# Quasi-isogeny groups of supersingular abelian surfaces via pro-étale
fundamental groups

Thibaud van den Hove  Thibaud van den Hove: Fachbereich Mathematik, TU-
Darmstadt, Germany<EMAIL_ADDRESS>
###### Abstract.

We consider a $J_{b}(\mathbb{Q}_{p})$-torsor on the supersingular locus of the
Siegel threefold constructed by Caraiani-Scholze, and show that it induces an
isomorphism between a free group on a finite number of generators, and the
group of self-quasi-isogenies of a supersingular abelian surface, respecting a
principal polarization and a prime-to-$p$ level structure. Along the way, we
classify certain pro-étale torsors in terms of the pro-étale fundamental
group, describe the category of geometric covers of non-normal schemes, and
use this to compute pro-étale fundamental groups of curves.

###### 1991 Mathematics Subject Classification:

Primary 14F35; Secondary 14H30, 11G10, 14K02

###### Contents

  1. 1 Introduction
  2. 2 The pro-étale fundamental group
  3. 3 Geometric covers of non-normal schemes
  4. 4 Computing the pro-étale fundamental group
  5. 5 Pro-étale torsors
  6. 6 The Siegel threefold

##  1\. Introduction

In [CS17, Proposition 1.13], Caraiani and Scholze construct a pro-étale
$J_{b}(\mathbb{Q}_{p})$-torsor on each Newton stratum $\mathscr{S}^{b}$ of
certain PEL Shimura varieties at hyperspecial level, where
$J_{b}(\mathbb{Q}_{p})$ is the locally profinite group of self-quasi-isogenies
of a certain $p$-divisible group, respecting extra structures. For connected
strata, these torsors induce a continuous morphism

$\pi_{1}^{\mathrm{pro\acute{e}t}}(\mathscr{S}^{b},x)\to
J_{b}(\mathbb{Q}_{p}),$

where $x$ is a geometric point of $\mathscr{S}^{b}$, and
$\pi_{1}^{\mathrm{pro\acute{e}t}}$ is the pro-étale fundamental group of Bhatt
and Scholze classifying geometric covers, as introduced in [BS15, §7]. The
main goal of this paper is to show that in the case of the basic stratum of
the Siegel threefold, this map is injective, and to determine its image. More
precisely, we compute the pro-étale fundamental group in this special case,
and the image of the map to $J_{b}(\mathbb{Q}_{p})$ consists exactly of those
quasi-isogenies coming from a quasi-isogeny of abelian surfaces. This gives
the following main theorem (cf. Remark 6.3, and Propositions 6.5 and 6.7).

######  Theorem 1.1.

Let $p\geq 3$ be prime, $N\geq 3$ be prime to $p$, and let $V_{N}$ be the
supersingular locus of the moduli space of principally polarized abelian
surfaces over $\overline{\mathbb{F}_{p}}$ with level-$N$ structure. Then the
$J_{b}(\mathbb{Q}_{p})$-torsor $\mathcal{P}_{N}$ on $V_{N}$ considered above
induces an isomorphism between a free group on a finite number of generators,
and the group of self-quasi-isogenies of a supersingular abelian surface,
respecting a principal polarization and a level-$N$ structure.

Part of this theorem is an instance of the fact that for the basic stratum of
general PEL Shimura varieties, the image of
$\pi_{1}^{\mathrm{pro\acute{e}t}}(\mathscr{S}^{b},x)\to J_{b}(\mathbb{Q}_{p})$
is related to p-adic uniformization of Shimura varieties as in [RZ96], cf.
[CS17, Remark 1.14], although we do not make use of $p$-adic uniformization to
determine this image. On the other hand, we do use it to get the corollary
that the corresponding Rapoport-Zink space is simply connected. Indeed, this
is the smallest cover of $V_{N}$ trivializing $\mathcal{P}_{N}$ (essentially
by definition), so that this follows from injectivity of
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to J_{b}(\mathbb{Q}_{p})$.

Before we mention other results we will use, let us sketch the strategy for
Theorem 1.1. We consider families of principally polarized supersingular
abelian surfaces with level structure over the projective line, constructed by
Moret-Bailly, which give morphisms
$\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}\to V_{N}$. By work of Katsura and
Oort, these morphisms are jointly surjective, and each normalize an
irreducible component of $V_{N}$. This will allow us to compute
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$, by Theorem 1.3 below. Using that
$\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}$ is simply connected, we then
study the morphism

$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to J_{b}(\mathbb{Q}_{p}),$

and use arguments on the level of abelian surfaces to show injectivity and
determine its image.

To compute $\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$, we will prove more
general results about the pro-étale fundamental group. To partially reduce the
problem to determining étale fundamental groups, we show that all geometric
covers of a scheme can be obtained by gluing finite étale covers of its
normalization. More precisely, we prove Theorem 1.2 below, cf. Theorem 3.4.
Here, we need some small topological assumption to ensure the pro-étale
fundamental group exists and behaves well. Moreover, we assume the normal
locus to be open, so that we can give the non-normal locus the structure of a
closed subscheme. This holds e.g. for quasi-excellent schemes [Gro65,
Corollaire 6.13.5]. On the other hand, by topological invariance of the pro-
étale fundamental group, we may assume reducedness without losing generality.
For any scheme $X$, we denote the category of geometric covers of $X$ by
$\operatorname{Cov}_{X}$. Let us also note that a similar result, with
different assumptions and proof, was obtained in [LYZ22, Lemma 2.17]: there,
the authors allow more general proper covers than just the normalization map,
but at the cost of assuming that the scheme is locally noetherian.

######  Theorem 1.2.

Let $X$ be a locally topologically noetherian, connected, reduced scheme,
whose normal locus is open. Denote by $\pi\colon X^{\nu}\to X$ its
normalization, $Y\subseteq X$ its non-normal locus, and let $Z=\pi^{-1}(Y)$,
where both $Y$ and $Z$ have the reduced closed subscheme structure. Then base
change induces an equivalence

$\operatorname{Cov}_{X}\cong\operatorname{Cov}_{X^{\nu}}\times_{\operatorname{Cov}_{Z}}\operatorname{Cov}_{Y}.$

The 2-fibre product
$\operatorname{Cov}_{X^{\nu}}\times_{\operatorname{Cov}_{Z}}\operatorname{Cov}_{Y}$
can be viewed as a descent data category for the covering $X^{\nu}\coprod Y\to
X$; as $Y\to X$ is a monomorphism, there is no need for cocycle conditions.

While Theorem 1.2 above does not allow us to compute the pro-étale fundamental
group in general, it suffices for the case of curves over separably closed
fields, which is enough for our purposes. In that case, we get the formula
below, cf. Theorem 4.4. Again, we note that a similar result was independently
obtained in [LYZ22, Theorem 2.27].

######  Theorem 1.3.

Let $X$ be a connected curve over a separably closed field $k$, with
irreducible components $X_{0},\ldots,X_{m}$. Choose geometric basepoints
$x_{i}$ in the normal locus of $X_{i}$ for each $i$, and for each $x\in X$,
let $b_{x}$ denote the number of branches at $x$. Then there is an isomorphism

$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x_{0})\cong
F_{b(X)-m}*^{N}\mathop{\Large{{\ast}}}_{i=0,\ldots,m}^{\quad
N}\pi_{1}^{\mathrm{\acute{e}t}}(X_{i}^{\nu},x_{i}),$

with $F_{b(X)-m}$ a discrete free group on $b(X)-m$ generators, and
$b(X)=\sum_{x\in X}(b_{x}-1)$.

Here, the notation $*^{N}$ denotes the Noohi coproduct, but for this
introduction, one can think of it as the usual topological coproduct, cf.
Remark 2.15. For curves over general fields, we can then use the _fundamental
exact sequence_ from [Lar19]. Let us also mention another method from Lara to
compute pro-étale fundamental groups, namely to use descent to deduce a
general Van Kampen type theorem, cf. [Lar19, Corollary 3.19]. Our method
however, will show that elements of the free group $F_{b(X)-m}$ can be thought
of as loops in $X$, recovering the topological intuition of fundamental
groups. Moreover, this interpretation will be useful while proving Theorem
1.1.

Finally, we need to know how the torsor $\mathcal{P}_{N}$ induces a morphism
of topological groups. This is an instance of the more general fact that the
pro-étale fundamental group classifies pro-étale $G$-torsors for certain
topological groups $G$, similarly to the case of topological and étale
fundamental groups. However, since pro-étale fundamental groups naturally live
in the category of Noohi groups, this classification holds for all Noohi
groups $G$ (and in particular for the locally profinite groups); cf. Theorem
5.3, which describes the whole groupoid of $G$-torsors.

######  Theorem 1.4.

Let $X$ be a locally topologically noetherian connected scheme with geometric
basepoint $x$, and $G$ a Noohi group. Then the isomorphism classes of pro-
étale $G$-torsors on $X$ are in bijection with
$\operatorname{Hom}_{\operatorname{cont}}(\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x),G)/\operatorname{Inn}(G)$.

Note that we cannot expect the theorem to hold for groups that are not Noohi.
This is similar to the case of étale fundamental groups, which are naturally
profinite, but where the analogue of this theorem fails already for infinite
discrete groups. As an example, the nodal curve over a separably closed field
has a universal geometric cover, which is an étale $\mathbb{Z}$-torsor. But
since the étale fundamental group of this nodal curve is
$\widehat{\mathbb{Z}}$, this would correspond to a non-trivial continuous
homomorphism $\widehat{\mathbb{Z}}\to\mathbb{Z}$, and such maps do not exist.
This is another aspect with respect to which the pro-étale fundamental group
behaves better than the étale version.

Outline. In Section 2, we recall the definitions and basic properties of the
pro-étale fundamental group and Noohi groups, that we will need in the rest of
the paper. In Section 3, we explain how one can obtain geometric covers of
non-normal schemes by gluing geometric covers of their normalizations, which
leads to Theorem 1.2. Specializing to the case of curves, we use this in
Section 4 to prove Theorem 1.3, and to compute some examples of pro-étale
fundamental groups of curves over non-separably closed fields. In Section 5,
we show Theorem 1.4, and in finally, in Section 6, we work out the concrete
example of a torsor on the basic stratum of the Siegel threefold, leading to
Theorem 1.1.

Acknowledgements First and foremost, I thank Johannes Anschütz for advising my
Master’s thesis at the University of Bonn, on which most of this work is
based. I also thank Mingjia Zhang for her suggestion to look at the example of
the Siegel threefold, and Peter Scholze for suggesting that the resulting map
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to J_{b}(\mathbb{Q}_{p})$ should be
injective, and that this is equivalent to the corresponding Rapoport-Zink
space being simply connected. Finally, I thank Torsten Wedhorn for helping me
realize I could weaken some of the assumptions from my thesis, Timo Richarz,
Jakob Stix and an anonymous referee for very helpful comments on earlier
versions of this paper, and Marcin Lara for pointing out a mistake in an
earlier draft.

Part of this work was done while supported financially by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement 101002592), and logistically by the
Deutsche Forschungsgemeinschaft (DFG), through the TRR 326 _Geometry and
Arithmetic of Uniformized Structures_ (project number 444845124).

##  2\. The pro-étale fundamental group

In this section, let us recall the definition of the pro-étale fundamental
group, and basic properties that we will need.

######  Definition 2.1.

  1. (1)

A morphism $f\colon Y\to X$ of schemes is _weakly étale_ if both $f$ and the
diagonal $\Delta_{f}:Y\to Y\times_{X}Y$ are flat.

  2. (2)

The _pro-étale site_ $X_{\mathrm{pro\acute{e}t}}$ of $X$ is defined as the
category of weakly étale $X$-schemes, with fpqc covers.

Let us denote the category of set-valued sheaves on
$X_{\mathrm{pro\acute{e}t}}$ by
$\operatorname{Shv}(X_{\mathrm{pro\acute{e}t}})$, and by
$\operatorname{Loc}_{X}$ the full subcategory of locally constant sheaves.

######  Definition 2.2.

For a scheme $X$ and a geometric basepoint $x$ of $X$, let
$\operatorname{ev}_{x}\colon\operatorname{Loc}_{X}\to\operatorname{Set}$ be
the evaluation functor. We define the _pro-étale fundamental group of $X$ at
the basepoint $x$_ as
$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x):=\operatorname{Aut}(\operatorname{ev}_{x})$.
We endow it with the topology coming from the compact-open topology on each
$\operatorname{Aut}(\operatorname{ev}_{x}(\mathcal{F}))$, for
$\mathcal{F}\in\operatorname{Loc}_{X}$.

For a topological group $G$, we denote by $G\operatorname{\text{-}Set}$ the
category of discrete sets with a continuous $G$-action, which comes with a
fibre functor

$F_{G}\colon G\operatorname{\text{-}Set}\to\operatorname{Set},$

given by the forgetful functor. The main theorem about the pro-étale
fundamental group is the following:

######  Theorem 2.3 ([BS15, Theorem 1.10]).

If $X$ is a locally topologically noetherian and connected scheme, then there
is an equivalence
$\operatorname{Loc}_{X}\cong\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)\operatorname{\text{-}Set}$,
compatible with fibre functors.

Note that some topological assumption on $X$ is necessary, as there exist
examples of connected schemes for which such an equivalence cannot hold, cf.
[BS15, Example 7.3.12]. It will also be useful to have a geometric
interpretation of locally constant pro-étale sheaves, similarly to the fact
that the finite locally constant étale sheaves are exactly those represented
by finite étale covers.

######  Definition 2.4.

A morphism $f\colon Y\to X$ of schemes is a _geometric cover_ if it is étale
and satisfies the valuative criterion for properness. We denote the category
of geometric covers of $X$ by $\operatorname{Cov}_{X}$.

######  Proposition 2.5 ([BS15, Lemma 7.3.9]).

If $X$ is locally topologically noetherian, then
$\operatorname{Loc}_{X}=\operatorname{Cov}_{X}$, as subcategories of
$\operatorname{Shv}(X_{\mathrm{pro\acute{e}t}})$.

As for both the topological and the étale fundamental groups, the pro-étale
fundamental group is independent of the choice of basepoint, up to (usually
non-canonical) isomorphism. This is an immediate corollary of the proof of
[BS15, Lemma 7.4.1], but we state it explicitly below, as we will need it
later on.

######  Lemma 2.6.

Let $X$ be a locally topologically noetherian connected scheme, and $x_{1}$,
$x_{2}$ two geometric points of $X$. Then there is an isomorphism

$\operatorname{ev}_{x_{1}}\cong\operatorname{ev}_{x_{2}}$

of functors $\operatorname{Loc}_{X}\to\operatorname{Set}$.

Another property of locally topologically noetherian schemes, is that any
étale morphism into such a scheme also has a locally topologically noetherian
source. This was shown in [BS15, Lemma 6.6.10], and it implies that any étale
morphism into a locally topologically noetherian scheme is quasi-separated. We
will use this fact without further mention in this paper.

For the rest of this section, let us fix some locally topologically noetherian
connected scheme $X$, and a geometric basepoint $x$ of $X$. As the étale
fundamental group $\pi_{1}^{\mathrm{\acute{e}t}}(X,x)$ is the automorphism
group of the fibre functor
$\operatorname{ev}_{x}:\operatorname{FEt}_{X}\to\operatorname{FSet}$, there is
a natural map
$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)\to\pi_{1}^{\mathrm{\acute{e}t}}(X,x)$.
These two fundamental groups are closely related, and even the same for nice
schemes:

######  Proposition 2.7 ([BS15, Lemma 7.4.3]).

The canonical continuous morphism
$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)\to\pi_{1}^{\mathrm{\acute{e}t}}(X,x)$
induces an isomorphism on profinite completions.

######  Proposition 2.8 ([BS15, Lemma 7.4.10]).

If $X$ is geometrically unibranch, the canonical morphism
$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)\to\pi_{1}^{\mathrm{\acute{e}t}}(X,x)$
is an isomorphism of topological groups.

######  Example 2.9 ([BS15, Paragraph before Definition 1.9]).

To show that the pro-étale fundamental group is a strictly finer invariant
than the étale fundamental group, consider the nodal curve $X$ over a
separably closed field $k$, obtained by identifying the points $0$ and
$\infty$ in $\mathbb{P}^{1}_{k}$. The connected geometric covers of $X$ are
then given by gluing copies of $\mathbb{P}^{1}_{k}$ along their points at $0$
and $\infty$. In particular there is a universal connected cover $X_{\infty}$,
for which $\operatorname{Aut}(X_{\infty}/X)\cong\mathbb{Z}$. So in this case,
we have

$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)\cong\mathbb{Z}\ncong\widehat{\mathbb{Z}}\cong\pi_{1}^{\mathrm{\acute{e}t}}(X,x).$

######  Remark 2.10.

In the example above, the pro-étale fundamental group is prodiscrete, and
hence agrees with the enlarged fundamental group introduced in [SGA3, Exp.
X.6]. However, in general the pro-étale fundamental group is a strictly finer
invariant, cf. [BS15, Lemma 7.4.6 and Example 7.4.9]. Another example is the
pushout
$\mathbb{G}_{m,\mathbb{C}}\coprod_{\operatorname{Spec}\mathbb{C}}\mathbb{G}_{m,\mathbb{C}}$,
where both maps $\operatorname{Spec}\mathbb{C}\to\mathbb{G}_{m,\mathbb{C}}$
are given by the unit map. This already appears in [Lar19, Example 4.5], but
will also follow from Theorem 4.4 as
$\pi_{1}^{\mathrm{\acute{e}t}}(\mathbb{G}_{m,\mathbb{C}},x)\cong\widehat{\mathbb{Z}}$
for any basepoint $x$.

Another useful property of the pro-étale fundamental group, is that just like
the étale fundamental group, it is a topological invariant:

######  Proposition 2.11 (Topological invariance).

A universal homeomorphism of locally topologically noetherian connected
schemes induces an isomorphism of pro-étale fundamental groups.

This can easily be shown using topological invariance of the étale site, and
the fact that separatedness and satisfying the existence part of the valuative
criterion are topological properties. We refer to [Lar19, Proposition 2.17]
for more details.

As a last topic in this section, let us briefly mention the kind of groups
that can appear. Throughout the rest of the paper, we will assume any
topological group is Hausdorff. (Recall that this is equivalent to being a
$T_{1}$-, or even a $T_{0}$-space. Indeed, in a $T_{0}$-group any point is
closed, as translations and inversion are homeomorphisms. But then the
diagonal is closed as the inverse image of the identity under the continuous
map $G\times G\to G:(x,y)\mapsto xy^{-1}$.)

######  Definition 2.12.

Let $G$ be a topological group, $F_{G}\colon
G\operatorname{\text{-}Set}\to\operatorname{Set}$ the forgetful functor, and
consider the group $\operatorname{Aut}(F_{G})$, topologized similarly as in
Definition 2.2. We say that $G$ is _Noohi_ if the natural map
$G\to\operatorname{Aut}(F_{G})$ is an isomorphism of topological groups.

Since the equivalence
$\operatorname{Loc}_{X}\cong\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)\operatorname{\text{-}Set}$
is compatible with the fibre functors $\operatorname{ev}_{x}$ and
$F_{\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)}$ to $\operatorname{Set}$, any pro-
étale fundamental group is Noohi. Below, a more intrinsic characterization of
Noohi groups is given in terms of Raikov completeness and the Raikov
completion $G^{*}$ of a topological group $G$, for which we refer to [AT08,
Section 3.6]. Note that the identity of any Noohi group has a basis of open
neighbourhoods given by open subgroups, as this holds for groups of the form
$\operatorname{Aut}(S)$ with the compact-open topology, where $S$ is a
discrete set.

######  Proposition 2.13 ([BS15, Proposition 7.1.5]).

If $G$ is a topological group where the identity has a basis of open
neighbourhoods given by open subgroups, then there is a natural isomorphism
$\operatorname{Aut}(F_{G})\cong G^{*}$. In particular, $G$ is Noohi if and
only if $G$ is Raikov complete.

######  Example 2.14.

Using this characterization, we see that any locally profinite group is Noohi.

######  Remark 2.15 ([BS15, Example 7.2.6]).

This characterization shows that the product of two Noohi groups is Noohi, so
that the category of Noohi groups admits products. And while it is not true
that the coproduct of Noohi groups is always Noohi, the category of Noohi
groups does admit coproducts: for two Noohi groups $G$ and $H$, it is given by
$\operatorname{Aut}(F_{G*H})$, where $G*H$ is the coproduct of topological
groups, and we denote it by $G*^{N}H$. Using infinite Galois theory, cf.
[BS15, Theorem 7.2.5], one sees that
$(G*H)\operatorname{\text{-}Set}\cong(G*^{N}H)\operatorname{\text{-}Set}$,
compatibly with both forgetful functors. For a different description of the
Noohi coproduct as the Raikov completion of $G*H$ endowed with a certain
topology (_not_ the coproduct topology), we refer to [Lav18, Corollary 1.14].

##  3\. Geometric covers of non-normal schemes

In this section, we show how to reduce the computation of the pro-étale
fundamental group to the case of a normal scheme, where it agrees with the
étale fundamental group by Proposition 2.8. A natural candidate for such a
normal scheme is the normalization $X^{\nu}$ of $X$. The idea is then similar
to Example 2.9: we would like show that any geometric cover of $X$ can be
obtained by gluing finite étale covers of $X^{\nu}$. However, as not every
scheme can be obtained by such a gluing process, we will start by
approximating $X$ by a scheme that can be obtained that way, and then show how
the pro-étale fundamental groups of $X$ and this other scheme relate. For this
section, let us fix a locally topologically noetherian connected scheme $X$,
which we may assume to be reduced by Proposition 2.11. Moreover, since the
gluing procedure mentioned above will be formalized using pushouts, which are
mostly only well-behaved when one morphism is a closed immersion, we will also
assume that the normal locus of $X$ is open in $X$, so that the non-normal
locus is closed. This holds for example for all quasi-excellent schemes, and
hence for all schemes locally of finite type over a field; cf. [Gro65, §6.13]
for this and other criteria.

######  Construction 3.1.

Consider the normalization map $\pi\colon X^{\nu}\to X$, let $Y\subseteq X$ be
the non-normal locus, and $Z:=\pi^{-1}(Y)$. We view both $Y$ and $Z$ as closed
subschemes of respectively $X$ and $X^{\nu}$, with the reduced subscheme
structure. Let us consider the pushout $X^{\nu}\coprod_{Z}Y$, which exists by
[Fer03, Théorème 7.1], as $Z\to X^{\nu}$ is a closed immersion, and $Z\to Y$
is integral. By the pushout property, there is a natural map $\varphi\colon
X^{\nu}\coprod_{Z}Y\to X$. While this map is not an isomorphism in general, it
is always a universal homeomorphism:

######  Lemma 3.2.

The natural map $\varphi\colon X^{\nu}\coprod_{Z}Y\to X$ is a universal
homeomorphism.

###### Proof.

By [Fer03, Théorème 7.1], the underlying topological space of
$X^{\nu}\coprod_{Z}Y$ is the pushout of the respective topological spaces,
$Y\to X^{\nu}\coprod_{Z}Y$ is a closed immersion, and $X^{\nu}\to
X^{\nu}\coprod_{Z}Y$ is an isomorphism away from $Y$. In particular, $\varphi$
is bijective and induces isomorphisms on residue fields, so by [Gro67,
Corollaire 18.12.11] we are left to show that $\varphi$ is integral. To show
that $\varphi$ is affine, we note that the normalization map $\pi$ is affine,
and for any affine open $U\subseteq X$, the preimage $\varphi^{-1}(U)$ is the
image of $U^{\nu}=\pi^{-1}(U)\subseteq X^{\nu}$ in $X^{\nu}\coprod_{Z}Y$, as
$X^{\nu}\to X^{\nu}\coprod_{Z}Y$ is surjective. But this image is just
$U^{\nu}\coprod_{Z_{U}}Y_{U}$, where $Y_{U}:=Y\times_{X}U$ and
$Z_{U}:=Z\times_{X}U$ are both affine. And as the pushout of affine schemes is
again affine (as the spectrum of the fibre product on the level of rings),
$\varphi$ is also affine. So we may assume that $X=\operatorname{Spec}A$, and
similarly $X^{\nu}\coprod_{Z}Y=\operatorname{Spec}A^{\prime}$ and
$X^{\nu}=\operatorname{Spec}A^{\nu}$, where we know that $A\to A^{\nu}$ is
integral. Now, $A^{\prime}$ is just the fibre product of $A^{\nu}$ along an
injection of rings, as $Z\to Y$ is a dominant morphism of reduced schemes. And
because fibre product of rings preserves injectivity, we see that
$A^{\prime}\to A^{\nu}$ is injective. Since $A^{\nu}$ is integral over $A$,
the same then holds for $A^{\prime}$, so that $\varphi$ is integral, and hence
a universal homeomorphism. ∎

In particular, by Proposition 2.11, the pro-étale fundamental groups of $X$
and $X^{\nu}\coprod_{Z}Y$ coincide.

######  Remark 3.3.

The reason we introduced the pushout $X^{\nu}\coprod_{Z}Y$ is that, although
it will still be non-normal in general, its singularities are better behaved
than those of $X$. And since the natural map $X^{\nu}\to X^{\nu}\coprod_{Z}Y$
is surjective and birational, it is a normalization morphism, so that taking
the pushout $X^{\nu}\coprod_{Z}Y$ is a weaker process than normalizing. In
fact, under certain assumptions on $X$, we can show that $X^{\nu}\coprod_{Z}Y$
is exactly the seminormalization of $X$, cf. Remark 4.2.

To simplify the notation, we will from now on write $\hat{X}$ instead of
$X^{\nu}\coprod_{Z}Y$. Let us now describe the category of geometric covers of
$\hat{X}$, similarly to [Sta, Tag 0ECL].

######  Theorem 3.4.

Base change induces an equivalence

$\operatorname{Cov}_{\hat{X}}\cong\operatorname{Cov}_{X^{\nu}}\times_{\operatorname{Cov}_{Z}}\operatorname{Cov}_{Y}.$

Recall from the introduction that one can view this 2-fibre product as a
descent category for the covering $X^{\nu}\coprod Y\to X$, and that no cocycle
condition is needed as $Y\to X$ is a monomorphism.

###### Proof.

First, we note that
$\operatorname{Cov}_{X^{\nu}}\times_{\operatorname{Cov}_{Z}}\operatorname{Cov}_{Y}$
can be identified with the category $\mathcal{C}$ of diagrams of the form

${X^{\prime}}$${Z^{\prime}}$${Y^{\prime}}$${X^{\nu}}$${Z}$${Y,}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{j^{\prime}}$$\scriptstyle{i^{\prime}}$$\scriptstyle{h}$$\scriptstyle{j}$$\scriptstyle{i}$

where the vertical morphisms are geometric covers and the two squares
cartesian, with the obvious morphisms between them. So it is enough to find an
equivalence $\operatorname{Cov}_{\hat{X}}\cong\mathcal{C}$. For a geometric
cover $W\to\hat{X}$, let $X^{\prime}:=X^{\nu}\times_{\hat{X}}W$, and similarly
$Y^{\prime}:=Y\times_{\hat{X}}W$ and $Z^{\prime}:=Z\times_{\hat{X}}W$. With
the obvious morphisms, this gives a diagram in $\mathcal{C}$, since being a
geometric cover is stable under base change, and because we have

$X^{\prime}\times_{X^{\nu}}Z=W\times_{\hat{X}}X^{\nu}\times_{X^{\nu}}Z\cong
W\times_{\hat{X}}Z=Z^{\prime},$

and similarly for $Y^{\prime}$.

Conversely, if we have a such a diagram in $\mathcal{C}$, we want to show the
pushout $X^{\prime}\coprod_{Z^{\prime}}Y^{\prime}$ exists, and is a geometric
cover of $X^{\nu}\coprod_{Z}Y=\hat{X}$. We will show this pushout exists under
the additional assumption that $\pi\colon X^{\nu}\to X$ is finite, and refer
to the first paragraph of the proof of [Sta, Tag 0ECK] for the general case.
By [Fer03, Théorème 7.1], we have to show that for any point $y^{\prime}\in
Y^{\prime}$, there is an open affine $U^{\prime}\subseteq X^{\prime}$ such
that $j^{\prime-1}(y^{\prime})$ is contained in $U^{\prime}$. If the three
vertical maps $f$, $g$ and $h$ in our diagram are the identity, this holds
because the normalization map is affine. Otherwise, note that as $X^{\nu}$ is
normal, $X^{\prime}$ is the disjoint union of finite étale covers of
$X^{\nu}$. And by our assumption that the normalization map is finite,
$Z^{\prime}\to Y^{\prime}$ is finite as well, so that $y^{\prime}$ only has
finitely many preimages in $Z^{\prime}$, which are contained in the disjoint
union $X^{\prime\prime}$ of finitely many components of $X^{\prime}$, each of
which is finite étale over $X^{\nu}$. So we can consider the image
$y=h(y^{\prime})$, an affine open $U\subseteq X^{\nu}$ containing $j^{-1}(y)$,
and then the inverse image $f_{\mid X^{\prime\prime}}^{-1}(U)\subseteq
X^{\prime\prime}$ is the open affine we are looking for.

Using [Sta, Tag 08KQ] (resp. [Sta, Tag 0ECK]), we find that
$X^{\prime}\coprod_{Z^{\prime}}Y^{\prime}\to X^{\nu}\coprod_{Z}Y$ is étale
(resp. separated), so we are left to show the existence part of the valuative
criterion. Fix a valuation ring $V$ with fraction field $K$, and consider a
commutative diagram as follows:

${\operatorname{Spec}K}$${X^{\prime}}$${X^{\prime}\coprod_{Z^{\prime}}Y^{\prime}}$${Y^{\prime}}$${\operatorname{Spec}V}$${X^{\nu}}$${\hat{X}}$${Y,}$

considering only the solid arrows at first. If the image of
$\operatorname{Spec}K$ in $X^{\prime}\coprod_{Z^{\prime}}Y^{\prime}$ is in the
normal locus, we can uniquely lift this map to a map $\operatorname{Spec}K\to
X^{\prime}$, and compose it to get a map $\operatorname{Spec}K\to X^{\nu}$.
Since $X^{\nu}\to\hat{X}$ is a normalization map, it is integral, and hence
universally closed. In particular, it satisifies the existence part of the
valuative criterion, so we can lift $\operatorname{Spec}V\to\hat{X}$ to a map
$\operatorname{Spec}V\to X^{\nu}$, and we get two dotted arrows as in the
diagram. Since $X^{\prime}\to X^{\nu}$ is a geometric cover by assumption, we
also get a lift $\operatorname{Spec}V\to X^{\prime}$, which we can compose
with $X^{\prime}\to X^{\prime}\coprod_{Z^{\prime}}Y^{\prime}$ to get the
desired lift. On the other hand, if the image of $\operatorname{Spec}K\to
X^{\prime}\coprod_{Z^{\prime}}Y^{\prime}$ is in the non-normal locus, we can
uniquely lift this map to $\operatorname{Spec}K\to Y^{\prime}$. And since
$Y\to\hat{X}$ is a closed immersion, the same argument as above gives a lift
$\operatorname{Spec}V\to X^{\prime}\coprod_{Z^{\prime}}Y^{\prime}$. We
conclude that $X^{\prime}\coprod_{Z^{\prime}}Y^{\prime}\to\hat{X}$ is a
geometric cover.

Both mappings on objects of $\mathcal{C}$ and $\operatorname{Cov}_{\hat{X}}$
can be upgraded to functors, and we want to show they are mutual quasi-
inverses. We will only check this on objects, as once we know this the case of
morphisms is easy. If we have a diagram $D$ in $\mathcal{C}$, take the
associated geometric cover, and then the diagram in $\mathcal{C}$ obtained by
fibre products, this new diagram will be isomorphic to $D$ by [Sta, Tag 07RU]
and cartesianness of the squares of $D$.

Conversely, let $W$ be a geometric cover of $\hat{X}$, and construct
$X^{\prime}$, $Y^{\prime}$ and $Z^{\prime}$ as above. There is a natural map
$X^{\prime}\coprod_{Z^{\prime}}Y^{\prime}\to W$, and we can check locally that
it is an isomorphism. To do this, let $A\to C\leftarrow B$ be a fibre diagram
of rings, and let $M$ be a flat ring over $A\times_{C}B$. Then we have an
exact sequence $0\to A\times_{C}B\to A\oplus B\to C$ of modules. Tensoring
this with $M$ over $A\times_{C}B$ gives another exact sequence, which realizes
$M$ as the fibre product of $M\otimes_{A\times_{C}B}A$ and
$M\otimes_{A\times_{C}B}B$ over $M\otimes_{A\times_{C}B}C$. Since this is just
the affine version of the construction above, we are done. ∎

##  4\. Computing the pro-étale fundamental group

Using Theorem 3.4 to compute $\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)$ can still
be difficult if $Y$ and $Z$ have complicated geometric covers. This happens to
a lesser extent if $Y$ and $Z$ consist of points, such as when $X$ is a curve,
and we will see that if $X$ is moreover defined over a separably closed field,
we will actually be able to find a formula for
$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)$. Note that we will not need any
separatedness or irreducibility assumptions on $X$, and by topological
invariance we do not lose any generality by assuming reducedness.

######  Assumption 4.1.

For this section, assume $X$ is connected, reduced, of finite type over a
field $k$, and that its non-normal locus is zero-dimensional, i.e., consists
of finitely many closed points.

######  Remark 4.2.

Under these assumptions, let us show that $X^{\nu}\coprod_{Z}Y$ is exactly the
seminormalization of $X$. Recall that this seminormalization is the initial
scheme $X^{s}$ with a universal homeomorphism to $X$ that induces isomorphisms
on residue fields. So we immediately get a morphism $\alpha\colon X^{s}\to
X^{\nu}\coprod_{Z}Y$. On the other hand, because of our assumption on $X$, we
also have a natural morphism $Y\to X^{s}$ such that the compositions $Z\to
Y\to X^{s}$ and $Z\to X^{\nu}\to X^{s}$ agree (recall that the normalization
always factors through the seminormalization, so that we also have a map
$X^{\nu}\to X^{s}$). The pushout property then gives a map $\beta\colon
X^{\nu}\coprod_{Z}Y\to X^{s}$. Since maps obtained by universal properties are
unique, the compositions $\alpha\circ\beta$ and $\beta\circ\alpha$ must be the
identity maps, so that $\alpha$ and $\beta$ are mutual inverses.

Let us recall the notion of _(geometric) branches_ of a point of a scheme:

######  Definition 4.3.

Let $T$ be a scheme which locally has finitely many irreducible components,
and $\pi\colon T^{\nu}\to T$ its normalization. For a point $t\in T$, one
defines:

  1. (1)

The number of _branches_ of $T$ at $t$ is the number of inverse images of $t$
in $T^{\nu}$.

  2. (2)

The number of _geometric branches_ of $T$ at $t$ is
$\sum_{t^{\nu}\in\pi^{-1}(t)}[k(t^{\nu}):k(t)]_{\text{sep}}$.

Clearly, for schemes of finite type over a separably closed field, the two
definitions agree. Note also that if we denote the number of geometric
branches at $x\in X$ by $b_{x}$, then by the assumption on our scheme $X$, the
number of points $x\in X$ for which $b_{x}>1$ is finite. In particular, we can
give sense to the infinite sum $\sum_{x\in X}(b_{x}-1)$. Finally, note that
since $X^{\nu}$ is the normalization of both $X$ and $\hat{X}$, and because
$\hat{X}\to X$ is a homeomorphism which induces isomorphisms on residue
fields, the number of (geometric) branches of $X$ at some point agrees with
the number of (geometric) branches at the corresponding point of $\hat{X}$.

These numbers of branches allow us to find a nice formula for the pro-étale
fundamental group of $X$, which we prove in the following theorem. This
generalizes a result obtained in [Lav18, Proposition 1.17] for the case of
projective normal crossing curves. It is also similar to [LYZ22, Theorem
2.27]; we note that both results were obtained independently. The picture to
keep in mind is that of the fundamental group of a graph of groups. Indeed, to
$X$ we can attach a graph of groups, where the vertices correspond to the
irreducible components of $X$, equipped with the étale fundamental group of
their normalization, and points on $X$ with multiple branches give rise to
edges, equipped with the trivial group. Then the formula below is similar to
[Ser80, Example 1 on p. 43], adapted to the setting of Noohi groups. Similar
computations have also appeared in [SGA1, Exp. XI, Corollaire 5.4] (and more
generally in [Sti06]) for the étale fundamental group, and a specific example
can be found in [Lar19, Example 3.25].

Recall that $*^{N}$ denotes the coproduct of Noohi groups, as in Remark 2.15.

######  Theorem 4.4.

Assume $k$ is separably closed, and let $X_{0},\ldots,X_{m}$ be the
irreducible components of $X$. Choose geometric basepoints $x_{i}$ of $X_{i}$
with closed image in the normal locus of $X$, for each $i$. Then there is an
isomorphism

$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x_{0})\cong
F_{b(X)-m}*^{N}\mathop{\Large{{\ast}}}_{i=0,\ldots,m}^{\quad
N}\pi_{1}^{\mathrm{\acute{e}t}}(X_{i}^{\nu},x_{i}),$

with $F_{b(X)-m}$ a discrete free group on $b(X)-m$ generators, and
$b(X)=\sum_{x\in X}(b_{x}-1)$.

###### Proof.

Since we are working over a separably closed field, we can simplify our
notation and not distinguish between a geometric point and its closed image.
Similarly, we will not distinguish a point of a scheme from its image under a
closed immersion.

By Proposition 2.11 and Lemma 3.2, we can replace $X$ by $\hat{X}$. By Remark
2.15, it is enough to show

$\operatorname{Cov}_{\hat{X}}\cong\big{(}F_{b(X)-m}*\mathop{\Large{{\ast}}}_{i=0,\ldots,m}\pi_{1}^{\mathrm{\acute{e}t}}(X_{i}^{\nu},x_{i})\big{)}\operatorname{\text{-}Set}.$

By Theorem 3.4, we can instead show that
$\big{(}F_{b(X)-m}*\mathop{\Large{{\ast}}}_{i=0,\ldots,m}\pi_{1}^{\mathrm{\acute{e}t}}(X_{i}^{\nu},x_{i})\big{)}\operatorname{\text{-}Set}$
is equivalent to $\mathcal{C}$, where $\mathcal{C}$ is the category of
diagrams appearing in the proof of the aforementioned theorem. To do this, we
make the following choices, using Lemma 2.6:

  * •

For each $z\in Z\subseteq X^{\nu}$ (i.e., for each $z\in X^{\nu}$ whose image
in $\hat{X}$ is not normal), fix a natural isomorphism
$F_{z}\colon\operatorname{ev}_{x_{i}}\xrightarrow{\cong}\operatorname{ev}_{z}$
of functors $\operatorname{Cov}_{X_{i}^{\nu}}\to\operatorname{Set}$, where $i$
is the index such that $z\in X_{i}^{\nu}$. By base change, this gives a
natural isomorphism
$\overline{F}_{z}\colon\operatorname{ev}_{x_{i}}\xrightarrow{\cong}\operatorname{ev}_{\overline{z}}$
of functors $\operatorname{Cov}_{\hat{X}}\to\operatorname{Set}$, with
$\overline{z}$ the image of $z$ in $\hat{X}$.

  * •

Rearranging the indices of the irreducible components of $\hat{X}$, we may
assume that for each $i=1,\ldots,m$, there is some $j<i$ such that
$\hat{X}_{i}\cap\hat{X}_{j}\neq\varnothing$. For any index $i$, fix some point
$y_{i}$ in such an intersection, and some points $y_{i}^{i}\in X_{i}^{\nu}$
and $y_{i}^{j}\in X_{j}^{\nu}$ lying above it. In particular, we get natural
isomorphisms
$F_{i}\colon\operatorname{ev}_{x_{0}}\cong\operatorname{ev}_{x_{i}}$ of
functors $\operatorname{Cov}_{\hat{X}}\to\operatorname{Set}$, by composing the
isomorphisms

$\operatorname{ev}_{x_{0}}\to\ldots\to\operatorname{ev}_{x_{j}}\xrightarrow{\overline{F}_{y_{i}^{j}}}\operatorname{ev}_{y_{i}}\xrightarrow{\overline{F}^{-1}_{y_{i}^{i}}}\operatorname{ev}_{x_{i}}.$

  * •

For each $x\in\hat{X}$ with $b_{x}>1$ that is not one of the $y_{i}$’s, choose
some $x^{\nu}\in X^{\nu}$ over $x$. And to simplify the notation later on, let
us denote $y_{i}^{j}$ by $y_{i}^{\nu}$, where $j<i$.

  * •

Finally, we fix a set of free generators $T$ of $F_{b(X)-m}$, and an
identification of $T$ with the set of $z\in Z$ which are not one of the
choices for $y_{i}^{i}$, $y_{i}^{j}$ or $x^{\nu}$ made above. (Note that the
number of such $z\in Z$ is exactly $b(X)-m$.)

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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

in $\mathcal{C}$, and denote the preimage of $X_{i}^{\nu}$ in $X^{\prime}$ by
$X^{\prime}_{i}$. Since $x_{0}$ lies in the normal locus of $\hat{X}$, the
fibre functor $F_{\mathcal{C}}$ of $\mathcal{C}$ (corresponding to the fibre
functor $\operatorname{ev}_{x_{0}}$ of $\operatorname{Cov}_{\hat{X}}$ under
the equivalence $\mathcal{C}\cong\operatorname{Cov}_{\hat{X}}$) sends this
diagram to the set of points of $X^{\prime}$ over $x_{0}$. Considering only
$X^{\prime}_{0}$, we see that $F_{\mathcal{C}}$ admits an obvious action by

$\pi_{1}^{\mathrm{pro\acute{e}t}}(X_{0}^{\nu},x_{0})\cong\pi_{1}^{\mathrm{\acute{e}t}}(X_{0}^{\nu},x_{0}).$

Similarly, using the fixed isomorphisms
$F_{i}\colon\operatorname{ev}_{x_{0}}\cong\operatorname{ev}_{x_{i}}$, we
obtain actions of

$\pi_{1}^{\mathrm{pro\acute{e}t}}(X_{i}^{\nu},x_{i})\cong\pi_{1}^{\mathrm{\acute{e}t}}(X_{i}^{\nu},x_{i})$

on $F_{\mathcal{C}}$. To define an action of $F_{b(X)-m}$ as well, it is
enough to specify $b(X)-m$ automorphisms of the functor $F_{\mathcal{C}}$. For
each $z\in T$ in $X_{i}^{\nu}$ with image $w\in\hat{X_{i}}$, we define the
following automorphism $t_{z}$: for a point $p\in F_{\mathcal{C}}(D)$,
consider $F_{z}(X^{\prime}_{i})\circ F_{i}(X^{\prime})(p)$. This is a point of
$Z^{\prime}$ lying over $z\in Z$. Let $q$ be the unique point of $Z^{\prime}$
which gets mapped to the same point in $Y^{\prime}$, but also gets mapped to
$w^{\nu}\in Z$. Then we define $t_{z}(p)=F_{j}(X^{\prime})^{-1}\circ
F_{w^{\nu}}(X^{\prime}_{j})^{-1}(q)$, where $j$ is the index such that
$w^{\nu}\in X_{j}^{\nu}$. This defines a natural automorphism of
$F_{\mathcal{C}}$, as $F_{i}$, $F_{j}$, $F_{z}$ and $F_{w^{\nu}}$ are natural
isomorphisms, and because the morphisms in $\mathcal{C}$ are induced by
morphisms of $\operatorname{Cov}_{X^{\nu}}$, $\operatorname{Cov}_{Y}$ and
$\operatorname{Cov}_{Z}$. (Note the slight abuse of notation we use by
composing $F_{z}$ and $F_{i}$, but since $x_{i}$ lies in the normal locus of
$\hat{X}$, this does not lead to any problems.)

Conversely, let $S$ be a set with a continuous
$\big{(}F_{b(X)-m}*\mathop{\Large{{\ast}}}_{i=0,\ldots,m}\pi_{1}^{\mathrm{\acute{e}t}}(X_{i}^{\nu},x_{i})\big{)}$-action.
Consider, for each $i$, the set $S$ with the restricted
$\pi_{1}^{\mathrm{\acute{e}t}}(X_{i}^{\nu},x_{i})\cong\pi_{1}^{\mathrm{pro\acute{e}t}}(X_{i}^{\nu},x_{i})$-action.
By the equivalence
$\pi_{1}^{\mathrm{pro\acute{e}t}}(X_{i}^{\nu},x_{i})\operatorname{\text{-}Set}\cong\operatorname{Cov}_{X_{i}^{\nu}}$
we get a geometric cover $X^{\prime}_{i}$ of $X_{i}^{\nu}$, and we define
$X^{\prime}:=\bigsqcup_{i=0}^{m}X^{\prime}_{i}$ and
$Z^{\prime}:=X^{\prime}\times_{X^{n}}Z$. Since we are working over a separably
closed field, both $Z$ and $Z^{\prime}$ are disjoint unions of copies of
$\operatorname{Spec}k$, and we have $Z^{\prime}\cong\coprod_{S}Z$. So to get a
diagram in $\mathcal{C}$, we have to take $Y^{\prime}=\coprod_{S}Y\to Y$, and
we are left to determine the map $Z^{\prime}\to Y^{\prime}$, i.e., how points
of $Z^{\prime}\subseteq X^{\prime}$ are glued together.

We start by gluing $X^{\prime}_{i}$’s and $X^{\prime}_{j}$’s together, for
$i\neq j$. We do this inductively on $i=1,\ldots,m$. For $i=1$, fix an
identification of $S$ with both the points of $X^{\prime}_{0}$ lying over
$x_{0}$ and the points of $X^{\prime}_{1}$ lying over $x_{1}$, compatibly with
the actions of $\pi_{1}^{\mathrm{\acute{e}t}}(X_{0}^{\nu},x_{0})$ and
$\pi_{1}^{\mathrm{\acute{e}t}}(X_{1}^{\nu},x_{1})$ respectively. We then map,
for each $s\in S$, the points $F_{y_{1}^{0}}(X^{\prime}_{0})(s)$ and
$F_{y_{1}^{1}}(X^{\prime}_{1})(s)$ to the same point in $Y^{\prime}$ (which
maps to the right point in $Y$), using the identification we just made. This
determines the images of points lying over $y_{1}^{0}$ and $y_{1}^{1}$, and we
can repeat this process for $i>1$.

For each $x\in\hat{X}$ with $b_{x}>1$ that was not of the form $y_{i}$, we had
fixed some $x^{\nu}\in X^{\nu}$ lying over $x$. Points of $Z^{\prime}$ lying
over these $x^{\nu}$ do not need to be glued to the points considered in the
previous paragraph, so one can simply choose their image in $Y^{\prime}$ (with
correct image in $Y$), such that they all map to distinct points.

Finally, consider some $z\in T$ with $z\in X_{l}^{\nu}$. Then $z$ is not of
the form $y_{i}^{i}$, $y_{i}^{j}$ or $x^{\nu}$. But any point $z^{\prime}$
lying over $z$ must be glued together with exactly one point lying over some
$x^{\nu}$ or some $y_{i}^{j}=y_{i}^{\nu}$ (let us denote both by $x^{\nu}\in
X_{j}^{\nu}$), and the images in $Y^{\prime}$ of these points are already
determined. To determine to which point $z^{\prime}$ must be glued, consider
the automorphism $t_{z}$ on $S$ associated to $z\in T\subseteq F_{b(X)-m}$.
Looking at the way that we constructed an automorphism associated to $z$
before, we see that $z^{\prime}$ must be glued to the point
$F_{x^{\nu}}(X^{\prime}_{j})\circ F_{j}(X^{\prime})\circ t_{z}\circ
F_{l}(X^{\prime})^{-1}\circ F_{z}(X^{\prime}_{l})^{-1}(z^{\prime})$.

The constructions above can naturally be upgraded to functors, which are
readily seen to be mutual quasi-inverses. This concludes the proof. ∎

######  Remark 4.5.

In a certain sense, the elements of
$F_{b(X)}\subseteq\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x_{0})$ correspond to
loops in $X$, which recovers some of the topological intuition for fundamental
groups.

######  Remark 4.6.

Even if $X$ is irreducible, it is relatively difficult to determine
$\pi_{1}^{\mathrm{\acute{e}t}}(X^{\nu},x)*^{N}F_{b(X)}$ completely: if
$\pi_{1}^{\mathrm{\acute{e}t}}(X^{\nu},x)$ is infinite profinite, then
$\pi_{1}^{\mathrm{\acute{e}t}}(X^{\nu},x)*F_{b(X)}$ will have some non-
discrete topology, so it might not be Noohi. However, we do know that the
natural map

$\pi_{1}^{\mathrm{\acute{e}t}}(X^{\nu},x)*F_{b(X)}\to\pi_{1}^{\mathrm{\acute{e}t}}(X^{\nu},x)*^{N}F_{b(X)}$

is injective. Indeed, since the latter is the automorphism group of
$\operatorname{Cov}_{\hat{X}}$, we have to show that any non-trivial element
$\psi\in\pi_{1}^{\mathrm{\acute{e}t}}(X^{\nu},x)*F_{b(X)}$ acts non-trivially
on the image under $\operatorname{ev}_{x}$ of some geometric cover of
$\hat{X}$. To construct such a cover, consider the (unique) reduced
decomposition of $\psi$ as a product of elements in
$\pi_{1}^{\mathrm{\acute{e}t}}(X^{\nu},x)$ and in $F_{b(X)}$, and let $Y\to
X^{\nu}$ be a connected finite étale cover on which some element of
$\pi_{1}^{\mathrm{\acute{e}t}}(X^{\nu},x)$ appearing in the decomposition of
$\psi$ acts non-trivially. (This is always possible unless $\psi$ is in the
image of

$F_{b(X)}\to\pi_{1}^{\mathrm{\acute{e}t}}(X^{\nu},x)*F_{b(X)}.$

In that case, taking $Y=X^{\nu}$ will suffice, as we assumed $\psi$ was non-
trivial.) Let $Y_{\infty}$ be the universal connected geometric cover of
$\hat{X}$ whose irreducible components are $Y$. It is then clear that $\psi$
acts non-trivially on $\operatorname{ev}_{x}(Y_{\infty})$.

######  Remark 4.7.

Simarly as for the étale fundamental group, there is a fundamental exact
sequence for the pro-étale fundamental group, cf. [Lar19, Theorem 4.14]: if
$Y$ is a geometrically connected scheme of finite type over $k$, then there is
a short exact sequence

$1\to\pi_{1}^{\mathrm{pro\acute{e}t}}(Y_{k^{\text{sep}}})\to\pi_{1}^{\mathrm{pro\acute{e}t}}(Y)\to\operatorname{Gal}(k^{\text{sep}}/k)\to
1$

of topological groups. In particular, using this together with our Theorem
4.4, we can determine the pro-étale fundamental group of a geometrically
connected curve, up to some extension problem. For simple cases however, we
can fully determine the pro-étale fundamental group, as the following examples
show.

######  Example 4.8.

[Lar19, Example 3.24] Consider a curve $X$ obtained by gluing the $k$-rational
points 0 and $\infty$ of the projective line $\mathbb{P}^{1}_{k}$ together.
(This is similar as in Example 2.9, except that we allow $k$ to be any field.)
For any étale field extension $k^{\prime}/k$, we get a geometric cover of $X$
by gluing copies of $\mathbb{P}^{1}_{k^{\prime}}$, either a finite or an
infinite number, by using Theorem 3.4. To show that, up to isomorphism, we can
get all the connected geometric covers of $X$ this way, we have to show we
cannot glue some $\mathbb{P}^{1}_{k_{1}}$ to some $\mathbb{P}^{1}_{k_{2}}$ if
$k_{1}\ncong k_{2}$. Indeed, there were such a geometric cover, it would
correspond to a diagram

${X^{\prime}}$${Z^{\prime}}$${Y^{\prime}}$${\mathbb{P}^{1}_{k}}$${\operatorname{Spec}k\bigsqcup\operatorname{Spec}k}$${\operatorname{Spec}k,}$

where $Z^{\prime}$ contains $\operatorname{Spec}k_{1}$ and
$\operatorname{Spec}k_{2}$ as clopen subschemes, which are mapped to the same
point of $Y^{\prime}$. But the right square of this diagram must be a fibre
square, which shows this can only happen if $k_{1}\cong k_{2}$. So we have
found all connected geometric covers of $X$. Let $x$ be a geometric point of
$X$. For the pro-étale fundamental group, note that it is generated by
$\mathbb{Z}$ and $\operatorname{Gal}(\overline{k}/k)$, where $\mathbb{Z}$ acts
by translation, and $\operatorname{Gal}(\overline{k}/k)$ by the usual Galois
action on the irreducible components. Indeed, we are only gluing points of
$\mathbb{P}^{1}_{k}$, and
$\pi_{1}^{\mathrm{\acute{e}t}}(\mathbb{P}^{1}_{k},x)\cong\operatorname{Gal}(\overline{k}/k)$.
Moreover, it is readily seen that these two actions commute with each other,
so that
$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)\cong\mathbb{Z}\times\operatorname{Gal}(\overline{k}/k)$,
with the product topology.

So far, we only considered curves whose normalization map induced isomorphisms
on residue fields. Let us see what can happen if this is not the case.

######  Example 4.9.

We want to consider a scheme $X$ which looks like the complex projective line,
except that the residue field at $\infty$ is $\mathbb{R}$ instead of
$\mathbb{C}$. Formally, we can construct this as
$\mathbb{P}^{1}_{\mathbb{C}}\coprod_{\operatorname{Spec}\mathbb{C}}\operatorname{Spec}\mathbb{R}$,
where
$\operatorname{Spec}\mathbb{C}\hookrightarrow\mathbb{P}^{1}_{\mathbb{C}}$ is
the inclusion of $\infty$. Using Theorem 3.4, we can see that, up to
isomorphism, $X$ has only two connected geometric covers: the identity, and a
scheme $Y$ gotten by gluing two copies of $\mathbb{P}^{1}_{\mathbb{C}}$
together along their respective points at infinity. This corresponds to the
diagram

${\mathbb{P}^{1}_{\mathbb{C}}\bigsqcup\mathbb{P}^{1}_{\mathbb{C}}}$${\operatorname{Spec}\mathbb{C}\bigsqcup\operatorname{Spec}\mathbb{C}}$${\operatorname{Spec}\mathbb{C}}$${\mathbb{P}^{1}_{\mathbb{C}}}$${\operatorname{Spec}\mathbb{C}}$${\operatorname{Spec}\mathbb{R}.}$

Indeed, since $\mathbb{P}^{1}_{\mathbb{C}}$ has no nontrivial geometric
covers, we see that any such diagram where the upper right scheme is not
connected will give rise the a geometric cover that is not connected. Since
$Y\to X$ is a cover of degree 2, we see that for any geometric point $x$ of
$X$, we have
$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)\cong\mathbb{Z}/2\mathbb{Z}$, with the
discrete topology.

######  Example 4.10.

Similarly as in the example above, consider a scheme $X$ which looks like the
complex projective line, but which has residue field $\mathbb{R}$ at two
points: 0 and $\infty$. While it is possible to determine all the connected
geometric covers, let us just mention that $X$ has a universal geometric cover
$Y$. It is obtained by taking countably many copies of
$\mathbb{P}^{1}_{\mathbb{C}}$, and gluing them together by repeatedly
identifying two points lying over $0\in X$, and two points over $\infty\in X$
(this is in contrast to Example 4.8, where we identified points lying over
$0\in\mathbb{P}^{1}_{k}$ with points over $\infty\in\mathbb{P}^{1}_{k}$). So
the pro-étale fundamental group of $X$ is discrete, and isomorphic to the
automorphism group of $Y$ over $X$. This automorphism group is generated by
translations, and the Galois action of $\mathbb{C}$ over $\mathbb{R}$. Since
this Galois action essentially reverses the order of the copies of
$\mathbb{P}^{1}_{\mathbb{C}}$, we see that
$\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x)\cong\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$
for any geometric point $x$ of $X$, where the action of
$\mathbb{Z}/2\mathbb{Z}$ is the unique non-trivial action.

######  Remark 4.11.

By [Sta, Tags 0C1S and 0C39], an integral scheme is (geometrically) unibranch
at a point if and only if the number of (geometric) branches at that point is
1. So the example above gives a scheme which is unibranch, but whose étale and
pro-étale fundamental group are not isomorphic. This shows that the
geometrically unibranch assumption from Proposition 2.8 cannot be weakened to
only unibranch.

##  5\. Pro-étale torsors

Although the pro-étale fundamental group of a scheme $X$ classifies the
geometric covers of $X$, we already had to determine the whole category of
geometric covers of $X$ to compute its pro-étale fundamental group. However,
the pro-étale fundamental group can also be used to classify more general
torsors, as we will show. Throughout this section, $X$ again denotes a locally
topologically noetherian connected scheme, and $x$ a geometric basepoint of
$X$.

######  Notation 5.1.

For any topological space $T$, denote by $\mathcal{F}_{T}$ the presheaf on
$X_{\mathrm{pro\acute{e}t}}$ given by $U\mapsto\operatorname{cont}(U,T)$. This
is a sheaf by [BS15, Lemma 4.2.12], and if $T$ is a topological group or
monoid, this gives $\mathcal{F}_{G}$ the structure of a pro-étale sheaf of
groups or monoids. For a topological group $G$, we denote by
$B\mathcal{F}_{G}(X_{\mathrm{pro\acute{e}t}})$ the groupoid of (right)
$\mathcal{F}_{G}$-torsors on $X_{\mathrm{pro\acute{e}t}}$, with equivariant
morphisms.

To construct such torsors later in this section, we will need to consider the
limit $\varprojlim_{U}G/U$ of topological spaces, where $U$ ranges over the
set of open subgroups of a Noohi group $G$. As these form a basis of open
neighbourhoods of $1\in G$, there is a natural injective map

$G\to\varprojlim_{U}G/U,$

but it is not surjective in general111This failure of surjectivity and the
counterexample were pointed out to me by Marcin Lara.: let $S$ be a discrete
set, and $G=\operatorname{Aut}(S)$ with the compact-open topology. Then $G$ is
a Noohi group by [BS15, Example 7.1.2], and a basis of open neighbourhoods of
$1\in\operatorname{Aut}(S)$ is given by the pointwise stabilizers $U_{F}$ of
finite subsets $F\subseteq S$. For each such $F$, there is a natural injection
$G/U_{F}\to\operatorname{map}(F,S)$, with image the injective maps $F\to S$.
Passing to the limit, we get a continuous injection

$\varprojlim_{F}G/U_{F}\to\varprojlim_{F}\operatorname{map}(F,S)=\operatorname{map}(S,S),$

and its image are again exactly the injective maps. In particular, if $S$ is
infinite, the map $G=\operatorname{Aut}(S)\to\varprojlim_{F}G/U_{F}$ is not
surjective.

What does hold, is that there is a natural homeomorphism
$\varprojlim_{U}G/U\cong\operatorname{End}(F_{G})$, with $F_{G}\colon
G\operatorname{\text{-}Set}\to\operatorname{Set}$ the forgetful functor, and
where $\operatorname{End}(F_{G})$ is topologized using the compact-open
topology, similarly as in Definition 2.2. This was shown in [Lep09,
Proposition 4.1.1], and the isomorphism is given by sending
$(g_{U})_{U}\in\varprojlim_{U}G/U$ to the endomorphism mapping $s\in S\in
G\operatorname{\text{-}Set}$ to $g_{G_{s}}\cdot s$, where $G_{s}\subseteq G$
is the stabilizer of $s$ in $S$. Moreover, the natural map
$G\to\varprojlim_{U}G/U$ can be identified with the inclusion
$\operatorname{Aut}(F_{G})\to\operatorname{End}(F_{G})$, so that the image of
this map are exactly the invertible elements of the monoid
$\operatorname{End}(F_{G})$.

######  Lemma 5.2.

Elements of $\operatorname{End}(F_{G})$ with a left or right inverse are
already invertible.

###### Proof.

Let $\varphi,\psi\in\operatorname{End}(F_{G})$ be such that
$\varphi\circ\psi=\operatorname{Id}_{F_{G}}$, and consider any $S\in
G\operatorname{\text{-}Set}$. Since the map $\varphi_{S}\colon S\to S$ is
surjective, it will be enough to show $\varphi_{S}$ is injective. Indeed, this
will imply each $\varphi_{S}$, and hence each $\psi_{S}=\varphi_{S}^{-1}$, is
bijective.

Let $(\alpha_{U})_{U}\in\varprojlim_{U}G/U$ be the element corresponding to
$\varphi$. Choosing for each open subgroup $U\subseteq G$ a representative
$g_{U}\in\alpha_{U}\subseteq G$ gives a net in $\operatorname{End}(F_{G})$,
which converges to $\varphi$. Now, there is a natural map

$\Phi\colon\operatorname{End}(F_{G})\to\operatorname{map}(S,S).$

If we equip the latter with the compact-open topology, this map is continuous
by definition of the topology on $\operatorname{End}(F_{G})$. In particular,
we have a net $(\Phi(g_{U}))_{U}$ in $\operatorname{map}(S,S)$, which
converges to $\varphi_{S}$. As a limit of injective (even bijective) maps into
a discrete set, we conclude that $\varphi_{S}$ is itself injective. ∎

The following theorem classifies pro-étale $\mathcal{F}_{G}$-torsors, for
Noohi groups $G$. Note that the case of profinite groups already appeared in
[BS15, Lemma 7.4.3], and one can use an argument similar to the proof of
[BS15, Lemma 7.4.7] to show the claim for locally profinite groups. Our
method, on the other hand, works for all Noohi groups, and does not need to
reduce to some simpler case. For two topological groups $G$ and $H$, let us
denote by $\underline{\operatorname{Hom}}_{\operatorname{cont}}(H,G)$ the
groupoid of continuous homomorphisms $H\to G$, where the morphisms between
$f_{1},f_{2}\colon H\to G$ are given by elements $g\in G$ conjugating $f_{1}$
into $f_{2}$.

######  Theorem 5.3.

If $G$ is a Noohi group, there is an equivalence

$B\mathcal{F}_{G}(X_{\mathrm{pro\acute{e}t}})\cong\underline{\operatorname{Hom}}_{\operatorname{cont}}(\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x),G).$

###### Proof.

By [BS15, Theorem 7.2.5 (2)],
$\underline{\operatorname{Hom}}_{\operatorname{cont}}(\pi_{1}^{\mathrm{pro\acute{e}t}}(X,x),G)$
is equivalent to the groupoid of functors
$G\operatorname{\text{-}Set}\to\operatorname{Loc}_{X}$ compatible with the
fibre functors $F_{G}$ and $\operatorname{ev}_{x}$. So it is enough to show
that $B\mathcal{F}_{G}(X_{\mathrm{pro\acute{e}t}})$ is equivalent with this
groupoid of functors.

For a pro-étale $\mathcal{F}_{G}$-torsor $\mathcal{P}$ on $X$, we get a
functor $G\operatorname{\text{-}Set}\to\operatorname{Loc}_{X}$ by sending
$S\in G\operatorname{\text{-}Set}$ to the contracted product
$\mathcal{P}\times^{\mathcal{F}_{G}}\mathcal{F}_{S}$, where the action of
$\mathcal{F}_{G}$ on $\mathcal{F}_{S}$ is the one induced by the action of $G$
on $S$. To show compatibility with the fibre functors, let us fix
$q\in\mathcal{P}_{x}$, which can be identified with the underlying set of $G$.
Then there is a natural isomorphism
$S\xrightarrow{\cong}\operatorname{ev}_{x}(\mathcal{P}\times^{\mathcal{F}_{G}}\mathcal{F}_{S})\colon
s\mapsto(q,s)$. Note that although we had to choose some $q\in\mathcal{P}_{x}$
to show this compatibility, the functor
$G\operatorname{\text{-}Set}\to\operatorname{Loc}_{X}$ itself is independent
of this choice.

On the other hand, suppose we have a functor $\Phi\colon
G\operatorname{\text{-}Set}\to\operatorname{Loc}_{X}$, compatible with the
fibre functors. For each $S\in G\operatorname{\text{-}Set}$, the action of
$\operatorname{End}(F_{G})$ on $\operatorname{ev}_{x}(\Phi(S))=S$ extends
uniquely to an action of $\mathcal{F}_{\operatorname{End}(F_{G})}$ on
$\Phi(S)$. Define a sheaf

$\tilde{\mathcal{P}}\in\operatorname{Shv}(X_{\mathrm{pro\acute{e}t}})\quad\text{as
}\varprojlim_{U}\Phi(G/U),$

where $U$ ranges over the open subgroups of $G$. Then the
$\mathcal{F}_{\operatorname{End}(F_{G})}$-action on each $\Phi(G/U)$ lifts to
a unique $\mathcal{F}_{\operatorname{End}(F_{G})}$-action on
$\tilde{\mathcal{P}}$. To construct a torsor, we may assume $X$ is affine, as
the general case will follow by gluing. In that case, $X$ admits an affine
w-contractible pro-étale cover $W$ by [BS15, Theorem 1.5]. Such w-contractible
schemes are defined by the property that any pro-étale cover over them splits.
In particular, any locally constant pro-étale sheaf over $W$ is already
constant, so that for any $T\in W_{\mathrm{pro\acute{e}t}}$, we have an
$\operatorname{End}(F_{G})$-equivariant isomorphism

$\tilde{\mathcal{P}}(T)=\varprojlim_{U}\Phi(G/U)(T)\cong\varprojlim_{U}\operatorname{cont}(T,G/U)=\operatorname{cont}(T,\varprojlim_{U}G/U).$
(5.1)

In particular, $\tilde{\mathcal{P}}(T)\cong\varprojlim_{U}G/U$ if $T$ is
connected. Now observe that by Lemma 5.2, the subset
$\operatorname{Aut}(F_{G})\subseteq\operatorname{End}(F_{G})$ can be
characterized as those elements $\varphi\in\operatorname{End}(F_{G})$ for
which $\operatorname{End}(F_{G})\cdot\varphi=\operatorname{End}(F_{G})$. In
particular, for connected $T\in W_{\mathrm{pro\acute{e}t}}$, the
$\operatorname{End}(F_{G})$-action allows us to recover which elements of
$\tilde{\mathcal{P}}(T)$ must correspond to elements of $G$, and for connected
$T^{\prime}\to T\in W_{\mathrm{pro\acute{e}t}}$, the restriction
$\tilde{\mathcal{P}}(T)\to\tilde{\mathcal{P}}(T^{\prime})$ preserves these
elements. This allows us to define the subpresheaf $\mathcal{P}$ of
$\tilde{\mathcal{P}}$ whose sections over some $Y\in
X_{\mathrm{pro\acute{e}t}}$ consist of those sections whose restriction to $T$
lies in $G\subseteq\tilde{\mathcal{P}}(T)$, for any connected $T\to Y$ with
$T\in W_{\mathrm{pro\acute{e}t}}$. Then $\mathcal{P}$ is a sheaf, the
$\mathcal{F}_{\operatorname{End}(F_{G})}$-action on $\tilde{\mathcal{P}}$
restricts to an $\mathcal{F}_{G}$-action on $\mathcal{P}$, and for any $T\in
W_{\mathrm{pro\acute{e}t}}$, we have an equivariant isomorphism
$\mathcal{P}(T)\cong\operatorname{cont}(T,G)$. In particular, $\mathcal{P}$
becomes isomorphic to $\mathcal{F}_{G}$ over $W$, which shows that
$\mathcal{P}$ is an $\mathcal{F}_{G}$-torsor.

These mappings can be upgraded to functors, which are readily seen to be
mutual quasi-inverses. ∎

In particular, taking isomorphism classes of these groupoids gives Theorem
1.4.

##  6\. The Siegel threefold

In this final section, we use our previous results to deduce the main theorem.
Let us fix a prime $p$, some integer $N\geq 3$ prime to $p$, and consider the
moduli space $\mathcal{A}_{2,1,N}$ of _principally polarized abelian surfaces
with level- $N$ structure_ over $\overline{\mathbb{F}_{p}}$. As it is three-
dimensional and corresponds to the Siegel Shimura datum, it is also known as
the _Siegel threefold_. The associated basic Newton stratum $V_{N}$ is exactly
the locus of supersingular abelian surfaces, which is one-dimensional, cf.
[Rap03, p.217]. To study this supersingular locus, we will construct families
of principally polarized supersingular abelian surfaces over
$\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}$. These families will then give
rise to maps $\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}\to V_{N}$, which will
allow us to get a grasp on the geometry of $V_{N}$. This construction is
originally due to Moret-Bailly, cf. [MB81], and works for all $p\geq 3$. So we
assume that $p\geq 3$, and refer to [MB79] for a similar construction when
$p=2$. For the rest of this section, we let
$S=\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}$, and any fibre product without
subscript will be over $\operatorname{Spec}\overline{\mathbb{F}_{p}}$.

######  Construction 6.1.

Choose two supersingular elliptic curves $E_{1}$ and $E_{2}$ over
$\overline{\mathbb{F}_{p}}$. They both admit a natural subgroup isomorphic to
$\alpha_{p}$, given by the kernel of their respective Frobenius maps; let us
fix such inclusions
$\alpha_{p}=\operatorname{Spec}\overline{\mathbb{F}_{p}}[\beta_{i}]/(\beta_{i}^{p})\hookrightarrow
E_{i}$. Consider the subgroup scheme
$H=V(Y\beta_{1}-X\beta_{2})\subset\alpha_{p}\times\alpha_{p}\times S$, where
$(X,Y)$ is a homogeneous coordinate of
$S=\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}$. Now we consider the quotient
$\mathfrak{X}:=(E_{1}\times E_{2}\times S)/H$, giving a diagram

${1}$${H}$${E_{1}\times E_{2}\times S}$${\mathfrak{X}}$${1}$${E_{1}\times
E_{2}}$${S}$$\scriptstyle{\Delta}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$$\scriptstyle{\pi}$$\scriptstyle{q}$

where the top row is exact. Clearly, $\mathfrak{X}$ is a family of
supersingular abelian surfaces over $S$, and we want to endow it with a
principal polarization. For this, consider any ample line bundle $\mathcal{L}$
on $E_{1}\times E_{2}$ which is symmetric (i.e.,
$i^{*}\mathcal{L}\cong\mathcal{L}$, with $i\colon E_{1}\times E_{2}\to
E_{1}\times E_{2}$ the inversion map) and satisfies
$K_{\mathcal{L}}\cong\alpha_{p}\times\alpha_{p}$, where
$K_{\mathcal{L}}:=\ker(A\to A^{\vee}\colon x\mapsto t_{x}^{*}\mathcal{L})$ and
$t_{x}$ is the translation by an element $x$ of $A$. Then, as $H\subseteq
K_{\operatorname{pr}_{1}^{*}(\mathcal{L})}$ and $H$ is totally isotropic for
the commutator pairing $e^{\operatorname{pr}_{1}^{*}(\mathcal{L})}\colon
K_{\operatorname{pr}_{1}^{*}(\mathcal{L})}\times
K_{\operatorname{pr}_{1}^{*}(\mathcal{L})}\to\mathbb{G}_{m,S}$ associated to
$\operatorname{pr}_{1}^{*}(\mathcal{L})$ (since this pairing is alternating,
the fibres of $H$ are isomorphic to $\alpha_{p}$, and there are no non-zero
homomorphisms $\alpha_{p}\to\mathbb{G}_{m}$), we see that
$\operatorname{pr}_{1}^{*}(\mathcal{L})$ descends to a unique line bundle
$\mathcal{M}$ on $\mathfrak{X}$, i.e.,
$\pi^{*}(\mathcal{M})\cong\operatorname{pr}_{1}^{*}(\mathcal{L})$. Moreover,
as $E_{1}\times E_{2}\times S\to\mathfrak{X}$ is an isogeny and $\mathcal{M}$
pulls back to an $S$-relatively ample line bundle, $\mathcal{M}$ is
$S$-relatively ample itself. And since both the polarization induced by
$\operatorname{pr}_{1}^{*}(\mathcal{L})$, and the isogeny $E_{1}\times
E_{2}\times S\to\mathfrak{X}$, have degree $p$, it follows that $\mathcal{M}$
induces a principal polarization on $\mathfrak{X}$.

######  Remark 6.2.

In fact, it does not matter which supersingular elliptic curves are chosen.
Indeed, due to a theorem of Deligne, a proof of which can be found in [Shi79,
Theorem 3.5], all products of $g\geq 2$ supersingular elliptic curves are
isomorphic. Moreover, since the inclusion
$\alpha_{p}\times\alpha_{p}\hookrightarrow E_{1}\times E_{2}$ is given by the
kernel of the Frobenius map, this is also independent of the chosen elliptic
curves.

Now, consider a principally polarized family $\mathfrak{X}\to S$ of
supersingular abelian surfaces as constructed above. This is an abelian
scheme, and since $N$ is prime to $p$, the $N$-torsion subgroup
$\mathfrak{X}[N]\subset\mathfrak{X}$ is finite étale over $S$. But since
$S\cong\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}$ is simply connected, any
finite étale cover is trivial, so that $\mathfrak{X}[N]$ is a disjoint union
of copies of $S$. Using this decomposition, we can easily put a level-$N$
structure on the family $\mathfrak{X}\to S$. By the moduli interpretation of
$V_{N}$, this then corresponds to a map $S\to V_{N}$. Moreover, the morphism
$S\to V_{N}$ is not constant, as $\mathfrak{X}\to S$ is not a constant family.
Indeed, by Remark 6.2, it is enough to show that if
$E/\overline{\mathbb{F}_{p}}$ is a supersingular elliptic curve, and
$H\subseteq E\times E\times S$ the subgroup from Construction 6.1, then
$(E\times E\times S)/H$ has fibres that are isomorphic to the product of two
supersingular elliptic curves, and fibres that are not. For the former, one
can just take $s=0\in S$, and the latter is shown in [Oor75, Introduction].
Finally, since both $S$ and $V_{N}$ are one-dimensional, we see that the image
of this map is an irreducible component of $V_{N}$.

######  Remark 6.3.

It was proven in [KO87, Theorems 2.1 and 2.3] that not only is every
irreducible component of $V_{N}$ the image of a map $S\to V_{N}$ as
constructed above, but also that these maps realize
$S=\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}$ as the normalization of such an
irreducible component. As $V_{N}$ is moreover connected, cf. [Oor01, Theorem
7.3 and Corollary 8.4], we can use Theorem 4.4 to conclude that the pro-étale
fundamental group of $V_{N}$ is a discrete free group on a finite number of
generators. In fact, one can determine the exact number of generators using
[KO87, Theorems 2.4, 5.1, and 5.3], but we will not need this. While we will
also not use it, it might be interesting to note that the non-normal points of
$V_{N}$ are exactly those whose corresponding abelian surface is isomorphic to
the product of two supersingular elliptic curves. Finally, what we will use
later, is that all singularities of $V_{N}$ are ordinary singularities. This
is proven in [Kob75, p193], and it implies that one can glue copies of
$S=\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}$ along closed points to get
$V_{N}$ itself, not just its seminormalization.

Now let us describe the pro-étale torsor we are interested in. While its
precise definition is a bit involved, we will not need it in the following, so
we content ourselves by giving the description below. Let us fix a geometric
basepoint

$x\colon\operatorname{Spec}\overline{\mathbb{F}_{p}}\to V_{N},$

with associated supersingular abelian surface $A_{x}$, principal polarization
$\lambda_{x}$, and level structure $\eta_{x}$. Recall that
$J_{b}(\mathbb{Q}_{p})$ is the group of self-quasi-isogenies of the
$p$-divisible group $A_{x}[p^{\infty}]$, respecting the induced polarization
up to a scalar in $\mathbb{Q}_{p}^{\times}$. As the notation suggests, these
are the $\mathbb{Q}_{p}$-valued points of an algebraic group, and hence are
naturally endowed with a locally profinite topology.

######  Definition 6.4 ([CS17, Proposition 4.3.13]).

There is a natural pro-étale $J_{b}(\mathbb{Q}_{p})$-torsor on $V_{N}$, which,
above any geometric point $y$ of $V_{N}$, parametrizes the quasi-isogenies
between $A_{x}[p^{\infty}]$ and $A_{y}[p^{\infty}]$, respecting the
polarizations up to a scalar in $\mathbb{Q}_{p}^{\times}$. We denote this
torsor by $\mathcal{P}_{N}$.

Since $J_{b}(\mathbb{Q}_{p})$ is locally profinite, we can use Theorem 5.3 to
get a map

$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to J_{b}(\mathbb{Q}_{p}),$

and it is the image of this map that we want to determine. Note that this map
is only defined up to inner automorphism of $J_{b}(\mathbb{Q}_{p})$, but such
inner automorphisms only conjugate the image anyway.

To describe this map, note that it depends on a choice of element in
$\operatorname{ev}_{x}(\mathcal{P}_{N})$. For this, we can simply consider the
natural quasi-isogeny

$A_{x}[p^{\infty}]\to A_{x}[p^{\infty}]$

respecting the polarizations, given by the identity. Now, since all connected
components of the normalization of $V_{N}$ are simply connected, the proof of
Theorem 4.4 allows us to describe an element of
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$ by a finite sequence
$(y_{0},y_{1},\ldots,y_{2m},y_{2m+1})$ of closed points of this normalization,
such that $y_{0}$ and $y_{2m+1}$ map to $x$ in $V_{N}$, such that any two
consecutive points of the form $(y_{2k},y_{2k+1})$ lie in the same connected
component, and such that any two consecutive points of the form
$(y_{2k-1},y_{2k})$ map to the same point of $V_{N}$. Elements of
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$ do not determine such sequences
uniquely, but each such sequence determines a unique element of
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$.

Let $\varphi\in\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$, and choose such a
sequence $(y_{0},\ldots,y_{2m+1})$ associated to $\varphi$. For each pair
$(y_{2k-1},y_{2k})$, let $A_{k}$ be the supersingular abelian surface
corresponding to their image in $V_{N}$. On the other hand, choose for each
pair $(y_{2k},y_{2k+1})$ a product of supersingular elliptic curves
$E_{k},E^{\prime}_{k}$ that, along with some line bundle $\mathcal{L}$ as in
Construction 6.1 and the right level structure, gives a family of principally
polarized abelian surfaces over $S$, corresponding to the connected component
of $y_{2k}$ and $y_{2k+1}$. Then each $A_{k}$ is in an obvious way a quotient
of $E_{k-1}\times E^{\prime}_{k-1}$ and $E_{k}\times E^{\prime}_{k}$, both by
a subgroup isomorphic to $\alpha_{p}$. We get the following sequence of maps:

${E_{0}\times E_{0}^{\prime}}$${\ldots}$${E_{m}\times
E_{m}^{\prime}}$${A_{x}}$${A_{1}}$${A_{m}}$${A_{x}.}$ (6.1)

Their composition in the category of abelian varieties up to isogeny gives a
quasi-isogeny $A_{x}\to A_{x}$, which only depends on $\varphi$, not on the
chosen sequence, and is also independent of the choices of supersingular
elliptic curves $E_{i}$ and $E^{\prime}_{i}$. Moreover, by the restrictions on
the $y_{i}$, this quasi-isogeny is compatible with both the polarization and
the level structure. Finally, it induces a quasi-isogeny $A_{x}[p^{\infty}]\to
A_{x}[p^{\infty}]$ of p-divisible groups, and hence an element of
$J_{b}(\mathbb{Q}_{p})$, which is exactly the image of $\varphi$.

We can say a few things about the quasi-isogeny $A_{x}\to A_{x}$ above:

  * •

It is of degree 1,

  * •

It is a $p$-power quasi-isogeny, and

  * •

It is compatible with both the polarization $\lambda_{x}$ and the level
structure $\eta_{x}$.

Our goal will be to show that any quasi-isogeny satisfying these conditions
will arise by a sequence as in (6.1), as this gives us a description of the
image of $\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to J_{b}(\mathbb{Q}_{p})$.
In fact, only the last condition is really necessary: any quasi-isogeny
preserving the principal polarization must have degree 1, and every prime-
to-$p$ quasi-isogeny that preserves the polarization and level structure is
already trivial.

######  Proposition 6.5.

The image of the map $\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to
J_{b}(\mathbb{Q}_{p})$ consists exactly of those elements of
$J_{b}(\mathbb{Q}_{p})$ that are induced by self-quasi-isogenies of $A_{x}$,
preserving the polarization $\lambda_{x}$ (not just up to a scalar) and level
structure $\eta_{x}$.

###### Proof.

We showed above that any element of
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$ maps to such a quasi-isogeny. Now,
fix a quasi-isogeny $\psi\colon A_{x}\to A_{x}$ respecting the polarization
and level structure. We want to factor $\psi$ into a sequence of isogenies
similar to (6.1). Since it is a $p$-power quasi-isogeny, there is some $n\geq
0$ such that $p^{n}\psi$ is an isogeny $A_{x}\to A_{x}$. So we can write
$\psi$ as the composition

$A_{x}\xleftarrow{[p^{n}]}A_{x}\xrightarrow{p^{n}\psi}A_{x},$

where the degrees of $[p^{n}]$ and $p^{n}\psi$ agree. Moreover, since these
degrees are powers of $p$, we can inductively quotient out subgroups of their
kernel of order $p$, to factor $\psi$ into a sequence of isogenies as follows,
all of which have degree $p$:

${A_{x}}$${A_{k}}$${A^{\prime}_{k}}$${A_{1}}$${A^{\prime}_{1}}$${A_{x}}$${A_{x}.}$

Note that since all abelian surfaces are supersingular and we are working over
$\overline{\mathbb{F}_{p}}$, all the kernels of these isogenies are isomorphic
to $\alpha_{p}$. Now, we want to obtain a factorization of $\psi$ into a
sequence of the following kind, where all the arrows are isogenies of degree
$p$:

${A^{\prime}_{1}}$${\ldots}$${A^{\prime}_{l+1}}$${A_{x}}$${A_{1}}$${A_{l}}$${A_{x}.}$
(6.2)

Again, we do this inductively, by rearranging subsequences of the form
$A_{i}\leftarrow A^{\prime}\rightarrow A_{j}$. If the kernels of the two maps
agree, then the composition $A_{i}\leftarrow A^{\prime}\rightarrow A_{j}$ is
an isomorphism, and we can replace it by a single abelian surface. If the two
kernels do not agree, they are still both isomorphic to $\alpha_{p}$. So we
can quotient out $A_{i}$ by the image of the kernel of $A^{\prime}\to A_{j}$,
and similarly take a quotient of $A_{j}$. As these two quotients agree, we get
a commutative diagram

${A^{\prime}}$${A_{i}}$${A_{j}}$${A^{\prime\prime},}$

and we can replace $A_{i}\leftarrow A^{\prime}\rightarrow A_{j}$ by $A_{i}\to
A^{\prime\prime}\leftarrow A_{j}$.

Finally, we want to show that a sequence of isogenies as (6.2) can arise from
an element in $\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$. For this, equip
each abelian surface $A_{i}$ and $A_{i}^{\prime}$ with the polarization and
level structure induced by the polarization and level structure of $A_{x}$.
Consider a subsequence of the form

$A_{i}\xleftarrow{f_{i}}A^{\prime}\xrightarrow{f_{j}}A_{j},$

and let $\lambda_{i}$ and $\lambda_{j}$ be the induced principal polarizations
on $A_{i}$ and $A_{j}$ respectively. Recall that we may assume
$\ker(f_{i})\neq\ker(f_{j})$. Since both $\ker(f_{i})$ and $\ker(f_{j})$ are
isomorphic to $\alpha_{p}$, [Oor75, Theorem 2] implies that $A^{\prime}$ is
isomorphic to the product of two supersingular elliptic curves. Note also that
$f_{i}^{*}\lambda_{i}=f_{j}^{*}\lambda_{j}$. We claim that
$\ker(f_{i}^{*}\lambda_{i})$ is isomorphic to $\alpha_{p}\times\alpha_{p}$.
Indeed, this is a finite subgroup scheme of a supersingular abelian surface of
degree $p^{2}$, so it must be isomorphic to either
$\alpha_{p}\times\alpha_{p}$ or a non-trivial extension of $\alpha_{p}$ by
itself. But such a non-trivial extension has only one subgroup isomorphic to
$\alpha_{p}$, while $\ker(f_{i}^{*}\lambda_{i})$ has at least two:
$\ker(f_{i})$ and $\ker(f_{j})$.

In particular, we can apply Construction 6.1 with the product of elliptic
curves $A^{\prime}$, the polarization $f_{i}^{*}\lambda_{i}$ and the right
level structure, to get a family of abelian surfaces over
$S=\mathbb{P}^{1}_{\overline{\mathbb{F}_{p}}}$ that specializes to $A_{i}$ and
$A_{j}$ at two points: these points are determined by the subgroups
$\ker(f_{i})$ and $\ker(f_{j})$ of
$\ker(f_{i}^{*}\lambda_{i})=\ker(f_{j}^{*}\lambda_{j})\cong\alpha_{p}\times\alpha_{p}$.
So each $A_{i}$ with the induced polarization and level structure corresponds
to a point of $V_{N}$, the $A^{\prime}_{j}$’s determine by the discussion
above a path in $V_{N}$ connecting all these points, and our assumption that
$\psi$ preserves $\lambda_{x}$ and $\eta_{x}$ assures us that this path is in
fact a loop. This gives us an element of
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$, which maps to $\psi\in
J_{b}(\mathbb{Q}_{p})$. ∎

######  Remark 6.6.

We have shown that the image $\Gamma$ of the map
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to J_{b}(\mathbb{Q}_{p})$
corresponding to $\mathcal{P}_{N}$ consists of the self-quasi-isogenies of
$A_{x}[p^{\infty}]$ that are induced by self-quasi-isogenies of $A_{x}$,
preserving the polarization $\lambda_{x}$ and the level structure $\eta_{x}$.
This agrees with a group defined in the proof of [RZ96, Theorem 6.23], which
is shown to be discrete. Using [RZ96, Theorem 6.30], it follows that one can
obtain the formal completion of $\mathcal{A}_{2,1,N}$ along $V_{N}$ as the
quotient of some Rapoport-Zink space by $\Gamma$. Note the slight inaccuracy
in [CS17, Remark 1.14] however, where it is stated that $\Gamma$ is a discrete
cocompact subgroup of $J_{b}(\mathbb{Q}_{p})$. This is not correct: all quasi-
isogenies in $\Gamma$ have degree one, so the degree map
$J_{b}(\mathbb{Q}_{p})\to\mathbb{Q}_{>0}$ factors through
$J_{b}(\mathbb{Q}_{p})/\Gamma$. But since the image of this degree map is not
compact, $J_{b}(\mathbb{Q}_{p})/\Gamma$ is not compact either, so that
$\Gamma\subset J_{b}(\mathbb{Q}_{p})$ is not cocompact. What does hold
however, is that $\Gamma\subset J_{b}(\mathbb{Q}_{p})$ is cocompact _modulo
the center_ of $J_{b}(\mathbb{Q}_{p})$, which is also the statement given in
[RZ96, Theorem 6.30].

Next, we show that the map $\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to
J_{b}(\mathbb{Q}_{p})$ is injective. In the proof, the following observation
will be used multiple times: let $G$ be a finite subgroup scheme of a
supersingular abelian variety over $\overline{\mathbb{F}_{p}}$, of degree
$p^{n}$ with $n\geq 1$. If $G$ does not contain a subgroup scheme isomorphic
to $\alpha_{p}\times\alpha_{p}$, then $G$ has only one subgroup scheme of
degree $p$.

######  Proposition 6.7.

The map $\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to J_{b}(\mathbb{Q}_{p})$
is injective.

###### Proof.

Consider a sequence of degree $p$ isogenies between principally polarized
abelian surfaces with level structure, of the form

${E_{0}\times E_{0}^{\prime}}$${\ldots}$${E_{m}\times
E_{m}^{\prime}}$${A_{x}}$${A_{1}}$${A_{m}}$${A_{x}}$ (6.3)

as in (6.1). Let us assume that the composition $A_{x}\to A_{x}$ is just the
identity, in which case we need to show this sequence determines the trivial
element of $\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$. Our goal will be to
find a different, strictly shorter sequence determining the same element of
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$, so that the proposition will
follow by induction. Consider the commutative diagram

${B_{m+2,0}}$${B_{m+1,0}}$${B_{m+1,1}}$${\iddots}$${\vdots}$${\ddots}$${B_{2,0}}$${B_{2,m-1}}$${B_{1,0}}$${B_{1,1}}$${B_{1,m-1}}$${B_{1,m}}$${A_{x}}$${A_{1}}$${\ldots}$${A_{m}}$${A_{x},}$
(6.4)

where we denote $E_{i}\times E_{i}^{\prime}$ by $B_{1,i}$, and the $B_{j,i}$
for $j>1$ are defined as the reductions of the obvious fibre products. We will
also denote $A_{i}$ by $B_{0,i}$ and $A_{x}$ by $B_{0,0}$, $B_{0,m+1}$, or
$A_{0}$, depending on the situation. One can inductively show that the
$B_{j,i}$’s are supersingular abelian surfaces: as reducedness implies
smoothness and isogenies are proper, we only need to show they are connected.
But this follows from [Sta, Tag 0377] and the fact that all our isogenies have
degree $1$ or $p$. For the rest of this proof, we will use the term _arrow_ to
denote a single isogeny of degree 1 or $p$ in (6.4) or (6.3), and use
_isogeny_ for compositions of such arrows. We want to show that there is some
$B_{j,i}$, with $j\in\\{1,2\\}$, such that the two arrows out of it have the
same kernel. Indeed, in that case two consecutive arrows in (6.3) will be
equal, so that removing them gives a strictly shorter sequence that determines
the same element of $\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)$.

Let us start by showing that there exists some $B_{j,i}$ with $j\geq 1$, such
that the two arrows out of it have the same kernel. This is clear if there is
some arrow in (6.4) of degree 1: then one can take some $B_{j,i}$, with $j\geq
1$ minimal such that one arrow out of $B_{j,i}$ has degree 1; minimality of
$j$ then implies both arrows out of $B_{j,i}$ have degree 1. So we may assume
all arrows have degree $p$. First, assume there is a composition of arrows
$B_{m+2,0}\to A_{i}$ (where $A_{x}=A_{0}$ is allowed) whose kernel does not
contain a subgroup of the form $\alpha_{p}\times\alpha_{p}$. Then the same
holds for any composition which starts at the other arrow going out of
$B_{m+2,0}$, but where the composed isogeny is the same; we can always find at
least one such composition by our assumption that the composed quasi-isogeny
$A_{x}\to A_{x}$ is the identity. But then the kernels of these composed
isogenies $B_{m+2,0}\to A_{i}$ contain only one subgroup of order $p$, so that
the arrows out of $B_{m+2,0}$ have the same kernel. On the other hand, if the
kernels of all isogenies $B_{m+2,0}\to A_{i}$ in (6.4) contain a subgroup
$\alpha_{p}\times\alpha_{p}$, choose a composition $B_{m+2,0}\to B_{j,i}$,
with $j\geq 1$ minimal such that the composed isogeny does not contain a
subgroup of the form $\alpha_{p}\times\alpha_{p}$. Then both $B_{j-1,i}$ and
$B_{j-1,i+1}$ can be obtained by quotienting out the image of
$\alpha_{p}\times\alpha_{p}\subseteq B_{m+2,0}$ in $B_{j,i}$, and in
particular, the two arrows out of $B_{j,i}$ have the same kernel. (Here, we
used that abelian surfaces have at most one subgroup scheme isomorphic to
$\alpha_{p}\times\alpha_{p}$.)

Now, consider some $B_{j,i}$ as above, so that the two arrows out of it have
the same kernel. As we want to have $j=1$ or $j=2$, let us assume that $j>2$;
again we may assume that all arrows with source $B_{j^{\prime},i^{\prime}}$
for $j^{\prime}<j$ have degree $p$, as otherwise we could already find a
smaller $j$ with the desired property. Note that $B_{j-1,i}$ and $B_{j-1,i+1}$
get canonically identified. If the two compositions $B_{j-1,i}\to B_{j-3,i+1}$
do not have kernel isomorphic to $\alpha_{p}\times\alpha_{p}$, then the
kernels of the arrows out of $B_{j-1,i}$ agree. A similar statement holds if
we replace $B_{j-1,i}$ by $B_{j-1,i+1}$. On the other hand, if the isogenies
$B_{j-1,i}\to B_{j-3,i+1}$ and $B_{j-1,i+1}\to B_{j-3,i+2}$ both have kernel
isomorphic to $\alpha_{p}\times\alpha_{p}$, then both $B_{j-3,i+1}$ and
$B_{j-3,i+2}$ can be obtained by quotienting out the image of (the unique
subgroup scheme) $\alpha_{p}\times\alpha_{p}\subseteq B_{j-1,i}=B_{j-1,i+1}$
in $B_{j-2,i+1}$. In particular, the arrows out of $B_{j-2,i+1}$ have the same
kernel. Continuing in this fashion, we can find $j\in\\{1,2\\}$ and a
$B_{j,i}$ with this property, concluding the proof. ∎

Let us end this paper with two immediate corollaries to Propositions 6.5 and
6.7:

######  Corollary 6.8.

Let $A$ be a supersingular abelian surface over $\overline{\mathbb{F}_{p}}$,
equipped with a principal polarization and level-$N$ structure. Then the group
of self-quasi-isogenies of $A$ respecting this extra structure is a free group
on a finite number of generators.

Our second corollary is concerned with Rapoport-Zink spaces as defined in
[RZ96]. Since we do not use these spaces in the rest of this paper, we omit
their definition, and refer to loc. cit. for the theory behind them. Let us
just mention that they are formal schemes over
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}=\operatorname{Spf}W(\overline{\mathbb{F}_{p}})$,
but by topological invariance, we can and do consider their reduced subschemes
instead, which live over $\operatorname{Spec}\overline{\mathbb{F}_{p}}$.
Moreover, recall from Remark 6.6 that by $p$-adic uniformization, $V_{N}$ is
canonically a quotient $\Gamma\backslash\mathcal{M}_{b}$, where
$\mathcal{M}_{b}$ is the Rapoport-Zink space corresponding to $V_{N}$, and
$\Gamma$ is the image of $\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to
J_{b}(\mathbb{Q}_{p})$, as before.

######  Corollary 6.9.

The Rapoport-Zink space $\mathcal{M}_{b}$ corresponding to the basic stratum
of the Siegel threefold is simply connected.

###### Proof.

Essentially by definition of both $\mathcal{M}_{b}$ and $\mathcal{P}_{N}$,
$\mathcal{M}_{b}$ is the smallest geometric cover of $V_{N}$ trivializing
$\mathcal{P}_{N}$. Hence, as in topology, $\mathcal{M}_{b}$ being simply
connected is equivalent to injectivity of the natural map
$\pi_{1}^{\mathrm{pro\acute{e}t}}(V_{N},x)\to\Gamma$. We conclude by
Proposition 6.7. ∎

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