# Central leaves on Shimura varieties with parahoric reduction

Jens Hesse Technische Universität Darmstadt<EMAIL_ADDRESS>
###### Abstract

We investigate the geometry of the special fiber of the integral model of a
Shimura variety with parahoric level at a given prime place.

To be more precise, we deal with the definition of central leaves in this
situation, their local closedness, and the relationship between the folations
for varying parahoric level. This is connected to the verification of axioms
for integral models formulated by He and Rapoport.

###### Contents

  1. 1 Background
    1. 1.1 Shimura data of Hodge type
    2. 1.2 Bruhat-Tits buildings
    3. 1.3 Alteration of the Hodge embedding
    4. 1.4 Siegel integral models
    5. 1.5 Construction of the integral model
    6. 1.6 Maps between Shimura varieties
    7. 1.7 Local structure of the integral model
      1. 1.7.1 Generizations and irreducible components
      2. 1.7.2 Normalization and completion
    8. 1.8 Hodge tensors and (lack of) moduli interpretation
      1. 1.8.1 The story for the $\mathbb{C}$-valued points
      2. 1.8.2 The story for the integral model
  2. 2 Central leaves in the case of parahoric reduction
    1. 2.1 Definition of central leaves
      1. 2.1.1 An alternative characterization of the central leaves
    2. 2.2 Local closedness of central leaves
      1. 2.2.1 Topological lemmas
      2. 2.2.2 Local closedness
    3. 2.3 Quasi-isogenies of $p$-divisible groups
    4. 2.4 Almost product structure
    5. 2.5 Change of parahoric level
      1. 2.5.1 The change-of-parahoric morphism
      2. 2.5.2 Newton-Igusa variety and change-of-parahoric

Introduction

Shimura varieties are objects of arithmetic geometry (namely varieties over
number fields) that naturally arise in the search for generalized, non-abelian
reciprocity laws (i.e., in the Langlands program) and as moduli spaces of
abelian varieties (with certain extra structures on them). One way of
approaching these objects is to try to understand their mod-$p$ reduction
(which has to be carefully defined first). Insofar as a moduli interpretation
in the above sense exists and continues to exist likewise for the mod-$p$
reduction111There need not be a _literal_ moduli interpretation, but in any
event the stratifications in question derive from a close connection to moduli
problems., it allows us to stratify the moduli space according to several
invariants of the abelian varieties parametrized, e.g., the isomorphism
classes of their $p$-torsion. (An important observation is that these
stratifications genuinely live in the characteristic $p$ world, making use of
Frobenius endomorphisms and so on.) This, very roughly, is the general theme
everything in this article revolves around.

More precisely, we will be dealing with Shimura varieties of Hodge type and
parahoric level structure, at some fixed prime $v\mid p$ of the number field
over which the Shimura variety is defined. Under some reasonably mild
assumptions, cf. 1.16, Kisin and Pappas [KP15] constructed a canonical
integral model for such a Shimura variety. We try to understand some aspects
of the geometry of the special fiber of said integral model, namely the
central leaves (roughly the patches where the isomorphism class of the
$p$-divisible group associated with the abelian variety is constant); their
being locally closed and how they vary as the (parahoric) level is varied.

Let us now go into more detail.

On the integral model $\mathscr{S}_{K}$ ($K$ parahoric level) we have a
“universal” abelian scheme (the quotation marks indicating that it is not
really universal for some moduli problem on $\mathscr{S}_{K}$, but it comes
from a universal abelian scheme via pullback) and we have various kinds of
Hodge tensors. We also have a “universal” isogeny chain of abelian schemes
tightly connected to the “universal” abelian scheme. We define the _naive
central leaves_ on the special fiber $\overline{\mathscr{S}}_{K}$ to be the
loci where the isomorphism type of the geometric fibers of the $p$-divisible
group associated with the “universal” abelian scheme (the “universal”
$p$-divisible group) is constant (alternatively, we may use the “universal”
isogeny chain). We arrive at the non-naive version by taking into account the
Hodge tensors. This is the content of section 2.1.

Next we show

######  Theorem A:

(Corollary 2.12) The central leaves are locally closed.

using a somewhat simpler construction than the one given in [HK17]. By
foundational work of Oort [Oor04] we already know the naive central leaves to
be locally closed. We show that the central leaves are open and closed inside
the naive central leaves. Some basic topological considerations allow us to
treat this question in perfected formal neighborhoods, allowing us to phrase
it as a question about $p$-divisible groups with crystalline Tate tensors; a
question which was answered by Hamacher [Ham17]. This forms section 2.2.

Then we consider (tensor-respecting) self-quasi-isogenies of $p$-divisible
groups (with tensors). The main take-away here is that, if we consider the
geometric fibers of the “universal” isogeny chain of $p$-divisible groups in
the above setting, then

######  Theorem B:

(Example 2.21) The self-quasi-isogenies are independent of the level.

In section 2.4 we recall the almost product structure (under an additional
technical assumption 2.23 concerning the Rapoport-Zink uniformization map,
which is satisfied e.g. if our reductive group over $\mathbb{Q}_{p}$ is
residually split222A reductive group $G/\mathbb{Q}_{p}$ is _residually split_
if it has the same rank as its base change to the maximal unramified extension
of $\mathbb{Q}_{p}$. The name “residually split” derives from the fact that in
this case the maximal reductive quotients of parahoric group schemes
associated with $G$ are split reductive. [Zho18]), which expresses a variant
of the Igusa variety (the _Newton-Igusa variety_ , with quasi-isogenies
instead of isomorphisms) as a product of the Igusa variety and the Rapoport-
Zink space.

We apply this to the question of how the central leaves behave under change of
the parahoric level. We begin by constructing the change-of-parahoric
morphisms on the Shimura varieties, Igusa varieties, Rapoport-Zink spaces and
Newton-Igusa varieties. The almost product isomorphism is compatible with
these morphisms (Lemma 2.38).

From this we can derive

######  Theorem C:

(Corollary 2.42) The map between Igusa varieties for varying parahoric level
is an isomorphism.

This implies in particular

######  Theorem D:

(Corollary 2.43) The change-of-parahoric map between central leaves is
surjective.

This is the most difficult part of the axioms on integral models He and
Rapoport give in [HR17]. Our proof is independent of the one by Rong Zhou in
[Zho18], even though we use some of the results he gave in the first version
of the cited preprint, which did not contain a proof of the surjectivity.
Positively answering another conjecture by He and Rapoport [HR17, Rmk. 3.4],
we show

######  Theorem E:

(Corollary 2.48) The change-of-parahoric map between central leaves is the
composition of a flat universal homeomorphism of finite type and a finite
Ã©tale morphism.

### Acknowledgements

This article essentially is an extract of my doctoral thesis [Hes20] (another
extract333In particular, there is a large overlap between the “Background”
sections of the two articles., dealing with the EKOR stratification, is
[Hes20a]). I thank Torsten Wedhorn for suggesting the topic of the
dissertation, his support and patiently answering my questions. Moreover I
thank Eva Viehmann and Paul Hamacher for their hospitality and helpful
discussions during a month-long stay in Munich at the TU München. I am also
grateful to Timo Richarz and Timo Henkel for numerous helpful discussions.

This research was supported by the Deutsche Forschungsgemeinschaft (DFG),
project number WE 2380/5.

##  1 Background

###  1.1 Shimura data of Hodge type

This article deals with aspects of the geometry of Shimura varieties (of Hodge
type), which are the (systems of) varieties associated with Shimura data (of
Hodge type).

######  Definition 1.1.

A _Shimura datum of Hodge type_ is a pair $(G,X)$, where $G$ is a reductive
algebraic group over $\mathbb{Q}$ and
$X\subseteq\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},G_{\mathbb{R}})$
is a $G(\mathbb{R})$-conjugacy class
($\mathbb{S}:=\operatorname{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}$
being the Deligne torus) subject to the following conditions:

  1. (1)

For $h\in X$, the induced Hodge structure
$\mathbb{S}\xrightarrow{h}G_{\mathbb{R}}\xrightarrow{\mathrm{Ad}}\operatorname{GL}(\operatorname{Lie}(G_{\mathbb{R}}))$
satisfies
$\operatorname{Lie}(G_{\mathbb{C}})=\operatorname{Lie}(G_{\mathbb{C}})^{-1,1}\oplus\operatorname{Lie}(G_{\mathbb{C}})^{0,0}\oplus\operatorname{Lie}(G_{\mathbb{C}})^{1,-1}$.

  2. (2)

$\operatorname{int}(h(i))\colon G^{\mathrm{ad}}_{\mathbb{R}}\to
G^{\mathrm{ad}}_{\mathbb{R}}$ is a Cartan involution, i.e., $\\{g\in
G^{\mathrm{ad}}(\mathbb{C})\;|\;gh(i)=h(i)\overline{g}\\}$ is compact. Another
way of phrasing this condition: Every finite-dimensional real representation
$V$ of $G^{\mathrm{ad}}_{\mathbb{R}}$ carries a
$G^{\mathrm{ad}}_{\mathbb{R}}$-invariant bilinear form $\varphi$ such that
$(u,v)\mapsto\varphi(u,h(i)v)$ is symmetric and positive definite. It is
enough to show that this holds for one _faithful_ finite-dimensional real
representation $V$.

  3. (3)

$G^{\mathrm{ad}}$ _cannot_ be non-trivially written as $G^{\mathrm{ad}}\cong
H\times I$ over $\mathbb{Q}$ with $\mathbb{S}\to
G_{\mathbb{R}}\xrightarrow{\mathrm{proj}}H_{\mathbb{R}}$ trivial.

  4. (4)

There exists an embedding
$(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$, where
$(\operatorname{GSp}(V),S^{\pm})$ is the Shimura datum associated with a
finite-dimensional symplectic $\mathbb{Q}$-vector space $V$ (see below). That
is, we have an embedding $G\hookrightarrow\operatorname{GSp}(V)$ of
$\mathbb{Q}$-group schemes such that the induced map
$\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},G_{\mathbb{R}})\hookrightarrow\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},\operatorname{GSp}(V_{\mathbb{R}}))$
restricts to a map $X\hookrightarrow S^{\pm}$.

######  Example 1.2.

Let $W$ be a finite-dimensional $\mathbb{R}$-vector space.

$\mathbb{R}$-group homomorphisms $\mathbb{S}\to\operatorname{GL}(W)$ then
correspond to Hodge decompositions of $W$, i.e., to decompositions
$W_{\mathbb{C}}=\oplus_{(p,q)\in\mathbb{Z}^{2}}W_{\mathbb{C}}^{p,q}$, such
that $W_{\mathbb{C}}^{p,q}$ is the complex conjugate of $W_{\mathbb{C}}^{q,p}$
for all $(p,q)\in\mathbb{Z}^{2}$. Under this correspondence,
$h\colon\mathbb{S}\to\operatorname{GL}(W)$ corresponds to the Hodge
decomposition $W_{\mathbb{C}}^{p,q}=\\{w\in W_{\mathbb{C}}\;|\;\forall
z\in\mathbb{S}(\mathbb{R})=\mathbb{C}^{\times}\colon
h(z)w=z^{-p}\bar{z}^{-q}w\\}$. Hodge decompositions of $W$ of type
$(-1,0)+(0,-1)$ correspond to complex structures on $W$: If
$h\colon\mathbb{S}\to\operatorname{GL}(W)$ yields such a Hodge decomposition,
then $h(i)$ gives an $\mathbb{R}$-endomorphism $J$ of $W$ with $J\circ
J=-\operatorname{id}_{W}$.

Let $V=(V,\psi)$ be a finite-dimensional symplectic $\mathbb{Q}$-vector space.
We say that a complex structure $J$ on $V_{\mathbb{R}}$ is positive (resp.
negative) if $\psi_{J}:=\psi_{\mathbb{R}}(\\_,J\\_)$ is a positive definite
(resp. negative definite) symmetric bilinear form on $V_{\mathbb{R}}$. Define
$S^{+}$ (resp. $S^{-}$) to be the set of positive (resp. negative) complex
structures on $(V_{\mathbb{R}},\psi_{\mathbb{R}})$ and $S^{\pm}:=S^{+}\sqcup
S^{-}$.

We can make this more concrete: A symplectic basis of
$(V_{\mathbb{R}},\psi_{\mathbb{R}})$ is a basis $e_{1},\dotsc,\allowbreak
e_{g},e_{-g},\dotsc,\allowbreak e_{-1}$, such that $\psi_{\mathbb{R}}$ is of
the form

$\begin{pmatrix}&\tilde{I}_{g}\\\ -\tilde{I}_{g}&\end{pmatrix}$

with respect to this basis, where $\tilde{I}_{g}=\begin{pmatrix}&&1\\\
&\iddots&\\\ 1&&\end{pmatrix}$ is the antidiagonal identity
matrix.444Occasionally (in particular when doing concrete matrix
calculations), it is more convenient to number the basis vectors
$1,\dotsc,g,-1,\dotsc,-g$ instead of $1,\dotsc,g,-g,\dotsc,-1$. Then the
standard symplectic form is given by $\left(\begin{smallmatrix}&I_{g}\\\
-I_{g}&\end{smallmatrix}\right)$, $I_{g}$ being the $g\times g$ identity
matrix.

Let $J$ be the endomorphism of $V_{\mathbb{R}}$ of the form

$\begin{pmatrix}&-\tilde{I}_{g}\\\ \tilde{I}_{g}&\end{pmatrix}$

with respect to this basis. Then $J\in S^{+}$ and what we have described is a
surjective map

$\\{\text{symplectic bases of
}(V_{\mathbb{R}},\psi_{\mathbb{R}})\\}\twoheadrightarrow S^{+}.$

In particular we see that
$\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}}):=\\{f\in\operatorname{GL}(V_{\mathbb{R}})\;|\;\psi_{\mathbb{R}}(f(\\_),f(\\_))=\psi_{\mathbb{R}}\\}$
(by virtue of acting simply transitively on the symplectic bases) acts
transitively on
$S^{+}\cong\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}})/\operatorname{SpO}(V_{\mathbb{R}},\psi_{\mathbb{R}},J)$
(where we define
$\operatorname{SpO}(V_{\mathbb{R}},\psi_{\mathbb{R}},J):=\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}})\cap
O(V_{\mathbb{R}},\psi_{J})=U((V_{\mathbb{R}},J),\psi_{J})$ for a fixed choice
of $J\in S^{+}$) and therefore the general symplectic group
$\operatorname{GSp}(V_{\mathbb{R}},\psi_{\mathbb{R}}):=\\{f\in\operatorname{GL}(V_{\mathbb{R}})\;|\;\psi_{\mathbb{R}}(f(\\_),f(\\_))=c\cdot\psi_{\mathbb{R}}\text{
for some }c\in\mathbb{R}^{\times}\\}$ acts transitively on $S^{\pm}$ (note
that the element of the form $e_{\pm i}\mapsto e_{\mp i}$ of
$\operatorname{GSp}(V_{\mathbb{R}},\psi_{\mathbb{R}})$ for any given choice of
symplectic basis $\left(e_{i}\right)_{i}$ permutes $S^{+}$ and $S^{-}$).

######  Definition 1.3.

Condition (1) of Definition 1.1 implies that the action of
$\mathbb{G}_{m,\mathbb{R}}$ (embedded in $\mathbb{S}$ in the natural way) on
$\operatorname{Lie}(G_{\mathbb{R}})$ is trivial, so that $h$ induces a
homomorphism
${w\colon\mathbb{G}_{m,\mathbb{R}}\to\operatorname{Cent}(G_{\mathbb{R}})}$.
This homomorphism is independent of the choice of $h\in X$ and is called the
_weight homomorphism_ of $(G,X)$.

Moreover, we denote by $\\{\mu\\}$ the the $G(\mathbb{C})$-conjugacy class of
the cocharacter
$\mu_{h}:=h\circ(\operatorname{id}_{\mathbb{G}_{m,\mathbb{C}}},1)\colon\mathbb{G}_{m,\mathbb{C}}\to\mathbb{G}_{m,\mathbb{C}}^{2}\cong\mathbb{S}_{\mathbb{C}}\to
G_{\mathbb{C}}$, where $h$ is as above. Obviously, the conjugacy class
$\\{\mu\\}$ is independent of the particular choice of $h\in X$.

######  Remark 1.4.

Let $L/\mathbb{Q}$ be a field extension such that $G_{L}$ contains a split
maximal torus $T$. Let $W:=\operatorname{Norm}_{G(L)}(T)/T$ be the Weyl group.
Then the natural map

$W\backslash\operatorname{Hom}_{L\text{-grp}}(\mathbb{G}_{m,L},T)\to
G(L)\backslash\operatorname{Hom}_{L\text{-grp}}(\mathbb{G}_{m,L},G_{L})$

is bijective.

Since the left hand side remains unchanged if we go from $L=\bar{\mathbb{Q}}$
(where as usual $\bar{\mathbb{Q}}$ denotes an algebraic closure of
$\mathbb{Q}$) to $L=\mathbb{C}$, we see that $\\{\mu\\}$ contains a
cocharacter defined over $\bar{\mathbb{Q}}$ and that we may then also consider
$\\{\mu\\}$ as a $G(\bar{\mathbb{Q}})$-conjugacy class.

######  Definition 1.5.

The _reflex field_ $\mathbf{E}=\mathbf{E}(G,X)$ of $(G,X)$ is the field of
definition of $\\{\mu\\}$, i.e., the fixed field in $\bar{\mathbb{Q}}$ of
$\\{\gamma\in\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\;|\;\gamma(\\{\mu\\})=\\{\mu\\}\\}$.

######  Example 1.6.

The reflex field of the Shimura datum
$(\operatorname{GSp}_{2g,\mathbb{Q}},S^{\pm})$ of Example 1.2 is $\mathbb{Q}$.
To wit, one of the cocharacters in the conjugacy class $\\{\mu\\}$ is

$\mu(z)=\left(\begin{smallmatrix}z&&&&&\\\ &\ddots&&&&\\\ &&z&&&\\\ &&&1&&\\\
&&&&\ddots&\\\ &&&&&1\end{smallmatrix}\right).$

######  Notation 1.7.

We denote the ring of (rational) adeles by
$\mathbb{A}:=\mathbb{A}_{\mathbb{Q}}$, the subring of finite adeles by
$\mathbb{A}_{f}:=\mathbb{A}_{\mathbb{Q},f}$ and the subring of finite adeles
away from some fixed prime $p$ by $\mathbb{A}_{f}^{p}$.

######  Definition and Remark 1.8.

Let $K\subseteq G(\mathbb{A}_{f})$ be a compact open subgroup. The _Shimura
variety of level $K$ associated with $(G,X)$_ is the double coset space

$\operatorname{Sh}_{K}(G,X):=G(\mathbb{Q})\backslash(X\times(G(\mathbb{A}_{f})/K)).$

A priori, this is just a set, but if $K$ is sufficiently small (i.e., “neat”
in the sense of [Bor69, Pin90]), $\operatorname{Sh}_{K}(G,X)$ can be
canonically written as a finite disjoint union of hermitian symmetric
domains.555If $K$ fails to be sufficiently small, one might very reasonably
argue that our definition of the Shimura variety of level $K$ really is the
definition of the _coarse_ Shimura variety and that one should be working with
stacks instead. Since we will only be interested in sufficiently small level,
this is inconsequential for us. In particular, this gives
$\operatorname{Sh}_{K}(G,X)$ the structure of a complex manifold.

In fact, by the theorem of Baily-Borel, this complex manifold attains the
structure of a quasi-projective complex variety in a canonical way.

By work of Deligne, Milne and Borovoi, this variety is defined already (and
again in a canonical way) over the reflex field $\mathbf{E}$. So in
particular, it is defined over a number field independent of $K$. This is
important when varying $K$ and it is the reason why we consider the whole
Shimura variety instead of its connected components over $\mathbb{C}$ on their
own. It is possible for the Shimura variety to have multiple connected
components over $\mathbb{C}$ while being connected over $\mathbf{E}$.

More detailed explanations may be found in [Mil05].

###  1.2 Bruhat-Tits buildings

Let $K$ be a complete discrete valuation field with ring of integers
$\mathcal{O}$, uniformizer $\varpi$ and perfect residue field
$\kappa:=\mathcal{O}/\varpi$.

######  Notation 1.9.

For a (connected) reductive group $G$ over $K$, we denote by
$\mathcal{B}(G,K)$ the extended (or enlarged) and by
$\mathcal{B}^{\mathrm{red}}(G,K)$ the reduced (i.e., non-extended) Bruhat-Tits
building of $G$ over $K$ [BT84]. Moreover,
$\mathcal{B}^{\mathrm{abstract}}(G,K)$ denotes the underlying abstract
simplicial complex.

######  Remark 1.10.

Let $V$ be a finite-dimensional $K$-vector space.

As described in [KP15, 1.1.9] (originally in [BT84a]), the points of
$\mathcal{B}(\operatorname{GL}(V),K)$ correspond to graded periodic lattice
chains $(\mathcal{L},c)$, i.e.,

  * •

$\emptyset\neq\mathcal{L}$ is a totally ordered set of full
$\mathcal{O}$-lattices in $V$ stable under scalar multiplication (i.e.,
$\Lambda\in\mathcal{L}\iff\varpi\Lambda\in\mathcal{L}$),

  * •

$c\colon\mathcal{L}\to\mathbb{R}$ is a strictly decreasing function such that
$c(\varpi^{n}\Lambda)=c(\Lambda)+n$.

######  Remark 1.11.

Fix such an $\mathcal{L}$ and let $\Lambda^{0}\in\mathcal{L}$. Then every
homothety class of lattices has a unique representative $\Lambda$ such that
$\Lambda\subseteq\Lambda^{0}$ and $\Lambda\not\subseteq\varpi\Lambda^{0}$.
Consider such representatives $\Lambda^{i}$ for all of the distinct homothety
classes of lattices that make up $\mathcal{L}$. Because $\mathcal{L}$ is
totally ordered and $\Lambda^{i}\not\subseteq\varpi\Lambda^{0}$, it follows
that $\Lambda^{i}\supseteq\varpi\Lambda^{0}$ for all $i$ and that
$\left\\{\Lambda^{i}/\varpi\Lambda^{0}\right\\}_{i}$ is a flag of non-trivial
linear subspaces of $\Lambda^{0}/\varpi\Lambda^{0}\cong\kappa^{n}$, where
$n:=\dim V$. Consequently, the number $r$ of homothety classes is in
$\\{1,\dotsc,n\\}$; it is called the _period length_ (or _rank_) of
$\mathcal{L}$. Numbering the $\Lambda^{i}$ in descending order we hence obtain
$r$ lattices $\Lambda^{0},\Lambda^{1},\dotsc,\Lambda^{r-1}$ such that

$\Lambda^{0}\supsetneqq\Lambda^{1}\supsetneqq\dotsb\supsetneqq\Lambda^{r-1}\supsetneqq\varpi\Lambda^{0}$
(1.12)

and $\mathcal{L}$ is given by the the strictly descending sequence of lattices

$\Lambda^{qr+i}=\varpi^{q}\Lambda^{i},\quad q\in\mathbb{Z},\;0\leq i<r.$

######  Remark 1.13.

Let $V$ be a finite-dimensional symplectic $K$-vector space.

$\mathcal{B}(\operatorname{GSp}(V),K)$ embeds into the subset of
$\mathcal{B}(\operatorname{GL}(V),K)$ consisting of those $(\mathcal{L},c)$
such that $\Lambda\in\mathcal{L}\implies\Lambda^{\vee}\in\mathcal{L}$.

Passing to the underlying abstract simplicial complexes means forgetting about
the grading $c$ and

$\mathcal{B}^{\mathrm{abstract}}(\operatorname{GSp}(V),K)=\\{\mathcal{L}\in\mathcal{B}^{\mathrm{abstract}}(\operatorname{GL}(V),K)\;|\;\Lambda\in\mathcal{L}\implies\Lambda^{\vee}\in\mathcal{L}\\}.$

If $\mathcal{L}\in\mathcal{B}^{\mathrm{abstract}}(\operatorname{GSp}(V),K)$
and $\\{\Lambda^{i}\\}_{i}$ is as in Remark 1.11, then there is an involution
$t\colon\mathbb{Z}\to\mathbb{Z}$ with
$\left(\Lambda^{i}\right)^{\vee}=\Lambda^{t(i)}$, $t(i+qr)=t(i)-qr$, and
$i<j\implies t(i)>t(j)$. So $-a:=t(0)>t(1)>\dotsb>t(r)=-a-r$, which implies
$t(i)=-i-a$. Thus $i_{0}-t(i_{0})=2i_{0}+a\in\\{0,1\\}$ for some unique
$i_{0}\in\mathbb{Z}$. Hence, upon renumbering the $\Lambda^{i}$, we may assume
that $a\in\\{0,1\\}$.

We therefore have

$\displaystyle\varpi\Lambda^{0}\subsetneqq\Lambda^{r-1}\subsetneqq\Lambda^{r-2}\subsetneqq\dotsb\subsetneqq\Lambda^{0}\subseteq\left(\Lambda^{0}\right)^{\vee}=\Lambda^{-a}\subsetneqq\left(\Lambda^{1}\right)^{\vee}=\Lambda^{-1-a}$
$\displaystyle\subsetneqq\dotsb\subsetneqq\left(\Lambda^{r-1}\right)^{\vee}=\Lambda^{-r+1-a}\subseteq\Lambda^{-r}=\varpi^{-1}\Lambda^{0}.$

###  1.3 Alteration of the Hodge embedding

######  Notation 1.14.

Let $E$ be a finite field extension of $\mathbb{Q}_{p}$.

Denote by $\breve{E}$ the completion of the maximal unramified extension of
$E$ (hence $\breve{E}=E\cdot\breve{\mathbb{Q}}_{p}$).

######  Remark 1.15.

If $E/\mathbb{Q}_{p}$ is unramified, then ${{\cal
O}_{\breve{E}}}=W(\bar{\mathbb{F}}_{p})$, $\bar{\mathbb{F}}_{p}$ denoting an
algebraic closure of $\mathbb{F}_{p}$ and
$W\colon\mathrm{Ring}\to\mathrm{Ring}$ being the ($p$-adic) Witt vectors
functor. This generalizes to the ramified case using _ramified Witt vectors_
instead, see e.g. [Haz78, Chap. IV, (18.6.13)] or [Ahs11, Chapter 1].

Let $(G,X)$ be a Shimura datum of Hodge type, let
$(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$ be an embedding as in
Definition 1.1 (4), and let $x\in\mathcal{B}(G,\mathbb{Q}_{p})$ be a point in
the Bruhat-Tits building of $G$ over $\mathbb{Q}_{p}$.

We consider the associated Bruhat-Tits scheme ${\cal G}_{x}$, i.e., the affine
smooth model of $G_{\mathbb{Q}_{p}}$ over $\mathbb{Z}_{p}$ such that ${\cal
G}_{x}(\breve{\mathbb{Z}}_{p})\subseteq G(\breve{\mathbb{Q}}_{p})$ is the
stabilizer of the facet of $x$ in ${\cal
B}(G,\breve{\mathbb{Q}}_{p})\overset{\text{\cite[cite]{[\@@bibref{}{landvogt}{}{},
Prop.\leavevmode\nobreak\
2.1.3]}}}{=}\mathcal{B}(G,\mathbb{Q}^{\mathrm{ur}}_{p})$. Let $K_{p}:={\cal
G}_{x}(\mathbb{Z}_{p})\subseteq G(\mathbb{Q}_{p})$ and let $K^{p}\subseteq
G(\mathbb{A}_{f}^{p})$ be a sufficiently small open compact subgroup. Define
$K:=K_{p}K^{p}\subseteq G(\mathbb{A}_{f})$.

######  Assumptions 1.16.

From now on, we will always make the following assumptions:

  * •

$\mathcal{G}_{x}=\mathcal{G}_{x}^{\circ}$ is connected.

  * •

$G$ splits over a tamely ramified extension of $\mathbb{Q}_{p}$.

  * •

$p\nmid\\#\pi_{1}(G^{\mathrm{der}})$.

######  Notation 1.17.

In order not to make notation overly cumbersome, we usually denote the base
change $G_{\mathbb{Q}_{p}}$ of $G$ to $\mathbb{Q}_{p}$ by $G$ again. (Later,
we will almost exclusively be dealing with $G_{\mathbb{Q}_{p}}$.)

Under the above assumptions, Kisin and Pappas construct in [KP15, section 1.2]
(building on [Lan00]) a toral $G(\breve{\mathbb{Q}}_{p})$\- and
$\operatorname{Gal}(\breve{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$-equivariant
embedding

$\iota\colon\mathcal{B}(G,\breve{\mathbb{Q}}_{p})\to\mathcal{B}(\operatorname{GL}(V),\breve{\mathbb{Q}}_{p}),$

restricting666In this case, this is obvious from Galois-equivariance, since
$\mathcal{B}(G,K)=\mathcal{B}(G,K^{\prime})^{\operatorname{Gal}(K^{\prime}/K)}$
if $K^{\prime}/K$ is an unramified extension. to a map
$\iota\colon\mathcal{B}(G,\mathbb{Q}_{p})\to\mathcal{B}(\operatorname{GL}(V),\mathbb{Q}_{p})$.

$\iota$ being a toral embedding means the following: $\iota$ is isometric
after a suitable normalization of the norm on
$\mathcal{B}(G,\breve{\mathbb{Q}}_{p})$; moreover it is the canonical
extension of a map
${\mathcal{B}^{\mathrm{red}}(G,\breve{\mathbb{Q}}_{p})\to\mathcal{B}^{\mathrm{red}}(\operatorname{GL}(V),\breve{\mathbb{Q}}_{p})}$
such that for each maximal $\breve{\mathbb{Q}}_{p}$-split torus $S\subseteq G$
there exists a maximal $\breve{\mathbb{Q}}_{p}$-split torus
$T\subseteq\operatorname{GL}(V)$ such that
$G\hookrightarrow\operatorname{GL}(V)$ maps $S$ into $T$ and $\iota$ restricts
to a map between the reduced apartments associated with $(G,S)$ and
$(\operatorname{GL}(V),T)$, respectively, compatible with translations.

This embedding $\iota$ depends on some choices: By assumption, there exists a
tamely ramified Galois extension $\tilde{K}/\mathbb{Q}_{p}$ with finite
inertia group such that $G_{\tilde{K}}$ is split reductive. Let
$H\to\operatorname{Spec}\mathbb{Z}_{p}$ be the split Chevalley form of $G$
over $\mathbb{Z}_{p}$. Then $\iota$ depends on the choice of

  * •

an isomorphism $G_{\tilde{K}}\cong H_{\tilde{K}}$,

  * •

a pinning
$\left(T,M,f,R,\Delta,\left(X_{\alpha}\right)_{\alpha\in\Delta}\right)$ of $H$
(cf. [SGA3, Exp. XXIII]777Since $\operatorname{Spec}\mathbb{Z}_{p}$ is
connected, some technicalities from _loc. cit._ disappear here.), which also
entails the following: Let $B$ be the Borel subgroup of $H$ corresponding to
$\Delta$. When we talk about roots, it is with respect to $T$, and when we
talk about positive roots and so on, it is with respect to $(T,B)$. We also
fix a hyperspecial vertex $x_{o}$ in $\mathcal{B}(H,\mathbb{Q}_{p})$ with
stabilizer $H(\mathbb{Z}_{p})$.

  * •

for every $\mathbb{Q}_{p}$-irreducible summand888Note that by reductivity and
$\operatorname{char}(\mathbb{Q}_{p})=0$, every finite-dimensional
representation of $G_{\mathbb{Q}_{p}}$ is completely reducible. $V_{i}$ of the
representation $G_{\mathbb{Q}_{p}}\to\operatorname{GL}(V)_{\mathbb{Q}_{p}}$ a
lattice $\Lambda_{i}=U(\mathfrak{n}^{-})v_{i}\subseteq V_{i}$, where
$v_{i}\neq 0$ is a highest weight vector of $V_{i}$ and $\mathfrak{n}^{-}$ is
the (strictly) negative root space inside the Lie algebra of $H$ over
$\mathbb{Z}_{p}$,

  * •

a grading $c_{\Lambda_{i}}+t_{i}$ of the lattice chain
$\\{p^{n}\Lambda_{i}\\}_{n\in\mathbb{Z}}$ given by real numbers
$t_{i}\in\mathbb{R}$. Here
${(c_{\Lambda_{i}}+t_{i})(p^{n}\Lambda_{i})}:=n+t_{i}$.

By [KP15, Lemma 2.3.3], we can (and do) arrange these choices in such a way
that $\iota$ factors

$\mathcal{B}(G,\breve{\mathbb{Q}}_{p})\xrightarrow{j}\mathcal{B}(\operatorname{GSp}(V),\breve{\mathbb{Q}}_{p})\to\mathcal{B}(\operatorname{GL}(V),\breve{\mathbb{Q}}_{p}),$

where the last map is the canonical toral embedding (whose definition will be
clear from Remark 1.13). Again, $j$ restricts to a map

$j\colon\mathcal{B}(G,\mathbb{Q}_{p})\hookrightarrow\mathcal{B}(\operatorname{GSp}(V),\mathbb{Q}_{p}).$
(1.18)

Let $y=(\mathcal{L},c)$ be the image of $x\in\mathcal{B}(G,\mathbb{Q}_{p})$
under the map (1.18) and let $\left(\Lambda^{i}\right)_{i},r,a$ be as in
Remark 1.13. We define
$N_{p}:=\operatorname{Stab}_{\operatorname{GSp}(V)(\mathbb{Q}_{p})}(\mathcal{L})$.

Consider the symplectic $\mathbb{Q}$-vector space

$V^{\S}:=\bigoplus_{i=-(r-1)-a}^{r-1}V$

(direct sum of symplectic spaces, i.e., if $\psi$ denotes the symplectic form
on $V$, then the symplectic form $\psi^{\S}$ on $V^{\S}$ is given by
$\bigoplus_{i=-(r-1)-a}^{r-1}\psi$) and the lattice in
$V^{\S}_{\mathbb{Q}_{p}}$

$\Lambda^{\S}:=\bigoplus_{i=-(r-1)-a}^{r-1}\Lambda^{i}.$

By replacing $\Lambda^{\S}$ by a homothetic lattice, we may assume that
$\Lambda^{\S}\subseteq\left(\Lambda^{\S}\right)^{\vee}$ (hence
$\left(\Lambda^{\S}\right)^{\vee\vee}=\Lambda^{\S}$).

We have a diagonal embedding

$\operatorname{GSp}(V)\hookrightarrow\operatorname{GSp}(V^{\S})$

and (with calligraphic script meaning that we talk about Bruhat-Tits group
schemes)

$\mathcal{GSP}(V)_{y}=\operatorname{Stab}_{\operatorname{GSp}(V)}(\mathcal{L})=\bigcap\operatorname{Stab}_{\operatorname{GSp}(V)}(\Lambda^{i})\subseteq\operatorname{GSp}(\Lambda^{\S})\subseteq\operatorname{GL}(\Lambda^{\S})\subseteq\mathcal{GL}(V)_{\Lambda^{\S}},$

so that upon replacing our original embedding
$G\hookrightarrow\operatorname{GSp}(V)$ by the embedding
$G\hookrightarrow\operatorname{GSp}(V)\hookrightarrow\operatorname{GSp}(V^{\S})$,
we can assume that the parahoric subgroup on the Siegel side is given as

$\operatorname{Stab}_{\operatorname{GL}(V^{\S})(\mathbb{Q}_{p})}(\Lambda^{\S})\cap\operatorname{GSp}(V^{\S})(\mathbb{Q}_{p}),$

i.e., that our level is essentially given by a single lattice, albeit one that
is (in general) not self-dual. This is [KP15, 2.3.15].

###  1.4 Siegel integral models

With notation as above let

$\displaystyle N_{p}$
$\displaystyle:=\operatorname{Stab}_{\operatorname{GSp}(V)(\mathbb{Q}_{p})}(\mathcal{L})\quad\text{(as
before)},$ $\displaystyle J_{p}$
$\displaystyle:=\operatorname{Stab}_{\operatorname{GL}(V^{\S})(\mathbb{Q}_{p})}(\Lambda^{\S})\cap\operatorname{GSp}(V^{\S})(\mathbb{Q}_{p}).$

Let $N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$ and
$J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$ be sufficiently
small open compact subgroups, and $N:=N_{p}N^{p}$, $J:=J_{p}J^{p}$.

In this subsection, we are going to describe integral models of
$\operatorname{Sh}_{N}(\operatorname{GSp}(V),S^{\pm})$ and of
$\operatorname{Sh}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ over
$\mathbb{Z}_{(p)}$ and relate the two.

######  Remark 1.19.

By [RZ96, Definition 6.9], the integral model
$\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})$ is given by the moduli
problem $(\mathbb{Z}_{(p)}\text{-scheme})\ni
S\mapsto\left\\{(A,\bar{\lambda},\eta^{p})\right\\}/{\scriptstyle\cong}$,
where:

  1. (a)

$A=\left(A_{\Lambda}\right)_{\Lambda\in\mathcal{L}}$ is an $\mathcal{L}$-set
of abelian schemes, i.e.,

     * •

for every $\Lambda\in\mathcal{L}$, an abelian $S$-scheme up to
$\mathbb{Z}_{(p)}$-isogeny $A_{\Lambda}$ (i.e., $A_{\Lambda}$ is an object of
the category $(\text{abelian }S\text{-schemes})\otimes\mathbb{Z}_{(p)}$, where
the category $\mathcal{A}\otimes R$ for $\mathcal{A}$ an preadditive category
and $R$ a ring has the same objects as $\mathcal{A}$ and
$\operatorname{Hom}_{\mathcal{A}\otimes
R}(X,Y)=\operatorname{Hom}(X,Y)\otimes_{\mathbb{Z}}R$ for all objects $X,Y$),

     * •

for every inclusion $\Lambda_{1}\subseteq\Lambda_{2}$ a
$\mathbb{Z}_{(p)}$-isogeny $\rho_{\Lambda_{2},\Lambda_{1}}\colon
A_{\Lambda_{1}}\to A_{\Lambda_{2}}$,

     * •

$\rho_{\Lambda_{3},\Lambda_{1}}=\rho_{\Lambda_{3},\Lambda_{2}}\circ\rho_{\Lambda_{2},\Lambda_{1}}$
if $\Lambda_{1}\subseteq\Lambda_{2}\subseteq\Lambda_{3}$ in $\mathcal{L}$,

     * •

the height of $\rho_{\Lambda_{2},\Lambda_{1}}$ is
$\log_{p}|\Lambda_{2}/\Lambda_{1}|$. Here $\rho_{\Lambda_{2},\Lambda_{1}}$
gives rise to a well-defined homomorphism of $p$-divisible groups, and what we
mean is that the kernel of this homomorphism (which is a finite locally free
commutative group scheme, which we also refer to simply as the kernel of
$\rho_{\Lambda_{2},\Lambda_{1}}$) is to have order
$|\Lambda_{2}/\Lambda_{1}|$.

     * •

For every $\Lambda\in\mathcal{L}$, there is an isomorphism (called
_periodicity isomorphism_) $\theta_{\Lambda}\colon A_{\Lambda}\to
A_{p\Lambda}$ such that
$\rho_{\Lambda,p\Lambda}\circ\theta_{\Lambda}=[p]\colon A_{\Lambda}\to
A_{\Lambda}$ is the multiplication-by-$p$ isogeny.

  2. (b)

$\bar{\lambda}\colon A\to\tilde{A}$ is a $\mathbb{Q}$-homogeneous principal
polarization, i.e., a $\underline{\mathbb{Q}^{\times}}$-orbit of a principal
polarization $\lambda\colon A\to\tilde{A}$. Here $\tilde{A}$ is the
$\mathcal{L}$-set of abelian schemes over $S$ up to prime-to-$p$ isogeny given
by $\tilde{A}_{\Lambda}:=(A_{\Lambda^{\vee}})^{\vee}$. And being a
polarization $\lambda$ means being a quasi-isogeny of $\mathcal{L}$-sets
$\lambda\colon A\to\tilde{A}$ such that

$A_{\Lambda}\xrightarrow{\lambda_{\Lambda}}\tilde{A}_{\Lambda}=(A_{\Lambda^{\vee}})^{\vee}\xrightarrow{\varrho_{\Lambda^{\vee},\Lambda}^{\vee}}(A_{\Lambda})^{\vee}$

is a polarization of $A_{\Lambda}$ for all $\Lambda$. If $\lambda_{\Lambda}$
can be chosen to be an isomorphism up to prime-to-$p$ isogeny for all
$\Lambda$, then we speak of a principal polarization. In that case, when
referring to $\lambda_{\Lambda}$, we mean a $\lambda_{\Lambda}$ which is an
isomorphism up to prime-to-$p$ isogeny.

  3. (c)

$\eta^{p}$ is a level-$N^{p}$-structure, i.e. (if $S$ is connected), it is a
$\pi_{1}(S,s)$-invariant $N^{p}$-orbit of symplectic similitudes
$\eta^{p}\colon V_{\mathbb{A}_{f}^{p}}\to H_{1}(A_{s},\mathbb{A}_{f}^{p})$
(where $s$ is some geometric basepoint and $H_{1}(A_{s},\mathbb{A}_{f}^{p})$
with its $\pi_{1}(S,s)$-action corresponds to the Tate
$\mathbb{A}_{f}^{p}$-module of $A$ (cf. [RZ96, 6.8]), which is a smooth
$\mathbb{A}_{f}^{p}$-sheaf). Note that this forces the abelian schemes
$A_{\Lambda}$ to be $(\dim_{\mathbb{Q}}V)$-dimensional.

######  Definition 1.20.

Set $\Lambda^{\S}_{\mathbb{Z}_{(p)}}:=\Lambda^{\S}_{\mathbb{Z}_{p}}\cap
V^{\S}_{\mathbb{Q}}=\prod_{i=-(r-1)-a}^{r-1}\Lambda_{\mathbb{Z}_{(p)}}^{i}$.
We choose a lattice $\Lambda^{\S}_{\mathbb{Z}}\subseteq V^{\S}$ such that
$\Lambda^{\S}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}=\Lambda^{\S}_{\mathbb{Z}_{(p)}}$
and $\Lambda^{\S}_{\mathbb{Z}}\subseteq(\Lambda^{\S}_{\mathbb{Z}})^{\vee}$.

######  Remark 1.21.

Set
$d:=\bigl{|}\left(\Lambda_{\mathbb{Z}}^{\S}\right)^{\vee}/\Lambda_{\mathbb{Z}}^{\S}\bigr{|}$.
By [Kis10, 2.3.3, 3.2.4], the integral model
$\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ is given by the
moduli problem $(\mathbb{Z}_{(p)}\text{-schemes})\ni
S\mapsto\left\\{(A^{\S},\lambda^{\S},\epsilon^{p})\right\\}/{\scriptstyle\cong}$,
where

  1. (a)

$A^{\S}$ is an abelian scheme over $S$ up to $\mathbb{Z}_{(p)}$-isogeny,

  2. (b)

$\lambda^{\S}\colon A^{\S}\to\left(A^{\S}\right)^{\vee}$ is a polarization of
degree $d$ (i.e., the polarization of the (well-defined) associated
$p$-divisible group has degree $d$),

  3. (c)

$\epsilon^{p}$ is a level-$J^{p}$-structure, i.e. (if $S$ is connected), it is
a $\pi_{1}(S,s)$-invariant $J^{p}$-orbit of symplectic similitudes
$\epsilon^{p}\colon V^{\S}_{\mathbb{A}_{f}^{p}}\to
H_{1}(A^{\S}_{s},\mathbb{A}_{f}^{p})$. Note that this forces the abelian
schemes $A^{\S}$ to be $(\dim_{\mathbb{Q}}V^{\S})$-dimensional.

This completes the descriptions of the moduli problems, and we turn to the
question of the relationship between the two. Consider (for appropriate
$N^{p},J^{p}$; see below) the morphism
$\chi\colon\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$
given on $S$-valued points by sending $(A,\bar{\lambda},\eta^{p})$ to
$(A^{\S},\lambda^{\S},\epsilon^{p})$, where

  1. (a)

$\displaystyle A^{\S}:=\prod_{i=-(r-1)-a}^{r-1}A_{\Lambda^{i}}$,

  2. (b)

$\displaystyle\lambda^{\S}:=\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{\Lambda^{i}}\right)$,

  3. (c)

$\epsilon^{p}$ is the product $\prod_{i=-(r-1)-a}^{r-1}\eta^{p}$, to be
interpreted as the product over $\eta^{p}\colon V_{\mathbb{A}_{f}^{p}}\to
H_{1}(A_{\Lambda^{i},s},\mathbb{A}_{f}^{p})\cong
H_{1}(A_{s},\mathbb{A}_{f}^{p})$, where the isomorphism
$H_{1}(A_{\Lambda^{i},s},\mathbb{A}_{f}^{p})\cong
H_{1}(A_{s},\mathbb{A}_{f}^{p})$ is by definition the identity for some fixed
$i=i_{0}$ and otherwise induced by the transition map
$\rho_{\Lambda^{i},\Lambda^{i_{0}}}$. We need that $N^{p}$ is mapped into
$J^{p}$ by $\operatorname{GSp}(V)\hookrightarrow\operatorname{GSp}(V^{\S})$
for this to make sense.

######  Lemma 1.22.

Let $S$ be a scheme, $\ell\neq p$ prime numbers. If $\ell$ does not appear as
a residue characteristic of $S$, then the Tate module functors

$\displaystyle H_{1}(\\_,\mathbb{Z}_{\ell})$
$\displaystyle\colon(\text{abelian }S\text{-schemes})\to(\text{\~{A}©tale
}\mathbb{Z}_{\ell}\text{-local systems on }S),$ $\displaystyle
H_{1}(\\_,\mathbb{Q}_{\ell})$ $\displaystyle\colon(\text{abelian
}S\text{-schemes})\to(\text{\~{A}©tale }\mathbb{Q}_{\ell}\text{-local systems
on }S)$

(cf. [Gro74, III, 5.4 and 6.2] for precise definitions) are faithful.

If only $p$ and $0$ appear as residue characteristics of $S$, then the Tate
module functor

$H_{1}(\\_,\mathbb{A}_{f}^{p})\colon(\text{abelian
}S\text{-schemes})\to(\text{\~{A}©tale }\mathbb{A}_{f}^{p}\text{-local systems
on }S)$

is faithful.

######  Proof:

First note that the statements about $H_{1}(\\_,\mathbb{Q}_{\ell})$ and
$H_{1}(\\_,\mathbb{A}_{f}^{p})$ follows from the statement about
$H_{1}(\\_,\mathbb{Z}_{\ell})$, which is why it is enough to only look at
$H_{1}(\\_,\mathbb{Z}_{\ell})$.

A homomorphism of abelian $S$-schemes $f\colon A\to B$ vanishes if and only if
it vanishes over every (geometric) fiber of $S$: Indeed, if it vanishes
fiberwise, then it is flat by the fiber criterion for flatness. Applying that
criterion again we see that the closed immersion and fiberwise isomorphism
$\ker(f)\hookrightarrow A$ is flat, which means that is an isomorphism.

This way we are reduced to the case where $R$ is an (algebraically closed)
field of characteristic different from $\ell$. In this setting the
faithfulness is well-known (the salient point being that the $\ell$-primary
torsion is dense). □

######  Lemma 1.23.

Let $H$ be a totally disconnected locally compact999By (our) definition,
locally compact implies Hausdorff. group (i.e., a locally profinite group) and
let $N\subseteq H$ be a compact subgroup. Then

$N=\bigcap_{\begin{subarray}{c}N\subseteq J\\\ J\subseteq H\text{ open compact
subgrp.}\end{subarray}}J.$

Note that this is (a variant of) a well-known theorem by van Dantzig if
$N=\\{1\\}$ [Dan36].

######  Proof:

We make use of the following fact [AT08, Prop. 3.1.7]: A Hausdorff space is
locally compact and totally disconnected if and only if the open compact sets
form a basis of the topology. (Van Dantzig’s theorem is the group version of
this, which talks only about a neighborhood basis of the identity and open
compact _subgroups_.)

First we show that $N$ is contained in some open compact subset $K\subseteq
H$. For every $x\in N$ choose a compact open neighborhood $x\in K_{x}\subseteq
H$. This is possible by the fact cited above. Then there is a finite subset
$I\subseteq N$ such that $N\subseteq\bigcup_{x\in I}K_{x}=:K$.

Next, for every $x\in N$ choose an open neighborhood of the identity $U_{x}$
such that $xU_{x}K\subseteq K$. With $N\subseteq U:=\bigcup_{x\in N}xU_{x}$ we
obtain $UK\subseteq K$. Replacing $U$ by $U\cap U^{-1}$, we may moreover
assume it is symmetric. The subgroup generated by $U$ is open (hence closed)
and contained in $K$, hence is an open compact subgroup.

Thus $N$ even is contained in an open compact sub _group_ ; in other words, we
may assume that $H$ is compact, i.e., is a profinite group.

Then $H/N$ is compact101010Hausdorff quotient spaces of compact spaces are
compact again, but for “locally compact” the analogous statement is not true
in general! and totally disconnected111111Take $x,y\in H$ such that $xN\neq
yN$. We show that any subspace $S\subseteq H/N$ containing both $xN$ and $yN$
is disconnected. Let $U\subseteq H/N$ be a neighborhood of $xN$ not containing
$yN$. Let $x\in V\subseteq\pi^{-1}(U)$ be open and compact, where $\pi\colon
H\to H/N$ is the projection. Then $yN\notin\pi(V)\subseteq H/N$ is open and
compact (hence closed) and we have $S=(\pi(V)\cap S)\sqcup S\setminus\pi(V)$
where both $\pi(V)\cap S$ and $S\setminus\pi(V)$ are open in $S$. This shows
that $S$ is disconnected. (i.e., is a Stone space). By the fact cited above,

$H/N\supseteq\\{1\\}=\bigcap_{L\subseteq H/N\text{ open compact subset}}L.$

Observe that the quotient map $H\to H/N$ is proper to deduce

$N=\bigcap_{\begin{subarray}{c}N\subseteq M\\\ M\subseteq H\text{ open compact
subset}\end{subarray}}M.$

Say $M$ is an open and compact subset of $H$ containing $N$. As we have shown
above, there is an open compact subgroup $J\subseteq H$ in between $N$ and
$M$, and this is all we need to complete the proof. □

######  Proposition 1.24.

For every compact open subgroup
$N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$

$\chi\colon\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$

is a well-defined morphism for all compact open subgroups $N^{p}\subseteq
J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$ and is a closed
immersion for all sufficiently small compact open subgroups $N^{p}\subseteq
J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$.

######  Proof:

The fact that it’s well-defined is clear from the construction.

To show the second statement, as in [Del71, Prop. 1.15], it is enough to show
that

$\mathscr{S}_{N_{p}N^{p}}(\operatorname{GSp}(V),S^{\pm})\to\varprojlim_{J^{p}}\mathscr{S}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$

is a closed immersion, i.e., a proper monomorphism.

We begin by proving that it is a monomorphism, i.e., injective on $S$-valued
points ($S$ arbitrary $\mathbb{Z}_{(p)}$-scheme). So, say
$(A_{1},\lambda_{1},\eta_{1}^{p})$ and $(A_{2},\lambda_{2},\eta_{2}^{p})$ both
map to $(A^{\S},\lambda^{\S},\epsilon_{J^{p}}^{p})$. That means precisely that
there is an isomorphism of abelian $S$-schemes up to
$\mathbb{Z}_{(p)}$-isogeny

$\phi\colon\prod_{i=-(r-1)-a}^{r-1}A_{1,\Lambda^{i}}\xrightarrow{\cong}\prod_{i=-(r-1)-a}^{r-1}A_{2,\Lambda^{i}}$

such that

$\phi^{\vee}\circ\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{2,\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{2,\Lambda^{i}}\right)\circ\phi=\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{1,\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{1,\Lambda^{i}}\right)$

and

$H_{1}(\phi,\mathbb{A}_{f}^{p})\circ\epsilon_{1,J^{p}}^{p}=\epsilon_{2,J^{p}}^{p}\mod{J^{p}}.$

We claim that $\phi$ comes from isomorphisms

$\phi_{i}\colon A_{1,\Lambda^{i}}\xrightarrow{\cong}A_{2,\Lambda^{i}}.$

Certainly there is but one candidate for $\phi_{i}$: define $\phi_{i}$ to be
the composition

$A_{1,\Lambda^{i}}\xrightarrow{\mathrm{incl}}\prod_{i=-(r-1)-a}^{r-1}A_{1,\Lambda^{i}}\xrightarrow{\phi}\prod_{i=-(r-1)-a}^{r-1}A_{2,\Lambda^{i}}\xrightarrow{\mathrm{proj}}A_{2,\Lambda^{i}}.$

Our claim then is that

$\phi=\prod_{i=-(r-1)-a}^{r-1}\phi_{i}.$

Apply $H^{1}(\\_,\mathbb{A}_{f}^{p})$ on both sides. For the left hand side,
we have

$H_{1}(\phi,\mathbb{A}_{f}^{p})=\epsilon_{2,J^{p}}^{p}\circ\left(\epsilon_{1,J^{p}}^{p}\right)^{-1}\mod{J^{p}}.$

and the right hand side of this equation is block diagonal. So

$H_{1}(\phi,\mathbb{A}_{f}^{p})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{A}_{f}^{p})\mod{J^{p}}.$

Since (by Lemma 1.23)

$N^{p}=\bigcap_{\begin{subarray}{c}N_{\ell}\subseteq J_{\ell}\\\
J_{\ell}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{Q}_{\ell})\text{ cpt. open
subgrp.}\end{subarray}}J_{\ell},$

it follows that (with $\ell\neq p$)

$H_{1}(\phi,\mathbb{Q}_{\ell})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{Q}_{\ell})\mod{N_{\ell}},$

hence (since $N_{\ell}$ acts block-diagonally) that
$H_{1}(\phi,\mathbb{Q}_{\ell})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{Q}_{\ell})$.

Since $H_{1}(\\_,\mathbb{Q}_{\ell})$ is faithful (Lemma 1.22), this implies
$\phi=\prod_{i=-(r-1)-a}^{r-1}\phi_{i}$, as desired.

Next, consider the extension by zero of
$\left(H_{1}(\rho_{1/2,\Lambda^{j},\Lambda^{i}},\mathbb{A}_{f}^{p})\right)_{i,j}$
(where for “$1/2$” either “$1$” or “$2$” can be plugged in) to a map
$H_{1}(A^{\S},\mathbb{A}_{f}^{p})\to H_{1}(A^{\S},\mathbb{A}_{f}^{p})$. Under
the isomorphism given by the $J^{p}$-level structure this corresponds, up to
the $J^{p}$-action, to the map $V^{\S}_{\mathbb{A}_{f}^{p}}\to
V^{\S}_{\mathbb{A}_{f}^{p}}$ given by mapping the $i$’th copy of
$V_{\mathbb{A}_{f}^{p}}$ identically to the $j$’th copy and the rest to zero.
Thus $\rho_{1/2,i,j}$ yield the same up to $J^{p}$ after applying
$H_{1}(\\_,\mathbb{A}_{f}^{p})$, hence they are equal in the
$\mathbb{Z}_{(p)}$-isogeny category.

Consequently, $\chi$ is a monomorphism.

For properness, we will use the valuative criterion. Let $R$ be a discrete
valuation ring with field of fractions $K$ and assume that a $K$-point
$A^{\S}=\prod_{i=-(r-1)-a}^{r-1}A_{\Lambda^{i}}$ with its additional
structures coming from $(A_{\Lambda^{i}})_{i}$ extends to an $R$-point
$\mathcal{A}^{\S}$. Consider the map $A^{\S}\to A_{\Lambda^{i_{0}}}\to A^{\S}$
where the first map is a projection and the second an inclusion. By the NÃ©ron
mapping property, this extends to a map $\mathcal{A}^{\S}\to\mathcal{A}^{\S}$.
Define $\mathcal{A}_{\Lambda^{i_{0}}}$ to be the image of this map.

The NÃ©ron mapping property also allows us to extend the transition isogenies
$\rho_{\Lambda^{i_{0}},\Lambda^{j_{0}}}\colon\allowbreak{A_{\Lambda^{j_{0}}}\to
A_{\Lambda^{i_{0}}}}$, $i_{0}\leq j_{0}$, the periodicity isomorphisms, and
the polarization.

Since $\pi_{1}(\operatorname{Spec}K)$ surjects onto
$\pi_{1}(\operatorname{Spec}R)$ (see [Stacks, Tag 0BQM]), extending the level
structure away from $p$ is trivial. □

###  1.5 Construction of the integral model

Let $\mathbf{E}$ be the reflex field of $(G,X)$ and $v\mid p$ a place of
$\mathbf{E}$.

As the first step towards the construction of the integral model, we define
$\mathscr{S}_{K}^{-}(G,X)\to\operatorname{Spec}\mathcal{O}_{\mathbf{E},(v)}$
to be the closure of $\operatorname{Sh}_{K}(G,X)$ in
$\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$
(or, equivalently by Proposition 1.24, in
$\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$).
By this we mean the topological closure with the reduced subscheme structure
or, equivalently (since $\operatorname{Sh}_{K}(G,X)$ is reduced), the flat
closure (in the sense of [EGA4, § 2.8]).

Then we define
$\mathscr{S}_{K}(G,X)\to\operatorname{Spec}\mathcal{O}_{\mathbf{E},(v)}$ to be
the normalization of
$\mathscr{S}_{K}^{-}(G,X)\to\operatorname{Spec}\mathcal{O}_{\mathbf{E},(v)}$.

This can be regarded as “the obvious definition”, but it is entirely non-
obvious and due to Kisin and Pappas that this really behaves as one would
expect from a canonical integral model. These expectations, properly explained
in [KP15], mainly concern the satisfaction of an extension property which is a
weaker version of the valuative criterion for properness, and the existence of
a local model diagram that essentially shows that the integral model is
Ã©tale-locally isomorphic to the local model described by Pappas and Zhu
[PZ13].

Define $E:=\mathbf{E}_{v}$, the $v$-adic completion of $\mathbf{E}$, and
denote by $\kappa$ the residue field of $E$, a finite extension of
$\mathbb{F}_{p}$. By abuse of notation, we also denote the base change of
$\mathscr{S}_{K}(G,X)$ to $\mathcal{O}_{E}$ by $\mathscr{S}_{K}(G,X)$ again.
By $\overline{\mathscr{S}}_{K}(G,X)$ we denote the base change of
$\mathscr{S}_{K}(G,X)$ to $\kappa$.

###  1.6 Maps between Shimura varieties

This section is based on [Zho18], where more detailed explanations may be
found. We let $K_{p}^{\prime}=\mathcal{G}^{\prime}(\mathbb{Z}_{p})$ be another
parahoric where $\mathcal{G}^{\prime}$ is a Bruhat-Tits group scheme with
$\mathcal{G}^{\prime}=(\mathcal{G}^{\prime})^{\circ}$. We assume that
$K_{p}\subseteq K_{p}^{\prime}$, i.e., that the facet (in the Bruhat-Tits
building) associated with $K_{p}^{\prime}$ is contained in the closure of the
one associated with $K_{p}$.

######  Theorem 1.25.

[Zho18, Theorem 7.1] For sufficiently small $K^{p}$ there exists a morphism

$\pi_{K_{p}K^{p},K_{p}^{\prime}K^{p}}\colon\mathscr{S}_{K_{p}K^{p}}(G,X)\to\mathscr{S}_{K_{p}^{\prime}K^{p}}(G,X).$

In moving towards a proof, let us first note that
$\operatorname{GSp}(V,\psi)\to\operatorname{GSp}(V^{\S},\psi^{\S})$ factors
through

$M:=\left\\{\left(g_{i}\right)_{i}\in\prod_{i=-(r-1)-a}^{r-1}\operatorname{GSp}(V,\psi)\;|\;c(g_{-(r-1)-a})=\dotsb=c(g_{r-1})\right\\},$

where $c\colon\operatorname{GSp}(V,\psi)\to\mathbb{G}_{m}$ is the
multiplicator homomorphism.

There is a natural $X_{M}$ that makes $(M,X_{M})$ into a Shimura datum. Define

$H_{p}:=\operatorname{Stab}_{M(\mathbb{Q}_{p})}(\Lambda^{\S}),\quad
J_{p}:=\operatorname{Stab}_{\operatorname{GSp}(V^{\S}_{\mathbb{Q}_{p}},\psi^{\S}_{\mathbb{Q}_{p}})}(\Lambda^{\S}).$

Then for sufficiently small $H^{p},J^{p}$, we obtain a closed immersion

$i\colon\operatorname{Sh}_{H_{p}H^{p}}(M,X_{M})\hookrightarrow\operatorname{Sh}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S},\psi^{\S}),S^{\S,\pm}).$

$\operatorname{Sh}_{H_{p}H^{p}}$ has a moduli interpretation over
$\mathbb{Z}_{(p)}$ (essentially: $\\#\\{-(r-1)-a,\dotsc,r-1\\}=2(r-1)+a+1$
abelian schemes up to prime-to-$p$ isogeny endowed with certain polarizations
and prime-to-$p$ level structure).

######  Proposition 1.26.

[Zho18, Prop. 7.2] If $J^{p}$ is sufficiently small, then $i$ extends to a
closed immersion

$i\colon\mathscr{S}_{H_{p}H^{p}}(M,X_{M})\to\mathscr{S}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S},\psi^{\S}),S^{\S,\pm}).$

Now consider the embedding
$i\colon\mathcal{B}(G,\mathbb{Q}_{p})\to\mathcal{B}(\operatorname{GSp}(V,\psi),\mathbb{Q}_{p})$.

We still have $K_{p}=\mathcal{G}(\mathbb{Z}_{p})$ with
$\mathcal{G}=\mathcal{G}_{x}$, $x\in\mathcal{B}(G,\mathbb{Q}_{p})$. Let
$\mathfrak{g}$ be the minimal facet in
$\mathcal{B}(\operatorname{GSp}(V,\psi),\mathbb{Q}_{p})$ containing $i(x)$. So
$\mathfrak{g}$ corresponds to some lattice chain
$\Lambda^{0}\supseteq\dotsb\supseteq\Lambda^{r-1}$ as above. We have

$\operatorname{Sh}_{K_{p}K^{p}}(G,X)\xrightarrow{\text{cl.
imm.}}\operatorname{Sh}_{H_{p}H^{p}}(M,X_{M})_{\mathbf{E}}\xrightarrow{\text{cl.
imm.}}\operatorname{Sh}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S},\psi^{\S}),S^{\S,\pm})_{\mathbf{E}}$

and $\mathscr{S}^{-}_{K}(G,X)$ is defined to be the closure of
$\operatorname{Sh}_{K_{p}K^{p}}(G,X)$ in
$\mathscr{S}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S},\psi^{\S}),S^{\S,\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$.

######  Corollary 1.27.

[Zho18, Cor. 7.3] $\mathscr{S}^{-}_{K}(G,X)$ also can be described as the
closure of $\operatorname{Sh}_{K_{p}K^{p}}(G,X)$ in
$\mathscr{S}_{H_{p}H^{p}}(M,X_{M})_{\mathcal{O}_{\mathbf{E},(v)}}$.

Let $\mathfrak{f}^{\prime}$ be a facet of
$\mathcal{B}(\operatorname{GSp}(V^{\S},\psi^{\S}),\mathbb{Q}_{p})$ in the
closure of $\mathfrak{g}$. Then $\mathfrak{f}^{\prime}$ corresponds to a “sub-
lattice chain” $\Lambda^{i_{1}}\supseteq\dotsb\supseteq\Lambda^{i_{s}}$,
$\left\\{i_{1},\dotsc,i_{s}\right\\}\subseteq\\{0,\dotsc,r-1\\}$. Defining
$M^{\prime},H_{p}^{\prime}$ as above, we naturally obtain morphisms $M\to
M^{\prime}$ and

$\omega_{H_{p}H^{p},H_{p}^{\prime}\left.H^{\prime}\right.^{p}}\colon\mathscr{S}_{H_{p}H^{p}}(M,X_{M})\to\mathscr{S}_{H^{\prime}_{p}\left.H^{\prime}\right.^{p}}(M^{\prime},X_{M^{\prime}})$

for suitable levels $H^{p},\left.H^{\prime}\right.^{p}$ away from $p$.

######  Sketch of Proof (of Theorem 1.25):

Let $\mathfrak{f}$ be the facet of $K_{p}$ (that is to say $\mathfrak{f}$ is
the minimal facet satisfying $x\in\mathfrak{f}$), $\mathfrak{f}^{\prime}$ that
of $K_{p}^{\prime}$. Then
$\mathfrak{f}\subseteq\overline{\mathfrak{f}^{\prime}}$.

Let $x^{\prime}\in\mathfrak{f}^{\prime}$ be so close to $x$ that
$\mathfrak{g}\subseteq\overline{\mathfrak{g}^{\prime}}$ in
$\mathcal{B}(\operatorname{GSp}(V,\psi),\mathbb{Q}_{p})$, where $\mathfrak{g}$
and $\mathfrak{g}^{\prime}$ denote the minimal facets containing $i(x)$ and
$i(x^{\prime})$ respectively.

The constructions from above yield:

$\mathscr{S}_{K_{p}K^{p}}^{-}(G,X)$$\mathscr{S}_{K^{\prime}_{p}\left.K^{\prime}\right.^{p}}^{-}(G,X)$$\mathscr{S}_{H_{p}^{\prime}\left.H^{\prime}\right.^{p}}(M^{\prime},X_{M^{\prime}})_{\mathcal{O}_{\mathbf{E},(v)}}$$\mathscr{S}_{H_{p}\left.H\right.^{p}}(M,X_{M})_{\mathcal{O}_{\mathbf{E},(v)}}$closed
imm.closed imm.$\omega_{H_{p}H^{p},H_{p}^{\prime}\left.H^{\prime}\right.^{p}}$
(1.28)

On the generic fiber we can complete this to a commutative square.

By Corollary 1.27, this implies that (1.28) also completes to a commutative
square. Now normalize. □

###  1.7 Local structure of the integral model

####  1.7.1 Generizations and irreducible components

Let $\mathscr{X}\to\operatorname{Spec}\mathcal{O}_{\breve{E}}$ be a flat
scheme locally of finite type; denote the special fiber by
$X\to\operatorname{Spec}\bar{\mathbb{F}}_{p}$ and the generic fiber by
$\mathcal{X}\to\operatorname{Spec}\breve{E}$. We assume that $\mathcal{X}$ is
locally integral (e.g. smooth).

For example, we can consider
$(\mathscr{X},X,\mathcal{X})=(\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}},\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}}\otimes_{{{\cal
O}_{\breve{E}}}}\bar{\mathbb{F}}_{p},\allowbreak{\operatorname{Sh}_{K}(G,X)\otimes_{E}\breve{E}})$.

Let $\bar{x}\in X(\bar{\mathbb{F}}_{p})$.

######  Lemma 1.29.

There is a generization $x$ of $\bar{x}$ which lies in the generic fiber
$\mathcal{X}$, and is a closed point in there, i.e., $x\in\mathcal{X}(L)$ for
a finite extension $L/\breve{E}$.

######  Definition 1.30.

We shall call such a point $x$ a _closed point generization_ of $\bar{x}$ for
short.

######  Proof:

Due to flatness (going-down) there is _some_ generization in the generic
fiber; call it $x_{0}$.

By [Stacks, Tag 053U] the following set is dense (and in particular non-empty)
in the closure of $\\{x_{0}\\}$ in $\mathcal{X}$:

$\left\\{x\in\mathscr{X}\;|\;x\text{ is a specialization of }x_{0}\text{ and a
closed point generization of }\bar{x}\right\\}.$

□

######  Lemma 1.31.

Notation as in the preceding lemma.

The specialization $x\leadsto\bar{x}$ can be realized by an ${\cal
O}_{L}$-valued point of $\mathscr{X}$.

######  Proof:

First off, by [EGA2, 7.1.9], it can be realized by a morphism
$\operatorname{Spec}R=\\{\eta,s\\}\to\mathscr{X}$ of ${{\cal
O}_{\breve{E}}}$-schemes, where $R$ is a discrete valuation ring such that
$L\cong\kappa(\eta)=\operatorname{Quot}(R)$ as field extensions of
$\kappa(x)$.

We hence get local homomorphisms of local rings ${{\cal
O}_{\breve{E}}}\to{\cal O}_{\mathscr{X},\bar{x}}\to R$.

Thus the discrete valuation on $L$ defined by $R$ extends the discrete
valuation on $\breve{E}$. But there is but one such extension and its
valuation ring is ${\cal O}_{L}$ (by definition). □

######  Lemma 1.32.

Mapping $x$ to the unique irreducible component of $\mathscr{X}$ that contains
$x$ establishes a surjection from the set of closed point generizations $x$ of
$\bar{x}$ to the set of irreducible components of $\mathscr{X}$ containing
$\bar{x}$.

######  Proof:

If $x_{0}\in\mathcal{X}$ is a generization of $\bar{x}$, then $x_{0}$ lies in
a unique irreducible component of $\mathscr{X}$ because $\mathcal{X}$ is
locally irreducible. Hence the map described above is well-defined.

Now for surjectivity: Given an irreducible component $C$ of $\mathscr{X}$
containing $\bar{x}$, let $x_{0}\in C$ be the generic point. Then $x_{0}$ must
be in the generic fiber (else we would be able to find a generization in the
generic fiber by going-down). Now go through the proof of Lemma 1.29 with this
particular choice of $x_{0}$. □

####  1.7.2 Normalization and completion

For reference, we collect some facts concerning the passage to normalization
and completion and in particular how it applies to integral models of Shimura
varieties.

######  Fact 1.33.

$\mathscr{S}^{-}_{K}(G,X)\to\operatorname{Spec}{{\cal O}_{\breve{E}}}$ is
quasi-projective, so in particular of finite type.

Hence $\mathscr{S}^{-}_{K}(G,X)$ and $\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}}$ are excellent.

As a consequence the normalization

$\mathscr{S}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}}\xrightarrow{\nu}\mathscr{S}^{-}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}}$

is finite. (This really is a normalization because normalization and
completion behave well together in the excellent case (to get from ${\cal
O}_{E^{\mathrm{ur}}}$ to ${{\cal O}_{\breve{E}}}$ and from ${\cal
O}_{\mathbf{E},(v)}$ to ${\cal O}_{E}$) and because normalization commutes
with base change along filtered colimits of smooth morphisms (to get from
${\cal O}_{E}$ to ${\cal O}_{E^{\mathrm{ur}}}$)).

We will always denote by $(\;)^{\sim}$ the integral closure of a ring in its
total ring of fractions.

$\bar{x}$ still denotes an $\bar{\mathbb{F}}_{p}$-valued point of
$\mathscr{S}^{-}_{K}(G,X)$. Let
$\nu^{-1}(\bar{x})=\\{\bar{y}=\bar{y}_{1},\dotsc,\bar{y}_{n}\\}$.

Let $\mathscr{S}^{-}:=\mathscr{S}^{-}_{K}(G,X)_{{{\cal O}_{\breve{E}}}}$ and
$\mathscr{S}:=\mathscr{S}_{K}(G,X)_{{{\cal O}_{\breve{E}}}}$.

######  Fact 1.34.

$\left(\nu_{*}{\cal
O}_{\mathscr{S}^{-}_{K}(G,X)}\right)_{\bar{x}}^{\wedge}=\prod_{j=1}^{n}\widehat{\cal
O}_{\mathscr{S},\bar{y}_{j}}$.

######  Fact 1.35.

By [Stacks, Tag 0C3B]: $\left(\nu_{*}{\cal
O}_{\mathscr{S}^{-}_{K}(G,X)}\right)_{\bar{x}}=\widetilde{\cal
O}_{\mathscr{S}^{-},\bar{x}}$.

######  Fact 1.36.

By [Stacks, Tag 035P]:
$\widetilde{A}=\prod_{\mathfrak{q}\in\operatorname{Min}(A)}(A/\mathfrak{q})^{\sim}$,
if $A$ is a reduced ring and $\\#\operatorname{Min}(A)<\infty$.

######  Fact 1.37.

$(\;)^{\sim}$ and $(\;)^{\wedge}$ commute in the case of excellence (see e.g.
[EGA4, 7.6.1, 7.8.3.1(vii)]).

For instance, $\left.{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\wedge}\right.^{\sim}=\left.{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\sim}\right.^{\wedge}$ (on the right hand side,
we have a completion of a ${\cal O}_{\mathscr{S}^{-},\bar{x}}$-module).

######  Fact 1.38.

In the case of excellence, the completion of a normal domain is a normal
domain (see e.g. [GW10, 12.50]).

Thus we have

$\displaystyle\prod_{\mathfrak{q}\in\operatorname{Min}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}})}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q})^{\sim}$
$\displaystyle\cong\left.\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\wedge}\right.^{\sim}\cong\left.\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\sim}\right.^{\wedge}$ (1.39)
$\displaystyle\cong\left(\nu_{*}{\cal
O}_{\mathscr{S}^{-}_{K}(G,X)}\right)_{\bar{x}}^{\wedge}\cong\prod_{j=1}^{n}\widehat{\cal
O}_{\mathscr{S},\bar{y}_{j}}$

and the rings $\widehat{\cal O}_{\mathscr{S},\bar{y}_{j}}$ are normal domains.
Hence we obtain a bijection

$\operatorname{Min}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}})\xleftrightarrow{\;1:1\;}\nu^{-1}(\bar{x})$

such that there exists a numbering $\mathfrak{q}_{1},\dotsc,\mathfrak{q}_{n}$
of the elements of $\operatorname{Min}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}})$ such that (1.39) restricts to an isomorphism

$\widehat{\cal O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}_{j}\cong\widehat{\cal
O}_{\mathscr{S},\bar{y}_{j}}$

(also see [EGA4, 7.6.2]).

Also:

$\displaystyle\left.\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}^{\sim}\right.^{\wedge}$
$\displaystyle\cong\left(\prod_{\mathfrak{q}\in\operatorname{Min}({\cal
O}_{\mathscr{S}^{-},\bar{x}})}({\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q})^{\sim}\right)^{\wedge}$
$\displaystyle\cong\prod_{\mathfrak{q}\in\operatorname{Min}({\cal
O}_{\mathscr{S}^{-},\bar{x}})}\left.\left({\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\right)^{\sim}\right.^{\wedge}$
$\displaystyle\cong\prod_{\mathfrak{q}\in\operatorname{Min}({\cal
O}_{\mathscr{S}^{-},\bar{x}})}\left.\left({\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\right)^{\wedge}\right.^{\sim}$
$\displaystyle\overset{\mathclap{\text{Artin-
Rees}}}{\cong}\prod_{\mathfrak{q}\in\operatorname{Min}({\cal
O}_{\mathscr{S}^{-},\bar{x}})}\left(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\widehat{\mathfrak{q}}\right)^{\sim},\quad\widehat{\mathfrak{q}}=\mathfrak{q}\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}.$

Now by [KP15, 2.1.2, 4.2.2] for all
$\mathfrak{q}\in\operatorname{Min}(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}})$ we have that

$\left(\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\right)^{\sim}=\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\cong
R_{G,\bar{x},\mathfrak{q}}=R_{G}$

is normal and $\widehat{\cal
O}_{\mathscr{S}^{-},\bar{x}}/\mathfrak{q}\cong\widehat{\cal
O}_{\mathscr{S},\bar{y}}$ for an appropriate choice of $\mathfrak{q}$ (i.e.,
of $x$). (The notation $R_{G}$ is from [KP15], where it is defined as the
formal local ring of the local model.)

###  1.8 Hodge tensors and (lack of) moduli interpretation

We discuss the moduli interpretation of Shimura varieties of Hodge type and
the partial extension of this interpretation to the integral model as
constructed in Section 1.5. Let $(G,X)$ be a Shimura datum of Hodge type, let
$(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$ be an embedding as in
Definition 1.1 (4).

####  1.8.1 The story for the $\mathbb{C}$-valued points

######  Lemma 1.40.

[Del82, Prop. 3.1] There exist numbers $n,r_{i},s_{i}\in\mathbb{Z}_{\geq 0}$
and tensors $t_{i}\in V^{\otimes r_{i}}\otimes(V^{*})^{\otimes s_{i}}$, $1\leq
i\leq n$, such that $G$ is the subgroup of $\operatorname{GSp}(V)$ fixing the
$t_{i}$.

######  Remark 1.41.

(See [Mil05, Section 7].) For $K$ a compact open subgroup of
$G(\mathbb{A}_{f})$, the set $\operatorname{Sh}_{K}(G,X)(\mathbb{C})$ of
Definition 1.8 has the following moduli interpretation: isomorphism classes of
triples $((W,h),\left(u_{i}\right)_{0\leq i\leq n},\eta K)$, where

  1. (a)

$(W,h)$ rational Hodge structure of type $(-1,0)+(0,-1)$,

  2. (b)

$\pm u_{0}$ polarization of $(W,h)$ (i.e., either $u_{0}$ or $-u_{0}$ is a
polarization),

  3. (c)

$u_{i}\in V^{\otimes r_{i}}\otimes(V^{*})^{\otimes s_{i}}$ for $1\leq i\leq
n$,

  4. (d)

$\eta K$ is a $K$-orbit of isomorphisms
$V\otimes\mathbb{A}_{f}\xrightarrow{\sim}W\otimes\mathbb{A}_{f}$, mapping
$\psi$ to a $\mathbb{A}_{f}^{\times}$-multiple of $u_{0}$, and every $t_{i}$
to $u_{i}$ ($i\geq 1$),

such that there exists an isomorphism $a\colon W\xrightarrow{\sim}V$ mapping
$u_{0}$ to a $\mathbb{Q}^{\times}$-multiple of $\psi$, every $u_{i}$ to
$t_{i}$ ($i\geq 1$), and $h$ to an element of $X$ (so
${}^{a}h:=\operatorname{GL}(a)\circ h\colon\mathbb{S}\to\operatorname{GL}(V)$
is in $X$).

######  Sketch of Proof:

Given $((W,h),\left(u_{i}\right)_{0\leq i\leq n},\eta K)$ and $a$ as above, we
consider the pair ${(^{a},a\circ\eta)}$. By assumption, ${}^{a}h\in X$ and
$a\circ\eta$ is a symplectic similitude fixing the $t_{i}$; hence
$(^{a}h,a\circ\eta)\in X\times G(\mathbb{A}_{f})$. The double quotient now
results precisely from the ambiguity in the choices of $a$ and $\eta\in\eta
K$. □

######  Remark 1.42.

Denote by $\mathbb{Q}(r)$ the rational Hodge structure of type $(-r,-r)$ with
underlying vector space $\mathbb{Q}$, and denote the multiplier character
$\operatorname{GSp}(V)\to\mathbb{G}_{m}$ as well as its restriction to $G$ by
$c$. Then for $h\in X$, $c\circ h\cong\mathbb{Q}(1)$ as rational Hodge
structure, and the symplectic form gives an isomorphism $V\cong
V^{*}\otimes\mathbb{Q}(1)$.

With this, we may also interpret a tensor $t\in V^{\otimes
r}\otimes(V^{*})^{\otimes s}$ as a multilinear map $V^{r+s}\to\mathbb{Q}$,
and, if $t$ is fixed by $G$, as a morphism
$(V,h)^{\otimes(r+s)}\to\mathbb{Q}(r)$ of Hodge structures. Provided $t\neq
0$, this implies $r+s=2r$, i.e. $r=s$.

######  Remark 1.43.

Since $A\mapsto H_{1}(A(\mathbb{C}),\mathbb{Q})$ yields an equivalence between
the category of complex abelian varieties up to isogeny and the category of
polarizable rational Hodge structures of type $(-1,0)+(0,-1)$, we may also
take $\operatorname{Sh}_{K}(G,X)(\mathbb{C})$ to be a moduli problem of
abelian varieties with extra structure: consider isomorphism classes of
triples $(A,\left(u_{i}\right)_{0\leq i\leq n},\eta K)$, where

  1. (a)

$A$ is a complex abelian variety up to isogeny,

  2. (b)

$\pm u_{0}$ is a polarization of the rational Hodge structure
$V:=H_{1}(A(\mathbb{C}),\mathbb{Q})$,

  3. (c)

$u_{1},\dotsc,u_{n}$ is as in Remark 1.41,

  4. (d)

$\eta K$ is a $K$-orbit of $\mathbb{A}_{f}$-linear isomorphisms
$V\otimes\mathbb{A}_{f}\to
V_{f}(A):=T_{f}(A)\otimes_{\mathbb{Z}}\mathbb{Q}=\left(\varprojlim
A(k)[n]\right)\otimes_{\mathbb{Z}}\mathbb{Q}$, mapping $\psi$ to a
$\mathbb{A}_{f}^{\times}$-multiple of $u_{0}$ and every $t_{i}$ to $u_{i}$
($i\geq 1$),

such that there exists an isomorphism $a\colon
H_{1}(A(\mathbb{C}),\mathbb{Q})\xrightarrow{\sim}V$ mapping $u_{0}$ to a
$\mathbb{Q}^{\times}$-multiple of $\psi$, $u_{i}$ to $t_{i}$ ($i\geq 1$) and
$h$ to an element of $X$.

######  Remark 1.44.

When rephrasing the moduli problem appropriately, the Shimura variety as an
$\mathbf{E}$-variety is given by such a moduli problem, see [Mil05, Section
14].

In particular, we have a universal abelian scheme
$\mathcal{A}\to\operatorname{Sh}_{K}(G,X)$.

####  1.8.2 The story for the integral model

Integral models of Hodge type Shimura varieties in general seem not to afford
nice straightforward moduli interpretations. Still, the moduli interpretation
of the $\mathbb{C}$-valued points extends to _some_ degree. First, we do have
a generalization of Lemma 1.40 as follows.

######  Proposition 1.45.

[Kis10, Prop. 1.3.2] Let $R$ be a discrete valuation ring of mixed
characteristic, $M$ a finite free $R$-module, and
$\mathcal{G}\subseteq\operatorname{GL}(M)$ a closed $R$-flat subgroup with
reductive generic fiber. Then $\mathcal{G}$ is defined by a finite collection
of tensors $\left(s_{\alpha}\right)_{\alpha}\subset M^{\otimes}$.

Here $M^{\otimes}$ is the direct sum of all the $R$-modules which can be
formed from $M$ by taking duals and (finite) tensor products.121212Kisin in
_loc. cit._ also allows for taking symmetric and exterior powers. As Deligne
pointed out, this is unnecessary. [Del11]

We return to the setting of Section 1.3; in particular we consider a Bruhat-
Tits scheme $\mathcal{G}:=\mathcal{G}_{x}$.

By the proposition just stated, we find a collection of tensors
$\left(s_{\alpha}\right)_{\alpha}\subset(\Lambda^{\S}_{\mathbb{Z}_{(p)}})^{\otimes}$
whose pointwise stabilizer is the Zariski closure $G_{\mathbb{Z}_{(p)}}$ of
$G$ in $\operatorname{GL}(\Lambda^{\S}_{\mathbb{Z}_{(p)}})$.

######  Remark 1.46.

We can for example consider, for every $j\in\\{-(r-1)-a,\dotsc,r-1\\}$, the
projection
$\operatorname{pr}_{j}\colon\Lambda_{\mathbb{Z}_{(p)}}^{\S}\to\Lambda_{\mathbb{Z}_{(p)}}^{j}$.

Since the $G$-action on
$\Lambda_{\mathbb{Z}_{(p)}}^{\S}=\prod_{i=-(r-1)-a}^{r-1}\Lambda_{\mathbb{Z}_{(p)}}^{i}$
is diagonal, the $\operatorname{pr}_{j}$ are fixed, i.e., we may count the
$\left(\operatorname{pr}_{j}\right)_{j}$ among the
$\left(s_{\alpha}\right)_{\alpha}$.

######  Lemma 1.47.

$G_{\mathbb{Z}_{(p)}}\otimes_{\mathbb{Z}_{(p)}}\mathbb{Z}_{p}={\cal G}$.

######  Proof:

First, $G_{\mathbb{Z}_{(p)}}\otimes_{\mathbb{Z}_{(p)}}\mathbb{Z}_{p}$ is the
Zariski closure $G_{\mathbb{Z}_{p}}$ of $G$ in
$\operatorname{GL}(\Lambda^{\S}_{\mathbb{Z}_{p}})$ and
$G_{\mathbb{Z}_{p}}\otimes\mathbb{Q}_{p}=G$: This follows from [GW10, Lemma
14.6].

According to [Stacks, Tag 056B], $G_{\mathbb{Z}_{p}}$ is the topological
closure of $G$ in $\operatorname{GL}(\Lambda^{\S}_{\mathbb{Z}_{p}})$ endowed
with the reduced subscheme structure. The special fiber therefore is reduced
as well, i.e. ($\mathbb{F}_{p}$ being perfect), geometrically reduced, i.e.
(being a group scheme), smooth. $G_{\mathbb{Z}_{p}}$ is flat over
$\mathbb{Z}_{p}$ by [GW10, Prop. 14.14]. $G_{\mathbb{Z}_{p}}$ obviously is of
finite presentation over $\mathbb{Z}_{p}$. Hence, $G_{\mathbb{Z}_{p}}$ is
smooth over $\mathbb{Z}_{p}$. It also is affine and is the stabilizer of
$\Lambda^{\S}_{\mathbb{Z}_{p}}$ (or rather,
$G_{\mathbb{Z}_{p}}(\mathbb{Z}_{p})\subseteq G(\mathbb{Q}_{p})$ is (and
analogously for $G_{\mathbb{Z}_{p}}(\breve{\mathbb{Z}}_{p})\subseteq
G(\breve{\mathbb{Q}}_{p})$)). □

######  Remarks 1.48.

  1. (1)

Obtaining an embedding of the Bruhat-Tits group scheme to which we can apply
Proposition 1.45 is the main reason for altering the Hodge embedding in the
way outlined in Section 1.3.

  2. (2)

We can also interpret the $s_{\alpha}$ as tensors in
$((\Lambda^{\S}_{\mathbb{Z}_{(p)}})^{*})^{\otimes}$ and that
$G_{\mathbb{Z}_{(p)}}$ also is the Zariski closure of $G$ in
$\operatorname{GL}((\Lambda^{\S}_{\mathbb{Z}_{(p)}})^{*})$ (using the
contragredient representation).

##### Hodge tensors: generic fiber.

######  Notation 1.49.

Recall the universal abelian scheme
$h\colon\mathcal{A}\to\operatorname{Sh}_{K}(G,X)$ from Remark 1.44.

We consider the local system
$\mathcal{L}_{B}:=R^{1}h_{*}^{\mathrm{an}}\underline{\mathbb{Q}}$ on
$\operatorname{Sh}_{K}(G,X)_{\mathbb{C}}^{\mathrm{an}}$ and the flat vector
bundle $\mathcal{V}_{\mathrm{dR}}:=R^{1}h_{*}\Omega^{\bullet}$ with Gauß-Manin
connection $\nabla$.

######  Lemma 1.50.

(See [CS17, Lemma 2.3.1].) $\mathcal{L}_{B}$ can be identified with the local
system of $\mathbb{Q}$-vector spaces on
$\operatorname{Sh}_{K}(G,X)^{\mathrm{an}}$ given by the
$G(\mathbb{Q})$-representation $V$ and the $G(\mathbb{Q})$-torsor

$p\colon X\times(G(\mathbb{A}_{f})/K)\to
G(\mathbb{Q})\backslash(X\times(G(\mathbb{A}_{f})/K)=\operatorname{Sh}_{K}(G,X)(\mathbb{C}).$

Said torsor is isomorphic to the $G(\mathbb{Q})$-torsor
$I:=\underline{\mathrm{Isom}}((V,s_{\alpha}),({\cal L}_{B},s_{\alpha,B}))$
that maps an open subset $U\subseteq\operatorname{Sh}_{K}(\mathbb{C})$ to

$I(U)=\\{\beta\colon V\times U\xrightarrow{\sim}\left.{\cal
L}_{B}\right|_{U}\;|\;\beta(s_{\alpha})=s_{\alpha,B}\\},$

where the $\left(s_{\alpha,B}\right)_{\alpha}\subset\mathcal{L}_{B}^{\otimes}$
are the global sections corresponding to the $G(\mathbb{Q})$-invariant tensors
$\left(s_{\alpha}\right)_{\alpha}$.

######  Remark 1.51.

By de Rham’s theorem,

$\mathcal{V}_{\mathrm{dR},\mathbb{C}}^{\mathrm{an}}\cong\mathcal{L}_{B}\otimes_{\mathbb{Q}}\mathcal{O}_{\operatorname{Sh}_{K}(G,X)^{\mathrm{an}}}.$

In particular the global sections
$\left(s_{\alpha,B}\right)_{\alpha}\subset\mathcal{L}_{B}^{\otimes}$ yield
flat global sections
$\left(s_{\alpha,\mathrm{dR}}\right)_{\alpha}\subset(\mathcal{V}_{\mathrm{dR},\mathbb{C}}^{\mathrm{an}})^{\otimes}$.
All such sections arise from sections in
$\mathcal{V}_{\mathrm{dR},\mathbb{C}}^{\otimes}$ (which we will denote the
same), see [Kis10, 2.2].

######  Remark 1.52.

Let $k/E$ be a field extension embeddable into $\mathbb{C}$, and choose an
embedding $\mathbb{Q}_{p}\hookrightarrow\mathbb{C}$ and an $E$-embedding
$\sigma\colon\bar{k}\hookrightarrow\mathbb{C}$. Let
$x\in\operatorname{Sh}_{K}(G,X)(k)$. By $p$-adic Hodge theory, the embedding
$\sigma$ gives rise to isomorphisms

$H^{1}_{\mathrm{dR}}(\mathcal{A}_{x}/k)\otimes_{k}\mathbb{C}\xrightarrow{\sim}H^{1}_{B}(\mathcal{A}_{x}(\mathbb{C}),\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}\xrightarrow{\sim}H^{1}_{\mathrm{\acute{e}t}}(\mathcal{A}_{x,\bar{k}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{C},$

and by proper base change $(\mathcal{L}_{B})_{x}:=x^{-1}\mathcal{L}_{B}\cong
H^{1}_{B}(\mathcal{A}_{x}(\mathbb{C}),\mathbb{Q})$.

######  Notation 1.53.

We denote by $s_{\alpha,B,x}$ the fiber of $s_{\alpha,B}$ at $x$, and by
$s_{\alpha,\mathrm{dR},x}$ and $s_{\alpha,\mathrm{\acute{e}t},x}$,
respectively, the images of $s_{\alpha,B,x}$ under the above isomorphisms. For
the Betti–Ã©tale comparison, we don’t need to go all the way to $\mathbb{C}$
and have $s_{\alpha,\mathrm{\acute{e}t},x}\in
H^{1}_{\mathrm{\acute{e}t}}(\mathcal{A}_{x,\bar{k}},\mathbb{Q}_{p})$.

######  Remark 1.54.

$(s_{\alpha,\mathrm{dR},x},s_{\alpha,\mathrm{\acute{e}t},x})$ is an absolute
Hodge cycle in the sense of [Del82], for every $\alpha$.

######  Lemma 1.55.

[Kis10, Lemma 2.2.1] The $\operatorname{Gal}(\bar{k}/k)$-action on
$H^{1}_{\mathrm{\acute{e}t}}(\mathcal{A}_{x,\bar{k}},\mathbb{Q}_{p})$ fixes
every $s_{\alpha,\mathrm{\acute{e}t},x}$ and factors through
$G(\mathbb{Q}_{p})$. Moreover, $s_{\alpha,\mathrm{dR},x}\in
H^{1}_{\mathrm{dR}}(\mathcal{A}_{x}/k)$.

In particular, $(s_{\alpha,\mathrm{dR},x},s_{\alpha,\mathrm{\acute{e}t},x})$
is independent of the choices made above.

######  Corollary 1.56.

[Kis10, Lemma 2.2.2] $s_{\alpha,\mathrm{dR}}$ is defined over $E$ for all
$\alpha$, i.e.,
$s_{\alpha,\mathrm{dR}}\in\mathcal{V}_{\mathrm{dR},E}^{\otimes}$.

##### Hodge tensors: special fiber.

######  Notation 1.57.

Denote by $\mathscr{A}_{\mathrm{Siegel}}$ the universal abelian scheme over
$\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$, cf. Remark 1.21, and
by $\mathscr{A}$ the pullback of $\mathscr{A}_{\mathrm{Siegel}}$ to
$\mathscr{S}_{K}(G,X)$. Then the pullback of $\mathscr{A}$ to
$\operatorname{Sh}_{K}(G,X)_{E}$ agrees with the pullback of $\mathcal{A}$ of
Remark 1.44.

We will occasionally call $\mathscr{A}$ the _“universal” abelian scheme_ (with
quotation marks).

Let $\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$ and let
$\mathbb{D}_{\bar{x}}:=\mathbb{D}(\mathscr{A}_{\bar{x}}[p^{\infty}])(W)$ be
the DieudonnÃ© module of the associated fiber of the “universal” abelian
scheme, $W=W(\bar{\mathbb{F}}_{p})=\breve{\mathbb{Z}}_{p}$. Choose a closed
point generization $x\in\operatorname{Sh}_{K}(G,X)(L)$, $L/E$ finite field
extension, in the sense of Definition 1.30. Note that $L$ can be embedded into
$\mathbb{C}_{p}\cong\mathbb{C}$. Then

$H^{1}_{\mathrm{\acute{e}t}}(\mathscr{A}_{x,\bar{E}},\mathbb{Z}_{p})\cong(T_{p}\mathscr{A}_{x,\bar{E}})^{*}\cong(T_{p}\mathscr{A}_{x,\bar{E}}^{*})(-1)\cong(\Lambda^{\S})^{*},$

allowing us to identify the tensors $s_{\alpha}$ with tensors
$s_{\alpha,\mathrm{\acute{e}t},x}\in
H^{1}(\mathscr{A}_{x,\bar{E}},\mathbb{Z}_{p})^{\otimes}$. Again, these tensors
can be shown to be $\operatorname{Gal}(\bar{E}/L)$-invariant. Now we have the
$p$-adic comparison isomorphism

$H_{\mathrm{\acute{e}t}}^{1}(A_{x,\bar{E}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}B_{\mathrm{cris}}\cong
H^{1}_{\mathrm{cris}}(\mathscr{A}_{\bar{x}}/W)\otimes_{W}B_{\mathrm{cris}}=\mathbb{D}(\mathscr{A}_{\bar{x}}[p^{\infty}])(W)\otimes_{W}B_{\mathrm{cris}},$
(1.58)

and by [KP15, 3.3.8], via this isomorphism, the
$s_{\alpha,\mathrm{\acute{e}t},x}$ also correspond to tensors
$s_{\alpha,0}:=s_{\alpha,0,\bar{x}}$ in $\mathbb{D}_{\bar{x}}^{\otimes}$. In
fact, we get an isomorphism

$(\Lambda^{\S})^{*}\otimes_{\mathbb{Z}_{p}}\breve{\mathbb{Z}}_{p}\cong\mathbb{D}_{\bar{x}}$
(1.59)

identifying $s_{\alpha}\otimes 1$ with $s_{\alpha,0,\bar{x}}$.

##### Globalizing crystalline tensors.

Above we constructed crystalline tensors fiberwise. This will not suffice to
understand the local geometry of the special fiber of the integral model.
Therefore we now “globalize” the tensors.

Let $\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$. As in Section
1.7.2, we write
$R_{G}:=R_{G,\bar{x}}:=\widehat{\mathcal{O}}_{\mathscr{S}_{K}(G,X)_{{\cal
O}_{\breve{E}}},\bar{x}}$. (So $\operatorname{Spf}R_{G}$ is the $\bar{x}$-adic
completion of $\mathscr{S}_{K}(G,X)_{{\cal O}_{\breve{E}}}$.)

######  Remark 1.60.

In [KP15, 3.2.12, 3.2.14] a construction of a “universal” deformation
$p$-divisible group

$\mathscr{G}_{R_{G}}\to\operatorname{Spec}R_{G}$ (1.61)

is given, characterized by its DieudonnÃ© display, to wit
$\mathrm{DDisp}(\mathscr{G}_{R_{G}})=\mathbb{D}_{\bar{x}}\otimes_{W}\hat{W}(R_{G})$
endowed with a natural DieudonnÃ© display structure.

$\mathscr{G}_{R_{G}}$ can be identified with the pullback of the $p$-divisible
group of the “universal” abelian scheme $\mathscr{A}$.

######  Remark 1.62.

$R_{G}$ is a complete local normal noetherian ring with perfect residue field.

######  Remark 1.63.

By [Lau14, Theorem B], the DieudonnÃ© crystal associated with
$\mathrm{DDisp}(\mathscr{G}_{R_{G}})$ coincides with the DieudonnÃ© crystal
$\mathbb{D}(\mathscr{G}_{R_{G}})$ of $\mathscr{G}_{R_{G}}$.

In fact,
$\mathrm{DDisp}(\mathscr{G}_{R_{G}})=\mathbb{D}(\mathscr{G}_{R_{G}})^{\vee}(\hat{W}(R_{G}))$
endowed with a natural DieudonnÃ© display structure, cf. [KP15, 3.1.7].

In particular we get tensors
$t_{\alpha,\bar{x}}^{\mathrm{def}}\in\mathbb{D}(\mathscr{G}_{R_{G}})^{\vee}(\hat{W}(R_{G}))^{\otimes}$
corresponding to $s_{\alpha,0,\bar{x}}\otimes
1\in(\mathbb{D}_{\bar{x}}\otimes_{W}\hat{W}(R_{G}))^{\otimes}$, and
accordingly tensors $t_{\alpha,\bar{x}}^{\mathrm{def}}\otimes
1\in\mathbb{D}(\mathscr{G}_{R_{G}})^{\vee}(W(R_{G}))^{\otimes}$.

######  Definition 1.64.

(See [Ham17, Definition 2.8].) Let $\mathscr{G}\to S$ be a $p$-divisible group
over a formally smooth $\mathbb{F}_{p}$-scheme $S$. Denote by
$\mathbb{D}(\mathscr{G})$ its contravariant DieudonnÃ© crystal of
$\mathscr{G}$ [BBM82, Def. 3.3.6], and by
$\mathds{1}:=\mathbb{D}(\underline{\mathbb{Q}_{p}/\mathbb{Z}_{p}}_{S})$ the
unit object in the tensor category of locally free
$\mathcal{O}_{S/\mathbb{Z}_{p},\mathrm{CRIS}}$-modules. Note that
$\mathbb{D}(\mathscr{G})$ comes with a Frobenius morphism
$\mathbb{D}(\mathscr{G})^{(p)}\to\mathbb{D}(\mathscr{G})$, and similar for
$\mathds{1}$.

  1. (1)

A _tensor_ $t$ of $\mathbb{D}(\mathscr{G})$ is a morphism
$\mathds{1}\to\mathbb{D}(\mathscr{G})^{\otimes}$ of locally free
$\mathcal{O}_{S/\mathbb{Z}_{p},\mathrm{CRIS}}$-modules.131313$(\;)^{\otimes}$
here (and also in point (2)) is defined as it is defined for $R$-modules in
Prop. 1.45.

  2. (2)

A tensor $t$ of $\mathbb{D}(\mathscr{G})$ is called a _crystalline Tate tensor
on $\mathscr{G}$_ if it induces a morphism of $F$-isocrystals
$\mathds{1}\to\mathbb{D}(\mathscr{G})[\tfrac{1}{p}]^{\otimes}$.

######  Remarks 1.65.

  1. (1)

To give a tensor as in the preceding definition therefore means the following:
For every $(U,T,i,\delta)$ with

    1. (a)

an $S$-scheme $U$,

    2. (b)

a $\mathbb{Z}_{p}$-scheme $T$ on which $p$ is locally nilpotent,

    3. (c)

a closed $\mathbb{Z}_{p}$-immersion $i\colon U\hookrightarrow T$,

    4. (d)

a pd-structure $\delta$ on the ideal in $\mathcal{O}_{T}$ defining the
immersion $i$, compatible with the canonical pd-structure on
$p\mathbb{Z}_{p}$,

functorially to give a morphism
$\mathds{1}(U,T,i,\delta)=\mathcal{O}_{T}(T)\to\mathbb{D}(\mathscr{G})(U,T,i,\delta)^{\otimes}$
of $\mathcal{O}_{T}(T)$-modules, i.e., functorially to give elements of
$\mathbb{D}(\mathscr{G})(U,T,i,\delta)^{\otimes}$.

  2. (2)

Assume $S=\operatorname{Spec}A$, where $A$ is an $\mathbb{F}_{p}$-algebra
which has a $p$-basis (e.g., $A$ perfect) _or_ which satisfies [Jon95,
(1.3.1.1)], the latter signifying the existence of an ideal $I\subseteq A$
such that

     * •

$A$ is noetherian and $I$-adically complete,

     * •

$A$ is formally smooth as a topological $\mathbb{F}_{p}$-algebra with the
$I$-adic topology,

     * •

$A/I$ contains a field with a finite $p$-basis and is finitely generated as an
algebra over this field.

Also fix a _lift_ $\tilde{A}$ of $A$ in the sense of [Jon95, Def. 1.2.1],
i.e., a $p$-adically complete $\mathbb{Z}_{p}$-flat ring $\tilde{A}$ together
with an isomorphism $\tilde{A}/p\tilde{A}\cong A$ and a ring endomorphism
$\sigma\colon\tilde{A}\to\tilde{A}$ such that $\sigma(a)\equiv
a^{p}\mod{p\tilde{A}}$. Note that if $A$ is perfect, then $\tilde{A}:=W(A)$
with the usual Frobenius lift works.

Then by [BM90, Prop. 1.3.3] and [Jon95, Cor. 2.2.3] the category of crystals
of quasi-coherent
$\mathcal{O}_{\operatorname{Spec}(A)/\mathbb{Z}_{p},\mathrm{CRIS}}$-modules is
equivalent to the category of $p$-adically complete $\tilde{A}$-modules $M$
endowed with an integrable topologically quasi-nilpotent connection
$\nabla\colon M\to M\otimes_{\tilde{A}}\hat{\Omega}_{\tilde{A}}^{1}$.

  3. (3)

In the setting of (2), say $(M,\nabla)$ is the image of
$\mathbb{D}(\mathscr{G})$ under the equivalence. Then
$M=\mathbb{D}(\mathscr{G})(\tilde{A})$, and to give a tensor
$\mathds{1}\to\mathbb{D}(\mathscr{G})^{\otimes}$ means to give a horizontal
section of $M^{\otimes}$.

######  Proposition 1.66.

[HK17, Prop. 3.3.1, Cor. 3.3.7] Denote by
$\mathscr{G}_{\overline{\mathscr{S}}^{\mathrm{perf}}}$ and
$\mathscr{G}_{\hat{\mathscr{S}}}$ the pullbacks of the $p$-divisible group
$\mathscr{A}[p^{\infty}]$ to the perfection141414This being the inverse
perfection of
$\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{E}}\bar{\mathbb{F}}_{p}$ in the
terminology of [BG18, Section 5].
$\overline{\mathscr{S}}^{\mathrm{perf}}:=(\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{E}}\bar{\mathbb{F}}_{p})^{\mathrm{perf}}$
and the $p$-adic completion $\hat{\mathscr{S}}$ of
$\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{E}}{{\cal O}_{\breve{E}}}$,
respectively.

For a $p$-divisible group $\mathscr{X}$ over a $p$-adic formal scheme
$\mathfrak{S}$ denote by $P(\mathscr{X})$ the locally free
$W(\mathcal{O}_{\mathfrak{S}})$-module given by
$P(\mathscr{X})(\operatorname{Spf}A)=\mathbb{D}(\mathscr{X}_{A})^{\vee}(W(A))$
for every open affine formal subscheme
$\operatorname{Spf}A\subseteq\mathfrak{S}$.

Then there exist tensors $\left(t_{\alpha}\right)_{\alpha}\subset
P(\mathscr{G}_{\hat{\mathscr{S}}})^{\otimes}$ whose pullback to
$P(\mathscr{G}_{R_{G},\bar{x}})^{\otimes}$ coincides with
$t_{\alpha,\bar{x}}^{\mathrm{def}}\otimes 1$ for all
$\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$.

Via pullback, these $t_{\alpha}$ yield crystalline Tate tensors
$(\bar{t}_{\alpha})_{\alpha}$ on
$\mathscr{G}_{\overline{\mathscr{S}}^{\mathrm{perf}}}$.

Moreover, if $x\in\mathscr{S}_{K}(G,X)(\mathcal{O}_{L})$, $L/\breve{E}$ finite
field extension, is a closed point generization of
$\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$, then
$s_{\alpha,\mathrm{\acute{e}t},x}\in(T_{p}\mathscr{A}_{x,\bar{E}})^{\otimes}$
gets identified with
$\bar{t}_{\alpha,\bar{x}}\in\mathbb{D}(\mathscr{A}_{\bar{x}}[p^{\infty}])(W(\bar{\mathbb{F}}_{p}))$
by the $p$-adic comparison isomorphism (1.58).

##  2 Central leaves in the case of parahoric reduction

We begin by giving some history. The foliation given by the central leaves was
introduced by Oort [Oor04, Oor09] in the setting of a $p$-divisible group over
a characteristic $p$ scheme (cf. Remark 2.2 below). As already indicated in
the introduction, the idea is to consider for every isomorphism class of
$p$-divisible groups over an algebraically closed field of characteristic $p$
the locus where the isomorphism class of the geometric fibers of the
$p$-divisible group is the given one. It is not at all readily apparent why
this would be a reasonable notion in geometrical or topological terms. The key
tool here is given by the slope filtrations introduced by Zink [Zin01a], and a
key insight is that there is a number $N\geq 0$ such that a $p$-divisible
group over an algebraically closed field of characteristic $p$ is in the
prescribed isomorphism class if and only if the analogous statement holds for
the $p^{N}$-torsion subgroups. For Shimura varieties of Hodge type and good
reduction, the central leaves were studied by Mantovan [Man04, Man05] (PEL
cases), Vasiu [Vas08] and Zhang [Zha15] (a good survey), among others.

He and Rapoport formulated a set of axioms that integral models for Shimura
varieties with parahoric level are supposed to satisfy in order for them to
merit the label “canonical”. Among these is having a notion of well-behaved
central leaves. This is what we investigate in the Hodge type situation. As
already mentioned, Zhou [Zho18] concurrently and independently also worked on
this and obtained very similar results in a different way, and we use some of
his results from the first version of the preprint, which didn’t yet address
the change-of-parahoric map between central leaves. Central leaves in the
parahoric level case also are subject in work of Hamacher and Kim [Ham17,
Kim19, HK17], but they do not deal with changing the parahoric level.

We still fix a Shimura datum $(G,X)$ of Hodge type, a parahoric subgroup
$K_{p}\subseteq G(\mathbb{Q}_{p})$ (associated with a Bruhat-Tits group scheme
$\mathcal{G}\to\operatorname{Spec}\mathbb{Z}_{p}$) and a sufficiently small
open compact subgroup $K^{p}\subseteq G(\mathbb{A}_{f}^{p})$. We also keep up
our standard assumptions 1.16.

###  2.1 Definition of central leaves

We use the notation for Hodge tensors, in particular
$\left(s_{\alpha}\right)_{\alpha}$ and $\left(s_{\alpha,0}\right)_{\alpha}$,
established in Section 1.8.2.

######  Definition 2.1.

Two points
$\bar{x}_{1},\bar{x}_{2}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$ lie in
the same _central leaf_ if there is an isomorphism
$\mathbb{D}_{\bar{x}_{1}}\cong\mathbb{D}_{\bar{x}_{2}}$ of DieudonnÃ© modules
that takes $s_{\alpha,0,\bar{x}_{1}}$ to $s_{\alpha,0,\bar{x}_{2}}$.

They lie in the same _naive central leaf_ if there is an arbitrary isomorphism
$\mathbb{D}_{\bar{x}_{1}}\cong\mathbb{D}_{\bar{x}_{2}}$ of DieudonnÃ© modules
(i.e., an isomorphism
$\mathscr{A}_{\bar{x}_{1}}[p^{\infty}]\cong\mathscr{A}_{\bar{x}_{1}}[p^{\infty}]$
of $p$-divisible groups).

######  Remark 2.2.

More generally, one can make analogous definitions when given any abelian
scheme $\mathcal{A}\to S$ over any $\mathbb{F}_{p}$-scheme $S$ (such that for
every point we are given tensors on the DieudonnÃ© module of the corresponding
abelian variety). Given two arbitrary points, one compares them after going to
a common algebraically closed extension of the residue fields, cf. [Oor04].

######  Notation 2.3.

Recall that we denote the completion of the maximal unramified extension
$\mathbb{Q}_{p}^{\mathrm{ur}}$ of $\mathbb{Q}_{p}$ by
$\breve{\mathbb{Q}}_{p}$, and its ring of integers accordingly by
$\breve{\mathbb{Z}}_{p}$. We set
$\breve{K}:=\breve{K}_{p}:=\mathcal{G}(\breve{\mathbb{Z}}_{p})$ and define
$\breve{K}_{\sigma}$ to be the graph of the Frobenius
$\sigma\colon\breve{K}\to\breve{K}$. So dividing out the action of
$\breve{K}_{\sigma}$ (which is mostly how $\breve{K}_{\sigma}$ will make an
appearance) means dividing out the action of $\breve{K}$ by
$\sigma$-conjugation.

######  Definition 2.4.

The central leaves are the fibers of the map

$\Upsilon=\Upsilon_{K}\colon\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})\to
G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$

given as follows: For $\bar{x}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$
there is an isomorphism $\beta\colon
V_{\mathbb{Z}_{p}}^{*}\otimes_{\mathbb{Z}_{p}}\breve{\mathbb{Z}}_{p}\cong\mathbb{D}_{\bar{x}}$
(equation (1.59)) sending $s_{\alpha}\otimes 1$ to $s_{\alpha,0,\bar{x}}$. We
hence can interpret the Frobenius on $\mathbb{D}_{\bar{x}}$ as an element of
$G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$, where dividing out
$\breve{K}_{\sigma}$ rids us of the ambiguity introduced by the choice of the
isomorphism $\beta$.

######  Notation 2.5.

We write

$\displaystyle G(\breve{\mathbb{Q}}_{p})$ $\displaystyle\to{}$ $\displaystyle
C(G):=G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$ $\displaystyle\to
B(G):=G(\breve{\mathbb{Q}}_{p})/G(\breve{\mathbb{Q}}_{p})_{\sigma},$
$\displaystyle b$ $\displaystyle\mapsto{}$
$\displaystyle[\mkern-2.0mu[b]\mkern-2.0mu]$ $\displaystyle\mapsto[b].$

####  2.1.1 An alternative characterization of the central leaves

Consider two points
$\bar{x}_{1},\bar{x}_{2}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$ in the
same central leaf, i.e., such that there is an isomorphism
$\mathbb{D}_{\bar{x}_{1}}\cong\mathbb{D}_{\bar{x}_{2}}$ of DieudonnÃ© modules
that takes $s_{\alpha,0,\bar{x}_{1}}$ to $s_{\alpha,0,\bar{x}_{2}}$.

$\bar{x}_{j}$ for $j=1,2$ has an associated isogeny chain of abelian schemes
in $\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})$, cf. Remark 1.19, and
$\mathbb{D}_{\bar{x}_{j}}=\mathbb{D}(\prod_{i=-(r-1)-a}^{r-1}A_{j,i})=\prod_{i=-(r-1)-a}^{r-1}\mathbb{D}(A_{j,i})$,
where $(A_{j,i})_{i}$ is the isogeny chain associated with $\bar{x}_{j}$ under
$\mathscr{S}_{K}(G,X)\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$,
cf. Remark 1.21 and Proposition 1.24. Note that
$A_{j}=\prod_{i=-(r-1)-a}^{r-1}A_{j,i}$ (plus extra structure) is the point
associated with $\bar{x}_{j}$ under
$\mathscr{S}_{K}(G,X)\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$.

######  Lemma 2.6.

Any tensor-preserving isomorphism
$\mathbb{D}_{\bar{x}_{1}}\cong\mathbb{D}_{\bar{x}_{2}}$ yields, for all $i$,
isomorphisms ${\mathbb{D}(A_{1,i})\cong\mathbb{D}(A_{2,i})}$.

######  Proof:

This follows immediately from Remark 1.46. □

We may thus rephrase our definition of central leaves as follows.

######  Lemma 2.7.

Let $\bar{x}_{1},\bar{x}_{2}\in\mathscr{S}_{K}(G,X)(\bar{\mathbb{F}}_{p})$
and, as above, denote by $A_{j,i}$ the abelian varieties in the associated
isogeny chains, $j=1,2$.

$\bar{x}_{1}$ and $\bar{x}_{2}$ lie in the same central leaf if and only if
there is an isomorphism between the associated rational DieudonnÃ© modules of
$A_{j,i}$ (which is common to all $A_{j,i}$, $j$ fixed, $i$ variable)
respecting the tensors and identifying the lattices $\mathbb{D}(A_{1,i})$ and
$\mathbb{D}(A_{2,i})$ for all $i$.

“Respecting the tensors” means the following: We get an induced identification
of the rational DieudonnÃ© modules of $A_{1}$ and $A_{2}$. This identification
has to preserve the tensors.

###  2.2 Local closedness of central leaves

####  2.2.1 Topological lemmas

We begin with some purely topological preliminaries which we shall use to
globalize statements about formal neighborhoods.

######  Lemma 2.8.

Let $X$ be a topological space with a subset $A\subseteq X$, and for $x\in X$
let $U_{x}\subseteq X$ be the set of generizations of $x$. Consider the
statements

  1. (i)

$A\cap U_{x}\subseteq U_{x}$ closed (resp. open) for all $x\in X$.

  2. (ii)

$A\cap U_{x}\subseteq U_{x}$ closed (resp. open) for all closed points $x\in
X$.

  3. (iii)

$A\subseteq X$ stable under specialization (resp. generization).

We have (i)$\implies$(ii),(iii), and if every $x\in X$ has a specialization in
$x$ that is a closed point, we also have (ii)$\implies$(iii).

######  Proof:

Denote the closure of $A$ in $X$ by $\operatorname{cl}_{X}(A)$.

For “(i)$\implies$(iii)” (with “closed” and “specialization”, respectively)
let $x\in A$ and $s\in\operatorname{cl}_{X}(\\{x\\})$. Then $s,x\in U_{s}$,
and $s\in\operatorname{cl}_{X}(\\{x\\})\cap
U_{s}=\operatorname{cl}_{U_{s}}(\\{x\\})$. By assumption $A\cap U_{s}\subseteq
U_{s}$ is stable under specialization, hence $s\in A\cap U_{s}\subseteq A$.

The rest is proven along similar lines. □

######  Example 2.9.

$\mathbb{N}\subseteq\mathbb{A}^{1}_{\mathbb{C}}$ satisfies property (i) from
the lemma but is not closed (hence not constructible).

######  Lemma 2.10.

Let $X$ be a sober topological space and let $A\subseteq X$ be stable under
both generization and specialization.

Then $A$ is a union of irreducible components of $X$.

If additionally $X$ only has finitely many irreducible components, $A$ even is
a union of connected components and is open and closed.

######  Proof:

Consider the unique generic points $\left\\{\eta_{i}\right\\}_{i\in I}$ of the
irreducible components of $X$. By assumption, we have, firstly, that if $A$
contains $\eta_{i}$ then it contains the entire irreducible component
$\overline{\\{\eta_{i}\\}}$, and, secondly, that if $A$ contains any one point
of $\overline{\\{\eta_{i}\\}}$, then it also contains $\eta_{i}$. Similarly,
if $A$ contains any one irreducible component $\overline{\\{\eta_{i}\\}}$, it
also contains all irreducible components that meet
$\overline{\\{\eta_{i}\\}}$. □

####  2.2.2 Local closedness

To show the local closedness of central leaves, we can content ourselves with
a construction in the spirit of Proposition 1.66 but somewhat simpler. Namely,
we pull back (1.61) to obtain

$\displaystyle\mathscr{G}_{\bar{x}}\to\operatorname{Spec}\bar{R}_{G}$
$\displaystyle:=\operatorname{Spec}R_{G}\otimes_{{{\cal
O}_{\breve{E}}}}\bar{\mathbb{F}}_{p}$
$\displaystyle=\operatorname{Spec}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)_{{{\cal
O}_{\breve{E}}}},\bar{x}}\otimes\bar{\mathbb{F}}_{p})=\operatorname{Spec}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}).$

The DieudonnÃ© display of $\mathscr{G}_{\bar{x}}$ then is

$\mathrm{DDisp}(\mathscr{G}_{\bar{x}})=\mathbb{D}(\mathscr{G}_{\bar{x}})^{\vee}(\hat{W}(\bar{R}_{G}))=\mathbb{D}_{\bar{x}}\otimes_{\breve{\mathbb{Z}}_{p}}\hat{W}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}).$

In particular we can consider the tensors $\left(s_{\alpha,0}\otimes
1\right)_{\alpha}$ on this DieudonnÃ© display and on
$\mathbb{D}(\mathscr{G}_{\bar{x}})^{\vee}(W(\bar{R}_{G}))$.

Thus we get crystalline Tate tensors
$\left(u_{\alpha,\bar{x}}\right)_{\alpha}$ on
$\mathbb{D}(\mathscr{G}_{\bar{x},\mathrm{perf}})$, where
$\mathscr{G}_{\bar{x},\mathrm{perf}}$ is the pullback of
$\mathscr{G}_{\bar{x}}$ to $\bar{R}_{G}^{\mathrm{perf}}$.

######  Theorem 2.11.

The central leaves on $\mathscr{S}_{K}(G,X)\otimes\bar{\mathbb{F}}_{p}$ are
open and closed in the naive central leaves.

######  Proof:

We consider the $p$-divisible group $\mathscr{G}_{\bar{x},\mathrm{perf}}$ over
$\widehat{\cal O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}^{\mathrm{perf}}$ together
with its crystalline Tate tensors. Note that perfection makes no difference
for topological considerations.

On the naive central leaves on $\operatorname{Spec}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}})=\operatorname{Spec}(\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}^{\mathrm{perf}})$ the
$p$-divisible group is geometrically fiberwise constant. By a lemma of
Hamacher [Ham17, 2.12], the tensors then are geometrically fiberwise constant
as well.

Using the fact that $\widehat{\cal O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}$ is noetherian, we obtain
that the central leaf on $\widehat{\cal O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}$ is closed and open in the
corresponding naive central leaf.

$\operatorname{Spec}{\cal O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}$ has the quotient topology
with respect to the fpqc covering $\operatorname{Spec}\widehat{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}\to\operatorname{Spec}{\cal
O}_{\mathscr{S}_{K}(G,X)\otimes_{{\cal
O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},\bar{x}}$, hence we also get the same
result after leaving away completion.

By Lemma 2.10 we conclude that we also get the same result for the central
leaves and naive central leaves on
$\mathscr{S}_{K}(G,X)\otimes\bar{\mathbb{F}}_{p}$. □

######  Corollary 2.12.

The central leaves on $\mathscr{S}_{K}(G,X)\otimes\bar{\mathbb{F}}_{p}$ are
locally closed.

######  Proof:

Because Oort [Oor04] has proven that the naive central leaves on
${\mathscr{S}_{K}(G,X)\otimes\bar{\mathbb{F}}_{p}}$ are locally closed, this
is an immediate consequence of the preceding theorem. □

######  Corollary 2.13.

In fact, Oort has shown that the naive central leaf is closed in the naive
Newton stratum. So our argument even goes to show that the central leaves are
closed in their respective Newton strata (naive and non-naive).

###  2.3 Quasi-isogenies of $p$-divisible groups

Let $b\in
G(\breve{\mathbb{Q}}_{p})=\mathcal{G}(\breve{\mathbb{Q}}_{p})\subseteq\operatorname{GL}(\Lambda^{\S})(\breve{\mathbb{Q}}_{p})$
and denote by $b^{\S}$ the image in
$\operatorname{GL}(V^{\S})(\breve{\mathbb{Q}}_{p})$.

Let $J_{b}$ denote the $\mathbb{Q}_{p}$-reductive group given on $R$-valued
points, $R\in(\mathbb{Q}_{p}\text{-alg})$, by

$J_{b}(R):=\left\\{g\in
G(R\otimes_{\mathbb{Q}_{p}}\breve{\mathbb{Q}}_{p})=\operatorname{Res}_{\breve{\mathbb{Q}}_{p}/\mathbb{Q}_{p}}(G)(R)\;|\;gb\sigma(g)^{-1}=b\right\\}$

(cf. [RZ96, Prop. 1.12] and [Kim19, Prop. 2.2.6]).

$J_{b}(\mathbb{Q}_{p})$ then naturally attains the structure of a locally
profinite group. We make it into a formal group scheme
$\underline{J_{b}(\mathbb{Q}_{p})}$ over
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}$; with
$\underline{J_{b}(\mathbb{Q}_{p})}(U)=\mathrm{Map}_{\mathrm{cont}}(U,J_{b}(\mathbb{Q}_{p}))$
for formal test schemes
$U\to\operatorname{Spf}\breve{\mathbb{Z}}_{p}$.151515Locally: If
$G=\varprojlim_{n}G_{n}$ is a profinite set (or even profinite group), then
$\underline{G}:=\varprojlim_{n}\underline{G_{n}}$ with $\underline{G_{n}}$
constant formal (group) scheme.

Let $\mathbb{X}_{b}=\mathbb{X}_{b,K}=\mathbb{X}_{b^{\S}}$ be a polarized
$p$-divisible group over $\bar{\mathbb{F}}_{p}$ with a distinguished
isomorphism between its DieudonnÃ© (symplectic) module and
$(\Lambda^{\S})_{\breve{\mathbb{Z}}_{p}}$ under which the Frobenius is
identified with $b$. The existence of such a $p$-divisible group depends on
$b$, but we _assume_ that $\mathbb{X}_{b}$ exists from now on (see also 2.35
(2) below). By DieudonnÃ© theory we obtain a bijection

$\displaystyle J_{b^{\S}}(\mathbb{Q}_{p})$
$\displaystyle:=\left\\{g\in\operatorname{GL}(\Lambda^{\S}\otimes\breve{\mathbb{Q}}_{p})\;|\;gb^{\S}\sigma(g)^{-1}=b^{\S}\right\\}$
$\displaystyle\cong\\{\text{(self-)quasi-isogenies of }\mathbb{X}_{b}\text{
(without polarization)}\\}$

(and similar with polarization (replace $\operatorname{GL}$ by
$\mathrm{(G)Sp}$)), and $J_{b}(\mathbb{Q}_{p})\subseteq
J_{b^{\S}}(\mathbb{Q}_{p})$ under this bijection corresponds to the _tensor-
preserving_ quasi-isogenies of $\mathbb{X}_{b}$.

We need to understand more than this; at least we will need to understand the
tensor-preserving quasi-isogenies of $(\mathbb{X}_{b})_{\Omega}$ for all
algebraically closed fields of characteristic $p$. Using internal hom
$p$-divisible groups first defined by Chai and Oort, Caraiani and Scholze
[CS17, Prop. 4.2.11] worked out the structure of the quasi-isogeny group
(albeit in a more special setting than ours; Kim [Kim19] generalized it to our
setting).

First of all, following [Kim19, section 3.2], we denote by
$\operatorname{Qisg}(\mathbb{X}_{b})$ the formal group scheme over
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}$ given by

$\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}\to\mathrm{Grp},\quad
R\mapsto\\{\text{quasi-isogenies of }(\mathbb{X}_{b})_{R/p}\\},$

where $\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ is the category of
$\breve{\mathbb{Z}}_{p}$-algebras on which $p$ is nilpotent (anti-equivalent
to the category of affine schemes over
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}$161616That is, affine schemes
together with a natural transformation of functors
$\mathrm{Ring}\to\mathrm{Set}$ from it to
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}$.).

######  Definition 2.14.

Let $R$ be a topological ring.

  1. (1)

$R$ is called _adic_ , if there is an ideal $I$ (called _ideal of definition_)
such that $\left\\{I^{n}\right\\}_{n\in\mathbb{N}}$ is a basis of
neighborhoods (or, equivalently, basis of _open_ neighborhoods) of $0$.

  2. (2)

$R$ is called _f-adic_ (or _Huber ring_), if there exists an open subring
$A_{0}\subseteq A$ that is adic with finitely generated ideal of definition.
Such a ring is called a _ring of definition_.

######  Definition 2.15.

Let $R$ be a ring of characteristic $p$.

  1. (1)

$R$ is _semiperfect_ , if the Frobenius endomorphism $\Phi\colon R\to R$ is
surjective.

  2. (2)

$R$ is _f-semiperfect_ , if it is semiperfect and the perfection
$R^{\flat}:=\varprojlim_{\Phi}R$ (with the inverse limit topology) is f-adic.

There is a functorial construction which assigns to every semiperfect ring $R$
its universal $p$-adically complete pd-thickening $A_{\mathrm{cris}}(R)$
[SW13, Prop. 4.1.3]. Set
$B_{\mathrm{cris}}^{+}(R):=A_{\mathrm{cris}}(R)[\frac{1}{p}]$. We have

$\mathbb{D}((\mathbb{X}_{b})_{R})(A_{\mathrm{cris}}(R))\cong\mathbb{D}(\mathbb{X}_{b})(\breve{\mathbb{Z}}_{p})\otimes_{\breve{\mathbb{Z}}_{p}}A_{\mathrm{cris}}(R)\cong\Lambda^{\S}\otimes_{\mathbb{Z}_{p}}A_{\mathrm{cris}}(R)$

by construction.

######  Assumption 2.16.

We will assume from now on that $[b]\in B(G)$ is _neutral acceptable_ in the
sense of [RV14, Def. 2.3], i.e., $[b]\in B(G,\\{\mu\\})$. (Of course this is a
completely harmless assumption with regard to the setting of Shimura
varieties.)

There exists an internal tensor-preserving quasi-endomorphism $p$-divisible
group:

######  Lemma 2.17.

[Kim19, Lemma 3.1.3] There is a $p$-divisible group $\mathcal{H}_{b}^{G}$ such
that for any f-semiperfect $\bar{\mathbb{F}}_{p}$-algebra $R$ there is a
natural $\mathbb{Q}_{p}$-linear isomorphism

$\tilde{\mathcal{H}}_{b}^{G}(R)\cong\operatorname{End}_{(s_{\alpha})}((\mathbb{X}_{b})_{R})\bigl{[}\tfrac{1}{p}\bigr{]},$

where $\tilde{\mathcal{H}}_{b}^{G}$ is the universal cover of
$\mathcal{H}_{b}^{G}$, and where on the right hand side we have the set of
$\gamma\in\operatorname{End}((\mathbb{X}_{b})_{R})[\frac{1}{p}]$ such that the
endomorphism of $\Lambda^{\S}\otimes_{\mathbb{Z}_{p}}B_{\mathrm{cris}}^{+}(R)$
induced by $\gamma$ preserves the tensors $s_{\alpha}\otimes 1$.

What we really need is an internal tensor-preserving quasi-isogeny
$p$-divisible group. Still following [Kim19], to this end we consider the
group sheaf $\operatorname{Qisg}(\mathbb{X}_{b})$ on
$\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ given by
$R\mapsto\\{\text{self-quasi-isogenies of }(\mathbb{X}_{b})_{R}\\}$. This can
be realized as a closed formal subscheme of $\tilde{\mathcal{H}}_{b}^{2}$
(where
$\mathcal{H}_{b}:=\mathcal{H}_{b^{\S}}=\mathcal{H}_{b^{\S}}^{\operatorname{GL}(V^{\S})}$
(no tensors)).

The closed formal subscheme
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})\subseteq\operatorname{Qisg}(\mathbb{X}_{b})$
over $\operatorname{Spf}\breve{\mathbb{Z}}_{p}$ then is defined by

$\operatorname{Qisg}_{G}(\mathbb{X}_{b})=\operatorname{Qisg}(\mathbb{X}_{b})\times_{\tilde{\mathcal{H}}_{b}^{2}}(\tilde{\mathcal{H}}_{b}^{G})^{2}.$

######  Proposition 2.18.

[Kim19, Prop 3.2.4] We have a natural map
$\underline{J_{b}(\mathbb{Q}_{p})}\to\operatorname{Qisg}_{G}(\mathbb{X}_{b})$,
which has a natural retraction

$\operatorname{Qisg}_{G}(\mathbb{X}_{b})\to\underline{J_{b}(\mathbb{Q}_{p})}$

with all fibers isomorphic to
$\operatorname{Spf}\breve{\mathbb{Z}}_{p}[\mkern-2.0mu[x_{1}^{p^{-\infty}},\dotsc,x_{d}^{p^{-\infty}}]\mkern-2.0mu]$
as formal schemes, where $d=\langle 2\rho,\nu_{[b]}\rangle$ with $2\rho$ the
sum of all the positive roots of $G_{\bar{\mathbb{Q}}_{p}}$ and $\nu_{[b]}$
the dominant171717This requires distinguishing a Weyl chamber as the positive
one; this arises from the choice of a convenient maximal torus in [Kim19, just
before Rmk. 2.1.4]. representative of the
$G(\breve{\mathbb{Q}}_{p})$-conjugacy class of $\nu_{b}\in
X_{*}(G_{\breve{\mathbb{Q}}_{p}})_{\mathbb{Q}}$.181818So $b\mapsto\nu_{[b]}$
is the _Newton map_ in the parlance of [Kot97, 256].

######  Corollary 2.19.

Let $R\in\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ be such that there is
no non-trivial continuous homomorphism of $\breve{\mathbb{Z}}_{p}$-algebras
$\breve{\mathbb{Z}}_{p}[\mkern-2.0mu[x^{p^{-\infty}}]\mkern-2.0mu]\to R$. Put
differently, there is no $r\in R$ having a compatible system of $p$-power
roots such that all power series with $\breve{\mathbb{Z}}_{p}$-coefficients in
$r$ and its $p$-power roots converge.

Then

$\operatorname{Qisg}_{G}(\mathbb{X}_{b})(R)\cong\underline{J_{b}(\mathbb{Q}_{p})}(R).$

######  Proof:

With $d$ as in the proposition, the assumption implies that there is no non-
trivial continuous homomorphism of $\breve{\mathbb{Z}}_{p}$-algebras
$\breve{\mathbb{Z}}_{p}[\mkern-2.0mu[x_{1}^{p^{-\infty}},\dotsc,x_{d}^{p^{-\infty}}]\mkern-2.0mu]\to
R$.

Define
$\operatorname{Qisg}_{G}^{\circ}(\mathbb{X}_{b}):=\ker(\operatorname{Qisg}_{G}(\mathbb{X}_{b})\to\underline{J_{b}(\mathbb{Q}_{p})})\cong\operatorname{Spf}\breve{\mathbb{Z}}_{p}[\mkern-2.0mu[x_{1}^{p^{-\infty}},\dotsc,x_{d}^{p^{-\infty}}]\mkern-2.0mu]$
and consider the split exact sequence

$0\to\operatorname{Qisg}_{G}^{\circ}(\mathbb{X}_{b})\to\operatorname{Qisg}_{G}(\mathbb{X}_{b})\to\underline{J_{b}(\mathbb{Q}_{p})}\to
0.$

□

######  Example 2.20.

Say $R\in\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ has characteristic $p$,
i.e., $R$ is a $\bar{\mathbb{F}}_{p}$-algebra and the $p$-adic topology is the
discrete topology. Let $r$ be an element of $R$ having a compatible system of
$p$-power roots such that all power series with
$\breve{\mathbb{Z}}_{p}$-coefficients in $r$ and its $p$-power roots converge.
Then $r$ must be nilpotent. Therefore, if
$R\in\operatorname{Nilp}_{\breve{\mathbb{Z}}_{p}}$ has characteristic $p$ and
is reduced (e.g., is a field), then the conditions of Corollary 2.19 are
satisfied.

In particular we deduce:

######  Example 2.21.

Let $L/\bar{\mathbb{F}}_{p}$ be a field extension. Then

$\operatorname{Qisg}_{G}(\mathbb{X}_{b})(L)\cong J_{b}(\mathbb{Q}_{p}).$

###  2.4 Almost product structure

######  Notation 2.22.

We shorten notation by writing

$\mathscr{S}:=\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{\mathbf{E},(v)}}{{\cal
O}_{\breve{E}}}\quad\text{and}\quad\overline{\mathscr{S}}:=\mathscr{S}_{K}(G,X)\otimes_{\mathcal{O}_{\mathbf{E},(v)}}\bar{\mathbb{F}}_{p},$

as well as

$\mathscr{S}^{\S}:=\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\pm,\S})\otimes_{\mathbb{Z}_{(p)}}\breve{\mathbb{Z}}_{p}\quad\text{and}\quad\overline{\mathscr{S}}^{\S}:={\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\pm,\S})\otimes_{\mathbb{Z}_{(p)}}\bar{\mathbb{F}}_{p}}.$

Moreover, we denote the central leaf associated with $b$ by
$\overline{C}^{b}:=\overline{C}^{b}_{K}:=\Upsilon_{K}^{-1}([\mkern-2.0mu[b]\mkern-2.0mu])\subseteq\overline{\mathscr{S}}$,
and by $\overline{C}^{b^{\S}}\subseteq\overline{\mathscr{S}}^{\S}$ the
corresponding central leaf in $\overline{\mathscr{S}}^{\S}$.

######  Assumption 2.23.

From now on, we assume that Axiom A from [HK17] holds.

######  Remark 2.24.

To explain Axiom A: Consider the _affine Deligne-Lusztig variety_

$X_{\mu}(b)_{K}(\bar{\mathbb{F}}_{p}):=\left\\{g\breve{K}_{p}\in
G(\breve{\mathbb{Q}}_{p})/\breve{K}_{p}\;|\;g^{-1}b\sigma(g)\in\breve{K}_{p}\mathrm{Adm}(\\{\mu\\})\breve{K}_{p}\right\\}$

(here $\mathrm{Adm}(\\{\mu\\})$ is the _admissible subset_ as defined in
[PRS13]).

Choose a point $\bar{x}\in\overline{\mathscr{S}}(\bar{\mathbb{F}}_{p})$ with a
quasi-isogeny $j\colon\mathbb{X}_{b}\to\mathscr{A}_{\bar{x}}[p^{\infty}]$
compatible with the extra structure.

We have a map $i^{\S}_{(\bar{x},j)}\colon
X_{\mu^{\S}}(b^{\S})(\bar{\mathbb{F}}_{p})\to\overline{\mathscr{S}}^{\S}(\bar{\mathbb{F}}_{p})$
given as follows: For $g\breve{J}_{p}\in
X_{\mu^{\S}}(b^{\S})(\bar{\mathbb{F}}_{p})$, there is a $p$-divisible group
$g\mathbb{X}_{b}$ isogenous to $\mathbb{X}_{b}$ with Frobenius on the
DieudonnÃ© module given by $g^{-1}b\sigma(g)$. The quasi-isogeny
$g\mathbb{X}_{b}\to\mathbb{X}_{b}\to\mathscr{A}_{\bar{x}}[p^{\infty}]$ lifts
to a quasi-isogeny of abelian schemes191919Idea: A lift of a quasi-isogeny
$y\colon A[p^{\infty}]\to X$, $A$ abelian scheme, $X$ $p$-divisible group, is
given by $A\to A/\ker(y)$.. We also get a polarization and a level structure
away from $p$ on the new abelian scheme, which is the image of
$g\breve{J}_{b}$ under $i^{\S}_{(\bar{x},j)}$.

The content of Axiom A (= Assumption 2.23) now is that

$X_{\sigma(\mu)}(b)_{K}(\bar{\mathbb{F}}_{p})\hookrightarrow
X_{\mu^{\S}}(b^{\S})(\bar{\mathbb{F}}_{p})\xrightarrow{i^{\S}_{(\bar{x},j)}}\overline{\mathscr{S}}^{\S}(\bar{\mathbb{F}}_{p})$

is to factor through $\overline{\mathscr{S}}^{-}(\bar{\mathbb{F}}_{p})$ such
that there is a unique lift $i\colon
X_{\sigma(\mu)}(b)_{K}(\bar{\mathbb{F}}_{p})\to\overline{\mathscr{S}}(\bar{\mathbb{F}}_{p})$
with $s_{\alpha,0,i(g\breve{K}_{p})}=s_{\alpha,0,\bar{x}}$ for all
$g\breve{K}_{p}\in X_{\sigma(\mu)}(b)_{K}(\bar{\mathbb{F}}_{p})$.

######  Remark 2.25.

Assumption 2.23 holds in the hyperspecial case [Kis17, Thm. 1.4.4], and it
holds if $G$ is residually split [Zho18, Prop. 6.4].

######  Remark 2.26.

With Assumption 2.23 in place, the central leaf $\overline{C}^{b}$ of $b$ is
non-empty by [HK17, Rmk. 4.3.1].

We review some constructions from the paper [HK17]. By
$\overline{\mathrm{Ig}}^{b^{\S}}$ we denote the (special fiber of the) Igusa
variety over $\overline{C}^{b^{\S}}$, cf. [CS17, Prop. 1.12]. It parameterizes
isomorphisms between the standard $p$-divisible group and the universal one,
and it is a perfect scheme over $\bar{\mathbb{F}}_{p}$. Define
$\overline{\mathrm{Ig}}^{b,\diamond}:=\overline{\mathrm{Ig}}^{b^{\S}}\times_{\overline{\mathscr{S}}^{\S,\mathrm{perf}}}\overline{\mathscr{S}}^{\mathrm{perf}}$.
Furthermore, $\mathfrak{Ig}^{b^{\S}},\mathfrak{Ig}^{b,\diamond}$ are the
unique flat lifts202020Locally: Let $R$ be a perfect
$\bar{\mathbb{F}}_{p}$-algebra. Then $W_{\mathcal{O}_{E}}(R)$ (ramified Witt
vectors, cf. Remark 1.15) is the unique $\varpi$-adically complete ${{\cal
O}_{\breve{E}}}$-flat algebra lifting it (cf. [Ahs11, Prop. 1.3.3]). of
$\overline{\mathrm{Ig}}^{b^{\S}}$ and $\overline{\mathrm{Ig}}^{b,\diamond}$
over $\operatorname{Spf}{{\cal O}_{\breve{E}}}$, given by the formula

$\mathfrak{Ig}(R):=\overline{\mathrm{Ig}}(R/\varpi)\quad\text{for all
}R\in\operatorname{Nilp}_{{{\cal O}_{\breve{E}}}}$ (2.27)

in both cases, where $\varpi$ is a uniformizer of ${{\cal O}_{\breve{E}}}$.

By definition, we have an isomorphism
$j\colon\mathbb{X}_{b}\times\overline{\mathrm{Ig}}^{b,\diamond}\xrightarrow{\cong}\mathscr{A}_{\overline{C}^{b}}[p^{\infty}]\times_{\overline{C}^{b}}\overline{\mathrm{Ig}}^{b,\diamond}$
and $\overline{\mathrm{Ig}}^{b}$ is by definition the locus of geometric
points of $\overline{\mathrm{Ig}}^{b,\diamond}$, where $j$ respects the
crystalline Tate tensors. This is a closed union of connected components of
$\overline{\mathrm{Ig}}^{b,\diamond}$ [HK17, Def./Lemma 5.1.1].

$\operatorname{Qisg}(\mathbb{X}_{b})$ acts on
$\overline{\mathrm{Ig}}^{b^{\S}}$ and $\mathfrak{Ig}^{b^{\S}}$,
$\overline{\mathrm{Ig}}^{b}\to\overline{\mathrm{Ig}}^{b^{\S}}$ is a closed
immersion with
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})_{\bar{\mathbb{F}}_{p}}$-stable image,
and $\mathfrak{Ig}^{b}\to\mathfrak{Ig}^{b^{\S}}$ is a closed immersion with
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})$-stable image [HK17, Prop. 5.1.2, Cor
5.1.3].

Let $\mathfrak{M}^{b^{\S}}\to\operatorname{Spf}\breve{\mathbb{Z}}_{p}$ be the
Rapoport-Zink space given by

$\mathfrak{M}^{b^{\S}}(R)=\left\\{(\mathscr{X},\rho)\;\middle|\;\begin{tabular}[]{@{}l@{}}$\mathscr{X}/R$
polarized $p$-divisible group and\\\ $\rho\colon\mathbb{X}_{b}\otimes
R/p\to\mathscr{X}\otimes R/p$ quasi-isogeny resp.
polarizations\end{tabular}\right\\}.$

Choose a point $\bar{x}\in\overline{\mathscr{S}}(\bar{\mathbb{F}}_{p})$ with a
quasi-isogeny $j\colon\mathbb{X}_{b}\to\mathscr{A}_{\bar{x}}[p^{\infty}]$
compatible with the extra structure.

$\mathfrak{M}^{b^{\S}}$ comes with the Rapoport-Zink uniformization map
$\Theta^{\S}\colon\mathfrak{M}^{b^{\S}}\to\mathscr{S}^{\S}$ depending on this
choice [RZ96, (6.3)]: Given $(\mathscr{X},\rho)$ as above, $\rho$ lifts to a
quasi-isogeny $\tilde{\rho}\colon\mathbb{X}_{b}\otimes R\to\mathscr{X}$, and
there is an abelian scheme $\mathscr{Y}$ with $p$-divisible group
$\mathscr{X}$ and a unique lift $\tilde{\mathscr{A}}_{\bar{x}}\otimes
R\to\mathscr{Y}$ of $\tilde{\rho}$ for a chosen lift
$\tilde{\mathscr{A}}_{\bar{x}}$ of $\mathscr{A}_{\bar{x}}$ to
$\breve{\mathbb{Z}}_{p}$. We also get a polarization and a level structure
away from $p$ on $\mathscr{Y}$, and $\mathscr{Y}$ with these extra structures
is the image of $(\mathscr{X},\rho)$ under $\Theta^{\S}$.

######  Remark 2.28.

The affine Deligne-Lusztig variety is the perfection of the special fiber of
the Rapoport-Zink space [Zhu17, Prop. 0.4]. Under this isomorphism,
$\Theta^{\S}$ corresponds to $i^{\S}$ from Remark 2.24.

Define
$\mathfrak{M}^{b,\diamond}:=\mathfrak{M}^{b^{\S}}\times_{\mathscr{S}^{\S}}\mathscr{S}$
and define $\Theta^{\diamond}\colon\mathfrak{M}^{b,\diamond}\to\mathscr{S}$ to
be the base change of $\Theta^{\S}$.

Let $\mathscr{X}^{\diamond}\to\mathfrak{M}^{b,\diamond}$ be the pullback of
the universal $p$-divisible group over $\mathfrak{M}^{b^{\S}}$. Then we have
two families of tensors on $\mathbb{D}(\mathscr{X}^{\diamond})[\tfrac{1}{p}]$
(or more precisely, two families of maps
$\mathfrak{M}^{b,\diamond}(\bar{\mathbb{F}}_{p})\ni\bar{y}\mapsto$ tensor on
$\mathbb{D}(\mathscr{X}^{\diamond}_{\bar{y}})[\tfrac{1}{p}]$):

  1. (1)

$\left(t_{\alpha}^{\diamond}\right)_{\alpha}$ obtained from
$\left(s_{\alpha}\right)_{\alpha}$ via
$\mathbb{D}(\mathbb{X}_{b}\otimes\mathfrak{M}^{b,\diamond})[\tfrac{1}{p}]\cong\mathbb{D}(\mathscr{X}^{\diamond})[\tfrac{1}{p}]$,
and

  2. (2)

for every $\bar{y}\in\mathfrak{M}^{b,\diamond}(\bar{\mathbb{F}}_{p})$,
$\left(u_{\alpha,\bar{y}}^{\diamond}\right)_{\alpha}$ with
$u_{\alpha,\bar{y}}^{\diamond}:=(\Theta^{\diamond})^{*}s_{\alpha,0,\Theta^{\diamond}(\bar{y})}$.

$\mathfrak{M}^{b}$ is defined to be the formal subscheme of
$\mathfrak{M}^{b,\diamond}$ corresponding to the locus where these agree.
Also, $\Theta\colon\mathfrak{M}^{b}\to\mathscr{S}$ is defined to be the
restriction of $\Theta^{\diamond}$. Moreover, $\mathfrak{M}^{b}$ has a natural
$\operatorname{Qisg}_{G}(\mathbb{X}_{b})$-action.

By [CS17, Â§ 4], there is an isomorphism

$\mathfrak{Ig}^{b^{\S}}\times\mathfrak{M}^{b^{\S}}\xrightarrow{\sim}\mathfrak{X}^{b^{\S}}$
(2.29)

with the _Newton-Igusa variety_

$\mathfrak{X}^{b^{\S}}(R)=\left\\{(A,\lambda,\eta^{p};\psi)\;\middle|\;\begin{tabular}[]{@{}l@{}}$(A,\lambda,\eta^{p})\in\mathscr{S}^{\S}(R)$,\\\
$\psi\colon(\mathbb{X}_{b},\lambda_{\mathbb{X}_{b}})\otimes
R/p\to(A[p^{\infty}],\lambda)\otimes R/p$ quasi-
isogeny\end{tabular}\right\\}.$

######  Remark 2.30.

Let us quickly recall how the isomorphism (2.29) works (suppressing
polarizations in the notation for simplicity). Let
$(\mathcal{A},\xi)\in\mathfrak{Ig}^{b^{\S}}(R)$ and
$(\mathscr{X},\rho)\in\mathfrak{M}^{b^{\S}}(R)$ be given. Consider the
composition

$\mathscr{X}\xrightarrow{\rho^{-1}}\mathbb{X}_{b}\otimes
R\xrightarrow{\xi}\mathcal{A}[p^{\infty}],$ (2.31)

where we denote a lift $\mathbb{X}_{b}\otimes
R/p\xrightarrow{\rho}\mathscr{X}\otimes R/p$ by $\rho$ again. Lift (2.31) to a
quasi-isogeny of abelian schemes $\mathcal{A}^{\prime}\to\mathcal{A}$. Then
$(\mathcal{A}^{\prime},\rho)\in\mathfrak{X}^{b^{\S}}(R)$ is our image point.

Define $\mathfrak{X}^{b}\subseteq\mathfrak{X}^{b^{\S}}$ to be the image of
$\mathfrak{Ig}^{b}\times\mathfrak{M}^{b}$ under the isomorphism (2.29). This
comes with a natural $\operatorname{Qisg}_{G}(\mathbb{X}_{b})$-action; cf.
[HK17, section 5.2].

We have canonical maps
$\pi_{\infty}^{\S}\colon\mathfrak{X}^{b^{\S}}\to\mathscr{S}^{\S}$ and
$\pi_{\infty}\colon\mathfrak{X}^{b}\to\mathscr{S}$.

######  Remark 2.32.

By [HK17, Thm. 5.2.6 (1)],
$\pi_{\infty,\bar{\mathbb{F}}_{p}}^{\mathrm{perf}}\colon\mathfrak{X}^{b,\mathrm{perf}}_{\bar{\mathbb{F}}_{p}}\to\overline{\mathscr{S}}^{\mathrm{perf}}$
represents the moduli problem

$\displaystyle(\text{perfect affine
}\overline{\mathscr{S}}^{\mathrm{perf}}\text{-schemes})$
$\displaystyle\to\mathrm{Set},$
$\displaystyle(\operatorname{Spec}(R)\xrightarrow{Q}\overline{\mathscr{S}}^{\mathrm{perf}})$
$\displaystyle\mapsto\begin{Bmatrix}\psi\colon(\mathbb{X}_{b})_{R}\to(\mathscr{G}_{\overline{\mathscr{S}}^{\mathrm{perf}}})_{Q}\text{
quasi-isogeny}\\\ \text{compatible with crystalline Tate-
tensors}\end{Bmatrix}$

with $\mathscr{G}_{\overline{\mathscr{S}}^{\mathrm{perf}}}$ and crystalline
Tate tensors on the right hand side as in Proposition 1.66.

###  2.5 Change of parahoric level

Now we consider the question of how the central leaves behave when the level
at $p$ is changed from $K_{p}$ to a larger $K_{p}^{\prime}$, where $K_{p}$ and
$K_{p}^{\prime}$ are associated with points
$x,x^{\prime}\in\mathcal{B}(G,\mathbb{Q}_{p})$ as described in Chapter 1.
Define $K:=K_{p}K^{p}$, as usual, and $K^{\prime}:=K_{p}^{\prime}K^{p}$. Let
$b\in G(\breve{\mathbb{Q}}_{p})$.

Let $(\mathcal{L},c)$ and $(\mathcal{L}^{\prime},c^{\prime})$ be the graded
lattice chains that are the images of $x$ and $x^{\prime}$, respectively,
under the embedding of buildings
$\iota\colon\mathcal{B}(G,\mathbb{Q}_{p})\hookrightarrow\mathcal{B}(\operatorname{GSp}(V),\mathbb{Q}_{p})$
described in Section 1.2. Note that the gradings $c$ and $c^{\prime}$ play no
role for our purposes.

######  Remarks 2.33.

  1. (1)

Observe that $K_{p}$ and $K_{p}^{\prime}$ depend only on the minimal facet
$\mathfrak{f}$ and $\mathfrak{f}^{\prime}$ with $x\in\mathfrak{f}$ and
$x^{\prime}\in\mathfrak{f}^{\prime}$, respectively. The inclusion
$K_{p}\subseteq K_{p}^{\prime}$ means precisely that
$\mathfrak{f}^{\prime}\subseteq\overline{\mathfrak{f}}$. We can move
$x^{\prime}$ arbitrarily close to $x$ without altering $K_{p}^{\prime}$. The
minimal facets $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ of
$\mathcal{B}(\operatorname{GSp}(V),\mathbb{Q}_{p})$ containing $\iota(x)$ and
$\iota(x^{\prime})$, respectively, do _not_ depend only on
$\mathfrak{f},\mathfrak{f}^{\prime}$. Still, for $x^{\prime}$ sufficiently
close to $x$, we may assume that
$\mathfrak{g}^{\prime}\subseteq\overline{\mathfrak{g}}$ (if it were not so, we
would not be able to move $\iota(x^{\prime})$ arbitrarily close to $\iota(x)$
— but recall that $\iota$ is continuous), i.e., that $\mathcal{L}^{\prime}$ is
a thinned out version of $\mathcal{L}$. Say $\mathcal{L}$ is given by

$\Lambda^{0}\supsetneqq\Lambda^{1}\supsetneqq\dotsb\supsetneqq\Lambda^{r-1}\supsetneqq
p\Lambda^{0}$

(as in (1.12)); then $\mathcal{L}^{\prime}$ is given by
$(\Lambda^{i_{j}})_{j}$ for some $0\leq i_{1}<i_{2}<\dotsb<i_{s}<r$.

  2. (2)

In proofs, we can often reduce to the case where “$\mathcal{L}$ and
$\mathcal{L}^{\prime}$ differ by one element”, which of course is supposed to
mean that $s=r-2$ in this notation.

######  Notation 2.34.

We define
$N_{p},N_{p}^{\prime},J_{p},J_{p}^{\prime},\Lambda^{\S},\left.\Lambda^{\prime}\right.^{\S},V^{\S},\left.V^{\prime}\right.^{\S}$
as in Section 1.4.

####  2.5.1 The change-of-parahoric morphism

As explained in Section 1.6, we obtain (for $K^{p}$ sufficiently small and
some compact open subgroup
$N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$) a commutative
diagram

$\mathscr{S}_{K_{p}K^{p}}(G,X)$$\mathscr{S}_{K^{\prime}_{p}\left.K\right.^{p}}(G,X)$$\mathscr{S}_{N_{p}N^{p}}(\operatorname{GSp}(V),S^{\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$$\mathscr{S}_{N_{p}^{\prime}\left.N\right.^{p}}(\operatorname{GSp}(V),S^{\pm})_{\mathcal{O}_{\mathbf{E},(v)}}$finitefinite$\pi_{K_{p}K^{p},K_{p}^{\prime}\left.K\right.^{p}}$$\pi_{N_{p}N^{p},N_{p}^{\prime}\left.N\right.^{p}}$

######  Notation 2.35.

  1. (1)

The change-of-parahoric map
$\pi_{K,K^{\prime}}:=\pi_{K_{p}K^{p},K_{p}^{\prime}\left.K\right.^{p}}$
restricts to a change-of-parahoric map
$\Upsilon_{K_{p}K^{p}}^{-1}([\mkern-2.0mu[b]\mkern-2.0mu])\to\Upsilon_{K_{p}^{\prime}K^{p}}^{-1}([\mkern-2.0mu[b]\mkern-2.0mu])$
between leaves, which we denote by
$\left.\pi_{K,K^{\prime}}\right|_{\Upsilon_{K}^{-1}([\mkern-2.0mu[b]\mkern-2.0mu])}$
or simply by $\pi_{K,K^{\prime}}$ again.

  2. (2)

We choose a base point
$\bar{x}_{b}\in\overline{C}^{b}_{K}(\bar{\mathbb{F}}_{p})$ and take its image
under $\pi_{K,K^{\prime}}$ as a base point
$\bar{x}_{b}^{\prime}\in\overline{C}^{b}_{K^{\prime}}(\bar{\mathbb{F}}_{p})$.
By $\mathbb{X}_{b,K}$ and $\mathbb{X}_{b,K^{\prime}}$, respectively, we denote
the corresponding polarized $p$-divisible groups.

  3. (3)

Denote by $\mathscr{A}_{K}$ the universal polarized abelian scheme over
$\overline{\mathscr{S}}_{K}$. By slight abuse of notation, we will also use
the same notation for its pullback to
$\overline{\mathscr{S}}_{K}^{\mathrm{perf}}$.

######  Remark 2.36.

We have morphisms $\mathbb{X}_{b,K}\to\mathbb{X}_{b,K^{\prime}}$ and, more
generally, $\mathscr{A}_{K}\to\mathscr{A}_{K^{\prime}}$ (lying over
$\overline{\mathscr{S}}_{K}\to\overline{\mathscr{S}}_{K^{\prime}}$). This
follows from the “thinning out” interpretation, Remark 2.33 (1).

####  2.5.2 Newton-Igusa variety and change-of-parahoric

######  Lemma 2.37.

There are natural change-of-parahoric morphisms

$\mathfrak{Ig}^{b}_{K}\to\mathfrak{Ig}^{b}_{K^{\prime}},\quad\mathfrak{M}^{b}_{K}\to\mathfrak{M}^{b}_{K^{\prime}},\quad\mathfrak{X}^{b}_{K}\to\mathfrak{X}^{b}_{K^{\prime}}.$

lying over the change-of-parahoric morphism
$\mathscr{S}_{K}\to\mathscr{S}_{K^{\prime}}$ and compatible with the Siegel
embeddings.

######  Proof:

Consider the universal isomorphism
$j\colon\mathbb{X}_{b,K}\times\overline{\mathrm{Ig}}_{K}^{b,\diamond}\xrightarrow{\cong}\mathscr{A}_{K,\overline{C}^{b}_{K}}[p^{\infty}]\times_{\overline{C}^{b}_{K}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}=\mathscr{A}_{K}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}$
and the diagram

$\mathbb{X}_{b,K}\times\overline{\mathrm{Ig}}^{b,\diamond}_{K}$$\mathscr{A}_{K}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}$$\mathbb{X}_{b,K^{\prime}}\times\overline{\mathrm{Ig}}^{b,\diamond}_{K}$$\mathscr{A}_{K^{\prime}}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K^{\prime}}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}$$\cong$$\cong$

The dashed arrow exists on $\overline{\mathrm{Ig}}^{b}_{K}$, i.e.,
$\mathbb{X}_{b,K^{\prime}}\times\overline{\mathrm{Ig}}^{b}_{K}\xrightarrow{\cong}\mathscr{A}_{K^{\prime}}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K^{\prime}}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b}$
exists. The reason is that the tensors are respected on
$\overline{\mathrm{Ig}}_{K}^{b}$ which in particular forces
$\mathbb{X}_{b,K}\times\overline{\mathrm{Ig}}^{b,\diamond}_{K}\xrightarrow{\cong}\mathscr{A}_{K}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b,\diamond}$
to be “diagonal”, similar to the proof of Proposition 1.24.

This isomorphism
$\mathbb{X}_{b,K^{\prime}}\times\overline{\mathrm{Ig}}^{b}_{K}\xrightarrow{\cong}\mathscr{A}_{K^{\prime}}[p^{\infty}]\times_{\overline{\mathscr{S}}_{K^{\prime}}^{\mathrm{perf}}}\overline{\mathrm{Ig}}_{K}^{b}$
now corresponds to a morphism
$\overline{\mathrm{Ig}}_{K}^{b}\to\overline{\mathrm{Ig}}_{K^{\prime}}^{b}$. By
construction (2.27) of $\mathfrak{Ig}^{b}$, we also get a corresponding
morphism $\mathfrak{Ig}_{K}^{b}\to\mathfrak{Ig}_{K^{\prime}}^{b}$.

For $\mathfrak{X}$ and $\mathfrak{M}$, it is very similar. □

######  Lemma 2.38.

The isomorphism $\mathfrak{Ig}^{b}\times\mathfrak{M}^{b}\cong\mathfrak{X}^{b}$
is compatible with change-of-parahoric, where the change-of-parahoric map for
$\mathfrak{Ig}^{b}\times\mathfrak{M}^{b}$ is by definition the product of
those for $\mathfrak{Ig}^{b}$ and $\mathfrak{M}^{b}$, respectively.

######  Proof:

All of $\mathfrak{Ig}^{b},\mathfrak{M}^{b},\mathfrak{X}^{b}$ are embedded into
the corresponding Siegel versions as closed formal subschemes, compatible with
change-of-parahoric. Hence we may assume that we are in the Siegel case.

In that case, the isomorphism has a very concrete description, cf. Remark
2.30, from which the lemma is clear. □

######  Definition 2.39.

By $\overline{\mathscr{S}}_{K}^{b}:=\overline{\mathscr{S}}_{K}^{[b]}$ we
denote the Newton stratum associated with ${[b]\in B(G)}$, i.e.,
$\bar{S}_{K,[b]}$ in the notation of [HR17].

######  Lemma 2.40.

Let $\Omega/\mathbb{F}_{p}$ be an algebraically closed field.

The action of $J_{b}(\mathbb{Q}_{p})$ on the fibers of
$\mathfrak{X}_{K}^{b}(\Omega)\to\overline{\mathscr{S}}^{b}_{K}(\Omega)$ is
simply transitive and the change-of-parahoric map

$\mathfrak{X}_{K}^{b}(\Omega)\to\mathfrak{X}_{K^{\prime}}^{b}(\Omega)$

is $J_{b}(\mathbb{Q}_{p})$-equivariant.

######  Proof:

The moduli description of Remark 2.32 gives us that the action is simply
transitive upon noting that

$\operatorname{Qisg}_{G}(\mathbb{X}_{b,K})(\Omega)\overset{\ref{QisgJbField}}{\cong}J_{b}(\mathbb{Q}_{p})\overset{\ref{QisgJbField}}{\cong}\operatorname{Qisg}_{G}(\mathbb{X}_{b,K^{\prime}})(\Omega).$

Equivariance follows from the description of the action of
$J_{b}(\mathbb{Q}_{p})$ and of the change-of-parahoric map in terms of lattice
chains in DieudonnÃ© theory: Given a point
$Q\colon\operatorname{Spec}\Omega\to\overline{\mathscr{S}}_{K}^{b}$, we
consider the map between the fiber of $Q$ under
$\mathfrak{X}^{b}_{K}(\Omega)\to\overline{\mathscr{S}}^{b}_{K}(\Omega)$ and
the fiber of the image of $Q$ under
$\mathfrak{X}^{b}_{K^{\prime}}(\Omega)\to\overline{\mathscr{S}}^{b}_{K^{\prime}}(\Omega)$.
We identify the lattice chain associated with $Q$ with a standard lattice
chain such that the Frobenius is identified with a $b^{\prime}\in
G(\breve{\mathbb{Q}}_{p})$, $b^{\prime}\equiv b\mod\breve{K}_{p,\sigma}$.
Elements of $J_{b}(\mathbb{Q}_{p})$ then act naturally on the common rational
DieudonnÃ© module. The action on the fiber of $Q$ is given by altering the
quasi-isogenies appearing in the description of that set by the quasi-isogeny
obtained this way. Passing from $K$ to $K^{\prime}$, i.e., from $Q$ to the
image of $Q$, means leaving out parts of the lattice chain and enlarging
$\breve{K}_{p}$ (so one still has the same $b^{\prime}$). □

######  Proposition 2.41.

The change-of-parahoric map

$\mathfrak{X}^{b}_{K,\bar{\mathbb{F}}_{p}}\to\mathfrak{X}^{b}_{K^{\prime},\bar{\mathbb{F}}_{p}}$

is surjective.

######  Proof:

We check that

$\mathfrak{X}^{b}_{K}(\Omega)\to\mathfrak{X}^{b}_{K^{\prime}}(\Omega)$

is surjective for every algebraically closed field $\Omega/\mathbb{F}_{p}$.

Considering the diagram

$\mathfrak{X}^{b}_{K}(\Omega)$$\mathfrak{X}^{b}_{K^{\prime}}(\Omega)$$\overline{\mathscr{S}}_{K}^{b}(\Omega)$$\overline{\mathscr{S}}_{K^{\prime}}^{b}(\Omega)$

this follows from the preceding lemma and the fact212121This follows simply
from the surjectiveness of
$\overline{\mathscr{S}}_{K}\to\overline{\mathscr{S}}_{K^{\prime}}$ and the
commutativity of the diagram
$\overline{\mathscr{S}}_{K}$$\overline{\mathscr{S}}_{K^{\prime}}$$B(G)$, non-
horizontal maps being the Newton maps. that the change-of-parahoric map
between Newton strata is surjective. □

######  Corollary 2.42.

The map
$\overline{\mathrm{Ig}}_{K}^{b}\to\overline{\mathrm{Ig}}_{K^{\prime}}^{b}$ is
an isomorphism.

######  Proof:

Lemma 2.38 and Proposition 2.41 imply that it is surjective.

Also, it is the restriction to closed subschemes of the isomorphism
$\overline{\mathrm{Ig}}_{K}^{b^{\S}}\to\overline{\mathrm{Ig}}_{K^{\prime}}^{b^{\S}}$.
Since all involved schemes are reduced, the corollary follows. □

######  Corollary 2.43.

The change-of-parahoric morphism between central leaves is surjective.

######  Proof:

We have a diagram

$\overline{\mathrm{Ig}}_{K}^{b}$$\overline{\mathrm{Ig}}_{K^{\prime}}^{b}$$\Upsilon^{-1}_{K}(b)$$\Upsilon^{-1}_{K^{\prime}}(b)$$\cong$

where we already know that all maps but the lower horizontal one are
surjective. □

######  Corollary 2.44.

The separable rank of $\Upsilon^{-1}_{K}(b)\to\Upsilon^{-1}_{K^{\prime}}(b)$,
i.e., the number of geometric points of the fibers, is finite and constant.

######  Proof:

We have

$\Upsilon^{-1}_{K}(b)\cong\overline{\mathrm{Ig}}_{K}^{b}/\operatorname{Aut}(\mathbb{X}_{b,K})\cong\overline{\mathrm{Ig}}_{K^{\prime}}^{b}/\operatorname{Aut}(\mathbb{X}_{b,K})\quad\text{and}\quad\Upsilon^{-1}_{K^{\prime}}(b)\cong\overline{\mathrm{Ig}}_{K^{\prime}}^{b}/\operatorname{Aut}(\mathbb{X}_{b,K^{\prime}}),$

so that all fibers are isomorphic to
$\operatorname{Aut}(\mathbb{X}_{b,K^{\prime}})/\operatorname{Aut}(\mathbb{X}_{b,K})$.
□