank of $J_{X_{1}}$ is zero” and “$r_{1}\geq 5$ when
$g_{1}=0$”, we obtain the first and second arrows by Proposition 3.6,
Proposition 2.3 and Remark 2.4, respectively. In other words, $\Phi$ induces
an element
$\tilde{f}_{\Phi}\in\mathrm{Isom}_{\mathrm{Set}}(\tilde{X}_{1}^{\mathrm{ab},p^{\prime}},\tilde{X}_{1}^{\mathrm{ab},p^{\prime}})$
such that the following diagram is commutative:

$\textstyle{\Pi^{(\mathrm{ab},p^{\prime})}_{U_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\Phi}}$$\textstyle{\Pi^{(\mathrm{ab},p^{\prime})}_{U_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Aut}_{\mathrm{Set}}(\tilde{X}_{1}^{\mathrm{ab},p^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}_{\Phi}\circ(-)\circ\tilde{f}_{\Phi}^{-1}}$$\textstyle{\mathrm{Aut}_{\mathrm{Set}}(\tilde{X}_{2}^{\mathrm{ab},p^{\prime}})}$
(4.5)

Let $f_{\Phi}:X_{1,\overline{k}_{1}}\to X_{2,\overline{k}_{2}}$ be the set-
isomorphism induced by $\tilde{f}_{\Phi}$. By Lemma 1.14, Proposition 3.6, and
Lemma 4.31, $\Phi$ induces a multiplicative-monoid-isomorphism
$K(\Pi_{U_{1}}^{(\mathrm{ab},p^{\prime})})\to
K(\Pi_{U_{2}}^{(\mathrm{ab},p^{\prime})})$. Let $L$, $L^{\prime}$ be finite
extensions of $k$ with $L\subset L^{\prime}$. Then Lemma 4.34 implies the
commutativity of the diagram:

$\textstyle{K(\Pi_{U_{1,L^{\prime}}}^{(\mathrm{ab},p^{\prime})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(\Pi_{U_{2,\rho(L^{\prime})}}^{(\mathrm{ab},p^{\prime})})}$$\textstyle{K(\Pi_{U_{1,L}}^{(\mathrm{ab},p^{\prime})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(\Pi_{U_{2,\rho(L)}}^{(\mathrm{ab},p^{\prime})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here, $\rho$ stands for the set-automorphism of finite extensions of $k$
induced by $\Phi^{0}$ (cf. Lemma 1.12). Hence we obtain the multiplicative-
monoid-isomorphism

$K(U_{1,\overline{k}})=\varinjlim_{L_{1}}K(\Pi_{U_{1,L_{1}}}^{(\mathrm{ab},p^{\prime})})\rightarrow\varinjlim_{L_{2}}K(\Pi_{U_{2,L_{2}}}^{(\mathrm{ab},p^{\prime})})=K(U_{2,\overline{k}})$
(4.6)

by taking projective limits, where $L_{i}$ runs over all finite extensions of
$k$. Moreover, the assumption “$r\geq 3$”, Lemma 4.323, and [20, Lemma 4.7]
(or [24, Lemma 2.15]) imply that the multiplicative-monoid-isomorphism (4.6)
is additive. Hence $\Phi$ induces a scheme-isomorphism
$\tilde{x}:X_{1,\overline{k}_{1}}\xrightarrow{\sim}X_{2,\overline{k}_{2}}$.
Since
$K(\Pi_{U_{1}}^{(\mathrm{ab},p^{\prime})})\xrightarrow{\sim}K(\Pi_{U_{2}}^{(\mathrm{ab},p^{\prime})})$
is a restriction of (4.6), we also obtain a scheme-isomorphism
$x:X_{1}\xrightarrow{\sim}X_{2}$. By Lemma 4.323, we obtain the scheme-
isomorphisms
$\tilde{u}:U_{1,\overline{k}_{1}}\xrightarrow{\sim}U_{2,\overline{k}_{2}}$,
$u:U_{1}\xrightarrow{\sim}U_{2}$, by restricting $\tilde{x}$, $x$ on
$U_{1,\overline{k}_{1}}$, $U_{1}$, respectively. By the construction,
$\tilde{x}$, $x$ and $\tilde{f}_{\Phi}$, $f_{\Phi}$ coincide on the closed
points of $X_{1,\overline{k}_{1}}$, $X_{1}$, respectively (cf. Remark 3.8).
Hence the image $(\Phi^{0},f_{\Phi})$ of $\Phi$ in
$\mathrm{Isom}(G_{k_{1}}\curvearrowright
X_{1,\overline{k}_{1}},G_{k_{2}}\curvearrowright X_{2,\overline{k}_{2}})$
coincides with $(\tilde{u},u)$. Therefore, the assertion holds. ∎

######  Corollary 4.6.

Let $i$, $k_{i}$, $p_{i}$, and $U_{i}$ be the same letters as in Theorem 4.5.
We assume the same assumptions in Theorem 4.5. Then the following hold:

  1. (1)

The following holds:

$U_{1}\xrightarrow[\mathrm{scheme}]{\sim}U_{2}\iff\Pi_{U_{1}}^{(2,p_{1}^{\prime})}\xrightarrow[]{\sim}\Pi^{(2,p_{2}^{\prime})}_{U_{2}}$

  2. (2)

Assume additionally that $k_{1}=k_{2}$, and we set $k=k_{1}(=k_{2})$. Then
there exists integer $n$ such that the following holds:

$U_{1}(n)\xrightarrow[k]{\sim}U_{2}\iff\Pi_{U_{1}}^{(2,p_{1}^{\prime})}\xrightarrow[G_{k}]{\sim}\Pi^{(2,p_{2}^{\prime})}_{U_{2}}$

###### Proof..

The assertion (1) follows from Theorem 4.5. We show (2). By (1), the right-
side isomorphism induces a scheme-isomorphism $U_{1}\xrightarrow{\sim}U_{2}$
and this induces an isomorphism $\mathrm{Fr}_{k}^{n}:k\xrightarrow{\sim}k$ for
some $n$. Hence, the composite morphism of
$U_{1}(n)\xrightarrow{\sim}U_{1}\xrightarrow{\sim}U_{2}$ is a $k$-isomorphism,
where the first isomorphism is defined by the fiber product of
$\mathrm{Fr}^{n}_{\mathrm{Spec}(k)}$ and structure morphism
$U_{1}\to\mathrm{Spec}(k)$. (Note that $k$ is perfect.) Therefore, (2) holds.
∎

######  Corollary 4.7.

Let $i$, $k_{i}$, $p_{i}$, $X_{i}$, $E_{i}$, $g_{i}$, $r_{i}$, and $U_{i}$ be
the same letters as in Theorem 4.5. We assume the same assumptions in Theorem
4.5. Then the following diagram is commutative:

$\textstyle{\mathrm{Isom}(\Pi^{(2,p_{1}^{\prime})}_{U_{1}},\Pi^{(2,p_{2}^{\prime})}_{U_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\mathcal{F}}}$$\textstyle{\mathrm{Isom}(U_{1,\overline{k}_{1}},U_{2,\overline{k}_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Isom}(\overline{\Pi}_{U_{1}}^{p^{\prime}_{1}},\overline{\Pi}^{p^{\prime}_{2}}_{U_{2}})/\mathrm{Inn}(\overline{\Pi}^{p^{\prime}_{2}}_{U_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Isom}(\overline{\Pi}_{U_{1}}^{\mathrm{ab},p_{1}^{\prime}},\overline{\Pi}^{\mathrm{ab},p^{\prime}_{2}}_{U_{2}})}$

Here, the lower horizontal arrows are the natural maps,
$\overline{\mathcal{F}}$ stands for the isomorphism induced by the proof of
Theorem 4.5, and the right-vertical arrow stands for the map induced by Lemma
1.12.

###### Proof..

Let $L$ be a finite extension of $k_{1}$ and $\rho$ the automorphism of the
set of finite extensions of $k_{1}$ induced by $\Phi^{0}$ (cf. Lemma 1.12). We
have the following maps:

$\displaystyle\mathrm{Isom}(U_{1,L},U_{2,\rho(L)})$ $\displaystyle\to$
$\displaystyle\mathrm{Isom}(\Pi_{U_{1,L}}^{(\mathrm{ab},p_{1}^{\prime})},\Pi^{(\mathrm{ab},p^{\prime}_{2})}_{U_{2,\rho(L)}})/\mathrm{Inn}(\Pi^{(\mathrm{ab},p^{\prime}_{2})}_{U_{2,\rho(L)}})$
$\displaystyle\to$
$\displaystyle\mathrm{Isom}((\Pi_{U_{1,L}}^{(p_{1}^{\prime})})^{\mathrm{ab}},(\Pi_{U_{2,\rho(L)}}^{(p_{2}^{\prime})})^{\mathrm{ab}})$

and
$\overline{\Pi}^{\mathrm{ab},p^{\prime}_{i}}_{U_{i}}\xrightarrow{\sim}(\varprojlim_{L}\Pi^{(\mathrm{ab},p^{\prime}_{i})}_{U_{i,L}})^{\mathrm{ab}}\xrightarrow{\sim}\varprojlim_{L}(\Pi_{U_{i,L}}^{(p^{\prime}_{i})})^{\mathrm{ab}}$.
Let $\Phi$ be an element of
$\mathrm{Isom}(\Pi^{(2,p_{1}^{\prime})}_{U_{1}},\Pi^{(2,p_{2}^{\prime})}_{U_{2}})$.
By the proof of Theorem 4.5, we get an element $\mathcal{F}_{L}(\Phi)$ of
$\mathrm{Isom}(U_{1,L},U_{2,\rho(L)})$ and the following diagram is
commutative by Lemma 4.34:

$\textstyle{U_{1,\overline{k}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\mathcal{F}}(\Phi)}$$\textstyle{U_{1,\overline{k}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{1,L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}_{L}(\Phi)}$$\textstyle{U_{1,\rho(L)}}$

Hence it is sufficient to show that $\Phi$ and $\mathcal{F}_{k}(\Phi)$ induce
the same element of
$\mathrm{Isom}((\Pi_{U_{1}}^{(p_{1}^{\prime})})^{\mathrm{ab}},(\Pi_{U_{2}}^{(p_{2}^{\prime})})^{\mathrm{ab}})$.
We write $\Phi^{\mathrm{ab}}$, $f_{k}(\Phi)$ for the images of $\Phi$,
$\mathcal{F}_{k}(\Phi)$ in
$\mathrm{Isom}((\Pi_{U_{1}}^{(p_{1}^{\prime})})^{\mathrm{ab}},(\Pi_{U_{2}}^{(p_{2}^{\prime})})^{\mathrm{ab}})$,
receptively. By Theorem 4.5, $\Phi$ and $f_{k}(\Phi)$ induce the same element
of $\mathrm{Isom}(G_{k_{1}}\curvearrowright
X_{1,\overline{k}_{1}},G_{k_{2}}\curvearrowright X_{2,\overline{k}_{2}})$. As
$\Phi^{\mathrm{ab}}$ preserves decomposition groups, it implies
$\Phi^{\mathrm{ab}}(D_{v})=f_{k}(\Phi)(D_{v})$ for each $v\in
U^{\mathrm{cl}}_{1}$, where $D_{v}$ is the decomposition group of
$(\Pi_{U_{1}}^{(p_{1}^{\prime})})^{\mathrm{ab}}$. We set
$s_{v}:G_{\kappa(v)}\xrightarrow{\sim}D_{v}$. By Lemma 1.14, we get
$\Phi^{\mathrm{ab}}(s_{v}(\mathrm{Fr}_{\kappa(v)}))=f_{k}(\Phi)(s_{v}(\mathrm{Fr}_{\kappa(v)}))$,
where $\mathrm{Fr}_{\kappa(v)}$ is the Frobenius element of $G_{\kappa(v)}$.
As we have $(\Pi_{U_{1}}^{(p_{1}^{\prime})})^{\mathrm{ab}}=\overline{\langle
s_{v}(\mathrm{Fr}_{\kappa(v)})\ |\ v\in U_{1}^{\mathrm{cl}}\rangle}$ by
Chebotarev’s density theorem, it implies $\Phi^{\mathrm{ab}}=f_{k}(\Phi)$.
Thus, the assertion holds. ∎

##  5 Specialization

In this section, we introduce the specialization morphism (cf. (5.2) below) of
smooth curves over henselian regular local rings. Since the specialization
morphism connects smooth curves over finitely generated fields and smooth
curves over finite fields, by using results of this section, we can generalize
Theorem 4.5 to the $m$-step solvable Grothendieck conjecture for hyperbolic
curves over finitely generated fields (cf. Proposition 5.5 below).

First, we consider the specialization morphism. Let $R$ be a henselian regular
local ring with the field of fractions $K$, $s\in\mathrm{Spec}(R)$ the closed
point, and $\eta\in\mathrm{Spec}(R)$ the generic point. Let
$(\mathfrak{X},\mathfrak{E})$ be a $smooth$ $curve$ of type $(g,r)$ over
$\mathrm{Spec}(R)$ and we set
$\mathfrak{U}\overset{\mathrm{def}}{=}\mathfrak{X}-\mathfrak{E}$.

######  Lemma 5.1.

In the above notation, the following sequence is exact for any point
$x\in\mathrm{Spec}(R)$:

$\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{x}},\overline{x}^{\prime})\to\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U},\overline{x}^{\prime})\to\pi_{1}^{\acute{e}\mathrm{t}}(\mathrm{Spec}(R),\overline{x})\to
1$

Here, $\overline{x}$ stands for a geometric point of $\mathrm{Spec}(R)$ over
$x$, $\overline{x}^{\prime}$ stands for a geometric points of
$\mathfrak{U}_{\overline{x}}$ over $\overline{x}$, and we use same notation
for the geometric point of $\mathfrak{U}$ induced by $\overline{x}^{\prime}$.

###### Proof..

Since
$\mathrm{Ker}(\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{x}},\overline{x}^{\prime})\twoheadrightarrow\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{X}_{\overline{x}},\overline{x}^{\prime}))$
and
$\mathrm{Ker}(\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U},\overline{x}^{\prime})\twoheadrightarrow\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{X},\overline{x}^{\prime}))$
are generated by the inertia subgroups of
$\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{x}},\overline{x}^{\prime})$,
$\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U},\overline{x}^{\prime})$,
respectively, the natural morphism
$\mathrm{Ker}(\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{x}},\overline{x}^{\prime})\twoheadrightarrow\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{X}_{\overline{x}},\overline{x}^{\prime}))\twoheadrightarrow\mathrm{Ker}(\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U},\overline{x}^{\prime})\to\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{X},\overline{x}^{\prime}))$
is surjective. Therefore, we may assume that $\mathfrak{U}$ is proper (i.e.,
$\mathfrak{U}=\mathfrak{X}$). Let $\mathfrak{X}\to
S^{\prime}\to\text{Spec}(R)$ be the stein factorization of the structure
morphism $\mathrm{pr}:\mathfrak{X}\to\text{Spec}(R)$. The smoothness of
$\mathrm{pr}$ implies that $\mathrm{pr}$ is flat, with geometrically reduced
fibers. Moreover, $\mathrm{pr}$ has geometrically connected fibers by the
definition of smooth curves. Therefore, $S^{\prime}\to\text{Spec}(R)$ is an
isomorphism, since it is finite, étale and each geometric fibers are reduced
and connected. Hence
$O_{\text{Spec}(R)}\xrightarrow{\sim}\mathrm{pr}_{*}(O_{\mathfrak{X}})$ holds.
Thus, the exactness of the assertion follows from [6, Exposé X Corollaire
1.4]. ∎

By the above lemma, we obtain the following commutative diagram of exact
sequences:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{\eta}},\overline{\eta}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\eta},\overline{\eta}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
G_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{\eta}},\overline{\eta}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U},\overline{\eta}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathrm{Spec}(R),\overline{\eta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{1}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{s}},\overline{s}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U},\overline{s}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathrm{Spec}(R),\overline{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{s}},\overline{s}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{s},\overline{s}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{G_{\kappa(s)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{1}$ (5.1)

Here, $\overline{\eta}$, $\overline{s}$ are geometric points over $\eta$, $s$,
respectively, $\overline{\eta}^{\prime}$, $\overline{s}^{\prime}$ are
geometric points of $\mathfrak{U}_{\overline{\eta}}$,
$\mathfrak{U}_{\overline{s}}$ that is on $\overline{\eta}$, $\overline{s}$,
respectively, $\kappa(s)$ stands for the residue field at $s$ of
characteristic $p_{s}$ $(\geq 0)$, and we use the same notation for the
geometric points of $\mathfrak{U}$ induced by $\overline{\eta}^{\prime}$,
$\overline{s}^{\prime}$. Note that the vertical arrows from the second line to
the third line are inner automorphisms defined by the path between
$\overline{\eta}^{\prime}$ and $\overline{s}^{\prime}$. By [6, Exposé X
Corollaire 3.9], the specialization morphism

$\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{\eta}},\overline{\eta}^{\prime})^{p_{s}^{\prime}}\to\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{s}},\overline{s}^{\prime})^{p_{s}^{\prime}}$
(5.2)

induced by (5.1) is an isomorphism. Therefore, by taking the geometrically
maximal $m$-step solvable and pro-prime-to-$p_{s}^{\prime}$ quotient of (5.1),
we obtain the following commutative diagram of exact sequences:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{\eta}},\overline{\eta}^{\prime})^{m,p_{s}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\eta},\overline{\eta}^{\prime})^{(m,p_{s}^{\prime})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\rho_{\mathfrak{U}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
G_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{\eta}},\overline{\eta}^{\prime})^{m,p_{s}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U},\overline{\eta}^{\prime})^{(m,p_{s}^{\prime})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathrm{Spec}(R),\overline{\eta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{s}},\overline{s}^{\prime})^{m,p_{s}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U},\overline{s}^{\prime})^{(m,p_{s}^{\prime})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathrm{Spec}(R),\overline{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{\overline{s}},\overline{s}^{\prime})^{m,p_{s}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\acute{e}\mathrm{t}}(\mathfrak{U}_{s},\overline{s}^{\prime})^{(m,p_{s}^{\prime})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{G_{\kappa(s)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\sim$

$\textstyle{1}$ (5.3)

Here, we define