# Local sections of arithmetic fundamental groups of $p$-adic curves

Mohamed Saïdi

We investigate sections of the arithmetic fundamental group $\pi_{1}(X)$ where
$X$ is either a smooth affinoid $p$-adic curve, or a formal germ of a $p$-adic
curve, and prove that they can be lifted (unconditionally) to sections of
cuspidally abelian Galois groups. As a consequence, if $X$ admits a
compactification $Y$, and the exact sequence of $\pi_{1}(X)$ splits, then
$\text{index}(Y)=1$. We also exhibit a necessary and sufficient condition for
a section of $\pi_{1}(X)$ to arise from a rational point of $Y$. One of the
key ingredients in our investigation is the fact, we prove in this paper in
case $X$ is affinoid, that the Picard group of $X$ is finite.

## §0. Introduction/Main results

This paper is motivated by the $p$-adic analog of the anabelian Grothendieck
section conjecture.

Let $p\geq 2$ be a prime number, $k/\mathbb{Q}_{p}$ a finite extension, and
$Y$ a proper, smooth, and geometrically connected hyperbolic $k$-curve. The
arithmetic fundamental group $\pi_{1}(Y)$ of $Y$ projects onto the Galois
group $G_{k}\overset\text{def}\to{=}\operatorname{Gal}(\bar{k}/k)$ of $k$. A
$k$-rational point $x:\operatorname{Spec}k\to Y$ gives rise, by functoriality
of fundamental groups, to a section $s_{x}:G_{k}\to\pi_{1}(Y)$ of the
projection $\pi_{1}(Y)\twoheadrightarrow G_{k}$. We shall refer to such a
section $s_{x}$ as geometric.

Question A. Is every section of the projection $\pi_{1}(Y)\twoheadrightarrow
G_{k}$ geometric?

In [Saïdi2], Theorem 2 in the Introduction, we established two necessary and
sufficient conditions for a group-theoretic section of the projection
$\pi_{1}(Y)\twoheadrightarrow G_{k}$ to be geometric. In [Hoshi] Hoshi
constructed a group-theoretic section $G_{k}\to\pi_{1}(Y)^{(p)}$ of the
projection $\pi_{1}(Y)^{(p)}\twoheadrightarrow G_{k}$ for a specific example
$Y$, where $\pi_{1}(Y)^{(p)}$ is the geometrically pro-$p$ quotient of
$\pi_{1}(Y)$, which is not geometric (i.e., doesn’t arise from a scheme
morphism $x:\operatorname{Spec}k\to Y$). The author is not aware of any
example of a $Y$ as above and a group-theoretic section of the projection
$\pi_{1}(Y)\twoheadrightarrow G_{k}$ which is not geometric.

Let $X$ be either a geometrically connected affinoid subspace of
$Y^{\operatorname{rig}}$, the rigid analytic curve associated to $Y$, or a
formal germ of $Y$ meaning
$X=\operatorname{Spec}(\hat{\Cal{O}}_{\Cal{Y},y}\otimes_{\Cal{O}_{k}}k)$ is
geometrically connected, where $\hat{\Cal{O}}_{\Cal{Y},y}$ is the completion
of the local ring $\Cal{O}_{\Cal{Y},y}$ of a model $\Cal{Y}$ of $Y$ over the
ring of valuation $\Cal{O}_{k}$ of $k$ at a closed point
$y\in\Cal{Y}^{\operatorname{cl}}$ (cf. Notations). Let $\pi_{1}(X)$ be the
étale fundamental group of $X$ which sits in the exact sequence (cf.
Notations)

$1\to\pi_{1}(X)^{\operatorname{geo}}\to\pi_{1}(X)\to
G_{k}\overset\text{def}\to{=}\operatorname{Gal}(\bar{k}/k)\to 1.$

A section $s:G_{k}\to\pi_{1}(X)$ of the projection
$\pi_{1}(X)\twoheadrightarrow G_{k}$ induces a section
$s_{Y}:G_{k}\to\pi_{1}(Y)$ of the projection $\pi_{1}(Y)\twoheadrightarrow
G_{k}$ [cf. Notations, diagram (0.1)] which we shall refer to as a local
section of the projection $\pi_{1}(Y)\twoheadrightarrow G_{k}$. A geometric
section is necessarily a local section as one easily verifies. This prompts
the following question, which motivates our study in this paper of local
sections of arithmetic fundamental groups of $p$-adic curves.

Question B. Is every local section of the projection
$\pi_{1}(Y)\twoheadrightarrow G_{k}$ geometric?

Motivated by Questions A and B we investigate sections of arithmetic
fundamental groups of affinoid $k$-curves and formal $p$-adic germs of curves.

Let $X$ be either a smooth and geometrically connected $k$-affinoid curve or a
formal $p$-adic germ (cf. Notations for precise definitions). Let
$\pi_{1}(X)^{\operatorname{geo},\operatorname{ab}}$ be the maximal abelian
quotient of $\pi_{1}(X)^{\operatorname{geo}}$ and
$\pi_{1}(X)^{(\operatorname{ab})}$ the geometrically abelian quotient of
$\pi_{1}(X)$ which sits in the exact sequence

$1\to\pi_{1}(X)^{\operatorname{geo},\operatorname{ab}}\to\pi_{1}(X)^{(\operatorname{ab})}\to
G_{k}\to 1.$

Similarly let
$G_{X}\overset\text{def}\to{=}\operatorname{Gal}(\overline{L}/L)$ be the
absolute Galois group of the function field $L$ of $X$ (see Notations for the
definition of $L$) which sits in the exact sequence (cf. $\S 1$)

$1\to G_{X}^{\operatorname{geo}}\to G_{X}\to G_{k}\to 1.$

Let $G_{X}^{\operatorname{geo},\operatorname{ab}}$ be the maximal abelian
quotient of $G_{X}^{\operatorname{geo}}$ and $G_{X}^{(\operatorname{ab})}$ the
geometrically abelian quotient of $G_{X}$ which sits in the exact sequence

$1\to G_{X}^{\operatorname{geo},\operatorname{ab}}\to
G_{X}^{(\operatorname{ab})}\to G_{k}\to 1.$

We have an exact sequence

$1\to\widetilde{\Cal{H}}_{X}\to
G_{X}^{(\operatorname{ab})}\to\pi_{1}(X)^{(\operatorname{ab})}\to 1$

where
$\widetilde{\Cal{H}}_{X}\overset\text{def}\to{=}\operatorname{Ker}[G_{X}^{(\operatorname{ab})}\twoheadrightarrow\pi_{1}(X)^{(\operatorname{ab})}]$.
In $\S 1$ we investigate the structure of the $G_{k}$-module
$\widetilde{\Cal{H}}_{X}$. We prove in Proposition 1.4 that
$\widetilde{\Cal{H}}_{X}$ is (canonically) isomorphic to $\prod_{x\in
X^{\operatorname{cl}}}\operatorname{Ind}_{k(x)}^{k}\hat{\mathbb{Z}}(1)$ where
the product is over all closed points of $X$ and $k(x)$ is the residue field
at $x$ (cf. loc. cit.).

The Galois group $G_{X}$ sits in an exact sequence

$1\to{\Cal{H}}_{X}\to G_{X}\to\pi_{1}(X)\to 1$

where
$\Cal{H}_{X}\overset\text{def}\to{=}\operatorname{Ker}[G_{X}\twoheadrightarrow\pi_{1}(X)]$.
Let $\Cal{H}_{X}^{\operatorname{ab}}$ be the maximal abelian quotient of
$\Cal{H}_{X}$ and $G_{X}^{(\text{c}-\operatorname{ab})}$ the geometrically
cuspidally abelian quotient of $G_{X}$ which sits in the exact sequence (cf.
loc. cit.)

$1\to\Cal{H}_{X}^{\operatorname{ab}}\to
G_{X}^{(\text{c}-\operatorname{ab})}\to\pi_{1}(X)\to 1.$

In $\S 2$ we investigate, in the framework of the theory of cuspidalisation of
sections of arithmetic fundamental groups (cf. [Saïdi1] and [Saïdi2]),
sections $s:G_{k}\to\pi_{1}(X)$ of the projection
$\pi_{1}(X)\twoheadrightarrow G_{k}$. Let $Y$ be a $k$-compactification of $X$
and $s_{Y}:G_{k}\to\pi_{1}(Y)$ the induced local section of the projection
$\pi_{1}(Y)\twoheadrightarrow G_{k}$ [cf. Notations for precise definitions
and the diagram (0.1) therein]. One of our main results is the following [cf.
Theorem 2.4 and Theorem 3.1(ii)].

###### Theorem A

(Lifting of sections to cuspidally abelian Galois groups). Let
$s:G_{k}\to\pi_{1}(X)$ be a section of the projection
$\pi_{1}(X)\twoheadrightarrow G_{k}$. The followings hold.

(i) There exists a section $s^{\text{c}-\operatorname{ab}}:G_{k}\to
G_{X}^{(\text{c}-\operatorname{ab})}$ of the projection
$G_{X}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which lifts the
section $s$, i.e., which inserts in the following commutative diagram

$\begin{CD}G_{k}@>{s^{\text{c}-\operatorname{ab}}}>{}>G_{X}^{(\text{c}-\operatorname{ab})}\\\
\Big{\|}@V{}V{}V\\\ G_{k}@>{s}>{}>\pi_{1}(X)\\\ \end{CD}$

where the right vertical map is the natural projection
$G_{X}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow\pi_{1}(X)$. In
particular, the set of sections of the projection
$G_{X}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which lift the
section $s$ is non-empty, and is (up to conjugation by elements of
$\Cal{H}_{X}^{\operatorname{ab}}$) a torsor under
$H^{1}(G_{k},\Cal{H}_{X}^{\operatorname{ab}})$.

(ii) Assume $Y$ is hyperbolic. Then the section $s_{Y}:G_{k}\to\pi_{1}(Y)$
induced by $s$ is uniformly orthogonal to $\operatorname{Pic}$ in the sense of
[Saïdi1], Definition 1.4.1.

The section $s$ is uniformly orthogonal to $\operatorname{Pic}$ (as in (ii)
above) means that the retraction map
$s^{*}:H^{2}(\pi_{1}(Y),\hat{\mathbb{Z}}(1))\overset\sim\to{\rightarrow}H^{2}_{\operatorname{et}}(Y,\hat{\mathbb{Z}}(1))\to
H^{2}(G_{k},\hat{\mathbb{Z}}(1))$, which is induced by the section $s$,
annihilates the Picard part of
$H^{2}_{\operatorname{et}}(Y,\hat{\mathbb{Z}}(1))$, and similarly for every
neighborhood $Y^{\prime}\to Y$ of the section $s$.

Theorem A(ii) implies that local sections of arithmetic fundamental groups of
hyperbolic $p$-adic curves satisfy condition (i) in [Saïdi2], Theorem 2 in the
Introduction. In this sense local sections are close to being geometric.
Establishing Theorem A(ii) was one of the main motivations for the author to
investigate local sections of arithmetic fundamental groups of $p$-adic
curves. Apart from local sections, and geometric sections, the author is not
aware (for the time being) of any examples of group-theoretic sections of
arithmetic fundamental groups of hyperbolic $p$-adic curves which are
orthogonal to $\operatorname{Pic}$.

As a consequence of Theorem A, and an observation of Esnault and Wittenberg on
geometrically abelian sections of $p$-adic curves, we deduce the following
(cf. Theorem 2.5).

###### Theorem B

Assume that $X$ admits a $k$-compatctification $Y$ (cf. Notations). If the
projection $\pi_{1}(X)\twoheadrightarrow G_{k}$ splits then
$\operatorname{index}(Y)=1$.

Theorem B asserts that the existence of local sections of arithmetic
fundamental groups of $p$-adic curves implies the existence of degree $1$
rational divisors. The link between sections of geometrically abelian Galois
groups and the existence of degree $1$ rational divisors has been investigated
in [Esnault-Wittenberg].

In $\S 3$ we assume that $X$ admits a $k$-compactification $Y$ (cf.
Notations). Let $\Pi_{Y}[X]$ be the étale fundamental group which classifies
finite covers $Y^{\prime}\to Y$ which only ramify at points of $Y$ not in $X$
(cf. 3.3, as well as Notations for the meaning of “not in $X$”). A section
$s:G_{k}\to\pi_{1}(X)$ of the projection $\pi_{1}(X)\twoheadrightarrow G_{k}$
induces naturally a section $s^{\dagger}:G_{k}\to\Pi_{Y}[X]$ of the projection
$\Pi_{Y}[X]\twoheadrightarrow G_{k}$. We say that the section $s$ is geometric
(relative to $Y$) if the image $s^{\dagger}(G_{k})$ is contained in a
decomposition group $D_{x}\subset\Pi_{Y}[X]$ associated to a rational point
$x\in Y(k)$ (cf. Definition 3.3.2). Further, we say that $s$ is admissible
(relative to $Y$) (cf. Definition 3.5.1) if for every open subgroup
$H\subset\Pi_{Y}[X]$ with $s^{\dagger}(G_{k})\subset H$, corresponding to (a
possibly ramified) cover $Y^{\prime}\to Y$, the following holds. Let
$G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ be the geometrically
cuspidally $1/p^{2}$-solvable Galois group of $Y^{\prime}$: i.e., the maximal
quotient $G_{Y^{\prime}}\twoheadrightarrow
H\twoheadrightarrow\pi_{1}(Y^{\prime})$ of the absolute Galois group
$G_{Y^{\prime}}$ of $Y^{\prime}$ such that
$\operatorname{Ker}[H\twoheadrightarrow\pi_{1}(Y^{\prime})]$ is abelian
annihilated by $p^{2}$ (cf. [Saïdi2], 3.1). There exists a section
$\tilde{s}_{Y^{\prime}}:G_{k}\to
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ of the projection
$G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow G_{k}$ [such
a section exists unconditionally (see discussion in 3.5)] satisfying the
following property:

For each open subgroup $F\subset
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ with
$\tilde{s}_{Y^{\prime}}(G_{k})\subset F$, corresponding to a (possibly
ramified) cover $Y^{\prime\prime}\to Y^{\prime}$ with $Y^{\prime\prime}$
geometrically connected, the class of
$\operatorname{Pic}^{1}_{Y^{\prime\prime}}$ in
$H^{1}(G_{k},\operatorname{Pic}^{0}_{Y^{\prime\prime}})$ is divisible by $p$.

Our main result in $\S 3$ is the following (cf. Theorem 3.5.2).

###### Theorem C

The section $s:G_{k}\to\pi_{1}(X)$ is geometric (relative to $Y$) if and only
if $s$ is admissible (relative to $Y$).

One of the key ingredients used in the proofs of the above results is the fact
that $\operatorname{Pic}(X)$ is finite. In the case where $X$ is a formal
$p$-adic germ this is established in [Saïdi2], Proposition 5.4, as a
consequence of a result of Shuji Saito (cf. loc. cit.). In case $X$ is
affinoid this is proven in $\S 4$ (cf. Proposition 4.1) and may be of interest
independently of the topics discussed in this paper. More precisely, we prove
the following.

###### Theorem D (Picard groups of affinoid $p$-adic curves)

Let $k$ be a $p$-adic local field (i.e., $k/\mathbb{Q}_{p}$ is a finite
extension) and $X=\operatorname{Sp}(A)$ a smooth and geometrically connected
$k$-affinoid curve. Then the Picard group $\operatorname{Pic}(X)$ is finite.

Finally in $\S 5$ we prove (cf. Proposition 5.1) a compactification result for
two dimensional complete local $p$-adic rings which is used in the proofs of
Proposition 1.2 and Proposition 2.2.

The results in $\S 4$ and $\S 5$ are used in this paper in $\S 2$ and $\S 3$,
none of the results in $\S 2$ and $\S 3$ is used in $\S 4$ or $\S 5$.

In this paper we worked with full arithmetic fundamental groups. Instead one
could consider a similar setting and work with geometrically pro-$p$
arithmetic fundamental groups and Galois groups as in [Saïdi2] (where one
considers geometrically pro-$\Sigma$ arithmetic fundamental groups and Galois
groups, $\Sigma$ being a set of primes containing $p$). In this geometrically
pro-$p$ (pro-$\Sigma$) setting one can prove analogs of Theorems A and C.

## Acknowledgment

I would like to thank Akio Tamagawa for several discussions we had on the
topic of this paper. I would like to thank the referee for his/her careful
reading of the paper and comments.

## Notations

The following notations will be used throughout this paper (unless we specify
otherwise).

$\bullet$ $p\geq 2$ is a prime number, $k$ is a $p$-adic local field (i.e.,
$k/\mathbb{Q}_{p}$ is a finite extension) with ring of valuation
$\Cal{O}_{k}$, uniformizer $\pi$, and residue field $F$. Thus $F$ is a finite
field of characteristic $p$.

$\bullet$ A proper, smooth, and geometrically connected $k$-curve $Y$ is
hyperbolic if $\text{genus}(Y)\geq 2$.

$\bullet$ For a profinite group $H$ we denote by $H^{\operatorname{ab}}$ the
maximal abelian quotient of $H$.

$\bullet$ Let

$1\to H^{\prime}\to H@>{\operatorname{pr}}>>G\to 1$

be an exact sequence of profinite groups. We will refer to a continuous
homomorphism $s:G\to H$ such that $\operatorname{pr}\circ
s=\operatorname{id}_{G}$ as a (group-theoretic) section of the above sequence,
or simply a section of the projection $\operatorname{pr}:H\twoheadrightarrow
G$.

$\bullet$ All scheme cohomology groups considered in this paper are étale
cohomology groups.

## Affinoid $p$-adic curves

$\bullet$ $X=\operatorname{Sp}A$ is a smooth and geometrically connected
affinoid $k$-curve. On occasions we will write, if there is no risk of
confusion, $X=\operatorname{Spec}A$ for the corresponding affine $k$-scheme.

$\bullet$ One can embed $X$ into a proper, smooth, and geometrically connected
rigid analytic curve $Y^{\operatorname{rig}}:X\hookrightarrow
Y^{\operatorname{rig}}$ so that $X$ is an open affinoid subspace of
$Y^{\operatorname{rig}}$ (cf. [Fresnel-Matignon], 2.6, Corollaire 2). Write
$Y$ for the algebraization of $Y^{\operatorname{rig}}$ via the rigid GAGA
functor which is a proper, smooth, and geometrically connected algebraic
$k$-curve. We will refer to $X$ as a $p$-adic affinoid curve (or simply an
affinoid) and $Y$ a $k$-compactification of $X$.

## Formal $p$-adic germs

$\bullet$ $A$ is a normal two dimensional complete local ring containing
$\Cal{O}_{k}$ with maximal ideal $\mathfrak{m}_{A}$ containing $\pi$ and
residue field $F=A/\mathfrak{m}_{A}$. Write
$A_{k}\overset\text{def}\to{=}A\otimes_{\Cal{O}_{k}}k=A[\frac{1}{\pi}]$ and
$X\overset\text{def}\to{=}\operatorname{Spec}A_{k}$. We assume $X$ is
geometrically connected and refer to $X$ as a formal $p$-adic germ.

$\bullet$ A ($k$-)compactification of $\operatorname{Spec}A$ is a proper and
flat relative $\Cal{O}_{k}$-curve $\Cal{Y}\to\operatorname{Spec}\Cal{O}_{k}$
with $\Cal{Y}$ normal,
$Y\overset\text{def}\to{=}\Cal{Y}\times_{\operatorname{Spec}\Cal{O}_{k}}\operatorname{Spec}k$
geometrically connected, $y\in\Cal{Y}^{\operatorname{cl}}$ is a closed point,
$\Cal{O}_{\Cal{Y},y}$ is the local ring of $\Cal{Y}$ at $y$,
$\hat{\Cal{O}}_{\Cal{Y},y}$ its completion, with an isomorphism
$\hat{\Cal{O}}_{\Cal{Y},y}\overset\sim\to{\rightarrow}A$. We have a natural
scheme morphism $X\to Y$. We shall refer to $Y$ as a $k$-compactification of
$X$. In $\S 5$ we prove the existence of such a compactification $X\to Y$
after possibly a finite extension of $k$ (cf. loc. cit., Proposition 5.1).

In what follows $X$ is either an affinoid $p$-adic curve or a formal $p$-adic
germ.

$\bullet$ We say that $X$ is hyperbolic if there exists a finite extension
$k^{\prime}/k$ such that
$X_{k^{\prime}}\overset\text{def}\to{=}\operatorname{Spec}(A\otimes_{k}k^{\prime})$
[resp.
$X_{k^{\prime}}\overset\text{def}\to{=}\operatorname{Sp}(A\otimes_{k}k^{\prime})$
if $X$ is affinoid] possesses a $k^{\prime}$-compactification $Y$ with $Y$
hyperbolic. There exists a finite extension $k^{\prime}/k$ and a finite
geometric étale cover $X^{\prime}\to X_{k^{\prime}}$ with $X^{\prime}$
geometrically connected and hyperbolic. This is Proposition 5.3 in case $X$ is
a formal $p$-adic germ and follows from [Saïdi3], Theorem A, in case $X$ is
affinoid.

$\bullet$ $\eta$ is a fixed choice of a geometric point of $X$ with values in
its generic point. Thus $\eta$ determines algebraic closures $\bar{k}$,
$\overline{L}$, of $k$, and $L\overset\text{def}\to{=}\operatorname{Fr}(A)$;
respectively. We have an exact sequence of fundamental groups

$1@>>>\pi_{1}(X,\eta)^{\operatorname{geo}}@>>>\pi_{1}(X,\eta)@>>>G_{k}@>>>1$

where $\pi_{1}(X,\eta)$ is the étale fundamental group of $X$ with geometric
point $\eta$ (cf. [Saïdi3], 2.1, for more details on the definition of
$\pi_{1}(X,\eta)$ in case $X$ is an affinoid),
$\pi_{1}(X,\eta)^{\operatorname{geo}}\overset\text{def}\to{=}\operatorname{Ker}[\pi_{1}(X,\eta)\twoheadrightarrow
G_{k}]$, and $G_{k}\overset\text{def}\to{=}\operatorname{Gal}(\bar{k}/k)$ is
the absolute Galois group of $k$.

In what follows $Y$ is a $k$-compactification of $X$.

$\bullet$ We have a commutative diagram of exact sequences of arithmetic
fundamental groups

$\begin{CD}1@>{}>{}>\pi_{1}(X,\eta)^{\operatorname{geo}}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\pi_{1}(Y_{\bar{k}},\bar{\eta})@>{}>{}>\pi_{1}(Y,\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
\end{CD}$ $None$

where $\pi_{1}(Y,\eta)$ [resp. $\pi_{1}(Y_{\bar{k}},\bar{\eta})$] is the étale
fundamental group of $Y$ (resp.
$Y_{\bar{k}}\overset\text{def}\to{=}Y\times_{\operatorname{Spec}k}{\operatorname{Spec}\bar{k}}$)
with geometric point $\eta$ (resp. $\bar{\eta}$ which is induced by $\eta$).
In case $X$ is an affinoid (resp. a formal $p$-adic germ) the middle vertical
map is induced by the rigid analytic morphism $X\to Y^{\operatorname{rig}}$
and the rigid GAGA functor (resp. the scheme morphism $X\to Y$).

$\bullet$ We write $X^{\operatorname{cl}}$ (resp. $Y^{\operatorname{cl}}$) for
the set of closed points of $X$ (resp. $Y$). For a closed point $x$ of $X$
(resp. $Y$) we write $k(x)$ for the residue field at $x$. Thus $k(x)$ is a
finite extension of $k$.

$\bullet$ We say that $x\in Y^{\operatorname{cl}}$ is not in X if $x$ is not
in the image of the scheme morphism $X\to Y$ if $X$ is a formal $p$-adic germ
or $x\notin X^{\operatorname{cl}}$ in case $X$ is affinoid. In case
$X=\operatorname{Spec}(\Cal{O}_{\Cal{Y},y}\otimes_{\Cal{O}_{k}}k)$ is a formal
$p$-adic germ the set of closed points of $Y$ not in $X$ is in one-to-one
correspondence with the set of closed points of $Y$ which do not specialise in
$y$ [cf. [Liu], $\S 10$, Proposition 1.40(a)].

Throughout sections $\S 1$, $\S 2$, and $\S 3$, $X$ will denote either an
affinoid $p$-adic curve or a formal $p$-adic germ. In $\S 3$ we will assume
$X$ admits a $k$-compactification $Y$ which is hyperbolic and fix a choice of
such a compactification throughout.

## §1. Geometrically abelian arithmetic fundamental groups

In this section we investigate the structure of various geometrically abelian
arithmetic fundamental groups and absolute Galois group associated to $X$. Let

$\pi_{1}(X,\eta)^{(\operatorname{ab})}\overset\text{def}\to{=}\pi_{1}(X,\eta)/\operatorname{Ker}[\pi_{1}(X,\eta)^{\operatorname{geo}}\twoheadrightarrow\pi_{1}(X,\eta)^{\operatorname{geo},\operatorname{ab}}]$

be the geometrically abelian fundamental group of $X$ [here
$\pi_{1}(X,\eta)^{\operatorname{geo},\operatorname{ab}}$ denotes the maximal
abelian quotient of $\pi_{1}(X,\eta)^{\operatorname{geo}}$].

###### Proposition 1.1

We use the above notations. The followings hold.

(i) Assume $X$ is an affinoid. For each prime number $\ell$ the
pro-$\ell$-Sylow subgroup of
$\pi_{1}(X,\eta)^{\operatorname{geo},\operatorname{ab}}$ is pro-$\ell$ abelian
free, of infinite rank if $\ell=p$, and finite (computable) rank otherwise
(see [Saïdi3], Theorem A, for the precise value of this rank in case $\ell\neq
p$).

(ii) Assume $X$ is a formal $p$-adic germ. For each prime number $\ell\neq p$
the pro-$\ell$-Sylow subgroup of
$\pi_{1}(X,\eta)^{\operatorname{geo},\operatorname{ab}}$ is pro-$\ell$ abelian
free of finite computable rank (see [Saïdi4], Theorem A, for the precise value
of this rank).

###### Demonstration Proof

Assertion (i) follows from [Saïdi3], Theorem A. [Note that the assumption in
loc. cit. that $X$ is the complement in a proper rigid analytic $k$-curve of
the disjoint union of finitely many $k$-rational open discs is satisfied after
a finite extension of $k$ (cf. [Fresnel-Matignon], 2.6, Théorème 6 and
Corollaire 1)]. Assertion (ii) follows from [Saïdi4], Theorem A. ∎

Let $S\overset\text{def}\to{=}\\{x_{1},\ldots,x_{n}\\}\subset
X^{\operatorname{cl}}$ be a finite set of closed points and write
$U\overset\text{def}\to{=}X\setminus S$ viewed as an open sub-scheme of $X$
(resp. $X=\operatorname{Spec}A$ in case $X$ is an affinoid). Let
$\pi_{1}(U,\eta)$ be the étale fundamental group of $U$ with geometric point
$\eta$ (cf. [Saïdi3], 2.1, for the definition of $\pi_{1}(U,\eta)$ in case $X$
is affinoid) which sits in the exact sequence

$1@>>>\pi_{1}(U,\eta)^{\operatorname{geo}}@>>>\pi_{1}(U,\eta)@>>>G_{k}@>>>1$

where
$\pi_{1}(U,\eta)^{\operatorname{geo}}\overset\text{def}\to{=}\operatorname{Ker}[\pi_{1}(U,\eta)\twoheadrightarrow
G_{k}]$ (cf. loc. cit. in case $X$ is affinoid). Let

$\pi_{1}(U,\eta)^{(\operatorname{ab})}\overset\text{def}\to{=}\pi_{1}(U,\eta)/\operatorname{Ker}[\pi_{1}(U,\eta)^{\operatorname{geo}}\twoheadrightarrow\pi_{1}(U,\eta)^{\operatorname{geo},\operatorname{ab}}]$

be the geometrically abelian fundamental group of $U$ [here
$\pi_{1}(U,\eta)^{\operatorname{geo},\operatorname{ab}}$ is the maximal
abelian quotient of $\pi_{1}(U,\eta)^{\operatorname{geo}}$]. We have an exact
sequence

$1\to\widetilde{\Delta}_{U}\to\pi_{1}(U,\eta)^{(\operatorname{ab})}\to\pi_{1}(X,\eta)^{(\operatorname{ab})}\to
1$ $None$

where
$\widetilde{\Delta}_{U}\overset\text{def}\to{=}\operatorname{Ker}[\pi_{1}(U,\eta)^{(\operatorname{ab})}\twoheadrightarrow\pi_{1}(X,\eta)^{(\operatorname{ab})}]=\operatorname{Ker}[\pi_{1}(U,\eta)^{\operatorname{geo},\operatorname{ab}}\twoheadrightarrow\pi_{1}(X,\eta)^{\operatorname{geo},\operatorname{ab}}]$
and the (surjective) map
$\pi_{1}(U,\eta)^{(\operatorname{ab})}\twoheadrightarrow\pi_{1}(X,\eta)^{(\operatorname{ab})}$
is induced by the natural projection
$\pi_{1}(U,\eta)\twoheadrightarrow\pi_{1}(X,\eta)$. Note that
$\widetilde{\Delta}_{U}$ has a natural structure of $G_{k}$-module.

###### Proposition 1.2

We use the above notations. There exists a natural isomorphism

$\prod_{i=1}^{n}\operatorname{Ind}_{k(x_{i})}^{k}\hat{\mathbb{Z}}(1)\overset\sim\to{\rightarrow}\widetilde{\Delta}_{U}$

of $G_{k}$-modules where the $(1)$ is a Tate twist.

###### Demonstration Proof

We have a natural surjective homomorphism
$\prod_{i=1}^{n}\operatorname{Ind}_{k(x_{i})}^{k}\hat{\mathbb{Z}}(1)\twoheadrightarrow\widetilde{\Delta}_{U}$
of $G_{k}$-modules mapping
$\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1)$ onto the
inertia subgroup [of $\pi_{1}(U,\eta)^{(\operatorname{ab})}$] at $x_{i}$, as
follows from the structure of inertia groups of Galois extensions of henselian
discrete valuation rings of residue characteristic zero. We show this map is
an isomorphism. To this end we can, without loss of generality, assume that
$X$ admits a $k$-compactification $Y$ (cf. Notations). Indeed, this holds for
$X$ affinoid (cf. loc. cit.), and holds after possibly replacing $k$ by a
finite field extension in case $X$ is a formal $p$-adic germ (cf. Proposition
5.1) which doesn’t alter the structure of $\widetilde{\Delta}_{U}$. We have a
commutative diagram of exact sequences

$\begin{CD}1@>{}>{}>\pi_{1}(X,\eta)^{\operatorname{geo},\operatorname{ab}}@>{}>{}>\pi_{1}(X,\eta)^{(\operatorname{ab})}@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\pi_{1}(Y_{\bar{k}},\bar{\eta})^{\operatorname{ab}}@>{}>{}>\pi_{1}(Y,\eta)^{(\operatorname{ab})}@>{}>{}>G_{k}@>{}>{}>1\\\
\end{CD}$ $None$

where
$\pi_{1}(Y,\eta)^{(\operatorname{ab})}\overset\text{def}\to{=}\pi_{1}(Y,\eta)/\operatorname{Ker}[\pi_{1}(Y_{\bar{k}},\bar{\eta})\twoheadrightarrow\pi_{1}(Y_{\bar{k}},\bar{\eta})^{\operatorname{ab}}]$
and the middle vertical map is induced by the natural homomorphism
$\pi_{1}(X,\eta)\to\pi_{1}(Y,\eta)$ [cf. Notations, diagram (0.1)].

Denote by $x_{i}^{\prime}$ the image of $x_{i}$ in $Y$, $\forall 1\leq i\leq
n$ [note that $k(x_{i})=k(x_{i}^{\prime})$]. Let $x_{0}^{\prime}\in
Y^{\operatorname{cl}}\setminus\\{x_{1}^{\prime},\ldots,x_{n}^{\prime}\\}$ be a
closed point which is not in the image of $X$ (cf. Notations). Write
$S^{\prime}\overset\text{def}\to{=}\\{x_{0}^{\prime},x_{1}^{\prime},\ldots,x_{n}^{\prime}\\}\subset
Y^{\operatorname{cl}}$ and $V\overset\text{def}\to{=}Y\setminus S^{\prime}$
which is an affine $k$-curve. Let $\pi_{1}(V,\eta)$ be the étale fundamental
group of $V$ with geometric point $\eta$ which sits in the exact sequence
$1@>>>\pi_{1}(V_{\bar{k}},\bar{\eta})@>>>\pi_{1}(V,\eta)@>>>G_{k}@>>>1,$ where
$\pi_{1}(V_{\bar{k}},\bar{\eta})$ is the étale fundamental group of
$V_{\bar{k}}\overset\text{def}\to{=}V\times_{k}\bar{k}$ with geometric point
$\bar{\eta}$ which is induced by $\eta$. Let
$\pi_{1}(V,\eta)^{(\operatorname{ab})}\overset\text{def}\to{=}\pi_{1}(V,\eta)/\operatorname{Ker}[\pi_{1}(V_{\bar{k}},\bar{\eta})\twoheadrightarrow\pi_{1}(V_{\bar{k}},\bar{\eta})^{\operatorname{ab}}]$
be the geometrically abelian fundamental group of $V$. We have a commutative
diagram of exact sequences

$\begin{CD}1@>{}>{}>\widetilde{\Delta}_{U}@>{}>{}>\pi_{1}(U,\eta)^{(\operatorname{ab})}@>{}>{}>\pi_{1}(X,\eta)^{(\operatorname{ab})}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
1@>{}>{}>\widetilde{\Delta}_{V}@>{}>{}>\pi_{1}(V,\eta)^{(\operatorname{ab})}@>{}>{}>\pi_{1}(Y,\eta)^{(\operatorname{ab})}@>{}>{}>1\\\
\end{CD}$ $None$

where
$\widetilde{\Delta}_{V}\overset\text{def}\to{=}\operatorname{Ker}[\pi_{1}(V,\eta)^{(\operatorname{ab})}\twoheadrightarrow\pi_{1}(Y,\eta)^{(\operatorname{ab})}]$.
The middle vertical map in diagram (1.3) is induced by the natural
homomorphism $\pi_{1}(U,\eta)\to\pi_{1}(V,\eta)$, which is induced by the
scheme morphism $X\to Y$ in case $X$ is a formal $p$-adic germ, and by the
rigid analytic morphism $X\to Y^{\operatorname{rig}}$ and the rigid GAGA
functor in case $X$ is affinoid (here we use the fact that $x_{0}^{\prime}$ is
not in the image of $X$). The right vertical map in diagram (1.3) is the
middle vertical map in diagram (1.2).

One has an exact sequence of $G_{k}$-modules (as follows from the well-known
structure of $\pi_{1}(V,\eta)^{(\operatorname{ab})}$, see for example the
discussion in [Saïdi5], $\S 0$)

$0\to\hat{\mathbb{Z}}(1)\to\prod_{i=0}^{n}\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1)\to\widetilde{\Delta}_{V}\to
0.$

Consider the composite homomorphism
$\tau:\prod_{i=1}^{n}\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1)@>>>\widetilde{\Delta}_{V}$
of $G_{k}$-modules:

$\prod_{i=1}^{n}\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1)\hookrightarrow\prod_{i=0}^{n}\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1)\twoheadrightarrow\widetilde{\Delta}_{V}$

where the first map is the natural embedding:
$(\beta_{1},\ldots,\beta_{n})\mapsto(0,\beta_{1},\ldots,\beta_{n})$ and the
second map is as in the above exact sequence. Thus $\tau$ is injective (cf.
above exact sequence). Consider the following commutative diagram

$\begin{CD}\prod_{i=1}^{n}\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1)@>{}>{}>\widetilde{\Delta}_{U}\\\
@V{}V{}V@V{}V{}V\\\
\prod_{i=0}^{n}\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1)@>{}>{}>\widetilde{\Delta}_{V}\\\
\end{CD}$

where the right vertical map is the one in diagram (1.3). The left vertical
and lower horizontal maps are as explained above, hence their composite is the
map $\tau$. The upper horizontal map is the natural projection
$\prod_{i=1}^{n}\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1)\twoheadrightarrow\widetilde{\Delta}_{U}$
mentioned at the start of the proof. This map is an isomorphism since it is
onto and it is injective as $\tau$ is. ∎

###### Definition Remark 1.3

With the notations in Proposition 1.2 and the proof therein assume that
$x^{\prime}_{0}\in Y(k)$ is a $k$-rational point. In this case
$\tau(\prod_{i=1}^{n}\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1))=\widetilde{\Delta}_{V}$,
the map $\widetilde{\Delta}_{U}\to\widetilde{\Delta}_{V}$ is an isomorphism,
and the right square in diagram (1.3) (cf. proof of Proposition 1.2) is
cartesian.

Let $G_{X}\overset\text{def}\to{=}\operatorname{Gal}(\overline{L}/L)$ [recall
$L\overset\text{def}\to{=}\operatorname{Fr}(A)$] which sits in the exact
sequences

$1\to G_{X}^{\operatorname{geo}}\to G_{X}\to G_{k}\to 1$

where
$G_{X}^{\operatorname{geo}}\overset\text{def}\to{=}\operatorname{Gal}(\overline{L}/L\bar{k})$,
and

$1\to\Cal{H}_{X}\to G_{X}\to\pi_{1}(X,\eta)\to 1$ $None$

where
$\Cal{H}_{X}\overset\text{def}\to{=}\operatorname{Ker}[G_{X}\twoheadrightarrow\pi_{1}(X,\eta)]$.
Let

$G_{X}^{(\operatorname{ab})}\overset\text{def}\to{=}G_{X}/\operatorname{Ker}(G_{X}^{\operatorname{geo}}\twoheadrightarrow
G_{X}^{\operatorname{geo},\operatorname{ab}})$

which we shall refer to as the geometrically abelian Galois group of $X$ (here
$G_{X}^{\operatorname{geo},\operatorname{ab}}$ is the maximal abelian quotient
of $G_{X}^{\operatorname{geo}}$). We have an exact sequence

$1\to\widetilde{\Cal{H}}_{X}\to
G_{X}^{(\operatorname{ab})}\to\pi_{1}(X,\eta)^{(\operatorname{ab})}\to 1$
$None$

where
$\widetilde{\Cal{H}}_{X}\overset\text{def}\to{=}\operatorname{Ker}[G_{X}^{(\operatorname{ab})}\twoheadrightarrow\pi_{1}(X,\eta)^{(\operatorname{ab})}]=\operatorname{Ker}[G_{X}^{\operatorname{geo},\operatorname{ab}}\twoheadrightarrow\pi_{1}(X,\eta)^{\operatorname{geo},\operatorname{ab}}]$.
Note that $\widetilde{\Cal{H}}_{X}$ has a natural structure of $G_{k}$-module.

###### Proposition 1.4

We use the above notations. There exists a natural isomorphism of
$G_{k}$-modules

$\prod_{x\in
X^{\operatorname{cl}}}\operatorname{Ind}_{k(x)}^{k}\hat{\mathbb{Z}}(1)\overset\sim\to{\rightarrow}\widetilde{\Cal{H}}_{X}$

where the product is over all closed points $x\in X^{\operatorname{cl}}$.

###### Demonstration Proof

This follows from Proposition 1.2 and the fact that
$\widetilde{\Cal{H}}_{X}\overset\sim\to{\rightarrow}\varprojlim_{U}\widetilde{\Delta}_{U}$
where $U=X\setminus S$; $S$ runs over all finite subsets of
$X^{\operatorname{cl}}$, and $\widetilde{\Delta}_{U}$ is as in the proof of
loc. cit.. (Note that
$G_{X}^{(\operatorname{ab})}\overset\sim\to{\rightarrow}\varprojlim_{U}\pi_{1}(U,\eta)^{(\operatorname{ab})}$
where the limit runs over all $U$ as above.) ∎

## §2. Cuspidally abelian arithmetic fundamental groups

In this section we investigate the problem of cuspidalisation of sections of
the projection $\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$. This problem has
been investigated in the case of proper and smooth hyperbolic $p$-adic curves
in [Saïdi1] and [Saïdi2]. We use the notations in $\S 0$ and $\S 1$.

Let $S\overset\text{def}\to{=}\\{x_{1},\ldots,x_{n}\\}\subset
X^{\operatorname{cl}}$ be a finite set of closed points and
$U\overset\text{def}\to{=}X\setminus S$ (cf. $\S 1$). Consider the exact
sequence

$1\to\Delta_{U}\to\pi_{1}(U,\eta)^{\operatorname{geo}}\to\pi_{1}(X,\eta)^{\operatorname{geo}}\to
1$

where
$\Delta_{U}\overset\text{def}\to{=}\operatorname{Ker}[\pi_{1}(U,\eta)^{\operatorname{geo}}\twoheadrightarrow\pi_{1}(X,\eta)^{\operatorname{geo}}]$.
The maximal abelian quotient $\Delta_{U}^{\operatorname{ab}}$ of $\Delta_{U}$
is a $\pi_{1}(X,\eta)^{\operatorname{geo}}$-module. Let
$\Delta_{U}^{\operatorname{cn}}$ be the maximal quotient of
$\Delta_{U}^{\operatorname{ab}}$ on which
$\pi_{1}(X,\eta)^{\operatorname{geo}}$ acts trivially. Define

$\pi_{1}(U,\eta)^{\operatorname{geo},\text{c}-\operatorname{ab}}\overset\text{def}\to{=}\pi_{1}(U,\eta)^{\operatorname{geo}}/\operatorname{Ker}(\Delta_{U}\twoheadrightarrow\Delta_{U}^{\operatorname{ab}})$

and

$\pi_{1}(U,\eta)^{\operatorname{geo},\text{c}-\operatorname{cn}}\overset\text{def}\to{=}\pi_{1}(U,\eta)^{\operatorname{geo}}/\operatorname{Ker}(\Delta_{U}\twoheadrightarrow\Delta_{U}^{\operatorname{cn}}).$

We shall refer to
$\pi_{1}(U,\eta)^{\operatorname{geo},\text{c}-\operatorname{ab}}$ (resp.
$\pi_{1}(U,\eta)^{\operatorname{geo},\text{c}-\operatorname{cn}}$) as the
cuspidally abelian (resp. cuspidally central) quotient of
$\pi_{1}(U,\eta)^{\operatorname{geo}}$. Further, define

$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}\overset\text{def}\to{=}\pi_{1}(U,\eta)/\operatorname{Ker}(\Delta_{U}\twoheadrightarrow\Delta_{U}^{\operatorname{ab}})$

and

$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}\overset\text{def}\to{=}\pi_{1}(U,\eta)/\operatorname{Ker}(\Delta_{U}\twoheadrightarrow\Delta_{U}^{\operatorname{cn}}).$

We shall refer to $\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}$ (resp.
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}$) as the (geometrically)
cuspidally abelian [resp. (geometrically) cuspidally central] quotient of
$\pi_{1}(U,\eta)$. We have the following commutative diagram of exact
sequences

$\begin{CD}1@>{}>{}>\Delta_{U}@>{}>{}>\pi_{1}(U,\eta)@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\Delta_{U}^{\operatorname{ab}}@>{}>{}>\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\Delta_{U}^{\operatorname{cn}}@>{}>{}>\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>1\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
1@>{}>{}>\widetilde{\Delta}_{U}@>{}>{}>\pi_{1}(U,\eta)^{(\operatorname{ab})}@>{}>{}>\pi_{1}(X,\eta)^{(\operatorname{ab})}@>{}>{}>1\end{CD}$
$None$

where the middle vertical maps are surjective, and the middle vertical map in
the lower diagram is induced by the natural surjective map
$\pi_{1}(U,\eta)^{\operatorname{geo},\text{c}-\operatorname{ab}}\twoheadrightarrow\pi_{1}(U,\eta)^{\operatorname{geo},\operatorname{ab}}$.
[Note that $\pi_{1}(X,\eta)^{\operatorname{geo}}$ acts trivially on the
quotient $\widetilde{\Delta}_{U}$ of $\Delta_{U}^{\operatorname{ab}}$.]

###### Lemma 2.1

We use the above notations. The homomorphism
$\Delta_{U}^{\operatorname{cn}}\to\widetilde{\Delta}_{U}$ in diagram (2.1) is
an isomorphism of $G_{k}$-modules. In particular, the lower right square in
diagram (2.1) is cartesian.

###### Demonstration Proof

The proof follows from Proposition 1.2 and the various definitions. More
precisely, there exists a natural surjective homomorphism
$\prod_{i=1}^{n}\operatorname{Ind}_{k(x_{i})}^{k}\hat{\mathbb{Z}}(1)\twoheadrightarrow\Delta_{U}^{\operatorname{cn}}$
[mapping $\operatorname{Ind}_{k(x_{i}^{\prime})}^{k}\hat{\mathbb{Z}}(1)$ onto
the inertia subgroup of $\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}$ at
$x_{i}$, as follows from the structure of inertia groups of Galois extensions
of henselian discrete valuation rings of residue characteristic zero] which
composed with the projection
$\Delta_{U}^{\operatorname{cn}}\twoheadrightarrow\widetilde{\Delta}_{U}$ is
the isomorphism
$\prod_{i=1}^{n}\operatorname{Ind}_{k(x_{i})}^{k}\hat{\mathbb{Z}}(1)\overset\sim\to{\rightarrow}\widetilde{\Delta}_{U}$
in Proposition 1.2 hence our assertion. ∎

Let $s:G_{k}\to\pi_{1}(X,\eta)$ be a section of the projection
$\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$.

###### Proposition 2.2

(Lifting of sections to cuspidally central arithmetic fundamental groups). We
use the above notations. There exists a section
$s_{U}^{\text{c}-\operatorname{cn}}:G_{k}\to\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}$
of the projection
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}\twoheadrightarrow G_{k}$ which
lifts the section $s$, i.e., which inserts in the following commutative
diagram

$\begin{CD}G_{k}@>{s_{U}^{\text{c}-\operatorname{cn}}}>{}>\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}\\\
\Big{\|}@V{}V{}V\\\ G_{k}@>{s}>{}>\pi_{1}(X,\eta)\\\ \end{CD}$

where the right vertical map is the natural projection
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}\twoheadrightarrow\pi_{1}(X,\eta)$.
In particular, the set of sections of the projection
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}\twoheadrightarrow G_{k}$ which
lift the section $s$ is non-empty, and is (up to conjugation by elements of
$\Delta_{U}^{\operatorname{cn}}$) a torsor under
$H^{1}(G_{k},\Delta_{U}^{\operatorname{cn}})$.

###### Demonstration Proof

Consider the commutative diagram of exact sequences

$\begin{CD}1@>{}>{}>\Delta_{U}^{\operatorname{cn}}@>{}>{}>E_{U}\overset\text{def}\to{=}E_{U}[s]@>{}>{}>G_{k}@>{}>{}>1\\\
\Big{\|}@V{}V{}V@V{s}V{}V\\\
1@>{}>{}>\Delta_{U}^{\operatorname{cn}}@>{}>{}>\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>1\end{CD}$

where the right square is cartesian. Thus the group extension $E_{U}$ is the
pull-back of the group extension
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}$ by the section $s$. The set
of (possible) splittings of the group extension $E_{U}$ is in one-to-one
correspondence with the set of sections of the projection
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}\twoheadrightarrow G_{k}$ which
lift the section $s$. We show the group extension $E_{U}$ splits.

To this end we can replace $k$ by a finite extension over which the points
$\\{x_{i}\\}_{i=1}^{n}$ are rational, and we can also assume $n=1$ (see the
argument at the start of the proof of Lemma 2.3.1 in [Saïdi1]). Further we can
replace $X$ by a neighbourhood $X^{\prime}$ of the section $s$: i.e., an étale
cover $X^{\prime}\to X$ corresponding to an open subgroup
$H=\pi_{1}(X^{\prime},\eta)\subset\pi_{1}(X,\eta)$ containing the image
$s(G_{k})$ of $s$. Indeed, if
$U^{\prime}\overset\text{def}\to{=}U\times_{X}X^{\prime}$ there exists a
commutative diagram of natural homomorphisms

$\begin{CD}\pi_{1}(U^{\prime},\eta)^{(\text{c}-\operatorname{cn})}@>{}>{}>\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}\\\
@V{}V{}V@V{}V{}V\\\ \pi_{1}(X^{\prime},\eta)@>{}>{}>\pi_{1}(X,\eta)\\\
\end{CD}$

where the upper horizontal map is induced by the natural map
$\pi_{1}(U^{\prime},\eta)\to\pi_{1}(U,\eta)$ [note
$\Delta_{U^{\prime}}=\Delta_{U}$ and
$\pi_{1}(X^{\prime},\eta)^{\operatorname{geo}}$ acts trivially on
$\Delta_{U}^{\operatorname{cn}}$], and the various maps in this diagram
commute with the projections onto $G_{k}$. The section $s$ induces a section
$\tilde{s}:G_{k}\to\pi_{1}(X^{\prime},\eta)$ of the projection
$\pi_{1}(X^{\prime},\eta)\twoheadrightarrow G_{k}$, and a lifting
$\tilde{s}_{U^{\prime}}^{\text{c}-\operatorname{cn}}:G_{k}\to\pi_{1}(U^{\prime},\eta)^{(\text{c}-\operatorname{cn})}$
of $\tilde{s}$ (as in the statement of Proposition 2.2) induces a lifting
$s_{U}^{\text{c}-\operatorname{cn}}:G_{k}\to\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}$
of $s$ as required (cf. above diagram). Now it follows from [Saïdi3], Theorem
A, in case $X$ is an affinoid, and Proposition 5.3 in this paper (cf. $\S 5$)
in case $X$ is a formal $p$-adic germ, that there exists (after possibly a
finite extension of $k$) a neighbourhood $X^{\prime}\to X$ of $s$ with
$X^{\prime}$ hyperbolic (cf. Notations). We can thus assume, without loss of
generality, that $X$ possesses a $k$-compactification $Y$ with $Y$ hyperbolic
and the set $S\overset\text{def}\to{=}\\{x\\}\subset X(k)$ consists of a
single $k$-rational point, in which case
$\Delta_{U}^{\operatorname{cn}}\overset\sim\to{\rightarrow}\hat{\mathbb{Z}}(1)$
as a $\pi_{1}(X,\eta)$-module (cf. Lemma 2.1 and Proposition 1.2).

Consider the following maps (here $X=\operatorname{Spec}A$ in case $X$ is
affinoid)

$H^{2}(\pi_{1}(X,\eta),\hat{\mathbb{Z}}(1))\hookrightarrow
H^{2}(X,\hat{\mathbb{Z}}(1))\leftarrow\operatorname{Pic}(X)$

where the map $H^{2}(\pi_{1}(X,\eta),\hat{\mathbb{Z}}(1))\hookrightarrow
H^{2}(X,\hat{\mathbb{Z}}(1))$ arises from the Cartan-Leray spectral sequence
and is injective (cf. [Serre], proof of Proposition 1), and the map
$\operatorname{Pic}(X)\rightarrow H^{2}(X,\hat{\mathbb{Z}}(1))$ is the cycle
class map arising from the Kummer exact sequence in étale topology. Let
$[\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}]\in
H^{2}(\pi_{1}(X,\eta),\hat{\mathbb{Z}}(1))$ be the class of the group
extension $\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}$. The image of
$[\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}]$ in
$H^{2}(X,\hat{\mathbb{Z}}(1))$ coincides with the image of the line bundle
$\Cal{O}(x)\in\operatorname{Pic}(X)$ via the Kummer map
$\operatorname{Pic}(X)\rightarrow H^{2}(X,\hat{\mathbb{Z}}(1))$. Indeed, this
follows from the following commutative diagram

$\begin{CD}H^{2}(\pi_{1}(X,\eta),\hat{\mathbb{Z}}(1))@>{}>{}>H^{2}(X,\hat{\mathbb{Z}}(1))@<{}<{}<\operatorname{Pic}(X)\\\
@A{}A{}A@A{}A{}A@A{}A{}A\\\
H^{2}(\pi_{1}(Y,\eta),\hat{\mathbb{Z}}(1))@>{}>{}>H^{2}(Y,\hat{\mathbb{Z}}(1))@<{}<{}<\operatorname{Pic}(Y)\\\
\end{CD}$

where the right and middle vertical maps are induced by the scheme morphism
$X\to Y$ if $X$ is a formal $p$-adic germ, and the rigid morphism $X\to
Y^{\operatorname{rig}}$ and the comparison theorems between étale cohomology
and rigid analytic étale cohomology in case $X$ is affinoid (cf. [Hansen],
Theorem 1.8 and Theorem 1.9). The right horizontal maps are the cycle class
maps arising from the Kummer exact sequence in étale topology and the left
lower horizontal map is an isomorphism arising from the Cartan-Leray spectral
sequence (cf. [Mochizuki], Proposition 1.1). The pull-back of the class
$[\pi_{1}(V,\eta)^{(\text{c}-\operatorname{cn})}]\in
H^{2}(\pi_{1}(Y,\eta),\hat{\mathbb{Z}}(1))$ in
$H^{2}(\pi_{1}(X,\eta),\hat{\mathbb{Z}}(1))$, where $V$ is the complement in
$Y$ of the image of $S=\\{x\\}$ [cf. [Saïdi1] 2.1.1 for the definition of
$\pi_{1}(V,\eta)^{(\text{c}-\operatorname{cn})}$], coincides with the class
$[\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}]$ (this follows from Lemma
2.1 and the various definitions). The class
$[\pi_{1}(V,\eta)^{(\text{c}-\operatorname{cn})}]\in
H^{2}(\pi_{1}(Y,\eta),\hat{\mathbb{Z}}(1))\overset\sim\to{\rightarrow}H^{2}(Y,\hat{\mathbb{Z}}(1))$
coincides with the image of the Chern class of the line bundle
$\Cal{O}(y)\in\operatorname{Pic}(Y)$ where $y\in Y(k)$ is the image of $x$
(cf. [Saïdi1] proof of Lemma 2.3.1). Thus, the image of
$[\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}]$ in
$H^{2}(X,\hat{\mathbb{Z}}(1))$ coincides with the image of the line bundle
$\Cal{O}(x)\in\operatorname{Pic}(X)$ via the cycle class map
$\operatorname{Pic}(X)\rightarrow H^{2}(X,\hat{\mathbb{Z}}(1))$ as claimed.

The Picard group $\operatorname{Pic}(X)$ is finite (cf. Theorem 4.1 in this
paper in case $X$ is affinoid, and [Saïdi2] Proposition 5.4 in case $X$ is a
formal $p$-adic germ). In particular, the image of
$[\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}]$ in
$H^{2}(X,\hat{\mathbb{Z}}(1))$ and hence the class
$[\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}]$ is a torsion element of
$H^{2}(\pi_{1}(X,\eta),\hat{\mathbb{Z}}(1))$. The class $[E_{U}]\in
H^{2}(G_{k},\hat{\mathbb{Z}}(1))$ of the group extension $E_{U}$ is the image
of $[\pi_{1}(U,\eta)^{(\text{c}-\operatorname{cn})}]$ under the retraction map
$H^{2}(\pi_{1}(X,\eta),\hat{\mathbb{Z}}(1))@>s^{\star}>>H^{2}(G_{k},\hat{\mathbb{Z}}(1))\overset\sim\to{\rightarrow}\hat{\mathbb{Z}}$
induced by $s$. Hence the class $[E_{U}]$ is trivial since $\hat{\mathbb{Z}}$
is torsion free, and the group extension $E_{U}$ splits. ∎

###### Theorem 2.3

(Lifting of sections to cuspidally abelian arithmetic fundamental groups). We
use the above notations. There exists a section
$s_{U}^{\operatorname{ab}}:G_{k}\to\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}$
of the projection
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which
lifts the section $s$, i.e., which inserts in the following commutative
diagram

$\begin{CD}G_{k}@>{s_{U}^{\text{c}-\operatorname{ab}}}>{}>\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}\\\
\Big{\|}@V{}V{}V\\\ G_{k}@>{s}>{}>\pi_{1}(X,\eta)\\\ \end{CD}$

where the right vertical map is the natural projection
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}\twoheadrightarrow\pi_{1}(X,\eta)$.
In particular, the set of sections of the projection
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which
lift the section $s$ is non-empty, and is (up to conjugation by elements of
$\Delta_{U}^{\operatorname{ab}}$) a torsor under
$H^{1}(G_{k},\Delta_{U}^{\operatorname{ab}})$.

###### Demonstration Proof

Let $\\{H\\}_{i\in I}$ be a projective system of open subgroups of
$\pi_{1}(X,\eta)$ containing $s(G_{k})$ such that $s(G_{k})=\bigcap_{i\in
I}H_{i}$. Thus, for $i\in I$, the open subgroup $H_{i}$ corresponds to an
étale finite cover $X_{i}\to X$ with $X_{i}$ geometrically connected and
$H_{i}$ is identified with $\pi_{1}(X_{i},\eta)$ which sits in the exact
sequence
$1\to\pi_{1}(X_{i},\eta)^{\operatorname{geo}}\to\pi_{1}(X_{i},\eta)\to
G_{k}\to 1$ (the geometric point; denote also $\eta$, of $X_{i}$ is induced by
the geometric point $\eta$ of $X$). Further, the section $s$ induces a section
$s_{i}:G_{k}\to\pi_{1}(X_{i},\eta)$ of the projection
$\pi_{1}(X_{i},\eta)\twoheadrightarrow G_{k}$. Let
$U_{i}\overset\text{def}\to{=}U\times_{X}X_{i}$ and
$\pi_{1}(U_{i},\eta)^{(\text{c}-\operatorname{cn})}$ the (geometrically)
cuspidally central arithmetic fundamental group of $U_{i}$ which sits in the
exact sequence
$1\to\Delta_{U_{i}}^{\operatorname{cn}}\to\pi_{1}(U_{i},\eta)^{(\text{c}-\operatorname{cn})}\to\pi_{1}(X_{i},\eta)\to
1$.

Consider the following commutative diagrams

$\begin{CD}1@>{}>{}>\Delta_{U}^{\operatorname{ab}}@>{}>{}>\Cal{E}_{U}@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V@V{s}V{}V\\\
1@>{}>{}>\Delta_{U}^{\operatorname{ab}}@>{}>{}>\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>1\\\
\end{CD}$

and for $i\in I$

$\begin{CD}1@>{}>{}>\Delta_{U_{i}}^{\operatorname{cn}}@>{}>{}>E_{U_{i}}@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V@V{s_{i}}V{}V\\\
1@>{}>{}>\Delta_{U_{i}}^{\operatorname{cn}}@>{}>{}>\pi_{1}(U_{i},\eta)^{(\text{c}-\operatorname{cn})}@>{}>{}>\pi_{1}(X_{i},\eta)@>{}>{}>1\\\
\end{CD}$

where the right squares are cartesian. Thus, $\Cal{E}_{U}$ (resp. $E_{U_{i}}$)
is the pull-back of the group extension
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}$ [resp.
$\pi_{1}(U_{i},\eta)^{(\text{c}-\operatorname{cn})}$] by the section $s$
(resp. $s_{i}$). There is a natural isomorphism
$\Delta_{U}^{\operatorname{ab}}=\varprojlim_{i\in
I}\Delta_{U_{i}}^{\operatorname{cn}}$ as follows from the facts that
$\Delta_{U}=\Delta_{U_{i}}$, $\forall i\in I$, and given a finite quotient
$\Delta_{U}^{\operatorname{ab}}\twoheadrightarrow H$ there exists $i\in I$
such that $\pi_{1}(X_{i},\eta)^{\operatorname{geo}}$ acts trivially on $H$.
Further, there is a natural isomorphism
$\Cal{E}_{U}\overset\sim\to{\rightarrow}\varprojlim_{i\in I}E_{U_{i}}$ (the
transition maps in the projective limit being surjective). The existence of a
section
$s_{U}^{\text{c}-\operatorname{ab}}:G_{k}\to\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}$
of the projection
$\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which
lifts the section $s$ is equivalent to the splitting of the group extension
$\Cal{E}_{U}$, and the set of those (possible) liftings
$s_{U}^{\text{c}-\operatorname{ab}}$ is in one-to-one correspondence with the
set of sections of the projection $\Cal{E}_{U}\twoheadrightarrow G_{k}$. The
natural projection $E_{U_{i}}\twoheadrightarrow G_{k}$ splits for all $i\in I$
(see proof of Proposition 2.2). We show the group extension $\Cal{E}_{U}$
splits.

Let $(P_{j})_{j\in J}$ be a projective system of quotients
$\Cal{E}_{U}\twoheadrightarrow P_{j}$, where $P_{j}$ sits in an exact sequence
$1\to F_{j}\to P_{j}\to G_{k}\to 1$ with $F_{j}$ finite, and
$\Cal{E}_{U}=\varprojlim_{j\in J}P_{j}$. [More precisely, write $\Cal{E}_{U}$
as a projective limit of finite groups $\\{\tilde{P}_{j}\\}_{j\in J}$ where
$\tilde{P}_{j}$ sits in an exact sequence $1\to F_{j}\to\tilde{P}_{j}\to
G_{j}\to 1$ with $G_{j}$ a quotient of $G_{k}$ and $F_{j}$ a quotient of
$\operatorname{Ker}(\Cal{E}_{U}\twoheadrightarrow G_{k})$. Let $1\to F_{j}\to
P_{j}\to G_{k}\to 1$ be the pull-back of the group extension $1\to
F_{j}\to\tilde{P}_{j}\to G_{j}\to 1$ by $G_{k}\twoheadrightarrow G_{j}$. Then
$\Cal{E}_{U}=\varprojlim_{j\in J}P_{j}$]. The set
$\operatorname{Sect}(G_{k},\Cal{E}_{U})$ of group-theoretic sections of the
projection $\Cal{E}_{U}\twoheadrightarrow G_{k}$ is naturally identified with
the projective limit $\varprojlim_{j\in J}\operatorname{Sect}(G_{k},P_{j})$ of
the sets $\operatorname{Sect}(G_{k},P_{j})$ of group-theoretic sections of the
projections $P_{j}\twoheadrightarrow G_{k}$, $j\in J$. The set
$\operatorname{Sect}(G_{k},P_{j})$ is non-empty, $\forall j\in J$. Indeed,
$P_{j}$ (being a quotient of $\Cal{E}_{U}$) is a quotient of $E_{U_{i}}$ for
some $i\in I$, this quotient $E_{U_{i}}\twoheadrightarrow P_{j}$ commutes with
the projections onto $G_{k}$, and we know the projection
$E_{U_{i}}\twoheadrightarrow G_{k}$ splits, hence the projection
$P_{j}\twoheadrightarrow G_{k}$ splits. Moreover, the set
$\operatorname{Sect}(G_{k},P_{j})$ is, up to conjugation by the elements of
$F_{j}$, a torsor under the group $H^{1}(G_{k},F_{j})$ which is finite since
$k$ is a $p$-adic local field [cf. [Neukirch-Schmidt-Wingberg],
(7.1.8)Theorem(iii)]. Thus, $\operatorname{Sect}(G_{k},P_{j})$ is a non-empty
finite set. The set $\operatorname{Sect}(G_{k},\Cal{E}_{U})$ is non-empty
being the projective limit of non-empty finite sets. This finishes the proof
of Theorem 2.3. ∎

Next let

$G_{X}^{(\text{c}-\operatorname{ab})}\overset\text{def}\to{=}G_{X}/\operatorname{Ker}(\Cal{H}_{X}\twoheadrightarrow\Cal{H}_{X}^{\operatorname{ab}})$

(cf. exact sequence (1.4) for the definition of $\Cal{H}_{X}$). Thus,
$G_{X}^{(\text{c}-\operatorname{ab})}=\varprojlim_{U}\pi_{1}(U,\eta)^{(\text{c}-\operatorname{ab})}$
where $U$ runs over all sub-schemes of $X$ as in Theorem 2.3.

###### Theorem 2.4

(Lifting of sections to cuspidally abelian Galois groups). We use the above
notations. There exists a section $s^{\text{c}-\operatorname{ab}}:G_{k}\to
G_{X}^{(\text{c}-\operatorname{ab})}$ of the projection
$G_{X}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which lifts the
section $s$, i.e., which inserts in the following commutative diagram

$\begin{CD}G_{k}@>{s^{\text{c}-\operatorname{ab}}}>{}>G_{X}^{(\text{c}-\operatorname{ab})}\\\
\Big{\|}@V{}V{}V\\\ G_{k}@>{s}>{}>\pi_{1}(X,\eta)\\\ \end{CD}$

where the right vertical map is the natural projection
$G_{X}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow\pi_{1}(X,\eta)$. In
particular, the set of sections of the projection
$G_{X}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which lift the
section $s$ is non-empty, and is (up to conjugation by elements of
$\Cal{H}_{X}^{\operatorname{ab}}$) a torsor under
$H^{1}(G_{k},\Cal{H}_{X}^{\operatorname{ab}})$.

###### Demonstration Proof

The proof follows, using the natural identification
$G_{X}^{\text{c}-\operatorname{ab}}\overset\sim\to{\rightarrow}\varprojlim_{U}\pi_{1}(U,\eta)^{\text{c}-\operatorname{ab}}$
[where $U$ runs over all sub-schemes of $X$ as in Theorem 2.3], from Theorem
2.3 and a similar argument in our context to the one used in the proof of
Theorem 2.3.5 in [Saïdi1]. Alternatively, one can use Theorem 2.3 and a
similar argument to the one used at the end of the proof of Theorem 2.3. ∎

The following is one of our main results in this section.

###### Theorem 2.5

Assume that $X$ admits a $k$-compatctification $Y$ (cf. Notations). If the
projection $\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$ splits then
$\operatorname{index}(Y)=1$.

###### Demonstration Proof

Assume that the projection $\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$ splits
and let $s:G_{k}\to\pi_{1}(X,\eta)$ be a section of this projection. By
Theorem 2.4 there exists a section $s^{\text{c}-\operatorname{ab}}:G_{k}\to
G_{X}^{(\text{c}-\operatorname{ab})}$ of the projection
$G_{X}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which lifts the
section $s$. The section $s^{\text{c}-\operatorname{ab}}$ induces naturally a
section $\tilde{s}:G_{k}\to G_{X}^{(\operatorname{ab})}$ of the projection
$G_{X}^{(\operatorname{ab})}\twoheadrightarrow G_{k}$ (see $\S 1$ for the
definition of $G_{X}^{(\operatorname{ab})}$ and note that
$G_{X}^{(\operatorname{ab})}$ is a quotient of
$G_{X}^{(\text{c}-\operatorname{ab})}$). Let
$G_{Y}\overset\text{def}\to{=}\operatorname{Gal}(\overline{K}/K)$ be the
absolute Galois group of the function field $K$ of $Y$ and
$G_{Y}^{(\operatorname{ab})}\overset\text{def}\to{=}G_{Y}/\operatorname{Ker}[\operatorname{Gal}(\overline{K}/K\bar{k})\twoheadrightarrow\operatorname{Gal}(\overline{K}/K\bar{k})^{\operatorname{ab}}]$
its geometrically abelian quotient. We have a commutative diagram

$\begin{CD}G_{X}^{(\operatorname{ab})}@>{}>{}>G_{k}\\\ @V{}V{}V\Big{\|}\\\
G_{Y}^{(\operatorname{ab})}@>{}>{}>G_{k}\\\ \end{CD}$

where the left vertical map is induced by the natural map $G_{X}\to G_{Y}$,
which is induced by the scheme morphism $X\to Y$ in case $X$ is a formal
$p$-adic germ, and by the rigid analytic morphism $X\to
Y^{\operatorname{rig}}$ and the rigid GAGA functor in case $X$ is affinoid.
The section $\tilde{s}:G_{k}\to G_{X}^{(\operatorname{ab})}$ induces a section
$s^{\dag}:G_{k}\to G_{Y}^{(\operatorname{ab})}$ of the projection
$G_{Y}^{(\operatorname{ab})}\twoheadrightarrow G_{k}$ (cf. above diagram). The
existence of the section $s^{\dag}$ implies that $\operatorname{index}(Y)=1$
as was observed by Esnault and Wittenberg (see [Esnault-Wittenberg] Remark
2.3(ii), and [Saïdi5] Theorem A for a more general result). ∎

## §3. Geometric sections of arithmetic fundamental groups

We investigate geometric sections of the projection
$\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$ (relative to a fixed
compactification of $X$). We use the notations in $\S 0$, $\S 1$, $\S 2$. We
further assume that $X$ possesses a $k$-compactification $Y$ with $Y$
hyperbolic (cf. Notations) which is fixed throughout $\S 3$.

Let

$s:G_{k}\to\pi_{1}(X,\eta)$

be a section of the projection $\pi_{1}(X,\eta)\twoheadrightarrow G_{k}$ fixed
throughout $\S 3$, which induces a (local) section

$s_{Y}:G_{k}\to\pi_{1}(Y,\eta)$

of the projection $\pi_{1}(Y,\eta)\twoheadrightarrow G_{k}$ [cf. diagram (0.1)
and $\S 0$].

We have an exact sequence

$1\to\Cal{I}_{Y}\to G_{Y}\to\pi_{1}(Y,\eta)\to 1$

where $G_{Y}=\operatorname{Gal}(\overline{K}/K)$ is the absolute Galois group
of the function field $K$ of $Y$ and
$\Cal{I}_{Y}\overset\text{def}\to{=}\operatorname{Ker}[G_{Y}\twoheadrightarrow\pi_{1}(Y,\eta)]$.
Let

$G_{Y}^{(\text{c}-\operatorname{ab})}\overset\text{def}\to{=}G_{Y}/\operatorname{Ker}(\Cal{I}_{Y}\twoheadrightarrow\Cal{I}_{Y}^{\operatorname{ab}}).$

Thus
$G_{Y}^{(\text{c}-\operatorname{ab})}=\varprojlim_{V}\pi_{1}(V,\eta)^{(\text{c}-\operatorname{ab})}$
where $V$ runs over all open sub-schemes of $Y$ [cf. [Saïdi1], 2.1.1, for the
definition of $\pi_{1}(V,\eta)^{(\text{c}-\operatorname{ab})}$].

###### Theorem 3.1

(Lifting of sections to cuspidally abelian Galois groups). We use the above
notations. The followings hold.

(i) There exists a section $s_{Y}^{\text{c}-\operatorname{ab}}:G_{k}\to
G_{Y}^{(\text{c}-\operatorname{ab})}$ of the projection
$G_{Y}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which lifts the
section $s_{Y}:G_{k}\to\pi_{1}(Y,\eta)$, i.e., which inserts in the following
commutative diagram

$\begin{CD}G_{k}@>{s_{Y}^{\text{c}-\operatorname{ab}}}>{}>G_{Y}^{(\text{c}-\operatorname{ab})}\\\
\Big{\|}@V{}V{}V\\\ G_{k}@>{s_{Y}}>{}>\pi_{1}(Y,\eta)\\\ \end{CD}$

where the right vertical map is the natural projection
$G_{Y}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow\pi_{1}(Y,\eta)$. In
particular, the set of sections of the projection
$G_{Y}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow G_{k}$ which lift the
section $s_{Y}$ is non-empty, and is (up to conjugation by elements of
$\Cal{I}_{Y}^{\operatorname{ab}}$) a torsor under
$H^{1}(G_{k},\Cal{I}_{Y}^{\operatorname{ab}})$.

(ii) The (local) section $s_{Y}:G_{k}\to\pi_{1}(Y,\eta)$ is uniformly
orthogonal to $\operatorname{Pic}$ in the sense of [Saïdi1], Definition 1.4.1.

###### Demonstration Proof

Assertion (i) follows from Theorem 2.4 and the fact that there exists a
natural homomorphism $G_{X}^{(\text{c}-\operatorname{ab})}\to
G_{Y}^{(\text{c}-\operatorname{ab})}$, induced by the natural homomorphism
$G_{X}\to G_{Y}$, which commutes with the projections to $G_{k}$. Assertion
(ii) follows from assertion (i) and Theorem 2.3.5 in [Saïdi1]. ∎

Consider the following push-out diagram

$\begin{CD}1@>{}>{}>\Cal{H}_{X}@>{}>{}>G_{X}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\Cal{H}_{X,1/p^{2}}@>{}>{}>G_{X}^{(1/p^{2}-\operatorname{sol})}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>1\\\
\end{CD}$

where $\Cal{H}_{X,1/p^{2}}$ is the maximal $1/p^{2}$-th solvable quotient of
$\Cal{H}_{X}$ and
$G_{X}^{(1/p^{2}-\operatorname{sol})}\overset\text{def}\to{=}G_{X}/\operatorname{Ker}(\Cal{H}_{X}\twoheadrightarrow\Cal{H}_{X,1/p^{2}})$.
Thus, $\Cal{H}_{X,1/p^{2}}$ is the maximal quotient of $\Cal{H}_{X}$ which is
abelian and annihilated by $p^{2}$ (cf. [Saïdi2], 1.2, for more details). We
have a commutative diagram of exact sequences

$\begin{CD}1@>{}>{}>\Cal{H}_{X,1/p^{2}}@>{}>{}>G_{X}^{(1/p^{2}-\operatorname{sol})}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>1\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
1@>{}>{}>\Cal{I}_{Y,1/p^{2}}@>{}>{}>G_{Y}^{(1/p^{2}-\operatorname{sol})}@>{}>{}>\pi_{1}(Y,\eta)@>{}>{}>1\\\
\end{CD}$ $None$

which is induced by the natural homomorphism $G_{X}\to G_{Y}$, where
$G_{Y}^{(1/p^{2}-\operatorname{sol})}$ is defined in a similar way to
$G_{X}^{(1/p^{2}-\operatorname{sol})}$. More precisely, $\Cal{I}_{Y,1/p^{2}}$
is the maximal quotient of $\Cal{I}_{Y}$ which is abelian and annihilated by
$p^{2}$ and
$G_{Y}^{(1/p^{2}-\operatorname{sol})}\overset\text{def}\to{=}G_{Y}/\operatorname{Ker}(\Cal{I}_{Y}\twoheadrightarrow\Cal{I}_{Y,1/p^{2}})$
is the geometrically cuspidally $1/p^{2}$-th step solvable quotient of $G_{Y}$
[cf. [Saïdi2], 3.1, recall the exact sequence $1\to\Cal{I}_{Y}\to
G_{Y}\to\pi_{1}(Y,\eta)\to 1$].

The following Proposition 3.2, item (i), is weaker than (and follows from)
Theorem 2.4, we state it in connection with Theorem 3.5.2 in this section.

###### Proposition 3.2 (Lifting of sections to cuspidally $1/p^2$-th step
solvable Galois groups)

We use the above notations. The followings hold.

(i) There exists a section $\tilde{s}:G_{k}\to
G_{X}^{(1/p^{2}-\operatorname{sol})}$ of the projection
$G_{X}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow G_{k}$ which lifts the
section $s:G_{k}\to\pi_{1}(X,\eta)$, i.e., which inserts in the following
commutative diagram

$\begin{CD}G_{k}@>{\tilde{s}}>{}>G_{X}^{(1/p^{2}-\operatorname{sol})}\\\
\Big{\|}@V{}V{}V\\\ G_{k}@>{s}>{}>\pi_{1}(X,\eta)\\\ \end{CD}$

where the right vertical map is the natural projection
$G_{X}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow\pi_{1}(X,\eta)$. In
particular, the set of sections of the projection
$G_{X}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow G_{k}$ which lift the
section $s$ is non-empty, and is (up to conjugation by elements of
$\Cal{H}_{X,1/p^{2}}$) a torsor under $H^{1}(G_{k},\Cal{H}_{X,1/p^{2}})$.

(ii) The section $\tilde{s}:G_{k}\to G_{X}^{(1/p^{2}-\operatorname{sol})}$ in
(i) induces a section $\tilde{s}_{Y}:G_{k}\to
G_{Y}^{(1/p^{2}-\operatorname{sol})}$ of the projection
$G_{Y}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow G_{k}$ which lifts the
section $s_{Y}:G_{k}\to\pi_{1}(Y,\eta)$. In particular, the (local) section
$s_{Y}:G_{k}\to\pi_{1}(Y,\eta)$ is uniformly orthogonal to
$\operatorname{Pic}$ mod-$p^{2}$ in the sense of [Saïdi2], Definition 3.4.1.

###### Demonstration Proof

Assertion (i) follows from Theorem 2.4 and the fact that there exists a
natural projection $G_{X}^{(\text{c}-\operatorname{ab})}\twoheadrightarrow
G_{X}^{(1/p^{2}-\operatorname{sol})}$ which commutes with the projections onto
$G_{k}$. Assertion (ii) follows from (i) and the fact that there exists a
natural homomorphism $G_{X}^{(1/p^{2}-\operatorname{sol})}\to
G_{Y}^{(1/p^{2}-\operatorname{sol})}$, induced by the homomorphism $G_{X}\to
G_{Y}$, which commutes with the projections onto $G_{k}$ (cf. diagram (3.1),
and [Saïdi2] Theorem 3.4.4). ∎

## 3.3

Write

$\Pi_{Y}[X]\overset\text{def}\to{=}\varprojlim_{T\subset Y\setminus
X}\pi_{1}(Y\setminus T,\eta)$

and

$\Pi_{Y}[X]^{\operatorname{geo}}\overset\text{def}\to{=}\varprojlim_{T\subset
Y\setminus X}\pi_{1}(Y\setminus T,\eta)^{\operatorname{geo}}$

where the limits are over all subsets $T$ consisting of finitely many closed
points of $Y$ not in $X$ (cf. Notations), $Y\setminus T$ is the corresponding
(affine if $T$ is non-empty) curve, and $\pi_{1}(Y\setminus
T,\eta)^{\operatorname{geo}}\overset\text{def}\to{=}\operatorname{Ker}[\pi_{1}(Y\setminus
T,\eta)\twoheadrightarrow G_{k}]$. We have the following commutative diagram
of exact sequences

$\begin{CD}1@>{}>{}>\pi_{1}(X,\eta)^{\operatorname{geo}}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\Pi_{Y}[X]^{\operatorname{geo}}@>{}>{}>\Pi_{Y}[X]@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\pi_{1}(Y_{\bar{k}},\bar{\eta})@>{}>{}>\pi_{1}(Y,\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
\end{CD}$ $None$

where the middle upper map is induced by the rigid analytic morphism $X\to
Y^{\operatorname{rig}}$ and the rigid GAGA functor in case $X$ is affinoid,
and the scheme morphism $X\to Y$ in case $X$ is a formal $p$-adic germ. The
left and middle lower vertical maps are the natural projections (they are
surjective).

###### Proposition 3.3.1

We use the above notations. The left and middle upper vertical maps in diagram
(3.2) are injective in the case $X$ is affinoid.

###### Demonstration Proof

The first assertion follows from Theorem A in [Saïdi3] (see the comments in
the proof of Proposition 1.1). The second assertion follows from the first and
the commutativity of the upper part in diagram (3.2). ∎

The section $s:G_{k}\to\pi_{1}(X,\eta)$ induces a section (denoted also $s$)

$s:G_{k}\to\Pi_{Y}[X]$

of the projections $\Pi_{Y}[X]\twoheadrightarrow G_{k}$ [cf. diagram (3.2)].

###### Definition Definition 3.3.2

We say that the section $s$ is geometric, relative to $Y$, if the image
$s(G_{k})$ of the section $s:G_{k}\to\Pi_{Y}[X]$ is contained in a
decomposition group $D_{x}\subset\Pi_{Y}[X]$ associated to a rational point
$x\in Y(k)$.

Note that if $s$ is geometric in the above sense, associated to $x\in Y(k)$,
then the (local) section $s_{Y}:G_{k}\to\pi_{1}(Y,\eta)$ of the projection
$\pi_{1}(Y,\eta)\twoheadrightarrow G_{k}$ induced by $s$ is geometric and is
associated to $x\in Y(k)$, i.e., $s_{Y}(G_{k})$ is contained in (hence equal
to) a decomposition group $D_{x}\subset\pi_{1}(Y,\eta)$ associated to $x$.

## 3.4

In this sub-section we assume that
$X=\operatorname{Spec}(A\otimes_{\Cal{O}_{k}}k)$ is a formal $p$-adic germ.

Let $\Cal{Y}\to\operatorname{Spec}\Cal{O}_{k}$ be a model of $Y$,
$y\in\Cal{Y}^{\operatorname{cl}}$ a closed point, and
$\hat{\Cal{O}}_{\Cal{Y},y}\overset\sim\to{\rightarrow}A$ an isomorphism (cf.
loc. cit.). Let
$\Cal{Y}_{F}\overset\text{def}\to{=}\Cal{Y}\times_{\operatorname{Spec}\Cal{O}_{k}}\operatorname{Spec}F$
be the special fibre of $\Cal{Y}$. Consider the following assumption (*):

(*) The gcd of the total multiplicities of the irreducible components of
$\Cal{Y}_{F}$ is $1$.

Let $\xi$ be a geometric point of $\Cal{Y}_{F}$ with values in the generic
point of an irreducible component $Y_{i_{0}}$ of $\Cal{Y}_{F}$. Thus $\xi$
determines an algebraic closure $\overline{F}$ of $F$. We have the following
commutative diagram of exact sequences

$\begin{CD}1@>{}>{}>\pi_{1}(X,\eta)^{\operatorname{geo}}@>{}>{}>\pi_{1}(X,\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\pi_{1}(Y_{\bar{k}},\bar{\eta})@>{}>{}>\pi_{1}(Y,\eta)@>{}>{}>G_{k}@>{}>{}>1\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
1@>{}>{}>\pi_{1}(\Cal{Y}_{\overline{F}},\bar{\xi})@>{}>{}>\pi_{1}(\Cal{Y}_{F},\xi)@>{}>{}>G_{F}@>{}>{}>1\\\
\end{CD}$ $None$

where the middle upper map is induced by the scheme morphism $X\to Y$, the
lower middle map (which is defined up to conjugation) is a specialisation map,
$\pi_{1}(\Cal{Y}_{F},\xi)$ [resp. $\pi_{1}(\Cal{Y}_{\overline{F}},\bar{\xi})$]
is the fundamental group of $\Cal{Y}$ (resp.
$\Cal{Y}_{\overline{F}}\overset\text{def}\to{=}\Cal{Y}\times_{\operatorname{Spec}\Cal{O}_{k}}\operatorname{Spec}\overline{F}$)
with geometric point $\xi$ (resp. $\bar{\xi}$ which is induced by $\xi$),
$G_{F}\overset\text{def}\to{=}\operatorname{Gal}(\overline{F}/F)$, and the
lower right vertical map is the natural projection $G_{k}\twoheadrightarrow
G_{F}$ (cf. [Saïdi6], diagram (0.1), and the discussion thereafter). The left
(hence also the middle) lower vertical map in diagram (3.3) is surjective
under the assumption (*) (cf. loc. cit. and the references therein).

The section $s:G_{k}\to\pi_{1}(X,\eta)$ induces the (local) section
$s_{Y}:G_{k}\to\pi_{1}(Y,\eta)$ of the projection
$\pi_{1}(Y,\eta)\twoheadrightarrow G_{k}$, as well as a homomorphism

$\tilde{s}:G_{k}\to\pi_{1}(\Cal{Y}_{F},\xi)$

obtained by composing the section $s_{Y}:G_{k}\to\pi_{1}(Y,\eta)$ with the
specialisation map $\pi_{1}(Y,\eta)\twoheadrightarrow\pi_{1}(\Cal{Y}_{F},\xi)$
in diagram (3.3).

###### Lemma 3.4.1

We use the above notations. The followings hold.

(i) The closed point $y\in\Cal{Y}^{\operatorname{cl}}$ is an $F$-rational
point.

(ii) The section $s_{Y}$ is unramified: the homomorphism
$\tilde{s}:G_{k}\to\pi_{1}(\Cal{Y}_{F},\xi)$ factors through $G_{F}$ and
induces a section $\bar{s}_{Y}:G_{F}\to\pi_{1}(\Cal{Y}_{F},\xi)$ of the
natural projection $\pi_{1}(\Cal{Y}_{F},\xi)\twoheadrightarrow G_{F}$.

(iii) The section $\bar{s}_{Y}:G_{F}\to\pi_{1}(\Cal{Y}_{F},\xi)$ in (ii) is
geometric and arises from the rational point $y$, i.e., arises from the
scheme-theoretic morphism $y:\operatorname{Spec}F\to\Cal{Y}_{F}$.

(iv) Assume that $\Cal{Y}$ is regular. Then condition (*) holds.

###### Demonstration Proof

Assertion (i) is clear (recall
$\hat{\Cal{O}}_{\Cal{Y},y}\overset\sim\to{\rightarrow}A$), it also follows
from (ii). We prove (ii).

We have a commutative diagram of scheme morphisms

$\begin{CD}X@>{}>{}>Y\\\ @V{}V{}V@V{}V{}V\\\
\operatorname{Spec}A@>{}>{}>\Cal{Y}\\\ @A{}A{}A@A{}A{}A\\\
\operatorname{Spec}(F)@>{y}>{}>\Cal{Y}_{F}\\\ \end{CD}$ $None$

where the lower horizontal morphism is induced by the closed point $y$ of
$\Cal{Y}_{F}$, and the lower vertical morphisms are closed immersions. This
diagram gives rise to a commutative diagram of homomorphisms between
fundamental groups

$\begin{CD}\pi_{1}(X,\eta)@>{}>{}>\pi_{1}(Y,\eta)\\\ @V{}V{}V@V{}V{}V\\\
\pi_{1}(\operatorname{Spec}A,\eta)@>{}>{}>\pi_{1}(\Cal{Y},\eta)\\\
@A{\tau_{y}}A{}A@A{\sigma}A{}A\\\
G_{F}@>{s_{y}}>{}>\pi_{1}(\Cal{Y}_{F},\xi)\\\ \end{CD}$ $None$

where the lower horizontal map is a section of the projection
$\pi_{1}(\Cal{Y}_{F},\xi)\twoheadrightarrow G_{F}$ arising from the
$F$-rational point $y\in\Cal{Y}_{F}$, and is defined up to conjugation, the
lower vertical maps are induced by the lower vertical maps in diagram (3.4)
(they are defined up to conjugation) and are isomorphisms (cf. [Grothendieck],
Exposé X, Théorème 2.1, for the right vertical map $\sigma$ being an
isomorphism). Further, the composite
$\psi:\pi_{1}(X,\eta)\to\pi_{1}(\operatorname{Spec}A,\eta)@>\tau_{y}^{-1}>>G_{F}@>s_{y}>>\pi_{1}(\Cal{Y}_{F},\xi)$
is the composite of the middle vertical maps in diagram (3.3) as follows from
the definition of the specialisation map
$\pi_{1}(Y,\eta)\to\pi_{1}(\Cal{Y}_{F},\xi)$: this map is the composite of the
maps
$\pi_{1}(Y,\eta)\to\pi_{1}(\Cal{Y},\eta)@>{\sigma^{-1}}>>\pi_{1}(\Cal{Y}_{F},\xi)$.
In particular, the homomorphism $\tilde{s}:G_{k}\to\pi_{1}(\Cal{Y}_{F},\xi)$
factors through $G_{F}$ and induces a section
$\bar{s}_{Y}:G_{F}\to\pi_{1}(\Cal{Y}_{F},\xi)$ of the natural projection
$\pi_{1}(\Cal{Y}_{F},\xi)\twoheadrightarrow G_{F}$. This shows (ii). The
section $\bar{s}_{Y}$ coincides (up to conjugation) with the section
$G_{F}@>s_{y}>>\pi_{1}(\Cal{Y}_{F},\xi)$ in diagram (3.5), hence is geometric
and arises from the $F$-rational point $y$ as claimed in (iii). The last
assertion follows from Theorem 2.5 and the well known fact that if $\Cal{Y}$
is regular then the gcd of the total multiplicities of the irreducible
components of $\Cal{Y}_{F}$ divides $\operatorname{index}(Y)$ (cf. for example
[Gabber-Liu-Lorenzini], Theorem 8.2 and Remark 8.6). ∎

###### Definition Remark 3.4.2

Assume that the morphism $\Cal{Y}\to\operatorname{Spec}\Cal{O}_{k}$ is smooth.
If $s$ is geometric, and arises from the rational point $x\in Y(k)$ (cf.
Definition 3.3.2), it follows from Lemma 3.4.1(iii) and the fact that
$\Cal{Y}_{F}$ is hyperbolic that the point $x$ specialises in $y$ necessarily
[cf. [Tamagawa], Proposition (2.8)(i)]. In particular, the point $x$ is the
image of a (unique) $k$-rational point $\tilde{x}\in X(k)$ via the morphism
$X\to Y$. The fact that $s_{Y}(G_{k})=D_{x}\subset\pi_{1}(Y,\eta)$ doesn’t
imply a priori that the image $s(G_{k})$ via the section
$s:G_{k}\to\pi_{1}(X,\eta)$ is contained in a decomposition group
$D_{\tilde{x}}\subset\pi_{1}(X,\eta)$ associated to $\tilde{x}$.

## 3.5

Let $H\subset\Pi_{Y}[X]$ be an open subgroup with $s(G_{k})\subset H$ [recall
$s:G_{k}\to\Pi_{Y}[X]$ is the section induced by $s:G_{k}\to\pi_{1}(X,\eta)$].
Thus, $H$ corresponds to a (possibly ramified) finite cover $Y^{\prime}\to Y$
with $Y^{\prime}$ geometrically connected. Let
$H^{\prime}\subset\pi_{1}(X,\eta)$ be the inverse image of $H$ via the
homomorphism $\pi_{1}(X,\eta)\to\Pi_{Y}[X]$ (cf. diagram (3.2)). Thus,
$H^{\prime}$ is an open subgroup of $\pi_{1}(X,\eta)$ containing the image of
the section $s:G_{k}\to\pi_{1}(X,\eta)$ and corresponds to an étale cover
$X^{\prime}\to X$ with $X^{\prime}$ geometrically connected. There is a
natural morphism $X^{\prime}\to(Y^{\prime})^{\operatorname{rig}}$ of rigid
analytic spaces in case $X$ is affinoid, and a natural scheme morphism
$X^{\prime}\to Y^{\prime}$ in case $X$ is a formal $p$-adic germ. The generic
point $\eta$ induces naturally a generic point (denoted also $\eta$) of
$X^{\prime}$ and $Y^{\prime}$. Further, we have a natural identification
$H^{\prime}=\pi_{1}(X^{\prime},\eta)$ and a natural homomorphism
$\pi_{1}(X^{\prime},\eta)\to\pi_{1}(Y^{\prime},\eta)$ which commutes with the
projections onto $G_{k}$.

The section $s:G_{k}\to\pi_{1}(X,\eta)$ induces naturally sections
$s^{\prime}:G_{k}\to\pi_{1}(X^{\prime},\eta)$ and
$s_{Y^{\prime}}:G_{k}\to\pi_{1}(Y^{\prime},\eta)$ of the natural projections
$\pi_{1}(X^{\prime},\eta)\twoheadrightarrow G_{k}$ and
$\pi_{1}(Y^{\prime},\eta)\twoheadrightarrow G_{k}$; respectively. The section
$s^{\prime}:G_{k}\to\pi_{1}(X^{\prime},\eta)$ lifts to a section
$\tilde{s}^{\prime}:G_{k}\to G_{X^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ of
the projection
$G_{X^{\prime}}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow G_{k}$ and
induces a section $\tilde{s}_{Y^{\prime}}:G_{k}\to
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ of the projection
$G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow G_{k}$ (cf.
Proposition 3.2). Let $F\subset G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$
be an open subgroup with $\tilde{s}_{Y^{\prime}}(G_{k})\subset F$. Thus $F$
corresponds to a (possibly ramified) finite cover $Y^{\prime\prime}\to
Y^{\prime}$ with $Y^{\prime\prime}$ geometrically connected. The generic point
$\eta$ induces naturally a generic point (denoted also $\eta$) of
$Y^{\prime\prime}$. Write
$\pi_{1}(Y^{\prime\prime},\eta)^{(1/p-\operatorname{sol})}$ for the
geometrically $1/p$-th step solvable quotient of
$\pi_{1}(Y^{\prime\prime},\eta)$ which sits in the following exact sequence

$1@>>>\pi_{1}(Y^{\prime\prime}_{\bar{k}},\bar{\eta})_{1/p}@>>>\pi_{1}(Y^{\prime\prime},\eta)^{(1/p-\operatorname{sol})}@>>>G_{k}@>>>1,$
$None$

where $\pi_{1}(Y^{\prime\prime}_{\bar{k}},\bar{\eta})_{1/p}$ is the maximal
$1/p$-th step solvable quotient of
$\pi_{1}(Y^{\prime\prime}_{\bar{k}},\bar{\eta})$ (cf. [Saïdi2], 1.2) and the
generic point $\bar{\eta}$ is induced by $\eta$. Thus
$\pi_{1}(Y^{\prime\prime}_{\bar{k}},\bar{\eta})_{1/p}$ is the maximal quotient
of $\pi_{1}(Y^{\prime\prime}_{\bar{k}},\bar{\eta})$ which is abelian and
annihilated by $p$ (cf. loc. cit.)

###### Definition Definition 3.5.1

We use the above notations. We say that the section $s$ is admissible,
relative to $Y$, if for every open subgroup $H\subset\Pi_{Y}[X]$ with
$s(G_{k})\subset H$, corresponding to (a possibly ramified) cover
$Y^{\prime}\to Y$, the following holds. There exists a section
$\tilde{s}_{Y^{\prime}}:G_{k}\to
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ of the projection
$G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow G_{k}$ [such
a section exists unconditionally (see above discussion)] satisfying the
following property: For each open subgroup $F\subset
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ with
$\tilde{s}_{Y^{\prime}}(G_{k})\subset F$, corresponding to a (possibly
ramified) cover $Y^{\prime\prime}\to Y^{\prime}$ with $Y^{\prime\prime}$
geometrically connected, the natural projection
$\pi_{1}(Y^{\prime\prime},\eta)^{(1/p-\operatorname{sol})}\twoheadrightarrow
G_{k}$ splits (cf. above discussion). Note that this latter condition is
equivalent to (cf. [Saïdi2] Lemma 3.4.8): The class of
$\operatorname{Pic}^{1}_{Y^{\prime\prime}}$ in
$H^{1}(G_{k},\operatorname{Pic}^{0}_{Y^{\prime\prime}})$ is divisible by $p$.

Our main result in this section is the following.

###### Theorem 3.5.2

We use the above notations. The section $s:G_{k}\to\pi_{1}(X,\eta)$ is
geometric relative to $Y$ (cf. Definition 3.3.2) if and only if $s$ is
admissible relative to $Y$ (cf. Definition 3.5.1).

###### Demonstration Proof

Assume first that the section $s:G_{k}\to\pi_{1}(X,\eta)$ is admissible
(relative to $Y$). We prove that $s$ is geometric (relative to $Y$). Using a
well-known limit argument due to Tamagawa [cf. [Tamagawa], Proposition
2.8(iv)] it suffices to show the following. For every open subgroup
$H\subset\Pi_{Y}[X]$ with $s(G_{k})\subset H$, corresponding to (a possibly
ramified) cover $Y^{\prime}\to Y$ with $Y^{\prime}$ hyperbolic,
$Y^{\prime}(k)\neq\emptyset$ holds. By assumption there exists a section
$\tilde{s}_{Y^{\prime}}:G_{k}\to
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ of the projection
$G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow G_{k}$
satisfying the condition in Definition 3.5.1. In [Saïdi2], 3.3, we defined a
certain quotient $G_{Y^{\prime}}\twoheadrightarrow
G_{Y^{\prime}}^{(p,2)}\twoheadrightarrow
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ of $G_{Y^{\prime}}$ (we refer
to loc. cit. for more details on the definition of $G_{Y^{\prime}}^{(p,2)}$).
Let $F\subset G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ be an open
subgroup with $\tilde{s}_{Y^{\prime}}(G_{k})\subset F$ corresponding to a
(possibly ramified) cover $Y^{\prime\prime}\to Y^{\prime}$ with
$Y^{\prime\prime}$ geometrically connected. By assumption the natural
projection
$\pi_{1}(Y^{\prime\prime},\eta)^{(1/p-\operatorname{sol})}\twoheadrightarrow
G_{k}$ splits (cf. Definition 3.5.1). This latter condition (for every $F$ as
above) implies that (in fact is equivalent to) the section
$\tilde{s}_{Y^{\prime}}:G_{k}\to
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ lifts to a section
$s_{Y^{\prime}}^{\dag}:G_{k}\to G_{Y^{\prime}}^{(p,2)}$ of the projection
$G_{Y^{\prime}}^{(p,2)}\twoheadrightarrow G_{k}$ (cf. [Saïdi2] Theorem 3.4.10
and Lemma 3.4.8). Further, the existence of the section
$s_{Y^{\prime}}^{\dag}:G_{k}\to G_{Y^{\prime}}^{(p,2)}$ as above implies that
$Y^{\prime}(k)\neq\emptyset$ by [Saïdi2], Proposition 4.6, as required.

Next, we assume that $s$ is geometric (relative to $Y$) and prove that $s$ is
admissible (relative to $Y$). By assumption $s(G_{k})$ is contained in
$D_{x}\subset\Pi_{Y}[X]$ where $D_{x}$ is a decomposition group associated to
a rational point $x\in Y(k)$. Let $H\subset\Pi_{Y}[X]$ be an open subgroup
with $s(G_{k})\subset H$ corresponding to (a possibly ramified) cover
$Y^{\prime}\to Y$. Then $Y^{\prime}(k)\neq\emptyset$. A rational point
$x^{\prime}\in Y^{\prime}(k)$ gives rise to a section
$\tilde{s}_{Y^{\prime}}:G_{k}\to
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ of the projection
$G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}\twoheadrightarrow G_{k}$. Let
$F\subset G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ be an open subgroup
with $\tilde{s}_{Y^{\prime}}(G_{k})\subset F$ corresponding to a (possibly
ramified) cover $Y^{\prime\prime}\to Y^{\prime}$ with $Y^{\prime\prime}$
geometrically connected. Then $Y^{\prime\prime}(k)\neq\emptyset$ holds since
the section $\tilde{s}_{Y^{\prime}}:G_{k}\to
G_{Y^{\prime}}^{(1/p^{2}-\operatorname{sol})}$ arises from the rational point
$x^{\prime}$ and $\tilde{s}_{Y^{\prime}}(G_{k})\subset F$. In particular, the
natural projection $\pi_{1}(Y^{\prime\prime},\eta)\twoheadrightarrow G_{k}$,
and a fortiori the projection
$\pi_{1}(Y^{\prime\prime},\eta)^{(1/p-\operatorname{sol})}\twoheadrightarrow
G_{k}$, splits. Thus $s$ is admissible as required. ∎

## §4. Picard groups of affinoid $p$-adic curves

The following is our main result in this section, it may be of interest
independently of the topics discussed in $\S 1$, $\S 2$, and $\S 3$.

###### Proposition 4.1

Let $X=\operatorname{Sp}(A)$ be a smooth and geometrically connected
$k$-affinoid curve. Then the Picard group $\operatorname{Pic}(X)$ is finite.

The rest of this section is devoted to the proof of Proposition 4.1.

Let $\Cal{X}=\operatorname{Spf}B$ be an excellent normal $\Cal{O}_{k}$-formal
scheme of finite type with generic fibre $X$, i.e., $A=B\otimes_{R}k$. Write
$\Cal{X}^{\operatorname{reg}}$ for the set of regular points of $\Cal{X}$.
Thus $\Cal{X}\setminus\Cal{X}^{\operatorname{reg}}=\\{z_{1},\ldots,z_{t}\\}$
consists of finitely many closed points of $\Cal{X}$. By Lipman’s theorem of
resolution of singularities for excellent $2$-dimensional schemes there exists
a birational and proper morphism $\lambda:\Cal{S}\to\Cal{X}$ with $\Cal{S}$
regular and
$\lambda^{-1}(\Cal{X}^{\operatorname{reg}})\to\Cal{X}^{\operatorname{reg}}$ an
isomorphism (cf. [Lipman]. Here we view $\Cal{X}$ as the ordinary affine
scheme $\operatorname{Spec}B$). For $n\geq 1$, write
$B_{n}\overset\text{def}\to{=}B/(\pi^{n})$,
$\Cal{X}_{n}\overset\text{def}\to{=}\operatorname{Spec}B_{n}$, and
$\Cal{S}_{n}\overset\text{def}\to{=}\Cal{S}\times_{\Cal{X}}\Cal{X}_{n}$.
Further denote
$\Cal{X}_{0}\overset\text{def}\to{=}\Cal{X}_{n}^{\operatorname{red}}$, and
$\Cal{S}_{0}\overset\text{def}\to{=}\Cal{S}_{n}^{\operatorname{red}}$. Thus
$\Cal{X}_{0}$ and $\Cal{S}_{0}$ are one dimensional reduced schemes over $F$.
Further, there exists a morphism $\lambda:\Cal{S}\to\Cal{X}$ as above with
$S_{0}$ a divisor with strict normal crossings (cf. [Cossart-Jannsen-Saito],
Corollary 0.4), which we assume from now on.

We have a surjective homomorphism
$\operatorname{Pic}(\Cal{X}^{\operatorname{reg}})\twoheadrightarrow\operatorname{Pic}(X)$.
To prove $\operatorname{Pic}(X)$ is finite it suffices to prove that
$\operatorname{Pic}(\Cal{X}^{\operatorname{reg}})$ is finite. For each
singular point $z_{i}$ of $\Cal{X}$ let
$E_{i}\overset\text{def}\to{=}\lambda^{-1}(z_{i})^{\operatorname{red}}$ and
$\\{D_{i,j}\\}_{1\leq j\leq n_{i}}$ the set of irreducible components of
$E_{i}$, $1\leq i\leq t$. Thus $E_{i}$ is a reduced proper curve over the
residue field $k(z_{i})$ at $z_{i}$ which is a finite field. We have an exact
sequence

$M\overset\text{def}\to{=}\oplus_{i=1}^{t}(\oplus_{j=1}^{n_{i}}\mathbb{Z})@>\beta>>\operatorname{Pic}(\Cal{S})\to\operatorname{Pic}(\Cal{X}^{\operatorname{reg}})\to
0$

where $\beta$ maps the copy of $\mathbb{Z}$ indexed by the pair $(i,j)$ to the
class of the divisor $D_{i,j}$. Further we have an isomorphism

$\operatorname{Pic}(\Cal{S})\overset\sim\to{\rightarrow}\varprojlim_{n\geq
1}\operatorname{Pic}(\Cal{S}_{n})$

(cf. [EGA III], première partie, Corollaire 5.1.6).

###### Lemma 4.2

We use notations as above. To prove that
$\operatorname{Pic}(\Cal{X}^{\operatorname{reg}})$ is finite it suffices to
prove the following two assertions:

(A) The cokernel of the composite map

$\phi_{n}:M\overset\text{def}\to{=}\oplus_{i=1}^{t}(\oplus_{j=1}^{n_{i}}\mathbb{Z})@>\beta>>\operatorname{Pic}(\Cal{S})\to\operatorname{Pic}(\Cal{S}_{n})$

is finite for $n\geq 1$.

(B) There exists $n_{0}>0$ such that the map

$\operatorname{Pic}(\Cal{S}_{n+1})\to\operatorname{Pic}(\Cal{S}_{n})$

is an isomorphism for $n>n_{0}$.

###### Demonstration Proof of Lemma 4.2

Follows from the above discussion and the fact that we have an exact sequence

$M\to\varprojlim_{n\geq 1}\operatorname{Pic}(\Cal{S}_{n})\to\varprojlim_{n\geq
1}\text{coker}(\phi_{n})\to 0$

where the first map is induced by the maps
$\phi_{n}:M\to\operatorname{Pic}(\Cal{S}_{n})$, $n\geq 1$, and
$\varprojlim_{n\geq 1}\text{coker}(\phi_{n})$ is finite if assertions (A) and
(B) are satisfied.

This finishes the proof of Lemma 4.2. ∎

The rest of this section is devoted to the proofs of assertions (A) and (B).

###### Demonstration Proof of assertion (A)

Let $\\{\eta_{r}\\}_{r=1}^{s}$ be the generic points of $\Cal{X}_{0}$,
$\rho:\Cal{S}_{0}^{\text{nor}}\to\Cal{S}_{0}$ the morphism of normalisation,
$\widetilde{E}_{i}\overset\text{def}\to{=}\rho^{-1}(E_{i})$, $1\leq i\leq t$,
and $H_{r}=\overline{\\{\eta_{r}\\}}$ the closure in
$\Cal{S}_{0}^{\text{nor}}$ of the (inverse image in $\Cal{S}_{0}$ of the)
generic point $\eta_{r}$ of $\Cal{X}_{0}$, $1\leq r\leq s$. Thus $H_{r}$ is a
connected affine normal one dimensional scheme over $F$. Let

$d:\operatorname{Pic}(\Cal{S}_{0})@>\rho^{*}>>\operatorname{Pic}(\Cal{S}_{0}^{\text{nor}})@>\text{deg}>>M=\oplus_{i=1}^{t}(\oplus_{j=1}^{n_{i}}\mathbb{Z})$

be the composite map where the first map is the pullback of line bundles via
the normalisation morphism $\rho:\Cal{S}_{0}^{\text{nor}}\to\Cal{S}_{0}$, and
the map deg is obtained by taking the degree of a line bundle on each
irreducible component $D_{i,j}$ of $E_{i}$.

###### Claim 1

$\text{ker}(d)$ is finite

###### Demonstration Proof of Claim 1

We have a commutative diagram of exact sequences

$\begin{CD}000\\\ @V{}V{}V@V{}V{}V@V{}V{}V\\\
0@>{}>{}>A_{1}@>{}>{}>\text{ker}(d)@>{}>{}>\ker(\deg)=\oplus_{r=1}^{s}\operatorname{Pic}(H_{r})\oplus(\oplus_{i=1}^{t}\operatorname{Pic}^{0}(\widetilde{E}_{i}))\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
0@>{}>{}>A_{2}@>{}>{}>\operatorname{Pic}(\Cal{S}_{0})@>{}>{}>\operatorname{Pic}(S_{0}^{\text{nor}})=\oplus_{r=1}^{s}\operatorname{Pic}(H_{r})\oplus(\oplus_{i=1}^{t}\operatorname{Pic}(\widetilde{E}_{i}))\\\
@V{d}V{}V@V{\text{deg}}V{}V\\\ M@=M\\\ \end{CD}$

where $A_{1}$ and $A_{2}$ are defined so that the above sequences are exact,
and $A_{2}$ is finite as follows from the facts that the sheaf
$\rho_{*}(\Cal{O}_{S_{0}^{\text{nor}}}^{\times})/\Cal{O}_{S_{0}}^{\times}$ is
a skyscraper sheaf and the residue fields at closed points of $\Cal{S}_{0}$
are finite fields. The kernel
$\text{ker}(\text{deg})=\oplus_{r=1}^{s}\operatorname{Pic}(H_{r})\oplus(\oplus_{i=1}^{t}\operatorname{Pic}^{0}(\widetilde{E}_{i}))$
of the right lower vertical map is finite:
$\operatorname{Pic}^{0}(\widetilde{E}_{i})$ is finite since
$\widetilde{E}_{i}$ is a proper and non-singular curve over a finite field,
and for $1\leq r\leq s$ it holds $\operatorname{Pic}(H_{r})$ is finite since
$H_{r}$ is an affine and normal $1$-dimensional scheme of finite type over the
finite field $F$. Indeed assume for simplicity that $H_{r}$ is geometrically
connected over $F$. Let $\ell/F$ be a finite extension such that
$U_{r}\overset\text{def}\to{=}H_{r}\times_{\operatorname{Spec}F}\operatorname{Spec}\ell$
admits a smooth and connected compactification $C_{r}$ with $(C_{r}\setminus
U_{r})(\ell)\neq\emptyset$. Let $U_{r}\to H_{r}$ be the canonical morphism and
$\operatorname{Pic}(H_{r})\to\operatorname{Pic}(U_{r})$ the induced map of
pull-back of line bundles. Then
$\operatorname{Ker}[\operatorname{Pic}(H_{r})\to\operatorname{Pic}(U_{r})]$ is
finite (cf. [Guralnick-Jaffe-Raskind], Theorem 1.8). Further, the map
$\operatorname{Pic}^{0}(C_{r})\to\operatorname{Pic}(U_{r})$ obtained by
restricting a degree $0$ line bundle on $C_{r}$ to $U_{r}$ is surjective [if
$x\in(C_{r}\setminus U_{r})(\ell)$ and $D\in\operatorname{Pic}(U_{r})$ has
degree $m$ then $D-mx\in\operatorname{Pic}^{0}(C_{r})$ restricts to $D$ on
$U_{r}$] hence $\operatorname{Pic}(U_{r})$ is finite since
$\operatorname{Pic}^{0}(C_{r})$ is finite. From the above it follows that
$\operatorname{Pic}(H_{r})$ is finite.

This finishes the proof of Claim 1. ∎

Consider the composite map

$\psi_{n}:\operatorname{Pic}(\Cal{S}_{n})\to\operatorname{Pic}(\Cal{S}_{0})@>d>>M=\oplus_{i=1}^{t}(\oplus_{j=1}^{n_{i}}\mathbb{Z}).$

###### Claim 2

$\text{ker}(\psi_{n})$ is finite

###### Demonstration Proof of Claim 2

First we prove that the kernel of the map
$\operatorname{Pic}(\Cal{S}_{n})\to\operatorname{Pic}(\Cal{S}_{n-1})$ is
finite for $n\geq 2$. Write $\Cal{I}_{n}$ for the sheaf of ideals of
$\Cal{O}_{\Cal{S}}$ defining $\Cal{S}_{n}$. We have an exact sequence of
sheaves on $\Cal{S}_{n}$:

$1\to
1+(\Cal{I}_{n-1}/\Cal{I}_{n})\to\Cal{O}_{S_{n}}^{\times}\to\Cal{O}_{S_{n-1}}^{\times}\to
1$

which induces an exact sequence in cohomology

$H^{1}(\Cal{S}_{n},1+(\Cal{I}_{n-1}/\Cal{I}_{n}))\to\operatorname{Pic}(\Cal{S}_{n})\to\operatorname{Pic}(\Cal{S}_{n-1})\to
H^{2}(\Cal{S}_{n},1+(\Cal{I}_{n-1}/\Cal{I}_{n})).$

Further the truncated exponential map $\alpha\mapsto 1+\alpha$ induces an
isomorphism of sheaves
$\Cal{I}_{n-1}/\Cal{I}_{n}\overset\sim\to{\rightarrow}1+(\Cal{I}_{n-1}/\Cal{I}_{n})$
[$(\Cal{I}_{n-1}/\Cal{I}_{n})^{2}=0$], hence
$H^{2}(\Cal{S}_{n},1+\Cal{I}_{n-1}/\Cal{I}_{n})=0$ and the map
$\operatorname{Pic}(\Cal{S}_{n})\to\operatorname{Pic}(\Cal{S}_{n-1})$ is
surjective. Moreover, $H^{1}(\Cal{S}_{n},\Cal{I}_{n-1}/\Cal{I}_{n})$ is
finite. Indeed $H^{1}(\Cal{S}_{n},\Cal{I}_{n-1}/\Cal{I}_{n})$ is a finitely
generated $B_{n}$-module with finite support since the morphism
$\lambda_{n}^{-1}(\Cal{Z}_{n}\setminus\\{z_{1},\ldots,z_{t}\\})\to\Cal{Z}_{n}\setminus\\{z_{1},\ldots,z_{t}\\}$
is affine and $R^{1}(\pi_{n})_{*}(\Cal{I}_{n-1}/\Cal{I}_{n})$ is the sheaf
associated to the $B_{n}$-module
$H^{1}(\Cal{S}_{n},\Cal{I}_{n-1}/\Cal{I}_{n})$, here
$\lambda_{n}:\Cal{S}_{n}\to\Cal{Z}_{n}$ is the proper morphism induced by
$\lambda$. This shows the kernel of the map
$\operatorname{Pic}(\Cal{S}_{n})\to\operatorname{Pic}(\Cal{S}_{n-1})$ is
finite for all $n\geq 2$. A similar argument shows that the kernel of the map
$\operatorname{Pic}(\Cal{S}_{1})\to\operatorname{Pic}(\Cal{S}_{0})$ is finite.
Hence, using Claim 1, $\text{ker}(\psi_{n})$ is finite.

This finishes the proof of Claim 2. ∎

In light of Claim 2, and in order to prove assertion (A), it suffices to prove
that the cokernel of the composite map

$M\overset\text{def}\to{=}\oplus_{i=1}^{t}(\oplus_{j=1}^{n_{i}}\mathbb{Z})@>\beta>>\operatorname{Pic}(\Cal{S})\to\operatorname{Pic}(\Cal{S}_{n})\to\operatorname{Pic}(\Cal{S}_{0})@>d>>M=\oplus_{i=1}^{t}(\oplus_{j=1}^{n_{i}}\mathbb{Z})$

is finite. The latter follows from the nondegeneracy of the intersection
pairing
$(\oplus_{j=1}^{n_{i}}\mathbb{Z})\times(\oplus_{j=1}^{n_{i}}\mathbb{Z})\to\mathbb{Z}$
on each fibre $E_{i}$ (cf. [Shafarevich], Lemma in page 69 and the discussion
in page 71 after this Lemma), $1\leq i\leq t$.

This finishes the proof of assertion (A). ∎

###### Demonstration Proof of assertion (B)

Let $\Cal{J}$ be an ample invertible $\Cal{O}_{S}$-ideal such that
$\text{Supp}(\Cal{O}_{S}/\Cal{J})=\Cal{S}_{0}$. The existence of such
$\Cal{J}$ follows from the facts that $H_{r}$ is affine (cf. Proof of
Assertion A), $1\leq r\leq s$, the intersection pairing
$(\oplus_{j=1}^{n_{i}}\mathbb{Z})\times(\oplus_{j=1}^{n_{i}}\mathbb{Z})\to\mathbb{Z}$
on each fibre $E_{i}$ is negative definite (cf. [Shafarevich], Lemma in page
69 and the discussion in page 71 after this Lemma), and the numerical
criterion of ampleness on curves. More precisely, $\forall\ 1\leq i\leq t$,
one can find a divisor $D=\sum_{j=1}^{n_{i}}m_{ij}D_{i,j}$ with $m_{i,j}<0$
and $D.D_{i,j}>0$ for all $1\leq j\leq n_{j}$.

For $m\geq 1$, let $\Cal{S}^{\prime}_{m}$ be the closed subscheme of $\Cal{S}$
defined by the sheaf of ideals $\Cal{J}^{m}$. To prove Assertion B it suffices
to prove that there exists $m_{0}>0$ such that the map

$\operatorname{Pic}(\Cal{S}^{\prime}_{m+1})\to\operatorname{Pic}(\Cal{S}^{\prime}_{m})$

is an isomorphism for any $m>m_{0}$. We have an exact sequence of shaves on
$\Cal{S}^{\prime}_{m+1}$:

$1\to\Cal{J}^{m}/\Cal{J}^{m+1}\to\Cal{O}_{S^{\prime}_{m+1}}^{\times}\to\Cal{O}_{S^{\prime}_{m}}^{\times}\to
1$

where the map
$\Cal{J}^{m}/\Cal{J}^{m+1}\to\Cal{O}_{S^{\prime}_{m+1}}^{\times}$ maps a local
section $\alpha$ to $1+\alpha$, which induces an exact sequence in cohomology

$H^{1}(\Cal{S}^{\prime}_{m+1},\Cal{J}^{m}/\Cal{J}^{m+1})\to\operatorname{Pic}(\Cal{S}^{\prime}_{m+1})\to\operatorname{Pic}(\Cal{S}^{\prime}_{m})\to
0.$

Now there exists $m_{0}>0$ such that
$H^{1}(\Cal{S}^{\prime}_{m+1},\Cal{J}^{m}/\Cal{J}^{m+1})=0$ if $m\geq m_{0}$
by [EGA III], première partie, Proposition 2.2.1.

This finishes the proof of assertion (B). ∎

This finishes the proof of Proposition 4.1. ∎

## §5. Compactification of formal germs of $p$-adic curves

In this section we use the following notations: $K$ is a complete discrete
valuation field with valuation ring $R$, uniformising parameter $\pi$, and
with perfect residue field $\ell\overset\text{def}\to{=}R/\pi R$. Further, $A$
is a two dimensional normal complete local ring containing $R$ with maximal
ideal $\mathfrak{m}_{A}$ containing $\pi$ and residue field
$\ell=A/{\mathfrak{m}}_{A}$. We assume that
$X\overset\text{def}\to{=}\operatorname{Spec}(A\otimes_{R}K)$ is geometrically
connected. Given a finite extension $L/K$ we write $\Cal{O}_{L}$ for the
valuation ring of $L$, $A_{L}\overset\text{def}\to{=}A\otimes_{\Cal{O}_{L}}L$,
$A_{\Cal{O}_{L}}\overset\text{def}\to{=}A\otimes_{R}{\Cal{O}_{L}}$, and
$A_{\Cal{O}_{L}}^{\operatorname{nor}}$ the normalisation of $A_{\Cal{O}_{L}}$
in its total ring of fractions.

###### Proposition 5.1 (Compactification of formal germs of $p$-adic curves)

We use the above notations. There exists a finite extension $L/K$, a flat,
proper, connected, and normal $\Cal{O}_{L}$-relative curve
$\Cal{Y}\to\operatorname{Spec}\Cal{O}_{L}$, a closed point $y\in\Cal{Y}$, and
an isomorphism
$\hat{\Cal{O}}_{\Cal{Y},y}\overset\sim\to{\rightarrow}A_{\Cal{O}_{L}}^{\operatorname{nor}}$
where $\hat{\Cal{O}}_{\Cal{Y},y}$ is the completion of the local ring
$\Cal{O}_{\Cal{Y},y}$ of $\Cal{Y}$ at $y$.

###### Demonstration Proof

By the main result in [Epp], Introduction, there exists a finite extension
$L/K$ with uniformising parameter $\pi_{L}$ such that
$A_{\Cal{O}_{L}}^{\operatorname{nor}}/\pi_{L}A_{\Cal{O}_{L}}^{\operatorname{nor}}$
is reduced. Note that $A_{\Cal{O}_{L}}^{\operatorname{nor}}$ is a normal two
dimensional complete local ring with perfect residue field (cf. [Bourbaki],
Chapter IX, $\S 4$, Lemma 1, and our assumption that $X$ is geometrically
connected). Without loss of generality we will assume that $A/\pi A$ is
reduced. We show there exists a proper, flat, connected, and normal relative
$R$-curve $\Cal{Y}\to\operatorname{Spec}R$, a closed point $y\in\Cal{Y}$, and
an isomorphism $\hat{\Cal{O}}_{\Cal{Y},y}\overset\sim\to{\rightarrow}A$.

First, $A/\pi A$ is a (reduced) one dimensional complete local ring with
residue field $\ell$, hence is isomorphic to a quotient
$\ell[[x_{1},\ldots,x_{t}]]/{\mathfrak{a}}$ of a formal power series ring
$\ell[[x_{1},\ldots,x_{t}]]$ over $\ell$ (cf. [Bourbaki], Chapitre IX, $\S
3$). It then follows from [Artin], Theorem 3.8, and basic facts on the theory
of algebraic curves, that there exists a proper and reduced connected (but not
necessarily irreducible) $\ell$-curve $Z$, a closed point $y\in Z$, and an
isomorphism $\hat{\Cal{O}}_{Z,y}\overset\sim\to{\rightarrow}A/\pi A$ where
$\hat{\Cal{O}}_{Z,y}$ is the completion of the local ring $\Cal{O}_{Z,y}$ of
$Z$ at $y$. Moreover, $Z$ is non-singular outside $y$. There exists a rational
function $f$ on $Z$ which defines a finite generically separable morphism
$f:Z\to\mathbb{P}^{1}_{\ell}$ such that $y=f^{-1}(\infty)$ (cf. [Harbater-
Stevenson], proof of Theorem 3). Thus, by considering the completion of the
morphism $f$ above $\infty$, we obtain a finite generically separable morphism
$\bar{g}:\operatorname{Spec}(A/\pi A)\to\operatorname{Spec}(\ell[[t]])$ where
$t$ is a local parameter at $\infty$. This morphism lifts to a finite morphism
$g:\operatorname{Spf}A\to\operatorname{Spf}(R[[T]])$ of formal schemes (cf.
loc. cit., Lemma 2). Let $\widetilde{Z}\to Z$ be the morphism of normalisation
and $\\{x_{1},\ldots,x_{m}\\}\subset\widetilde{Z}$ the pre-image of $y$. There
is a one-to-one correspondence between the set
$\\{\mathfrak{p}_{1},\ldots,\mathfrak{p}_{m}\\}\subset\operatorname{Spec}A$ of
prime ideals of height $1$ containing $\pi$ and the set
$\\{x_{1},\ldots,x_{m}\\}$, $\mathfrak{p}_{i}$ corresponds to $x_{i}$, $1\leq
i\leq m$. The composite morphism $\widetilde{Z}\to Z\to\mathbb{P}^{1}_{\ell}$
induces, by completion above $\infty$, finite separable morphisms
$\bar{g}_{i}:\operatorname{Spec}\operatorname{Fr}(\hat{\Cal{O}}_{\widetilde{Z},x_{i}})\to\operatorname{Spec}\ell((t))$
where $\operatorname{Fr}(\hat{\Cal{O}}_{\widetilde{Z},x_{i}})$ is the fraction
field of the completion $\hat{\Cal{O}}_{\widetilde{Z},x_{i}}$ of the local
ring $\Cal{O}_{\widetilde{Z},x_{i}}$ of $\widetilde{Z}$ at $x_{i}$, $1\leq
i\leq m$ (with the above notations $t=T\mod\pi$).

Consider the formal closed unit disc $D=\operatorname{Spf}R<\frac{1}{T}>$ with
parameter $\frac{1}{T}$ and its special fibre
$D_{\ell}=\operatorname{Spec}\ell[\frac{1}{t}]$
($D_{\ell}\overset\sim\to{\rightarrow}\mathbb{A}^{1}_{\ell}$). By a result of
Gabber and Katz (cf. [Katz], Main Theorem 1.4.1) there exists, for $1\leq
i\leq m$, a finite cover $\bar{h}_{i}:C_{i}\to D_{\ell}$ with $C_{i}$
connected, which only (tamely) ramifies above the point $\frac{1}{t}=0$ and
such that the completion of $\bar{h}_{i}$ above $t=0$ is generically
isomorphic to the cover
$\bar{g}_{i}:\operatorname{Spec}\operatorname{Fr}(\hat{\Cal{O}}_{\widetilde{Z},x_{i}})\to\operatorname{Spec}\ell((t))$.
Using formal patching techniques (cf. [Saïdi4], 1.2) one can lift the covers
$\bar{h}_{i}$ to finite covers $h_{i}:Y_{i}\to D$ which only ramify above the
point $\frac{1}{T}=0$, $1\leq i\leq m$ [outside $\frac{1}{T}=0$ the existence
of such a lifting follows from the theorems of lifting of étale covers (cf.
[Grothendieck], Exposé I, Corollaire 8.4). In a formal neighbourhood of
$\frac{1}{T}=0$ such a lifting is possible under the tameness condition: étale
locally near $\frac{1}{t}$ the cover $\bar{h}_{i}$ is defined by an equation
$y^{s}=\frac{1}{t^{e}}$, where $s\geq 1$ is an integer prime to the
characteristic of $\ell$, and one lifts to the cover defined by
$Y^{s}=\frac{1}{T^{e}}$]. For $1\leq i\leq m$, let
$\hat{A}_{\mathfrak{p}_{i}}$ be the completion of the localisation
$A_{\mathfrak{p}_{i}}$ of $A$ at ${\mathfrak{p}}_{i}$. Thus,
$\hat{A}_{\mathfrak{p}_{i}}$ is a complete discrete valuation ring with
uniformising parameter $\pi$ (recall $A/\pi A$ is reduced) and residue field
$\operatorname{Fr}(\hat{\Cal{O}}_{\widetilde{Z},x_{i}})$. Let $B$ be the
completion of the localisation of $R[[T]]$ at $\pi$. Thus, $B$ is a complete
discrete valuation ring with residue field $\ell((t))$. The finite cover
$g:\operatorname{Spf}A\to\operatorname{Spf}(R[[T]])$ induces, by pull-back to
$\operatorname{Spf}B$, finite covers
$g_{i}:\operatorname{Spf}\hat{A}_{\mathfrak{p}_{i}}\to\operatorname{Spf}B$
which (by construction) lift the covers
$\bar{g}_{i}:\operatorname{Spec}\operatorname{Fr}(\hat{\Cal{O}}_{\widetilde{Z},x_{i}})\to\operatorname{Spec}\ell((t))$,
$1\leq i\leq m$. Further, the cover $h_{i}:Y_{i}\to D$ induces, by pull-back
to $\operatorname{Spf}B$, a finite cover
$\tilde{h}_{i}:\operatorname{Spf}B_{i}\to\operatorname{Spf}B$ which by
construction lifts the cover
$\bar{g}_{i}:\operatorname{Spec}\operatorname{Fr}(\hat{\Cal{O}}_{\widetilde{Z},x_{i}})\to\operatorname{Spec}\ell((t))$.
Thus, the covers $\tilde{h}_{i}:\operatorname{Spf}B_{i}\to\operatorname{Spf}B$
and $g_{i}:\operatorname{Spf}\hat{A}_{\mathfrak{p}_{i}}\to\operatorname{Spf}B$
are isomorphic since $\bar{g}_{i}$ is generically separable. Using formal
patching techniques (cf. loc. cit.) one can patch the covers
$g:\operatorname{Spf}A\to\operatorname{Spf}(R[[T]])$ and $h_{i}:Y_{i}\to D$,
$1\leq i\leq m$, to construct a finite cover $\Cal{Y}\to\mathbb{P}^{1}_{R}$ in
the category of formal schemes with $\Cal{Y}$ normal, connected, proper, and
flat over $\operatorname{Spf}R$. The special fibre
$\Cal{Y}_{\ell}\overset\text{def}\to{=}\Cal{Y}\times_{\operatorname{Spec}R}\operatorname{Spec}\ell$
of $\Cal{Y}$ consists of $m$ irreducible components which intersect at the
point $y$ and is (by construction) non-singular outside $y$. The formal curve
$\Cal{Y}$ is algebraic by formal GAGA and (by construction)
$\hat{\Cal{O}}_{\Cal{Y},y}\overset\sim\to{\rightarrow}A$ as required. ∎

###### Definition Remark 5.2

Proposition 5.1 asserts the existence, after possibly a finite extension of
$K$, of a proper $R$-curve $\Cal{Y}$ and a closed point
$y\in\Cal{Y}^{\operatorname{cl}}$ such that
$\hat{\Cal{O}}_{\Cal{Y},y}\overset\sim\to{\rightarrow}A$. The special fibre
$\Cal{Y}_{\ell}\overset\text{def}\to{=}\Cal{Y}\times_{\operatorname{Spec}R}\operatorname{Spec}\ell$
of $\Cal{Y}$ consists of $m_{y}\overset\text{def}\to{=}m$ (cf. the proof of
Proposition 5.1 for the definition of $m$) irreducible components
$\\{C_{1},\ldots,C_{m}\\}$ which intersect at $y$, $\Cal{Y}_{\ell}$ is non-
singular outside $y$, and the normalisation morphism
$C_{i}^{\operatorname{nor}}\to C_{i}$ is a homeomorphism, $1\leq i\leq m$. In
fact one can, assuming the existence of a compactification of
$\operatorname{Spec}A$ as in Proposition 5.1, construct such a
compactification $\Cal{Y}$ of $\operatorname{Spec}A$ with the additional
property that
$C_{i}^{\operatorname{nor}}\overset\sim\to{\rightarrow}\mathbb{P}^{1}_{\ell}$,
$\forall 1\leq i\leq m$ (cf. [Saïdi4], Remark 3.1).

###### Proposition 5.3

We use the above notations. There exists a finite extension $L/K$ and a finite
morphism
$\operatorname{Spec}B\to\operatorname{Spec}A_{\Cal{O}_{L}}^{\operatorname{nor}}$
with $B$ local, normal, hyperbolic (cf. Notations), and the morphism
$\operatorname{Spec}B_{L}\to\operatorname{Spec}A_{L}$ is geometric and étale.

###### Demonstration Proof

This follows easily from Proposition 5.1, Remark 5.2, and Theorem 3 in
[Saïdi4]. ∎

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Mohamed Saïdi

College of Engineering, Mathematics, and Physical Sciences

University of Exeter

Harrison Building

North Park Road

EXETER EX4 4QF

United Kingdom

M.Saidiexeter.ac.uk