$\mathrm{Sh}_{{\sf K}}(\mathbf{T},\\{h\\})_{E}=\bigsqcup_{i\in
I}\textup{Spec}\,(E_{i})$ (3.13)

for some finite index set $I$. In fact, each field $E_{i}$ is the same field,
and is an unramified extension of $\mathbb{Q}_{p}$ (see [7, Lemma 4.2] and the
proof of [7, Lemma 4.8]).

Set $T=\mathbf{T}_{\mathbb{Q}_{p}}$. As discussed in §2.1.3, there is a unique
parahoric model $\mathcal{T}$ of $T$, given by the connected component
$(\mathcal{T}^{\mathrm{lft}})^{\circ}$ of the Néron model
$\mathcal{T}^{\mathrm{lft}}$ of $T$.

######  Proposition 3.22.

Suppose $(\mathbf{T},\\{h\\},\mathcal{T})$ is parahoric Shimura datum of toral
type, and let $\mathfrak{d}=(\mathbf{G}_{1},\mathbf{X}_{1},\mathcal{G}_{1})$
be a very good parahoric Shimura datum of Hodge type which is well-adapted to
$(\mathbf{T},\\{h\\},\mathcal{T})$. Then,

$\displaystyle\mathscr{S}_{\sf
K}^{\mathfrak{d}}(\mathbf{T},\\{h\\})=\bigsqcup_{i\in
I}\textup{Spec}\,(\mathcal{O}_{E_{i}}),$

for any neat compact open subgroup
$\mathsf{K}^{p}\subseteq\mathbf{T}(\mathbb{A}_{f}^{p})$ and with ${\sf K}={\sf
K}_{p}{\sf K}^{p}$. The same holds true for $\mathscr{S}_{\sf
K}^{{\varnothing}}(\mathbf{T},\\{h\\})$ if $(\mathbf{T},\\{h\\})$ is of Hodge
type.

###### Proof.

First, observe that as $(\mathbf{G}_{1},\mathbf{X}_{1})$ is adapted to the
datum $(\mathbf{T},\\{h\\})$, we have an isogeny
$\mathbf{G}_{1}^{\textup{der}}\to\mathbf{T}^{\textup{der}}=\\{e\\}$, and hence
$\mathbf{G}_{1}^{\textup{der}}$ is trivial, since it is connected. Thus,
$\mathbf{G}_{1}$ is a torus.101010Indeed, if $H$ is a commutative reductive
group over a field $K$, and $T$ is a maximal torus of $H$, then $H=Z_{H}(T)=T$
(see [37, Corollary 17.84]). Consider an arbitrary neat compact open subgroup
${\sf K}^{p}_{1}\subseteq\mathbf{G}_{1}(\mathbb{A}_{f}^{p})$.

Choose a parahoric Hodge embedding
$\iota\colon(\mathbf{G}_{1},\mathbf{X}_{1},\mathcal{G}_{1})\to(\mathbf{GSp}(\mathbf{V}),\mathfrak{h}^{\pm},\mathrm{GSp}(\Lambda))$
which respects stabilizers, and a neat compact open subgroup ${\sf
L}_{1}^{p}\subseteq\mathrm{GSp}(\mathbf{V}\otimes{Q}\mathbb{A}_{f}^{p})$ such
that $\iota({\sf K}_{1,p}\mathsf{K}^{p}_{1})\subseteq\mathrm{GSp}(\Lambda){\sf
L}_{1}^{p}$, inducing a closed embedding

$\mathrm{Sh}_{{\sf
K}_{1,p}\mathsf{K}^{p}_{1}}(\mathbf{G}_{1},\mathbf{X}_{1})\to\mathrm{Sh}_{\mathrm{GSp}(\Lambda){\sf
L}_{1}^{p}}(\mathbf{GSp}(\mathbf{V}),\mathfrak{h}^{\pm})_{\mathbf{E}_{1}}.$
(3.14)

Note that $\boldsymbol{\mathscr{S}}_{\mathrm{GSp}(\Lambda){\sf
L}_{1}^{p}}(\mathbf{GSp}(\mathbf{V}),\mathfrak{h}^{\pm})$ is flat, separated,
and finite over $\mathbb{Z}_{(p)}$. Thus, considering (3.13), and using
notation from Step 2 of §3.3, it follows from Lemma 3.23 below that
$\boldsymbol{\mathscr{S}}^{{\varnothing}}_{\widetilde{\mathsf{K}}_{p,1}{\sf
K}^{p}_{1}}(\mathbf{G}_{1},\mathbf{X}_{1})$ and thus
$\boldsymbol{\mathscr{S}}^{{\varnothing}}_{\mathsf{K}_{p,1}{\sf
K}^{p}_{1}}(\mathbf{G}_{1},\mathbf{X}_{1})$ are disjoint unions of semi-local
localizations of $\mathcal{O}_{\mathbf{F}}$ for some finite extension
$\mathbf{F}$ of $\mathbf{E}$.

This implies that, using notation now from Step 3 of §3.3, that
$\boldsymbol{\mathscr{S}}_{\mathsf{K}_{p,1}}^{{\varnothing}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathcal{O}_{E}}^{+}$
is either the spectrum of a field or of a semi-local Dedekind domain. Thus,
from the construction in Step 4 of §3.3 we deduce that for each neat compact
open $\mathsf{K}^{p}\subseteq\mathbf{G}(\mathbb{A}_{f}^{p})$ one has that
$\mathscr{S}_{{\sf
K}_{p}\mathsf{K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})$ is either the
disjoint union of spectra of extensions of $E$, or the disjoint union of
spectra of the rings of integers of finite extensions of $E$. That said, the
former is impossible as it is then clear that such a model cannot satisfy the
extension property from (2) of Theorem 3.12, and so the latter must hold. The
fist claim follows. Moreover, the second claim follows from similar arguments.

∎

Following [47, Tag 035E], if $X$ is a scheme with the property that every
quasi-compact open has finitely many connected components, then we denote the
normalization of $X$ by $X^{\nu}$.

######  Lemma 3.23.

Let $\mathcal{O}$ be an excellent Dedekind domain and $\mathscr{X}$ be a
separated, finite type, flat $\mathcal{O}$-scheme. Let $x$ is a closed point
of $\mathscr{X}_{\eta}$ and let $\overline{x}$ be the Zariski closure of $x$
in $\mathscr{X}$. Then, $\overline{x}^{\nu}$ is an open subscheme of
$\textup{Spec}\,(\mathcal{O}^{\prime})$ for a Dedekind domain
$\mathcal{O}^{\prime}$ finite flat over $\mathcal{O}$.

###### Proof.

Let $\overline{X}$ be a compactification of
$\mathscr{X}\to\textup{Spec}\,(\mathcal{O})$ (see [47, Tag 0ATT] and [47, Tag
0F41]). Replacing $\overline{\mathscr{X}}$ with the closed suscheme cut out by
the $\mathcal{O}$-torsion in $\mathcal{O}_{\mathscr{X}}$, we may assume that
$\overline{\mathscr{X}}$ is flat over $\mathcal{O}$ (see [17, Proposition
14.14]).

Let $Z$ denote the Zariski closure of $x$ in $\overline{\mathscr{X}}$. We
claim that $Z^{\nu}=\textup{Spec}\,(\mathcal{O}^{\prime})$ for a Dedekind ring
$\mathcal{O}^{\prime}$ finite flat over $\mathcal{O}$. As
$\overline{x}=\mathscr{X}\cap Z$, the lemma will then follow from [47, Tag
035K]. To prove the claim, we observe that $Z$ is evidently irreducible and
reduced, and thus integral. But, as $Z\to\mathrm{Spec}(\mathcal{O})$ is
dominant, we deduce that it is flat by [17, Proposition 14.14]. But, since
$\overline{\mathscr{X}}$ is a proper $\mathcal{O}$-scheme, so is $Z$. Thus, by
[47, Tag 0D4J] we deduce that $Z_{s}$ is zero dimensional. Thus,
$Z\to\textup{Spec}\,(\mathcal{O})$ is proper and quasi-finite, and so finite
(see [47, Tag 02OG]). It follows that $Z^{\nu}$ is also an integral scheme and
$Z^{\nu}\to\textup{Spec}\,(\mathcal{O})$ is dominant and finite (using [47,
Tag 035R]) over $\mathcal{O}$ and so flat again by [17, Proposition 14.14].
Write $Z^{\nu}=\textup{Spec}\,(\mathcal{O}^{\prime})$, then
$\mathcal{O}^{\prime}$ is $1$-dimensional Noetherian normal domain, and so a
Dedekind domain as desired. ∎

Given Proposition 3.22, we see that the definition of
$\mathscr{S}_{\mathsf{K}}^{\mathfrak{d}}=(\mathbf{T},\\{h\\})$ is independent
of the choice of $\mathfrak{d}$. We denote by
$\mathscr{S}_{\mathsf{K}}(\mathbf{T},\\{h\\})$ the common object.

####  3.4.2. Functoriality for ad-isomorphisms

Suppose that
$\alpha\colon(\mathbf{G}_{1},\mathbf{X}_{1},\mathcal{G}_{1})\to(\mathbf{G},\mathbf{X},\mathcal{G})$
is an ad-isomorphism of parahoric Shimura datum. Take a very good parahoric
Shimura datum $\mathfrak{d}=(\mathbf{G}_{2},\mathbf{X}_{2},\mathcal{G}_{2})$
of Hodge type well-adapted to
$(\mathbf{G}_{1},\mathbf{X}_{1},\mathcal{G}_{1})$. Then, since $\alpha$ is an
ad-isomorphism, one sees that $\mathfrak{d}$ is also well-adapted to
$(\mathbf{G},\mathbf{X},\mathcal{G})$ in the obvious way.

######  Proposition 3.24.

There exists a unique quotient-finite étale morphism

$\alpha\colon\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})\to\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E_{1}}},$

which is equivariant for the map
$\alpha\colon\mathbf{G}_{1}(\mathbb{A}_{f}^{p})\to\mathbf{G}(\mathbb{A}_{f}^{p})$
and models

$\alpha\colon\mathrm{Sh}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})\to\mathrm{Sh}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{E_{1}}.$

Here by _quotient-finite étale_ we mean that if
$\mathsf{K}_{1}^{p}\subseteq\mathbf{G}_{1}(\mathbb{A}_{f}^{p})$ and
$\mathsf{K}^{p}\subseteq\mathbf{G}(\mathbb{A}_{f}^{p})$ are neat compact open
subgroups such that $\alpha(\mathsf{K}_{1}^{p})\subseteq\mathsf{K}^{p}$, then
the induced map

$\alpha\colon\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}^{p}_{1}}(\mathbf{G}_{1},\mathbf{X}_{1})\to\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}^{p}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E_{1}}},$

is a finite étale morphism.

###### Proof of Proposition 3.24.

The fact that such a map is unique is clear by the separatedness of $\alpha$
and the flatness of
$\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}^{p}_{1}}(\mathbf{G}_{1},\mathbf{X}_{1})$,
which guarantees that its generic fiber is Zariski dense (e.g., see [17,
Proposition 14.14]).

To see the existence, we note that by definition we have

$\mathscr{S}_{\mathsf{K}_{p,1}}^{\mathfrak{d}}(\mathbf{G}_{1},\mathbf{X}_{1})=\left(\left[\mathscr{S}_{\mathsf{K}_{p,2}}^{{\varnothing}}(\mathbf{G}_{2},\mathbf{X}_{2})^{+}_{\mathcal{O}_{E_{1}}}\times\mathscr{A}(\bm{\mathcal{G}}_{1})\right]/\mathscr{A}(\bm{\mathcal{G}}_{2})^{\circ}\right)^{J_{1}},$

and

$\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E_{1}}}=\left(\left[\mathscr{S}_{\mathsf{K}_{p,2}}^{{\varnothing}}(\mathbf{G}_{2},\mathbf{X}_{2})^{+}_{\mathcal{O}_{E_{1}}}\times\mathscr{A}(\bm{\mathcal{G}})\right]/\mathscr{A}(\bm{\mathcal{G}}_{2})^{\circ}\right)^{J}.$

Now, by Lemma 3.15 and Lemma 3.16 we get morphisms

$\alpha\colon\mathscr{A}(\bm{\mathcal{G}}_{1})\to\mathscr{A}(\bm{\mathcal{G}}),\quad\alpha\colon\mathscr{A}(\bm{\mathcal{G}}_{1})^{\circ}\to\mathscr{A}(\bm{\mathcal{G}}),\
\text{ and }\
\alpha\colon\mathscr{E}(\bm{\mathcal{G}}_{1})\to\mathscr{E}(\bm{\mathcal{G}}).$

Because $\alpha(\mathsf{K}_{p,1})\subseteq\mathsf{K}_{p}$, we have a map of
sets $\alpha\colon J_{1}\to J$. Since the isomorphism from (3.5) is natural in
these operations, we deduce the existence of a map

$\left(\left[\mathscr{S}_{\mathsf{K}_{p,2}}^{{\varnothing}}(\mathbf{G}_{2},\mathbf{X}_{2})^{+}_{\mathcal{O}_{E_{1}}}\times\mathscr{A}(\bm{\mathcal{G}}_{1})\right]/\mathscr{A}(\bm{\mathcal{G}}_{2})^{\circ}\right)^{J_{1}}\to\left(\left[\mathscr{S}_{\mathsf{K}_{p,2}}^{{\varnothing}}(\mathbf{G}_{2},\mathbf{X}_{2})^{+}_{\mathcal{O}_{E_{1}}}\times\mathscr{A}(\bm{\mathcal{G}})\right]/\mathscr{A}(\bm{\mathcal{G}}_{2})^{\circ}\right)^{J},$

equivariant for
$\alpha\colon\mathbf{G}_{1}(\mathbb{A}_{f}^{p})\to\mathbf{G}(\mathbb{A}_{f}^{p})$.
That the generic fiber of this map agrees with

$\alpha\colon\mathrm{Sh}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{E_{1}}\to\mathrm{Sh}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{E_{1}}.$

follows as in [26, Lemma 4.6.13].

Finally, we show that $\alpha$ is quotient-finite étale. Let
$\mathscr{S}_{1}^{+}$ denote a connected component of
$\mathscr{S}^{\mathfrak{d}}_{{\sf K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})$
mapping to a connected component $\mathscr{S}^{+}$ of
$\mathscr{S}^{\mathfrak{d}}_{{\sf
K}_{p}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E_{1}}}$. By construction, we
have

$\displaystyle\mathscr{S}_{2}^{+}=\mathscr{S}^{{\varnothing}}_{{\sf
K}_{p,2}}(\mathbf{G}_{2},\mathbf{X}_{2})^{+}_{\mathcal{O}_{E_{1}}}/\Delta_{1}\quad\text{and}\quad\mathscr{S}^{+}=\mathscr{S}^{{\varnothing}}_{{\sf
K}_{p,2}}(\mathbf{G}_{2},\mathbf{X}_{2})^{+}_{\mathcal{O}_{E_{1}}}/\Delta,$

where

$\displaystyle\Delta_{1}=\ker(\mathscr{A}(\bm{\mathcal{G}}_{2})^{\circ}\to\mathscr{A}(\mathbf{\bm{\mathcal{G}}}))\quad\text{and}\quad\Delta=\ker(\mathscr{A}(\bm{\mathcal{G}}_{2})^{\circ}\to\mathscr{A}(\bm{\mathcal{G}})),$

(see also the arguments of [26, Corollary 4.6.15]). Thus $\mathscr{S}^{+}$ is
the quotient of $\mathscr{S}_{1}^{+}$ by the finite group $\Delta/\Delta_{1}$,
and the result follows. ∎

In particular, the proposition implies functoriality for morphisms
$(\mathbf{T}_{1},\\{h_{1}\\},\mathcal{T}_{1})\to(\mathbf{T},\\{h\\},\mathcal{T})$
of parahoric Shimura data of toral type.

####  3.4.3. Functoriality for abelianizations

Suppose now that $(\mathbf{G},\mathbf{X},\mathcal{G})$ is a parahoric Shimura
datum of abelian type. Consider the parahoric Shimura datum
$(\mathbf{G}^{\textup{ab}},\mathbf{X}^{\textup{ab}},\mathcal{G}^{\textup{ab}})$,
where $\mathbf{X}^{\textup{ab}}$ is the result of post-composing any element
$h$ of $\mathbf{X}$ with the canonical map
$\delta:\mathbf{G}\to\mathbf{G}^{\textup{ab}}$. We have the obvious associated
morphism of parahoric Shimura data

$\alpha\colon(\mathbf{G},\mathbf{X},\mathcal{G})\to(\mathbf{G}^{\textup{ab}},\mathbf{X}^{\textup{ab}},\mathcal{G}^{\textup{ab}}).$

Write ${\sf K}_{p}^{\textup{ab}}=\mathcal{G}^{\textup{ab}}(\mathbb{Z}_{p})$.

######  Proposition 3.25.

There exists a unique

$\alpha\colon\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})\to\mathscr{S}_{\mathsf{K}_{p}^{\textup{ab}}}(\mathbf{G}^{\textup{ab}},\mathbf{X}^{\textup{ab}})_{\mathcal{O}_{E}},$

which is equivariant for the map
$\alpha\colon\mathbf{G}_{1}(\mathbb{A}_{f}^{p})\to\mathbf{G}(\mathbb{A}_{f}^{p})$
and models

$\alpha\colon\mathrm{Sh}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{E_{1}}\to\mathrm{Sh}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{E_{1}}.$

###### Proof.

Given the descriptions of $\mathrm{Sh}_{{\sf
K}_{p}^{\textup{ab}}}(\mathbf{G}^{\textup{ab}},\mathbf{X}^{\textup{ab}})$ and
$\mathscr{S}_{{\sf
K}_{p}^{\textup{ab}}}(\mathbf{G}^{\textup{ab}},\mathbf{X}^{\textup{ab}})_{\mathcal{O}_{E}}$
from (3.13) and Proposition 3.22, respectively, the result follows from Lemma
3.26. More precisely, it suffices to construct maps at every finite level.
But, by Proposition 3.22, the Kisin–Pappas–Zhou model $S^{\prime}$ for
$(\mathbf{G}^{\textup{ab}},\mathbf{X}^{\textup{ab}})$ base-changed to
$\mathcal{O}_{E}$ is finite flat over $S=\textup{Spec}\,(\mathcal{O}_{E})$. On
the other hand, the finite levels $X$ of the Kisin–Pappas–Zhou models $X$ for
$(\mathbf{G},\mathbf{X})$ are normal and quasi-projective and flat over $S$.
Thus may apply Lemma 3.26 to the map at finite levels on the generic fiber. ∎

######  Lemma 3.26.

Suppose that $S$ is a scheme and $U$ is an open subscheme of $S$ which is
schematically dense and for which the inclusion $U\hookrightarrow S$ is quasi-
compact. Given a diagram

${X_{U}}$${X}$${S^{\prime}_{U}}$${S^{\prime}}$${U}$${S}$$\scriptstyle{\alpha}$$\scriptstyle{f}$$\scriptstyle{g}$

with $X$ a normal scheme, $f$ quasi-compact, separated, and flat, and $g$
quasi-compact, flat, and integral, there exists a unique arrow dashed arrow as
in the diagram above which makes the top square Cartesian.

###### Proof.

By [47, Tag 081H], $X_{U}\subseteq X$ is schematically dense, so the
uniqueness follows from [47, Tag 01RH]. In turn, by uniqueness, we may
localize on $S$ and $X$ to prove existence. Using that $g$ is affine, we may
then assume without loss of generality that $S=\textup{Spec}\,(A)$,
$S^{\prime}=\textup{Spec}\,(A^{\prime})$ and $A\to A^{\prime}$ is integral.
Moreover, localizing on $X$ we may further assume $X=\textup{Spec}\,(B)$ with
$B$ a normal domain. We then have the following diagram of rings

${A}$${A^{\prime}}$${B}$${\mathcal{O}(U)}$${\mathcal{O}(S_{U}^{\prime})}$${\mathcal{O}(X_{U})}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{\alpha}$

The center and right-hand vertical arrows are injections by flatness of $f$
and $g$ along with [47, Tag 081H] and [47, Tag 01RE].

We claim that $\alpha(A^{\prime})\subseteq B$. Indeed, let $a^{\prime}\in
A^{\prime}$. Then by integrality of $A\to A^{\prime}$, there exists a monic
polynomial $p(T)$ in $A[T]$ such that $p(a^{\prime})=0$. But then
$\alpha(a^{\prime})$ in $\mathcal{O}(X_{U})$ satisfies the monic polynomial
$f(p)(T)$ in $B[T]$. Thus $\alpha(a^{\prime})$ must lie in $B$ by normality of
$B$.

It follows that we have a map
$\textup{Spec}\,(\alpha):\textup{Spec}\,(B)\to\textup{Spec}\,(A^{\prime})$. We
claim that the diagram

${X_{U}}$${X=\textup{Spec}\,(B)}$${S^{\prime}_{U}}$${S^{\prime}=\textup{Spec}\,(A^{\prime})}$$\scriptstyle{\alpha}$$\scriptstyle{\textup{Spec}\,(\alpha)}$

is Cartesian. For this it is enough to show that
$\textup{Spec}\,(\alpha)\big{|}_{X_{U}}=\alpha$. Since $U\hookrightarrow X$ is
quasi-compact, $U$ is quasi-compact, and therefore the same is true of
$S_{U}^{\prime}$ and $X_{U}$ by quasi-compactness of $f$ and $g$. Since
$\textup{Spec}\,(\alpha)$ and $\alpha$ induce the same map
$\mathcal{O}(S^{\prime}_{U})\to\mathcal{O}(X_{U})$, the result now follows
from [47, Tag 01P9]. ∎

####  3.4.4. Two constructions for very good Hodge-type data

We record here the following basic compatibility between the constructions
made in §3.3. Let $\mathfrak{d}=(\mathbf{G},\mathbf{X},\mathcal{G})$ be a very
good parahoric Shimura datum of Hodge type.

######  Proposition 3.27.

There is a canonical $\mathbf{G}(\mathbb{A}^{p}_{f})$-equivariant
identification

$\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathscr{S}^{{\varnothing}}_{\mathsf{K}_{p}},(\mathbf{G},\mathbf{X}),$

extending the identity in the generic fiber.

###### Proof.

We first construct a $\mathbf{G}(\mathbb{A}_{f}^{p})$-equivariant map

$\boldsymbol{\mathscr{S}}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}\to\boldsymbol{\mathscr{S}}_{\mathsf{K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}.$

Choosing a connected component $\mathbf{X}^{+}$ of $\mathbf{X}$, we obtain an
inclusion

$\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}^{+}\hookrightarrow\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}},$

which extends by the action of $\mathscr{A}(\bm{\mathcal{G}})$ to

$\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}^{+}\times\mathscr{A}(\bm{\mathcal{G}})\to\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime p}_{(v)}}$
(3.15)

via $(s,a)\mapsto s\cdot a$. One checks that (3.15) is equivariant for the
action of
$\mathscr{A}(\bm{\mathcal{G}})\rtimes\mathscr{E}_{\mathbf{E}^{\prime}}(\bm{\mathcal{G}})$
on $\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}^{+}\times\mathscr{A}(\bm{\mathcal{G}})$, and it induces

$[\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}^{+}\times\mathscr{A}(\bm{\mathcal{G}})]/\mathscr{A}(\bm{\mathcal{G}})^{\circ}\to\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}.$ (3.16)

Indeed, this follows by the definition of the action of
$\mathscr{A}(\bm{\mathcal{G}})\rtimes\mathscr{E}_{\mathbf{E}^{\prime}}(\bm{\mathcal{G}})$
on the product $\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}^{+}\times\mathscr{A}(\bm{\mathcal{G}})$ via (3.9) and (3.11), along
with the fact that the composition

$\mathscr{A}(\bm{\mathcal{G}})^{\circ}\to\mathscr{E}_{\mathbf{E}^{\prime}}(\bm{\mathcal{G}})\to\mathscr{E}(\bm{\mathcal{G}})\to\mathscr{A}(\bm{\mathcal{G}})$

is the inclusion
$\mathscr{A}(\bm{\mathcal{G}})^{\circ}\hookrightarrow\mathscr{A}(\bm{\mathcal{G}})$.
In turn, (3.16) determines a
$\mathscr{A}(\bm{\mathcal{G}})\ast_{\mathscr{A}(\bm{\mathcal{G}})^{\circ}}\mathscr{E}_{\mathbf{E}^{\prime}}(\bm{\mathcal{G}})$-equivariant,
and hence $\mathbf{G}(\mathbb{A}_{f})\times\textup{Gal}(\mathbf{E}^{\prime
p}/\mathbf{E})$-equivariant, map

$\boldsymbol{\mathscr{S}}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}=\left([\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}^{+}\times\mathscr{A}(\bm{\mathcal{G}})]/\mathscr{A}(\bm{\mathcal{G}})^{\circ}\right)^{J}\to\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}.$ (3.17)

The map (3.17) is birational, since it is an isomorphism on generic fibers by
[26, Lemma 4.6.13], and, by its definition, it is quasi-finite. By
$\mathbf{G}(\mathbb{A}_{f}^{p})$-equivariance, we may first pass to the map
(3.17) after taking the quotient by any neat compact open subgroup ${\sf
K}^{p}$ of $\mathbf{G}(\mathbb{A}_{f}^{p})$. In that case, the target is
normal, so it follows that the quotient of (3.17) by ${\sf K}^{p}$ is an open
immersion on connected components by Lemma 3.21 and [17, Corollary 12.88].
Passing to the limit, we deduce that (3.17) is an open immersion. Moreover,
(3.17) is surjective by part (2) of Theorem 3.12111111Indeed, it is enough to
show (3.17) is surjective on special fibers. An
$\overline{\mathbb{F}}_{p}$-point of $\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime p}_{(v)}}$
factors through an $R$-point of $\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}$, where $R$ is a discrete valuation ring of characteristic $(0,p)$
(see [47, Tag 0CM2]). This, in turn, induces an $R[\nicefrac{{1}}{{p}}]$-point
of $\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime
p}_{(v)}}$. We can pull this point back along the isomorphism on generic
fibers, and the result lifts to an $R$-point of
$\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime p}_{(v)}}$
by part (2) of Theorem 3.12. This maps to the original
$\overline{\mathbb{F}}_{p}$-point of $\boldsymbol{\mathscr{S}}_{{\sf
K}_{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime p}_{(v)}}$
by separatedness, as they generically agree., so it is an isomorphism.

∎

####  3.4.5. Kisin–Pappas–Zhou integral models for some products

Let $\mathfrak{d}=(\mathbf{G},\mathbf{X},\mathcal{G})$ be a very good
parahoric Shimura datum of Hodge type, and that
$(\mathbf{T},\\{h\\},\mathcal{T})$ is a parahoric Shimura datum of toral type.
Write $\mathbf{E}^{\prime}$ for the compositum of the reflex fields of the two
Shimura data, which is the reflex field of
$(\mathbf{G}\times\mathbf{T},\mathbf{X}\times\\{h\\})$, and set
$\mathsf{M}_{p}=\mathcal{T}(\mathbb{Z}_{p})$.

######  Proposition 3.28.

There is a canonical
$\mathbf{G}(\mathbb{A}^{p}_{f})\times\mathbf{T}(\mathbb{A}_{f}^{p})$-equivariant
identification

$\mathscr{S}^{\mathfrak{d}}_{{\sf
K}_{p}\times\mathsf{M}_{p}}(\mathbf{G}\times\mathbf{T},\mathbf{X}\times\\{h\\})\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathscr{S}_{{\sf
K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\prime}}}\times\mathscr{S}_{\mathsf{M}_{p}}(\mathbf{T},\\{h\\})_{\mathcal{O}_{E^{\prime}}}.$

###### Proof.

The projection map $\mathbf{G}\times\mathbf{T}\to\mathbf{G}$ is an ad-
isomorphism, so by Proposition 3.24, we have a quotient-finite map

$\mathscr{S}^{\mathfrak{d}}_{{\sf
K}_{p}\times\mathsf{M}_{p}}(\mathbf{G}\times\mathbf{T},\mathbf{X}\times\\{h\\})\to\mathscr{S}_{{\sf
K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\prime}}}$
(3.18)

extending the morphism on the generic fibers which is equivariant for the
projection
$(\mathbf{G}\times\mathbf{T})(\mathbb{A}_{f}^{p})\to\mathbf{G}(\mathbb{A}_{f}^{p})$.
On the other hand, the map of reductive groups
$\mathbf{G}\times\mathbf{T}\to\mathbf{T}$ factors through
$(\mathbf{G}\times\mathbf{T})^{\textup{ab}}$, which is a torus. Hence
Proposition 3.24 and Proposition 3.25 combine to furnish us with a map

$\mathscr{S}^{\mathfrak{d}}_{{\sf
K}_{p}\times\mathsf{M}_{p}}(\mathbf{G}\times\mathbf{T},\mathbf{X}\times\\{h\\})\to\mathscr{S}_{\mathsf{M}_{p}}(\mathbf{T},\\{h\\})_{\mathcal{O}_{E^{\prime}}}$

extending the morphism on generic fibers, which is equivariant for
$(\mathbf{G}\times\mathbf{T})(\mathbb{A}_{f}^{p})\to\mathbf{T}(\mathbb{A}_{f}^{p})$.
From these, we obtain the map

$\mathscr{S}^{\mathfrak{d}}_{{\sf
K}_{p}\times\mathsf{M}_{p}}(\mathbf{G}\times\mathbf{T},\mathbf{X}\times\\{h\\})\to\mathscr{S}_{{\sf
K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\prime}}}\times\mathscr{S}_{\mathsf{M}_{p}}(\mathbf{T},\\{h\\})_{\mathcal{O}_{E^{\prime}}},$
(3.19)

which is equivariant for
$(\mathbf{G}\times\mathbf{T})(\mathbb{A}_{f}^{p})\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathbf{G}(\mathbb{A}_{f}^{p})\times\mathbf{T}(\mathbb{A}_{f}^{p})$.
The map (3.19) is an isomorphism on generic fibers, and is quasi-finite at
each finite level because (3.18) is quotient-finite. It follows that (3.19) is
an isomorphism by the argument at the end of the proof of Proposition 3.27. ∎

##  4\. The Pappas–Rapoport conjecture

In this section we state the Pappas–Rapoport conjecture and recall the cases
in which it has been previously proven. We also state our main theorem and
applications.

As in §3.3, throughout this section we often use the following notation and
data:

     * •

$(\mathbf{G},\mathbf{X})$ is a Shimura datum,

     * •

$\mathbf{Z}$ is the center of $\mathbf{G}$,

     * •

$\mathbf{E}$ is the reflex field of $(\mathbf{G},\mathbf{X})$,

     * •

$v$ is a $p$-adic place of $\mathbf{E}$,

     * •

$E=\mathbf{E}_{v}$,

     * •

$k_{E}$ is the residue field of $E$,

     * •

$G=\mathbf{G}_{\mathbb{Q}_{p}}$,

     * •

$\mathcal{G}=\mathcal{G}_{x}^{\circ}$ is a parahoric model of $G$,

     * •

$\mathsf{K}_{p}\vcentcolon=\mathcal{G}(\mathbb{Z}_{p})$,

     * •

${\sf K}^{p}\subseteq\mathbf{G}(\mathbb{A}^{f}_{p})$ is a neat compact open
subgroup,

     * •

${\sf K}={\sf K}_{p}{\sf K}^{p}$.

###  4.1. The Pappas–Rapoport conjecture

We now recall the formulation of the Pappas–Rapoport conjecture as given in
[42] and extended in [7].

####  4.1.1. The group $\mathcal{G}^{c}$

In constrast to the case of Shimura varieties of Hodge type discussed in [42],
to obtain shtukas on arbitrary Shimura varieties of abelian type, one must
consider a modification $\mathcal{G}^{c}$ of the parahoric group
$\mathcal{G}$.

Let $\mathbf{T}$ be a multiplicative group over $\mathbb{Q}$. We denote by
$\mathbf{T}_{\mathrm{ac}}$ the anti-cuspidal part of $\mathbf{T}$ as in [28,
Definition 1.5.4]. More precisely, if $\mathbf{T}_{a}$ denotes the largest
anisotropic subtorus of $\mathbf{T}$, then $\mathbf{T}_{\mathrm{ac}}$ is the
largest subtorus of $\mathbf{T}_{a}$ whose base change to $\mathbb{R}$
contains the maximal split subtorus of $(\mathbf{T}_{a}){R}$.

We write ${\mathbf{G}}^{c}$ for the quotient
${\mathbf{G}}^{c}/\mathbf{Z}_{\mathrm{ac}}$, and if $\mathsf{H}$ is a subgroup
of ${\mathbf{G}}(\mathbb{A}_{f})$, we write $\mathsf{H}^{c}$ for the image of
$\mathsf{H}$ under
${\mathbf{G}}(\mathbb{A}_{f})\to{\mathbf{G}}^{c}(\mathbb{A}_{f})$. We also
denote by $G^{c}$ the base change of ${\mathbf{G}}^{c}$ to $\mathbb{Q}_{p}$.
The following lemma will be used below.

######  Lemma 4.1.

If $\alpha\colon(\mathbf{G}_{1},\mathbf{X}_{1})\to(\mathbf{G},\mathbf{X})$ is
a morphism of Shimura data then
$\alpha(\mathbf{Z}_{1,\mathrm{ac}})\subseteq\mathbf{Z}_{\mathrm{ac}}$ and so
$\alpha$ induces a map $\alpha\colon\mathbf{G}_{1}^{c}\to\mathbf{G}^{c}$.
Consequently, if $(\mathbf{G},\mathbf{X})$ is of Hodge type, then the natural
maps $\mathbf{G}\to\mathbf{G}^{c}$ and
$\mathbf{G}^{\textup{ab}}\to(\mathbf{G}^{\textup{ab}})^{c}$ are isomorphisms.

###### Proof.

The first claim is [21, Lemma 4.7]. The second claim then follows from the
first by embedding an arbitrary Hodge type datum $(\mathbf{G},\mathbf{X})$
into one of Siegel type $(\mathbf{G}_{1},\mathbf{X}_{1})$, where it can be
explicitly checked that $\mathbf{Z}_{1,\mathrm{ac}}$ is trivial. For the claim
concerning $\mathbf{G}^{\textup{ab}}$, we observe that the map
$\delta\colon\mathbf{G}\to\mathbf{G}^{\textup{ab}}$ restricts to a surjection
$\delta\colon\mathbf{Z}\to\mathbf{G}^{\textup{ab}}$ (e.g., see [37, Example
19.25]). From here, it’s easy to check that $\delta$ surjects
$\mathbf{Z}_{\mathrm{ac}}$ onto $\mathbf{G}^{\mathrm{ab}}_{\mathrm{ac}}$, and
the claim follows. ∎

We set $\mathcal{G}^{c}$ to be the parahoric model of $\mathbf{G}$ induced by
$\mathcal{G}$ in the sense of §2.1.5. By construction, we see that $G\to
G^{c}$ extends to a morphism of group $\mathbb{Z}_{p}$-schemes
$\mathcal{G}\to\mathcal{G}^{c}$. For a conjugacy class of cocharacters
$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$ of $G_{\overline{\mathbb{Q}}_{p}}$, we denote by
$\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}^{c}$ the image of this conjugacy class under
$G_{\overline{\mathbb{Q}}_{p}}\to G^{c}_{\overline{\mathbb{Q}}_{p}}$.

####  4.1.2. The shtuka over $\mathrm{Sh}_{\sf K}(\mathbf{G},\mathbf{X})_{E}$.

We now recall the existence of a “universal shtuka” which lives over the
Shimura variety $\mathrm{Sh}_{\sf K}(\mathbf{G},\mathbf{X})_{E}$, as in [42,
§4.1] and [7, §4.2].

For a normal compact open subgroup ${\sf K}_{p}^{\prime}\subseteq{\sf K}_{p}$,
we temporarily write ${\sf K}^{\prime}$ for the product ${\sf
K}_{p}^{\prime}{\sf K}^{p}$. Then the transition morphism

$\displaystyle\pi_{\mathsf{K}^{\prime},\mathsf{K}}\colon\mathrm{Sh}_{{\sf
K}^{\prime}}(\mathbf{G},\mathbf{X})_{E}\to\mathrm{Sh}_{\sf
K}(\mathbf{G},\mathbf{X})_{E}$ (4.1)

is a finite étale Galois cover with Galois group
$\mathsf{K}_{p}/(\mathsf{K}_{p}^{\prime}\mathbf{Z}(\mathbb{\mathbb{Q}})^{-}_{\mathsf{K}})$,
where $\mathbf{Z}(\mathbb{\mathbb{Q}})^{-}_{\mathsf{K}}$ is the closure of
$\mathbf{Z}(\mathbb{Q})\cap\mathsf{K}$ in $\mathsf{K}$ (see [28, §1.5.8]). We
consider the infinite-level Shimura variety

$\displaystyle\mathrm{Sh}_{{\sf
K}^{p}}(\mathbf{G},\mathbf{X})_{E}\vcentcolon=\varprojlim_{{\sf
K}_{p}^{\prime}\subseteq{\sf K}_{p}}\mathrm{Sh}_{{\sf K}_{p}^{\prime}{\sf
K}^{p}}(\mathbf{G},\mathbf{X})_{E},$

which exists as an $E$-scheme as each $\pi_{\mathsf{K}^{\prime},\mathsf{K}}$
is affine (see [47, Tag 01YX]), and forms a
$\mathsf{K}_{p}/\mathbf{Z}(\mathbb{Q})_{\mathsf{K}}^{-}$-torsor for the pro-
etale site (in the sense of [5]) on
$\mathrm{Sh}_{\mathsf{K}}(\mathbf{G},\mathbf{X})_{E}$.

The map $\mathsf{K}_{p}\to\mathcal{G}^{c}(\mathbb{Z}_{p})$ factorizes through
$\mathsf{K}_{p}/\mathbf{Z}(\mathbb{Q})_{\mathsf{K}}^{-}$ (see [28, §1.5.8]).
Thus, as in [7, §4.2] or [21, §2.1.5 and §4.3], from the
$\mathsf{K}_{p}/\mathbf{Z}(\mathbb{Q})_{\mathsf{K}}^{-}$-torsor

$\displaystyle\mathrm{Sh}_{{\sf
K}^{p}}({\mathbf{G}},X)_{E}\to\mathrm{Sh}_{{\sf K}_{p}{\sf
K}^{p}}({\mathbf{G}},X)_{E},$

one constructs a $\mathcal{G}^{c}(\mathbb{Z}_{p})$ torsor $\mathbb{P}_{\sf K}$
on the pro-étale site (see [45]) of $\mathrm{Sh}_{\sf
K}({\mathbf{G}},X)^{\textup{an}}_{E}$.

######  Proposition 4.2 ([42, Proposition 4.1.4] and [7, Proposition 4.4]).

There exists a $\mathcal{G}^{c}$-shtuka $\mathscr{P}_{{\sf K},E}$ over
$\mathrm{Sh}_{\sf K}(\mathbf{G},\mathbf{X})_{E}^{\lozenge}\to\textup{Spd}(E)$
with one leg bounded by
$\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{h}^{c}$ which is associated to $\mathbb{P}_{\sf K}$ in the
sense of [42, §2.5]. Moreover, if
$\mathsf{K}^{\prime}=\mathsf{K}_{p}{\mathsf{K}^{p}}^{\prime}$ and $g$ is in
$\mathbf{G}(\mathbb{A}_{f}^{p})$ is such that
$g^{-1}\mathsf{K}^{p}g\subseteq{\mathsf{K}^{p}}^{\prime}$, there exists
compatible isomorphisms

$t_{\mathsf{K},\mathsf{K}^{\prime}}(g)^{\ast}(\mathscr{P}_{\mathsf{K}^{\prime},E})\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathscr{P}_{\mathsf{K},E}.$
(4.2)

####  4.1.3. Integral local Shimura varieties

Let us define (see [49, Definition 24.1.1]), a _parahoric local Shimura datum_
to be a triple $(\mathcal{G},b,\mu)$ where

     * •

$\mathcal{G}$ is a parahoric group $\mathbb{Z}_{p}$-scheme with generic fiber
$G$,

     * •

$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$ is a conjugacy class of minuscule cocharacters of
$G_{\overline{\mathbb{Q}}_{p}}$,

     * •

and $b$ is an element of $G(\breve{\mathbb{Q}}_{p})$ inducing an element of
$B(G,\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}^{-1})$ (see [44, Definition 2.3]).

The reflex field of $(\mathcal{G},b,\mu)$, usually denoted $E$, is the field
of definition of
$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$. A morphism
$f\colon(\mathcal{G}_{1},b_{1},\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{1})\to(\mathcal{G},b,\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{})$ of parahoric local Shimura datum is is a morphism of
$\mathbb{Z}_{p}$-group schemes $f\colon\mathcal{G}_{1}\to\mathcal{G}$ carrying
$\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{1}$ to
$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$ and $b_{1}$ to $b$. If such a morphism exists, then $E_{1}\supseteq
E$. We say that $f$ is an _ad-isomorphism_ if $f\colon G_{1}\to G$ is an ad-
isomorphism.

Given a parahoric local Shimura datum
$(\mathcal{G},b,\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{})$ with reflex field $E$, we obtain a presheaf

$\mathcal{M}_{\mathcal{G},b,{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}}^{\mathrm{int}}\colon\mathbf{Perf}_{\mathcal{O}_{\breve{E}}}\to\mathbf{Set}$

assigning to any perfectoid space $S$ in characteristic $p$ the set of
isomorphism classes of tuples
$(S^{\sharp},\mathscr{P},\phi_{\mathscr{P}},i_{r})$, where:

     * •

$S^{\sharp}$ is the untilt of $S$ over $\mathcal{O}_{\breve{E}}$ associated
with $S\to\textup{Spd}(\mathcal{O}_{\breve{E}})$,

     * •

$(\mathscr{P},\phi_{\mathscr{P}})$ is a $\mathcal{G}$-shtuka on $S$ with one
leg along $S^{\sharp}$ which is bounded by
$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$ (see [42, Definition 2.4.3]),

     * •

and $i_{r}$ is an isomorphism of $\mathcal{G}$-torsors
$\mathcal{G}\big{|}_{\mathcal{Y}_{[r,\infty)}(S)}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathscr{P}\big{|}_{\mathcal{Y}_{[r,\infty)}(S)}$
for large enough $r$, under which $\phi_{\mathscr{P}}$ is identified with
$b\times\mathrm{Frob}_{S}$ (see [49, §25.1]).

By [49, §25.1] and [13, Proposition 2.23], the presheaf
$\mathcal{M}_{\mathcal{G},b,{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}}^{\mathrm{int}}$ is a small $v$-sheaf (in the sense of [46,
Definition 12.1]), which is called the integral local Shimura variety
associated to the local Shimura datum
$(\mathcal{G},b,\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{})$.

If $k_{E}$ denotes the residue field of $E$, then by [14, Proposition 2.30],
we have a natural identification

$\mathcal{M}_{\mathcal{G},b,{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}}^{\mathrm{int}}(\textup{Spd}(\bar{k}_{E}))=X_{\mathcal{G}}(b,\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}^{-1})(\bar{k}_{E}),$ (4.3)

where
$X_{\mathcal{G}}(b,\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}^{-1})$ denotes the affine Deligne-Lusztig variety associated to
the triple
$(\mathcal{G},b,\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}^{-1})$ (see [42, Definition 3.3.1]). For any point $x$ of
$\mathcal{M}^{\mathrm{int}}_{\mathcal{G},b,\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}(\textup{Spd}(\bar{k}_{E}))$ we write
$(\mathcal{M}^{\mathrm{int}}_{\mathcal{G},b,{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}})^{\wedge}_{/x}$ for the formal completion of
$\mathcal{M}_{\mathcal{G},b,{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}}^{\mathrm{int}}$ at $x$ in the sense of [14].

####  4.1.4. The conjecture of Pappas and Rapoport

We now state the Pappas–Rapoport conjecture. For details, see [42, §4.2] and
[7, §4.3].

Suppose we have a normal flat $\mathcal{O}_{E}$-model $\mathscr{S}_{\sf K}$ of
$\mathrm{Sh}_{\sf K}(\mathbf{G},\mathbf{X})_{E}$, equipped with an extension
$\mathscr{P}_{\sf K}$ of the $\mathcal{G}^{c}$-shtuka $\mathscr{P}_{{\sf
K},E}$ to a $\mathcal{G}^{c}$-shtuka defined over $\mathscr{S}_{\sf
K}^{\lozenge/}$. Note that $\mathscr{P}_{\sf K}$ is necessarily bounded by
$\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{h}^{c}$ by [7, Lemma 2.1].

For any point $x$ of $\mathscr{S}_{\sf K}(\bar{k}_{E})$, the pullback
$x^{\ast}\mathscr{P}_{\sf K}$ defines a $\mathcal{G}^{c}$-shtuka over
$\textup{Spd}(\bar{k}_{E})$. In turn, by [42, Example 2.4.9],
$x^{\ast}\mathscr{P}_{\sf K}$ determines a pair $(\mathcal{P}_{x},\phi_{x})$
consisting of a $\mathcal{G}^{c}$-torsor $\mathcal{P}_{x}$ over
$\textup{Spec}\,(W(\bar{k}_{E}))$ along with an isomorphism

$\displaystyle\phi_{x}:\phi^{\ast}(\mathcal{P})[1/p]\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathcal{P}_{x}[1/p].$

Here $\phi$ denotes the Frobenius homomorphism for $W(\bar{k}_{E})$. After
choosing a trivialization of $\mathcal{P}_{x}$, we obtain an element $b_{x}$
in $G^{c}(\breve{\mathbb{Q}}_{p})$, and changing the trivialization will
change $b_{x}$ to some $\sigma$-conjugate of $b_{x}$. Therefore we obtain a
well-defined conjugacy class $[b_{x}]$ in $B(G^{c})$.

Since the shtuka $\mathscr{P}_{\sf K}$ is bounded by
$\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{h}^{c}$, the same holds for $(\mathcal{P}_{x},\phi_{x})$, so
the pair $(\mathcal{P}_{x},\textup{id})$ determines a point $x_{0}$ of
$X_{\mathcal{G}}^{c}(b_{x},(\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}^{c}_{h})^{-1})(\bar{k}_{E})$. Hence
$X_{\mathcal{G}}(b_{x},\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{h}^{-1})(\bar{k}_{E})$ is nonempty, so by [18, Theorem A], it
follows that $[b_{x}]$ belongs to
$B(G^{c},(\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{h}^{c})^{-1})$. Then the triple
$(\mathcal{G}^{c},b_{x},\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{h}^{c})$ defines a parahoric local Shimura datum, and we can
consider the integral local Shimura variety
$\mathcal{M}^{\mathrm{int}}_{\mathcal{G}^{c},b_{x},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}_{h}^{c}}$ along with the base point $x_{0}$ (which makes
sense by (4.3)).

######  Definition 4.3 ([42, Conjecture 4.2.2] and [7, Conjecture 4.5]).

Consider a system $\\{\mathscr{S}_{{\sf K}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ of normal flat $\mathcal{O}_{E}$-models $\mathscr{S}_{\sf K}$ of
$\mathrm{Sh}_{\sf K}(\mathbf{G},\mathbf{X})$ for ${\sf K}={\sf K}_{p}{\sf
K}^{p}$ with ${\sf K}^{p}$ varying over all sufficiently small compact open
subgroups of $\mathbf{G}(\mathbb{A}_{f}^{p})$.

We say the system $\\{\mathscr{S}_{\sf K}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ is an canonical integral model for $\\{\mathrm{Sh}_{{\sf
K}}(\mathbf{G},\mathbf{X})\\}_{{\sf K}^{p}}$ if the following properties are
satisfied.

    1. (i)

For every discrete valuation ring $R$ of characteristic $(0,p)$ over
$\mathcal{O}_{E}$,

$\displaystyle\mathrm{Sh}_{{\sf
K}_{p}}(\mathbf{G},\mathbf{X})(R[1/p])=\left(\varprojlim_{{\sf
K}^{p}}\mathscr{S}_{\sf K}(\mathbf{G},\mathbf{X})\right)(R).$ (4.4)

    2. (ii)

For every sufficiently small ${\sf
K}^{p}\subseteq{\mathbf{G}}(\mathbb{A}_{f}^{p})$ and ${{\sf
K}^{\prime}}^{p}\subseteq{\mathbf{G}}(\mathbb{A}_{f}^{p})$, and element $g$ of
${\mathbf{G}}(\mathbb{A}_{f}^{p})$ with $g^{-1}{{\sf K}}^{p}g\subseteq{\sf
K}^{p}$, there are finite étale morphisms

$t_{\mathsf{K},\mathsf{K}^{\prime}}(g)\colon\mathscr{S}_{{\sf
K}}(\mathbf{G},\mathbf{X})\to\mathscr{S}_{{\sf
K}^{\prime}}(\mathbf{G},\mathbf{X})$

extending the corresponding maps on the generic fiber.

    3. (iii)

The $\mathcal{G}^{c}$-shtuka $\mathscr{P}_{{\sf K},E}$ extends to a
$\mathcal{G}^{c}$-shtuka $\mathscr{P}_{\sf K}$ on $\mathscr{S}_{\sf
K}(\mathbf{G},\mathbf{X})^{\lozenge/}$ for every sufficiently small ${\sf
K}^{p}\subseteq\mathbf{G}(\mathbb{A}_{f}^{p})$.

    4. (iv)

Consider $x$ in $\mathscr{S}_{\sf K}(\mathbf{G},\mathbf{X})(\bar{k}_{E})$ with
corresponding element $b_{x}$ of $G(\breve{\mathbb{Q}}_{p})$, and let $x_{0}$
be the natural base point in
$\mathcal{M}_{\mathcal{G}^{c},b_{x},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}_{h}^{c}}^{\mathrm{int}}(\bar{k}_{E})$. Then there is an
isomorphism of $v$-sheaves

$\displaystyle\Theta_{x}:\left(\mathcal{M}^{\mathrm{int}}_{\mathcal{G}^{c},b_{x},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{h}^{c}}}\right)^{\wedge}_{/x_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\left(\mathscr{S}_{\sf
K}(\mathbf{G},\mathbf{X})^{\wedge}_{/x}\right)^{\lozenge},$ (4.5)

such that $\Theta_{x}^{\ast}(\mathscr{P}_{\sf K})$ agrees with the universal
$\mathcal{G}^{c}$-shtuka $\mathscr{P}^{\mathrm{univ}}$ on
$\mathcal{M}_{\mathcal{G}^{c},b_{x},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{h}^{c}}}^{\mathrm{int}}$.

We observe the following uniqueness properties concerning a system of models
satisfying the Pappas–Rapoport conjecture.

######  Proposition 4.4 (cf. [42, Corollary 2.7.10 and Theorem 4.2.4]).

An canonical integral model of $\\{\mathrm{Sh}_{{\sf
K}}(\mathbf{G},\mathbf{X})\\}_{{\sf K}^{p}}$ and the $\mathcal{G}^{c}$-shtukas
$\mathscr{P}_{\mathsf{K}}$ on these canonical integral models, are unique up
to unique isomorphism $($if they exist$)$.

The following is then what we refer to as the _Pappas–Rapoport conjecture_.

######  Conjecture 4.5 (Pappas-Rapoport).

For any parahoric Shimura datum $(\mathcal{G},\mathbf{G},\mathbf{X})$, there
exists an canonical integral model for $\\{\mathrm{Sh}_{\sf
K}(\mathbf{G},\mathbf{X})\\}_{{\sf K}^{p}}$.

Conjecture 4.5 is known in the following cases, where we have elected to use
our notation from §3.3 for maximal clarity.

######  Theorem 4.6 ([42, Theorem 4.5.2] and [10, Corollary 4.1.5]).

For any parahoric Shimura datum $(\mathbf{G},\mathbf{X},\mathcal{G})$ of Hodge
type, the system
$\\{\mathscr{S}_{\mathsf{K}}^{{\varnothing}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ satisfies the conditions of the Pappas–Rapoport conjecture.

######  Remark 4.7.

In [42], Pappas and Rapoport prove the conjecture under the stubborn technical
assumption that ${\sf K}_{p}$ is a stabilizer parahoric, i.e., where ${\sf
K}_{p}=\mathcal{G}(\mathbb{Z}_{p})$ for a parahoric group scheme $\mathcal{G}$
which is also the Bruhat–Tits stabilizer group scheme of a point in the
extended Bruhat–Tits building of $G$. This assumption frequently holds; for
example every parahoric $\mathbb{Z}_{p}$-model of $G$ is of this form if
$\pi_{1}(G)_{I}$ is torsion free. That says, it eliminates many examples of
Shimura varieties of abelian type (e.g., many cases of type $D{H}$). The
theorem was subsequently extended to arbitrary Shimura data of Hodge type in
[10].

######  Theorem 4.8 ([7, Theorem A]).

If $(\mathbf{G},\mathbf{X},\mathcal{G})$ is a parahoric Shimura datum of toral
type, then $\\{\mathscr{S}_{\mathsf{K}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ satisfies the conditions of the Pappas–Rapoport conjecture.

###### Proof.

This follows from [7, Theorem A] along with Proposition 3.22. ∎

######  Theorem 4.9 ([21, Proposition 5.35]).

If $(\mathbf{G},\mathbf{X},\mathcal{G})$ is a parahoric Shimura datum of
abelian type where $\mathcal{G}$ is reductive, then the canonical integral
models
$\\{\mathscr{S}_{\mathsf{K}}^{{\mathrm{Kis}}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ from _[22]_ satisfy the conditions of the Pappas–Rapoport conjecture.

###  4.2. Statement of the main result and applications

We now state our main result and give some applications to the study of
Kisin–Pappas–Zhou models.

######  Theorem 4.10.

Let $(\mathbf{G},\mathbf{X},\mathcal{G})$ be an acceptable parahoric Shimura
data of abelian type, and let $\mathfrak{d}$ be a well-adapted very good
parahoric Shimura datum of Hodge type. Then, the system
$\\{\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ satisfies the conditions of the Pappas–Rapoport conjecture.

As an immediate implication, and as an opportunity to reemphasize the mildness
of the acceptability condition, we record the following corollary.

######  Corollary 4.11.

Suppose that $p>3$. Then, the Pappas–Rapoport conjecture holds for all
parahoric Shimura data of abelian type.

In addition, utilizing Proposition 4.4, and the previously known cases of the
Pappas–Rapoport conjecture stated above, we can deduce the following omnibus
independence result where, again, we are using notation from §3.3.

######  Corollary 4.12.

Suppose that $(\mathbf{G},\mathbf{X},\mathcal{G})$ is an acceptable parahoric
Shimura datum of abelian type. Then, the following statements are true.

    1. $($1$)$

The system $\\{\mathscr{S}_{\mathsf{K}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ is canonically independent of the choice of $\mathfrak{d}$.

    2. $($2$)$

If $(\mathbf{G},\mathbf{X},\mathcal{G})$ is of Hodge type, then the system
$\\{\mathscr{S}^{{\varnothing}}_{\mathsf{K}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ is canonically isomorphic to the system
$\\{\mathscr{S}_{\mathsf{K}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ for any $\mathfrak{d}$.

    3. $($3$)$

If $\mathcal{G}$ is reductive then the system
$\\{\mathscr{S}_{\mathsf{K}}^{{\mathrm{Kis}}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ is canonically isomorphic to the system
$\\{\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}}(\mathbf{G},\mathbf{X})\\}_{{\sf
K}^{p}}$ for any $\mathfrak{d}$.

Given this independence result, the notation
$\\{\mathscr{S}_{\mathsf{K}}(\mathbf{G},\mathbf{X})\\}_{{\sf K}^{p}}$ and
$\mathscr{S}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})$ for any of the systems
of models discussed in §3.3 is unambiguous.

######  Remark 4.13.

One implication of Corollary 4.12 is the fact that the more naively
constructed models
$\mathscr{S}_{\mathsf{K}^{p}}^{{\varnothing}}(\mathbf{G},\mathbf{X})$ possess
a local model diagram (e.g., see (3) of [29, Theorem 5.2.12]) even if [29,
Condition (5.1.11.1)] does not hold. Indeed, this follows by combining [29,
Theorem 5.2.12] and (2) of Corollary 4.12.

Lastly, we observe that Theorem 4.10 implies functoriality for the systems
$\\{\mathscr{S}_{\mathsf{K}}(\mathbf{G},\mathbf{X})\\}_{{\sf K}^{p}}$.

######  Theorem 4.14.

Suppose that
$\alpha\colon(\mathbf{G}_{1},\mathbf{X}_{1},\mathcal{G}_{1})\to(\mathbf{G},\mathbf{X},\mathcal{G})$
is a morphism of acceptable parahoric Shimura data of abelian type. Then,
there exists a unique morphism

$\alpha\colon\mathscr{S}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})\to\mathscr{S}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E_{1}}},$

which is equivariant for the map
$\alpha\colon\mathbf{G}_{1}(\mathbb{A}_{f}^{p})\to\mathbf{G}(\mathbb{A}_{f}^{p})$,
and whose generic fiber recovers the map

$\alpha\colon\mathrm{Sh}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{E_{1}}\to\mathrm{Sh}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{E_{1}}.$

###### Proof.

This follows from the arguments of [42, Corollary 4.3.2], and [10, Corollary
4.0.9]. We replicate the proof here for completeness.

We first observe, as in [42, Proof of Corollary 4.3.2], that the product

$\mathscr{S}^{\prime\prime}\vcentcolon=\mathscr{S}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})\times_{\textup{Spec}\,\mathcal{O}_{E_{1}}}\mathscr{S}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E_{1}}}$

determines an integral model of $\mathrm{Sh}_{{\sf K}_{p,1}\times{\sf
K}_{p}}(\mathbf{G}_{1}\times\mathbf{G},\mathbf{X}_{1}\times\mathbf{X})$ which
satisfies Conjecture 4.5. Then by [42, Theorem 4.3.1], the graph of $\alpha$
determines a unique morphism of integral models $\mathscr{S}_{{\sf
K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})\to\mathscr{S}^{\prime\prime}$ which
extends the graph of $\alpha$ on the generic fiber. We obtain the desired
morphism by composition with the projection
$\mathscr{S}^{\prime\prime}\to\mathscr{S}_{{\sf
K}_{p}}(\mathbf{G},\mathbf{X})_{E_{1}}$. ∎

######  Remark 4.15.

We remark that Theorem 4.14 is also proved in [50], by establishing a modified
version of Conjecture 4.5 using $\mathcal{G}^{\textup{ad}}$ in place of
$\mathcal{G}^{c}$; see Remark 1.1.

##  5\. The proof of the main result

This final section is devoted to the proof of our main theorem (Theorem 4.10).
This will require the development of multiple ancillary concepts and results
that we hope will be useful in other contexts. We fix notation as at the
beginning of §4.

###  5.1. An auxiliary Kisin–Pappas–Zhou model

A useful heuristic for Shimura data of abelian type is that they are ‘spanned’
by two (largely orthogonal) extremes: Shimura data of Hodge type and Shimura
data of toral type. Finding a precise formalism to realize this heuristic,
especially when their integral models are part of the picture, is somewhat
subtle. That said, this was achieved at hyperspecial level by Lovering in
[31]. The goal of this subsection is to extend this formalism to the
Kisin–Pappas–Zhou models of §3.

####  5.1.1. The Lovering construction

We begin by recalling the underlying Shimura datum as defined in [31, §4.6].
Let us fix an extension $\mathbf{E}^{\prime}$ of $\mathbf{E}$. Define then the
group

$\mathbf{B}_{\mathbf{E}^{\prime}}\vcentcolon=\mathbf{G}\times_{\delta,\mathbf{G}^{\textup{ab}},r}\mathbf{T}_{\mathbf{E}^{\prime}},$

where
$\mathbf{T}_{\mathbf{E}^{\prime}}\vcentcolon={\mathrm{Res}}_{\mathbf{E}^{\prime}/\mathbb{Q}}\,\,\mathbb{G}_{m,\mathbf{E}^{\prime}}$,
$\delta$ is the canonical map $\mathbf{G}\to\mathbf{G}^{\textup{ab}}$, and $r$
is obtained as the composition

${\mathrm{Res}}_{\mathbf{E}^{\prime}/\mathbb{Q}}\,\,\mathbb{G}_{m,\mathbf{E}^{\prime}}\xrightarrow{{\mathrm{Res}}_{\mathbf{E}^{\prime}/\mathbb{Q}}\,\,\mu_{h}}{\mathrm{Res}}_{\mathbf{E}^{\prime}/\mathbb{Q}}\,\mathbf{G}_{\mathbf{E}^{\prime}}^{\textup{ab}}\xrightarrow{N}\mathbf{G}^{\textup{ab}}.$

Here
$\mu_{h}\colon\mathbb{G}_{m,\mathbf{E}^{\prime}}\to\mathbf{G}^{\textup{ab}}_{\mathbf{E}^{\prime}}$
denotes the composition $\delta\circ\mu$ for any element $\mu$ of
$\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{h}$, which is defined over $\mathbf{E}^{\prime}$ by the
definition of reflex field and independent of choice since
$\mathbf{G}^{\textup{ab}}_{\mathbb{C}}$ is abelian, and $N$ denotes the
natural norm map. Observe that
$\mathbf{B}_{\mathbf{E}^{\prime}}^{\textup{der}}=\mathbf{G}^{\textup{der}}$,
so we have a short exact sequence

$1\to\mathbf{G}^{\textup{der}}\to\mathbf{B}_{\mathbf{E}^{\prime}}\to\mathbf{T}_{\mathbf{E}^{\prime}}\to
1.$

Furthermore, we have an exact sequence

$1\to\mathbf{B}_{\mathbf{E}^{\prime}}\to\mathbf{G}\times\mathbf{T}_{\mathbf{E}^{\prime}}\xrightarrow{(x,y)\mapsto\delta^{-1}(x)r(y)}\mathbf{G}^{\textup{ab}}\to
1,$

and thus $\mathbf{B}_{\mathbf{E}^{\prime}}$ is reductive (see [37, Corollary
21.53]).

Let $\\{\tau\\}$ be the set of archimedean places of $\mathbf{E}^{\prime}$,
and let $\\{[\tau]\\}$ denote the set of equivalences classes under the
relation $\tau\sim\overline{\tau}$. Let
$\tau_{0}\colon\mathbf{E}^{\prime}\to\mathbb{C}$ be the natural inclusion
which defines an element $[\tau_{0}]$ of $\\{[\tau]\\}$. Then, we have that
$\mathbf{E}^{\prime}\otimes{Q}\mathbb{R}$ is isomorphic to
$\prod_{[\tau]}\mathbf{E}^{\prime}_{[\tau]}$, where
$\mathbf{E}^{\prime}_{[\tau]}$ is $\mathbb{R}$ or $\mathbb{C}$ if $\tau$ is
real or complex, respectively. We define

$h_{\mathbf{T}}\colon\mathbb{S}\to(\mathbf{T}_{\mathbf{E}^{\prime}}){R}=\prod_{[\tau]}{\mathrm{Res}}_{\mathbf{E}^{\prime}_{\tau}/\mathbb{R}}\,\,\mathbb{G}_{m,\mathbf{E}^{\prime}_{[\tau]}},$

to be the unique morphism having trivial projection
$h_{\mathbf{T}_{\mathbf{E}^{\prime}},[\tau]}$ to every $[\tau]$ not equal to
$[\tau_{0}]$, and satisfying

$h_{\mathbf{T}_{\mathbf{E}^{\prime}},[\tau_{0}]}(z)=\begin{cases}z\overline{z}&\mbox{if}\quad\tau_{0}\text{
is real},\\\ z&\mbox{if}\quad\tau_{0}\text{ is complex},\end{cases}$

for $z$ in $\mathbb{S}(\mathbb{R})=\mathbb{C}^{\times}$.

One may show (see [31, Proposition 4.6.5]) that for any element $h$ of
$\mathbf{X}$, the pair

$(h,h_{\mathbf{T}_{\mathbf{E}^{\prime}}})\colon\mathbb{S}\to\mathbf{G}_{\mathbb{R}}\times(\mathbf{T}_{\mathbf{E}^{\prime}})_{\mathbb{R}},$
(5.1)

factorizes through $(\mathbf{B}_{\mathbf{E}^{\prime}})_{\mathbb{R}}$, that the
resulting $\mathbf{B}_{\mathbf{E}^{\prime}}(\mathbb{R})$-conjugacy class
$\mathbf{Y}_{\mathbf{E}^{\prime}}$ is independent of the choice of $h$, and
that the pair
$(\mathbf{B}_{\mathbf{E}^{\prime}},\mathbf{Y}_{\mathbf{E}^{\prime}})$ is a
Shimura datum with reflex field $\mathbf{E}^{\prime}$. Moreover, there is a
commutative diagram of Shimura data:

${{(\mathbf{B}_{\mathbf{E}^{\prime}},\mathbf{Y}_{\mathbf{E}^{\prime}})}}$${{(\mathbf{T}_{\mathbf{E}^{\prime}},\\{h_{\mathbf{T}}\\})}}$${{(\mathbf{G},\mathbf{X})}}$${{(\mathbf{G}^{\textup{ab}},\mathbf{X}^{\textup{ab}}),}}$
(5.2)

where $\mathbf{B}_{\mathbf{E}^{\prime}}\to\mathbf{G}$,
$\mathbf{B}_{\mathbf{E}^{\prime}}\to\mathbf{T}_{\mathbf{E}^{\prime}}$, and
$\mathbf{G}\to\mathbf{G}^{\textup{ab}}$ are the natural maps, and the map
$\mathbf{T}_{\mathbf{E}^{\prime}}\to\mathbf{G}^{\textup{ab}}$ is the norm map.
We call $(\mathbf{B}_{\mathbf{E}^{\prime}},\mathbf{Y}_{\mathbf{E}^{\prime}})$
the _Lovering construction_ applied to $(\mathbf{G},\mathbf{X})$. When
$\mathbf{E}^{\prime}=\mathbf{E}$, we simplify $\mathbf{T}_{\mathbf{E}}$,
$\mathbf{B}_{\mathbf{E}}$, and $\mathbf{Y}_{\mathbf{E}}$ to $\mathbf{T}$,
$\mathbf{B}$, and $\mathbf{Y}$, respectively.

We now record the following pleasant functoriality property of the Lovering
construction.

######  Proposition 5.1 ([31, Lemma 4.6.6]).

Let $(\mathbf{G}_{1},\mathbf{X}_{1})$ be another Shimura datum and let
$\mathbf{E}^{\prime}$ be a field containing the compositum
$\mathbf{E}_{1}\mathbf{E}$. Suppose that
$f\colon\mathbf{G}_{1}^{\textup{der}}\to\mathbf{G}^{\textup{der}}$ is an
isogeny which induces an isomorphism
$f^{\mathrm{ad}}\colon(\mathbf{G}_{1}^{\textup{ad}},\mathbf{X}_{1}^{\textup{ad}})\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}(\mathbf{G}^{\textup{ad}},\mathbf{X}^{\textup{ad}})$.
Then there exists a unique map
$g\colon\mathbf{B}_{1,\mathbf{E}^{\prime}}\to\mathbf{B}$ filling in the
following diagram

${1}$${{\mathbf{G}_{1}^{\textup{der}}}}$${{\mathbf{B}_{1,\mathbf{E}^{\prime}}}}$${{\mathbf{T}_{1,\mathbf{E}^{\prime}}}}$${1}$${1}$${{\mathbf{G}^{\textup{der}}}}$${{\mathbf{B}}}$${{\mathbf{T}}}$${1,}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{N}$

where $N$ is the natural norm map. Moreover, $g$ induces a morphism of Shimura
data
$(\mathbf{B}_{1,\mathbf{E}^{\prime}},\mathbf{Y}_{1,\mathbf{E}^{\prime}})\to(\mathbf{B},\mathbf{Y})$.

####  5.1.2. Relationship to Kisin–Pappas–Zhou models

We now explain how the Lovering construction fits into the theory of
Kisin–Pappas–Zhou models.

To begin, let us fix a very good parahoric Shimura of Hodge type
$\mathfrak{d}=(\mathbf{G}_{1},\mathbf{X}_{1},\mathcal{G}_{1})$ well-adapted to
$(\mathbf{G},\mathbf{X},\mathcal{G})$, and let us explicitly denote by $f$ the
implicitly defined isogeny
$\mathbf{G}_{1}^{\textup{der}}\to\mathbf{G}^{\textup{der}}$. We further set
$\mathbf{E}^{\prime}=\mathbf{E}\mathbf{E}_{1}$.

Consider the Lovering constructions $(\mathbf{B},\mathbf{Y})$ and
$(\mathbf{B}_{1,\mathbf{E}^{\prime}},\mathbf{Y}_{1,\mathbf{E}^{\prime}})$. For
notational simplicitly we shorten
$(\mathbf{B}_{1,\mathbf{E}^{\prime}},\mathbf{Y}_{1,\mathbf{E}^{\prime}})$ to
$(\mathbf{B}_{1}^{\prime},\mathbf{Y}_{1}^{\prime})$, and similarly for other
attendant notation. Observe that both of these Shimura data are of abelian
type and, in fact, $(\mathbf{G}_{1},\mathbf{X}_{1})$ is an associated very
good parahoric Shimura datum of Hodge type adapted to both. Indeed, this
follows since

${\mathbf{B}^{\prime}_{1}}^{\textup{der}}=\mathbf{G}_{1}^{\textup{der}}\ \
\text{ and }\ \ \mathbf{B}^{\textup{der}}=\mathbf{G}^{\textup{der}},$ (5.3)

and so we may consider the isogenies

$\mathbf{G}_{1}^{\textup{der}}\xrightarrow{\mathrm{id}}{\mathbf{B}^{\prime}_{1}}^{\textup{der}}\
\ \text{ and }\ \
\mathbf{G}_{1}^{\textup{der}}\xrightarrow{f}\mathbf{B}^{\textup{der}},$

which produce the desired isomorphisms on adjoint Shimura data (cf.
Proposition 2.4).

We would like to upgrade this setup to the level of parahoric Shimura data. To
this end, observe that by (5.3) and Lemma 2.5 one deduces that there are
natural identifications

$\mathscr{B}(B,\mathbb{Q}_{p})\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathscr{B}(G\times
T,\mathbb{\mathbb{Q}}_{p}),\ \ \text{ and }\ \
\mathscr{B}(B_{1}^{\prime},\mathbb{Q}_{p})\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathscr{B}(G_{1}\times
T_{1}^{\prime},\mathbb{\mathbb{Q}}_{p}).$

Thus, we may associate parahoric models $\mathcal{B}$ and
$\mathcal{B}_{1}^{\prime}$ to our choice of parahoric models $\mathcal{G}$ and
$\mathcal{G}_{1}$, parahoric models, respectively. In fact, it follows from
Proposition 2.15

$\mathcal{B}^{\prime}=\mathcal{G}\times_{\mathcal{G}^{\textup{ab}}}\mathcal{T},\
\ \text{ and }\ \
\mathcal{B}_{1}^{\prime}=\mathcal{G}_{1}\times_{\mathcal{G}_{1}^{\textup{ab}}}\mathcal{T}_{1}^{\prime},$

where $\mathcal{T}$ and $\mathcal{T}_{1}^{\prime}$ are the unique parahoric
models of $T$ and $T_{1}^{\prime}$, respectively.

Now, applying (5.2) and Proposition 5.1 one produces a diagram of Shimura data

${{(\mathbf{T}_{1}^{\prime},\\{h_{\mathbf{T}_{1}}\\})}}$${{(\mathbf{G}_{1}^{\textup{ab}},\mathbf{X}_{1}^{\textup{ab}})}}$${{(\mathbf{B}_{1}^{\prime},\mathbf{Y}^{\prime}_{1})}}$${{(\mathbf{G}_{1},\mathbf{X}_{1})}}$${{(\mathbf{B},\mathbf{Y})}}$${{(\mathbf{G},\mathbf{X}).}}$

In particular, we obtain a diagram of $\mathbf{E}^{\prime}$-schemes

${{\mathrm{Sh}_{\mathsf{M}^{\prime}_{p,1}}(\mathbf{T}_{1}^{\prime},\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathbf{E}^{\prime}}}}$${{\mathrm{Sh}_{\mathsf{N}_{p,1}}(\mathbf{G}_{1}^{\mathrm{ab}},\mathbf{X}_{1}^{\mathrm{ab}})_{\mathbf{E}^{\prime}}}}$${{\mathrm{Sh}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}^{\prime}_{1})_{\mathbf{E}^{\prime}}}}$${{\mathrm{Sh}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathbf{E}^{\prime}}}}$${{\mathrm{Sh}_{\mathsf{L}_{p}}(\mathbf{B},\mathbf{Y})_{\mathbf{E}^{\prime}}}}$${{\mathrm{Sh}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{\mathbf{E}^{\prime}}}}$
(5.4)

each map equivariant for the natural map of groups. Here, we write
$\mathsf{L}_{p}=\mathcal{B}(\mathbb{Z}_{p})$ and
$\mathsf{L}^{\prime}_{p,1}=\mathcal{B}_{1}^{\prime}(\mathbb{Z}_{p})$,
$\mathsf{M}^{\prime}_{p,1}=\mathcal{T}_{1}^{\prime}(\mathbb{Z}_{p})$, and
$\mathsf{N}_{p,1}=\mathcal{G}_{1}^{\textup{ab}}(\mathbb{Z}_{p})$. We then have
the following relationship between the Lovering construction and
Kisin–Pappas–Zhou models.

######  Proposition 5.2.

There exists a unique diagram

${{\mathscr{S}_{\mathsf{M}^{\prime}_{p,1}}(\mathbf{T}_{1}^{\prime},\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}}}}$${{\mathscr{S}_{\mathsf{N}_{p,1}}(\mathbf{G}_{1}^{\mathrm{ab}},\mathbf{X}_{1}^{\mathrm{ab}})_{\mathcal{O}_{E}}}}$${{\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}^{\prime}_{1},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}}$${{\mathscr{S}^{{\varnothing}}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathcal{O}_{E}}}}$${{\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}_{p}}(\mathbf{B},\mathbf{Y})}}$${{\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})}}$

whose maps are equivariant for the appropriate maps of groups, and which
models (5.4) pulled back to the completion of $\mathbf{E}^{\prime}$ at a prime
above $v$.

###### Proof.

The existence of an equivariant map

${\mathscr{S}_{\mathsf{M}^{\prime}_{p,1}}(\mathbf{T}^{\prime}_{1},\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}}}\to{\mathscr{S}_{\mathsf{N}_{p,1}}(\mathbf{G}_{1}^{\mathrm{ab}},\mathbf{X}_{1}^{\mathrm{ab}})_{\mathcal{O}_{E}}}$

modeling the map on the generic fiber follows from Proposition 3.22, while the
existence of the equivariant map

${\mathscr{S}^{{\varnothing}}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathcal{O}_{E}}}\to{\mathscr{S}_{\mathsf{N}_{p,1}}(\mathbf{G}_{1}^{\mathrm{ab}},\mathbf{X}_{1}^{\mathrm{ab}})_{\mathcal{O}_{E}}}$

modeling the map on the generic fiber follows from Proposition 3.25. To
construct the equivariant maps

${\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}^{\prime}_{1},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}\to{\mathscr{S}^{{\varnothing}}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathcal{O}_{E}}},\
\text{ and }\
{\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}^{\prime}_{1},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}\to{\mathscr{S}_{\mathsf{M}^{\prime}_{p,1}}(\mathbf{T}^{\prime}_{1},\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}}},$

modeling the maps on the generic fiber, it suffices by Proposition 3.28 and
Proposition 3.27 to construct a map

${\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}^{\prime}_{1},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}\to\mathscr{S}_{\mathsf{K}_{p,1}\times\mathsf{M}^{\prime}_{p,1}}(\mathbf{G}_{1}\times\mathbf{T}^{\prime}_{1},\mathbf{X}_{1}\times\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}}.$

Because the map
$\mathbf{B}^{\prime}_{1}\to\mathbf{G}_{1}\times\mathbf{T}^{\prime}_{1}$ is an
isomorphism on derived subgroups, we see from Proposition 2.4 that it is an
ad-isomorphism, and so the claim follows from Proposition 3.24. The fact that
this constructed square commutes may be checked on the generic fiber, where it
is true by construction.

The existence of an equivariant map

${\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}^{\prime}_{1},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}\to{\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}_{p}}(\mathbf{B},\mathbf{Y})}$

follows from Proposition 3.24 as the induced map
${\mathbf{B}^{\prime}_{1}}^{\textup{der}}\to\mathbf{B}^{\textup{der}}$ is
identified with the isogeny
$f\colon\mathbf{G}_{1}^{\textup{der}}\to\mathbf{G}$ and so Proposition 2.4
applies. We similarly deduce the existence of a map

${\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}_{p}}(\mathbf{B},\mathbf{Y})}\to{\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})},$

using the fact that $\mathbf{B}^{\textup{der}}\to\mathbf{G}^{\textup{der}}$ is
an isomorphism, along with Proposition 3.24 and Proposition 2.4. ∎

###  5.2. Construction of the $\mathcal{G}^{c}$-shtuka

In this section we establish the following theorem.

######  Theorem 5.3.

There exists a unique $\mathcal{G}^{c}$-shtuka $\mathscr{P}_{\mathsf{K}_{p}}$
on $\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})$
bounded by
$\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{h}^{c}$, which is compatible with the
$\mathbf{G}(\mathbb{A}_{f}^{p})$-action and which models
$\mathscr{P}_{\mathsf{K}_{p},E}$.

By [42, Corollary 2.7.10], there can be at most one shtuka $\mathscr{P}_{{\sf
K}_{p}}$ modeling $\mathscr{P}_{{\sf K}_{p},E}$. Moreover, such a shtuka is
automatically bounded by
$\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{h}^{c}$ by [7, Lemma 2.1]. Thus, it suffices to construct a
$\mathcal{G}^{c}$-shtuka $\mathscr{P}_{\mathsf{K}_{p}}$ which is compatible
with the $\mathbf{G}(\mathbb{A}_{f}^{p})$-action and which models
$\mathscr{P}_{\mathsf{K}_{p},E}$. In fact, it suffices to construct a model
with $\mathbf{G}(\mathbb{A}_{f}^{p})$-action of
$\mathscr{P}_{\mathsf{K}_{p},E^{\prime}}$ for any discrete algebraic extension
$E^{\prime}$ of $E$. Indeed, by [42, Corollary 2.7.10] one may lift the
descent datum of $\mathscr{P}_{\mathsf{K}_{p},E^{\prime}}$ relative to
$\mathscr{P}_{\mathsf{K}_{p},E}$ to an integral descent datum, which is
effective by [49, Proposition 19.5.3].

The proof of the existence of such a model will occupy the rest of this
subsection, and will be carried out in several steps.

#### Step 1: Group Theory

We begin by establishing the following group-theoretic result which will be
useful in our construction.

######  Proposition 5.4.

There is a canonical identification

${\mathcal{B}_{1}^{\prime}}^{c}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathcal{G}_{1}\times_{\mathcal{G}_{1}^{\textup{ab}}}{\mathcal{T}^{\prime}_{1}}^{c}.$

###### Proof.

Let us begin by verifying that there is a canonical isomorphism

${\mathbf{B}^{\prime}_{1}}^{c}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathbf{G}_{1}\times_{\mathbf{G}_{1}^{\textup{ab}}}{\mathbf{T}^{\prime}_{1}}^{c}.$
(5.5)

Note here that we are using Lemma 4.1 to identify $\mathbf{G}_{1}$ and
$\mathbf{G}_{1}^{\mathrm{ab}}$ with $\mathbf{G}_{1}^{c}$ and
$(\mathbf{G}_{1}^{\mathrm{ab}})^{c}$, respectively.

Observe that since
$\mathbf{B}_{1}^{\prime}\to\mathbf{G}_{1}\times\mathbf{T}^{\prime}_{1}$ is an
ad-isomorphism, we may use Proposition 2.4 to deduce that

$Z(\mathbf{B}_{1}^{\prime})=Z(\mathbf{G}_{1})\times_{\mathbf{G}_{1}^{\textup{ab}}}\mathbf{T}_{1}^{\prime}.$

We then employ the following simple lemma.

######  Lemma 5.5.

Let

${{\mathbf{T}}}$${{\mathbf{T}_{1}}}$${{\mathbf{T}_{2}}}$${{\mathbf{T}_{3},}}$$\scriptstyle{\lrcorner}$

be a Cartesian diagram, where $\mathbf{T}_{i}$, for $i=1,2,3$, are
multiplicative groups over $\mathbb{Q}$, and such that
$\mathbf{T}_{1}^{\circ}\to\mathbf{T}_{3}^{\circ}$ is surjective. Then,

$\mathbf{T}_{\mathrm{ac}}=(\mathbf{T}_{1})_{\mathrm{ac}}\times_{(\mathbf{T}_{2})_{\mathrm{ac}}}(\mathbf{T}_{3})_{\mathrm{ac}}.$

###### Proof.

This follows easily from Proposition 2.14, and the observation that for a
containment $A\subseteq B$ of multiplicative groups over some field $F$, one
has that $A_{a}=(B_{a}\cap A)^{\circ}$ and $A_{s}=(B_{s}\cap A)^{\circ}$,
where the subscripts $a$ and $s$ denote the maximal anisotropic and split
subtori, respectively. Namely, one applies this with $A=\mathbf{T}$, and
$B=\mathbf{T}_{1}\times\mathbf{T}_{2}$. ∎

Using this, and the fact that fiber products of groups commute with quotients,
we easily deduce the existence of an isomorphism as in (5.5). Note that
$\mathcal{B}_{1}^{c}$ is a parahoric group by definition. The same is true of
$\mathcal{G}_{1}\times_{\mathcal{G}_{1}^{\textup{ab}}}{\mathcal{T}^{\prime}_{1}}^{c}$.
Indeed, this follows from Proposition 2.15, since $G_{1}^{\textup{der}}$ is
$R$-smooth (recall that we assume
$(\mathbf{G}_{1},\mathbf{X}_{1},\mathcal{G}_{1})$ is very good) and thus
Proposition 2.11 applies, implying that
$\ker(\mathcal{G}_{1}\to\mathcal{G}^{\textup{ab}})$ is
$\mathcal{G}^{\textup{der}}$ which is is smooth with connected fibers. To
conclude, it now suffices to show that $\mathcal{B}_{1}^{c}$ and
$\mathcal{G}_{1}\times_{\mathcal{G}_{1}^{\textup{ab}}}{\mathcal{T}^{\prime}_{1}}^{c}$
correspond to matching points under the bijection

$\mathscr{B}({B^{\prime}_{1}}^{c},\mathbb{Q}_{p})\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathscr{B}({G}_{1}\times_{{G}_{1}^{\mathrm{ab}}}{T_{1}^{\prime}}^{c},\mathbb{Q}_{p}),$

coming from (5.5).

We have the following commutative diagram of equivariant bijections (utilizing
a combination of Lemma 2.5 and Proposition 2.15 several times)

${{\mathscr{B}({G}_{1}\times_{{G}_{1}^{\mathrm{ab}}}{T_{1}^{\prime}}^{c})}}$${{\mathscr{B}({G}_{1}\times{T_{1}^{\prime}}^{c},\mathbb{Q}_{p})}}$${{\mathscr{B}({G}_{1}\times{T}_{1}^{\prime},\mathbb{Q}_{p})}}$${{\mathscr{B}({B_{1}^{\prime}}^{c},\mathbb{Q}_{p})}}$${{\mathscr{B}({B}^{\prime}_{1},\mathbb{Q}_{p}).}}$

Tracing through the definitions, we see that the claim follows as the points
we consider ultimately can be induced from the same point $(x,\ast)$ in the
top-right corner, where $\mathcal{G}_{1}=\mathcal{G}_{x}^{\circ}$. Here we are
implicitly using that the formation of the (reduced) Bruhat–Tits building
commutes with products. ∎

#### Step 2: Pullback and Descent to the Fiber Product

Let us now recall the commutative diagram

${{\mathscr{S}_{\mathsf{M}^{\prime}_{p,1}}(\mathbf{T}_{1}^{\prime},\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}}}}$${{\mathscr{S}_{\mathsf{N}_{p,1}}(\mathbf{G}_{1}^{\mathrm{ab}},\mathbf{X}_{1}^{\mathrm{ab}})_{\mathcal{O}_{E}}}}$${{\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}^{\prime}_{1},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}}$${{\mathscr{S}^{{\varnothing}}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathcal{O}_{E}}}}$${{\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}_{p}}(\mathbf{B},\mathbf{Y})}}$${{\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})}}$$\scriptstyle{a}$$\scriptstyle{e}$$\scriptstyle{b}$$\scriptstyle{d}$$\scriptstyle{c}$

from Proposition 5.2, with the arrows labeled for convenience, and where the
arrow $e$ is defined to make the lower triangle commute.

Utilizing Theorem 4.6 and Theorem 4.8 there exist a
$\mathbf{G}_{1}(\mathbb{A}_{f}^{p})$-equivariant $\mathcal{G}_{1}$-shtuka
$\mathscr{P}_{\mathsf{K}_{p,1}}$ and a
$\mathbf{T}^{\prime}_{1}(\mathbb{A}_{f}^{p})$-equivariant
${\mathcal{T}^{\prime}_{1}}^{c}$-shtuka
$\mathscr{P}_{\mathsf{M}^{\prime}_{p,1}}$ on
${\mathscr{S}^{{\varnothing}}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathcal{O}_{E}}}$
and
${\mathscr{S}_{\mathsf{M}^{\prime}_{p,1}}(\mathbf{T}^{\prime}_{1},\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}}}$,
respectively. We then obtain a $\mathcal{G}_{1}$-shtuka and a
${\mathcal{T}^{\prime}_{1}}^{c}$-shtuka on
${\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}_{p,1}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}$,
given by $a^{\ast}\mathscr{P}_{\mathsf{K}_{p,1}}$ and
$d^{\ast}\mathscr{P}_{\mathsf{M}^{\prime}_{p,1}}$, respectively.

We claim that $a^{\ast}\mathscr{P}_{\mathsf{K}_{p,1}}$ and
$d^{\ast}\mathscr{P}_{\mathsf{M}^{\prime}_{p,1}}$ are both equivariant
relative to $\mathbf{B}^{\prime}_{1}(\mathbb{A}_{f}^{p})$. Indeed, this
follows from the equivariance of these maps, as for an element $g$ of
$\mathbf{B}_{1}^{\prime}(\mathbb{A}_{f}^{p})$ we have natural isomorphisms

$[g]^{\ast}a^{\ast}\mathscr{P}_{\mathsf{K}_{p,1}}\simeq
a^{\ast}([a(g)]^{\ast}\mathscr{P}_{\mathsf{K}_{p,1}})\simeq
a^{\ast}\mathscr{P}_{\mathsf{K}_{p,1}},$

using the $\mathbf{G}_{1}(\mathbb{A}_{f}^{p})$-equivariance of
$\mathscr{P}_{\mathsf{K}_{p,1}}$, and similarly in the case of
$d^{\ast}\mathscr{P}_{\mathsf{M}^{\prime}_{p,1}}$.

Our plan is now to use Corollary 2.17 to glue the shtukas
$a^{\ast}\mathscr{P}_{{\sf K}_{p,1}}$ and
$d^{\ast}\mathscr{P}_{\mathsf{M}^{\prime}_{p,1}}$ in order to obtain a
$\mathcal{B}_{1}^{\prime c}$-shtuka on
$\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}_{1}^{\prime})_{\mathcal{O}_{E}}$
modeling the $\mathcal{B}_{1}^{\prime c}$-shtuka
$\mathscr{P}_{\mathsf{L}^{\prime}_{p,1},E}$ living over the generic fiber
$\mathrm{Sh}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}_{1}^{\prime})_{E^{\prime}}$,
see Proposition 4.2.

######  Lemma 5.6.

Let $\delta$ be the natural map
$\mathcal{G}_{1}\to\mathcal{G}_{1}^{{\textup{ab}}}$, and let
$N\colon{\mathcal{T}^{\prime}_{1}}^{c}\to\mathcal{G}_{1}^{\textup{ab}}$ be the
$($map induced by$)$ the norm. Then there is a
$\mathbf{B}^{\prime}_{1}(\mathbb{A}_{f}^{p})$-equivariant isomorphism of
$\mathcal{G}_{1}^{{\textup{ab}}}$-torsors

$\theta\colon\delta_{\ast}a^{\ast}\mathscr{P}_{\mathsf{K}_{p,1}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}N_{\ast}d^{\ast}\mathscr{P}_{\mathsf{M}_{p,1}}.$

###### Proof.

To construct such an isomorphism, it suffices construct compatible
isomorphisms over
$\mathscr{S}_{{\mathsf{L}^{\prime}_{1}}^{p}}^{\mathfrak{d}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}$,
for each neat compact open subgroup
${\mathsf{L}^{\prime}_{1}}^{p}\subseteq\mathbf{B}^{\prime}_{1}(\mathbb{A}_{f}^{p})$.
Furthermore, by [42, Corollary 2.7.10], it suffices to construct such
compatible isomorphisms on
$\mathrm{Sh}_{\mathsf{L}_{p,1}^{\prime}{\mathsf{L}^{\prime}_{1}}^{p}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}^{\prime}_{1})_{E}$.

The shtukas $\mathscr{P}_{\mathsf{K}_{p,1},E}$ and
$\mathscr{P}_{\mathsf{M}_{p,1}}$ are functorially constructed from the
corresponding étale realization functors $\mathbb{P}_{\mathsf{K}_{p,1},E}$ and
$\mathbb{P}_{\mathsf{M}_{p,1}}$, respectively, in a way compatible with
pullbacks along morphisms of Shimura varieties and pushforwards along
morphisms of groups. Standard compatabilities of étale realizations (c.f.,
[21, Equation (4.3.2)]) imply that there are natural isomorphisms

$\delta_{\ast}a^{\ast}\mathbb{P}_{\mathsf{K}_{p,1}}\simeq(b\circ
a)^{\ast}\mathbb{P}_{\mathsf{N}_{p,1}}\simeq(c\circ
d)^{\ast}\mathbb{P}_{\mathsf{N}_{p,1}}\simeq
N_{\ast}d^{\ast}\mathbb{P}_{\mathsf{M}_{p,1}},$

from where the claim follows. From this we deduce the existence of canonical
isomormorphisms

$\delta_{\ast}a^{\ast}\mathscr{P}_{\mathsf{K}_{p,1}}\simeq(b\circ
a)^{\ast}\mathscr{P}_{\mathsf{N}_{p,1}}\simeq(c\circ
d)^{\ast}\mathscr{P}_{\mathsf{N}_{p,1}}\simeq
N_{\ast}d^{\ast}\mathscr{P}_{\mathsf{M}_{p,1}},$

where $\mathscr{P}_{\mathsf{N}_{p,1}}$ is the unique model of
$\mathscr{P}_{\mathsf{N}_{p,1},E}$ from Theorem 4.8. The resulting composition
$\theta:\delta_{\ast}a^{\ast}\mathscr{P}_{{\sf
K}_{p,1}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}N_{\ast}d^{\ast}\mathscr{P}_{\mathsf{M}_{p,1}}$
is the desired isomorphism. ∎

######  Proposition 5.7.

There exists a $\mathcal{B}_{1}^{\prime c}$-shtuka
$\mathscr{P}_{\mathsf{L}^{\prime}_{p,1}}$ on
$\mathscr{S}_{\mathsf{L}^{\prime}_{p,1}}^{\mathfrak{d}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}_{1}^{\prime})_{\mathcal{O}_{E}}$
which models $\mathscr{P}_{\mathsf{L}^{\prime}_{p,1},E}$.

###### Proof.

Using Lemma 5.6, we may apply Corollary 2.17 and Proposition 5.4 to deduce the
construction of a $\mathcal{B}_{1}^{\prime c}$-shtuka
$\mathscr{P}_{\mathsf{L}^{\prime}_{p,1}}$ on
${\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}^{\prime}_{1},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}$.
By construction this shtuka has an equivariant action of
$\mathbf{B}_{1}^{\prime}(\mathbb{A}_{f}^{p})$. As in the last paragraph, to
show $\mathscr{P}_{\mathsf{L}^{\prime}_{p,1}}$ models the
${\mathcal{B}_{1}^{\prime}}^{c}$-shtuka
$\mathscr{P}_{\mathsf{L}^{\prime}_{p,1},E}$, it suffices to show that there is
an isomorphism

$\mathbb{P}_{\mathsf{K}_{p,1}}\times_{\theta}\mathbb{P}_{\mathsf{M}_{p,1}}\simeq\mathbb{P}_{\mathsf{L}^{\prime}_{p,1}}.$

Here we are applying Proposition 2.16, which makes sense when one views these
local systems as torsors as in [21, §2.1.1]. But, in view of Proposition 5.4
this is an elementary calculation (cf. [32, §3.4.4]). ∎

#### Step 3: Descending to
$\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}}$

We now show that there is a $\mathbf{G}(\mathbb{A}_{f}^{p})$-equivariant
$\mathcal{G}^{c}$-shtuka on
$\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{E^{\mathrm{ur}}}$
which models $\mathscr{P}_{\mathsf{K}_{p},E^{\mathrm{ur}}}$. As previously
noted, this is sufficient to prove Theorem 5.3. Let us begin by considering
the $\mathcal{G}^{c}$-shtuka

$\mathscr{P}^{\prime}\vcentcolon=\mathscr{P}_{\mathsf{L}^{\prime}_{p,1},\mathcal{O}_{E^{\mathrm{ur}}}}\times^{{\mathcal{B}_{1}^{\prime}}^{c}}\mathcal{G}^{c},$

on
$\mathscr{S}_{{\mathsf{L}^{\prime}_{1}}^{p}}^{\mathfrak{d}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E^{\mathrm{ur}}}}$.

Choose a connected component ${\mathscr{S}^{\prime}}^{+}$ of
$\mathscr{S}_{{\mathsf{L}^{\prime}_{1}}^{p}}^{\mathfrak{d}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E^{\mathrm{ur}}}}$,
and let $\mathscr{S}^{+}$ be the connected component of
$\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}}$
containing its image under the map

$\mathscr{S}_{{\mathsf{L}^{\prime}_{1}}^{p}}^{\mathfrak{d}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E^{\mathrm{ur}}}}\to\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}}.$
(5.6)

Set ${\mathscr{P}^{\prime}}^{+}$ to be the pullback of $\mathscr{P}^{+}$ to
${\mathscr{S}^{\prime}}^{+}$.

######  Lemma 5.8.

The $\mathcal{G}^{c}$-shtuka ${\mathscr{P}^{\prime}}^{+}$ descends uniquely to
a $\mathcal{G}^{c}$-shtuka $\mathscr{P}^{+}$ on $\mathscr{S}^{+}$, which
models the restriction of $\mathscr{P}_{\mathsf{K}_{p},E^{\mathrm{ur}}}$ to
$\mathscr{S}^{+}_{E^{\mathrm{ur}}}$

###### Proof.

As noted in the proof of Proposition 3.24, the map
${\mathscr{S}^{\prime}}^{+}\to\mathscr{S}^{+}$ is a quotient by the group

$\Delta=\ker(\mathscr{A}(\bm{\mathcal{G}}_{1})^{\circ}\to\mathscr{A}(\bm{\mathcal{G}}))/\ker(\mathscr{A}(\bm{\mathcal{G}}_{1})^{\circ}\to\mathscr{A}(\bm{\mathcal{B}}^{\prime}_{1})),$

which acts through a finite quotient asn in [22, §E.6]. Thus, by [49,
Proposition 19.5.3], it suffices to show that there is a $\Delta$-action on
${\mathscr{P}^{\prime}}^{+}$ which gives a descent datum. Arguing as in Step
2, it suffices to verify that there is a $\Delta$-descent datum on the
$\mathcal{G}^{c}(\mathbb{Z}_{p})$-local system ${\mathbb{P}^{\prime}}^{+}$ on
${\mathscr{S}^{\prime}}^{+}_{E^{\mathrm{ur}}}$, whose definition is, mutatis
mutandis, the same as that of ${\mathscr{P}^{\prime}}^{+}$. But, this is clear
as the $\mathcal{G}^{c}(\mathbb{Z}_{p})$-local system $\mathbb{P}^{\prime}$ on
$\mathrm{Sh}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}^{\prime}_{1},\mathbf{Y}^{\prime}_{1})_{E^{\mathrm{ur}}}$
is the pullback of a $\mathcal{G}^{c}(\mathbb{Z}_{p})$-local system on
$\mathrm{Sh}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{E^{\mathrm{ur}}}$ by
[21, Equation (4.3.2)].

Thus we obtain a $\mathcal{G}^{c}$-shtuka $\mathscr{P}^{+}$ on
$\mathscr{S}^{+}$. That this shtuka models the restriction of
$\mathscr{P}_{\mathsf{K}_{p},E^{\mathrm{ur}}}$ to
$\mathscr{S}^{+}_{E^{\mathrm{ur}}}$ is clear by design. Indeed,
$\mathscr{P}^{+}$ is isomorphic to the $\mathcal{G}^{c}$-shtuka obtained as
the descent of ${\mathscr{P}^{\prime}}^{+}$ with respect to the
$\Delta$-action which, as noted before, agrees with that of the restriction to
$\mathscr{S}^{+}_{E^{\mathrm{ur}}}$ of the pullback of
$\mathscr{P}_{\mathsf{K}_{p},E^{\mathrm{ur}}}$ along the map in (5.6). ∎

Using the $\mathscr{A}(\bm{\mathcal{G}})$-equivariant bijection

$\pi_{0}(\mathrm{Sh}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{{E^{\mathrm{ur}}}})\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\pi_{0}({\mathscr{S}}^{\mathfrak{d}}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E\mathrm{ur}}}),$
(5.7)

(see Lemma 3.20 and Lemma 3.21) and [26, Lemma 4.6.13], the group
$\mathscr{A}(\bm{\mathcal{G}})$ acts transitively on
$\pi_{0}(\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}})$.
Thus, for any component $\mathscr{T}^{+}$ of
$\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}}$
we may choose some $g$ in $\mathscr{A}(\bm{\mathcal{G}})$ such that

$[g]\colon\mathscr{S}^{+}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\mathscr{T}^{+}.$

We define

$\mathscr{P}_{\mathsf{K}_{p},\mathcal{O}_{E^{\mathrm{ur}}},\mathscr{T}^{+}}\vcentcolon=[g]^{\ast}\mathscr{P}^{+}.$

To see that this is canonically independent of the choice of $g$ we again note
by [42, Corollary 2.7.10] that it suffices to verify this on the generic
fiber. There, the statement follows from the fact that $\mathscr{P}^{+}$
models the restriction of $\mathscr{P}_{\mathsf{K}_{p},E^{\mathrm{ur}}}$ to
$\mathscr{S}^{+}_{E^{\mathrm{ur}}}$, and the
$\mathscr{A}(\bm{\mathcal{G}})$-equivariance of
$\mathscr{P}_{\mathsf{K}_{p},E^{\mathrm{ur}}}$. Moreover, by the same logic we
see that
$\mathscr{P}_{\mathsf{K}_{p},\mathcal{O}_{E^{\mathrm{ur}}},\mathscr{T}^{+}}$
models the restriction of $\mathscr{P}_{\mathsf{K}_{p,1},E^{\mathrm{ur}}}$ to
$\mathscr{T}^{+}_{E^{\mathrm{ur}}}$.

For each neat compact open subgroup
$\mathsf{K}^{p}\subseteq\mathbf{G}(\mathbb{A}_{f}^{p})$ and each connected
component $\mathscr{T}_{\mathsf{K}^{p}}^{+}$ of
$\mathscr{S}_{\mathsf{K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}}$
we may choose any connected component $\mathscr{T}^{+}$ of
$\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}}$
mapping to it. The fact here that the map

$\pi_{0}(\mathscr{S}_{\mathsf{K}_{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}})\to\pi_{0}(\mathscr{S}_{\mathsf{K}_{p}\mathsf{K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}})$

is surjective follows from Lemma 3.20.

######  Lemma 5.9.

The shtuka $\mathscr{P}_{{\sf
K}_{p},\mathcal{O}_{E^{\mathrm{ur}}},\mathscr{T}^{+}}$ descends along the map
$\mathscr{T}^{+}\to\mathscr{T}_{{\sf K}^{p}}^{+}$ to a shtuka
$\mathscr{P}_{{\sf K}^{p},\mathcal{O}_{E^{\mathrm{ur}}},\mathscr{T}^{+}_{{\sf
K}^{p}}}$ which models the restriction of $\mathscr{P}_{{\sf
K}^{p},E^{\mathrm{ur}}}$ to $\mathscr{T}^{+}_{{\sf K}^{p},E^{\mathrm{ur}}}$.

###### Proof.

The map $\mathscr{T}^{+}\to\mathscr{T}^{+}_{\mathsf{K}^{p}}$ is a profinite
étale cover, and thus by [49, Proposition 19.5.3] to show that
$\mathscr{P}_{\mathsf{K}_{p},\mathcal{O}_{E^{\mathrm{ur}}},\mathscr{T}^{+}}$
descends along this map, it suffices to produce descent data. Again by [42,
Corollary 2.7.10], this data can be obtained using the descent data for the
restriction of $\mathscr{P}_{\mathsf{K}_{p,1},E^{\mathrm{ur}}}$ to
$\mathscr{T}^{+}_{E^{\mathrm{ur}}}$ relative to the map
$\mathscr{T}^{+}_{E^{\mathrm{ur}}}\to\mathscr{T}^{+}_{\mathsf{K}^{p},E^{\mathrm{ur}}}$.

It is clear by the definition of
$\mathscr{P}_{\mathsf{K}_{p},\mathcal{O}_{E^{\mathrm{ur}}},\mathscr{T}^{+}}$,
[22, Lemma 2.2.5], and the equivariance of the map
$\mathscr{T}^{+}\to\mathscr{T}^{+}_{\mathsf{K}^{p}}$ that this descent is
independent of all choices, and hence we obtain
$\mathscr{P}_{\mathsf{K}^{p},\mathcal{O}_{E^{\mathrm{ur}}},\mathscr{T}^{+}_{\mathsf{K}^{p}}}$.
We conclude by observing that
$\mathscr{P}_{\mathsf{K}^{p},\mathcal{O}_{E^{\mathrm{ur}}},\mathscr{T}^{+}_{\mathsf{K}^{p}}}$
models the restriction of $\mathscr{P}_{\mathsf{K}^{p},E^{\mathrm{ur}}}$ to
$\mathscr{T}^{+}_{\mathsf{K}^{p},E^{\mathrm{ur}}}$ since, by definition, they
have the same descent data relative to the map
$\mathscr{T}^{+}_{E^{\mathrm{ur}}}\to\mathscr{T}^{+}_{\mathsf{K}^{p},E^{\mathrm{ur}}}$.
∎

Finally, we can construct the desired $\mathcal{G}^{c}$-shtuka on
$\mathscr{S}_{{\sf K}_{p}{\sf
K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}}$.

######  Construction 5.10.

We set

$\mathscr{P}_{\mathsf{K}^{p},\mathcal{O}_{E^{\mathrm{ur}}}}=\bigsqcup_{\mathscr{T}_{\mathsf{K}^{p}}^{+}\in\pi_{0}(\mathscr{S}_{{\sf
K}_{p}{\sf
K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}})}\mathscr{P}_{\mathsf{K}^{p},\mathcal{O}_{E^{\mathrm{ur}}},\mathscr{T}_{\mathsf{K}^{p}}^{+}}.$

More precisely, let $k_{E}$ denote the residue field of $E$. Then, as
$\pi_{0}(\mathscr{S}_{{\sf K}_{p}{\sf
K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}})$
is finite, the set of objects $S$ of $\mathbf{Perf}_{k_{E}}$ such that
$S\to(\mathscr{S}_{{\sf K}_{p}{\sf
K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}})^{\lozenge/}$
factorizes through some $(\mathscr{T}_{\mathsf{K}^{p}}^{+})^{\lozenge/}$ is a
basis of $\mathbf{Perf}_{k_{E}}$, and on such an object we define the
$\mathcal{G}^{c}$-shtuka over $S$ to be that determined by the (unique) map
$S\to(\mathscr{T}_{\mathsf{K}^{p}}^{+})^{\lozenge/}$.

###### Proof of Theorem 5.3.

It is clear from our definition of
$\mathscr{P}_{\mathsf{K}^{p},\mathcal{O}_{E^{\mathrm{ur}}}}$ that the
collection $\\{\mathscr{P}_{{\sf
K}^{p},\mathcal{O}_{E^{\mathrm{ur}}}}\\}_{{\sf K}^{p}}$ forms a compatible
family of $\mathcal{G}^{c}$-shtukas over
$\mathscr{S}_{\mathsf{K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E^{\mathrm{ur}}}}$.
Also, by design, $\mathscr{P}_{\mathsf{K}^{p},\mathcal{O}_{E^{u}r}}$ models
$\mathscr{P}_{\mathsf{K}^{p},E^{\mathrm{ur}}}$ since it does so on each
connected component. Theorem 5.3 follows. ∎

###  5.3. The completion of the proof

We are now prepared to prove Theorem 4.10. Given Theorem 3.12 and Theorem 5.3
it remains to verify the following proposition.

######  Proposition 5.11.

For any neat compact open subgroup
$\mathsf{K}^{p}\subseteq\mathbf{G}(\mathbb{A}_{f}^{p})$, and $x$ in
$\mathscr{S}_{\mathsf{K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})(\overline{k}_{E})$,
there exists an isomorphism of $v$-sheaves

$\Theta_{x}\colon\left(\mathcal{M}^{\mathrm{int}}_{\mathcal{G}^{c},b_{x},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}^{c}_{h}}}\right)^{\wedge}_{/x_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\left(\mathscr{S}_{\mathsf{K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})^{\wedge}_{/x}\right)^{\lozenge},$

such that $\Theta_{x}^{\ast}(\mathscr{P}_{\mathsf{K}^{p}})$ is isomorphic to
$\mathscr{P}^{\mathrm{univ}}$, with notation as in Definition 4.3.

The key to constructing such an isomorphism is the following result of
Pappas–Rapoport.

######  Proposition 5.12 ([42, Proposition 5.3.1]).

Suppose that
$f\colon(\mathcal{G}_{1},\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}_{1},b_{1})\to(\mathcal{G},b,\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{})$ is an ad-isomorphism of local Shimura data, and that $x_{1}$
is an element of
$\mathcal{M}^{\mathrm{int}}_{\mathcal{G}_{1},b_{1},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{1}}}(\overline{k}_{E_{1}})$ with image $x$ in
$\mathcal{M}^{\mathrm{int}}_{\mathcal{G},b,{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}}(\overline{k}_{E})$ $($note that
$\overline{k}_{E_{1}}=\overline{k}_{E}$$)$. Then the induced map

$f\colon\left(\mathcal{M}^{\mathrm{int}}_{\mathcal{G}_{1},b_{1},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{1}}}\right)^{\wedge}_{/x_{1}}\to\left(\mathcal{M}^{\mathrm{int}}_{\mathcal{G},b,{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}}}\right)^{\wedge}_{/x}\times_{\textup{Spd}(\mathcal{O}_{E})}\textup{Spd}(\mathcal{O}_{E^{\mathrm{ur}}_{1}}),$

is an isomorphism, and there is an identification

$f^{\ast}(\mathscr{P}_{\mathcal{G}}^{\mathrm{univ}})\simeq\mathscr{P}_{\mathcal{G}_{1}}^{\mathrm{univ}}\times^{\mathcal{G}_{1}}\mathcal{G}.$

###### Proof of Proposition 5.11.

Consider the commutative diagram

${{\mathscr{S}_{\mathsf{M}^{\prime}_{p,1}}(\mathbf{T}_{1}^{\prime},\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}}}}$${{\mathscr{S}_{\mathsf{N}_{p,1}}(\mathbf{G}_{1}^{\mathrm{ab}},\mathbf{X}_{1}^{\mathrm{ab}})_{\mathcal{O}_{E}}}}$${{\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}^{\prime}_{p,1}}(\mathbf{B}^{\prime}_{1},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}}$${{\mathscr{S}^{{\varnothing}}_{\mathsf{K}_{p,1}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathcal{O}_{E}}}}$${{\mathscr{S}^{\mathfrak{d}}_{\mathsf{L}_{p}}(\mathbf{B},\mathbf{Y})}}$${{\mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_{p}}(\mathbf{G},\mathbf{X})}}$$\scriptstyle{a}$$\scriptstyle{e}$$\scriptstyle{b}$$\scriptstyle{d}$$\scriptstyle{c}$
(5.8)

from Proposition 5.2, where each map is equivariant for the relevant group
homomorphism. Choose compact open subgroups

${\mathsf{L}^{\prime}}^{p}\subseteq\mathbf{B}_{1}^{\prime}(\mathbb{A}_{f}^{p}),\quad\mathsf{K}^{p}\subseteq\mathbf{G}(\mathbb{A}_{f}^{p}),\quad{\mathsf{M}^{\prime}}^{p}_{1}\subseteq\mathbf{T}^{\prime}_{1}(\mathbb{A}_{f}^{p}),\
\text{ and }\ \mathsf{K}^{p}_{1}\subseteq\mathbf{G}_{1}(\mathbb{A}_{f}^{p}),$

such that
$(a,e)({\mathsf{L}^{\prime}}^{p}_{1})\subseteq\mathsf{K}_{1}^{p}\times{\mathsf{M}^{\prime}_{1}}^{p}$
and $e({\mathsf{L}^{\prime}}^{p}_{1})\subseteq\mathsf{K}^{p}$. We then obtain
the diagram

${{\mathscr{S}_{{\mathsf{L}^{\prime}_{1}}^{p}}^{\mathfrak{d}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}^{\prime}_{1})_{\mathcal{O}_{E}}}}$${{\mathscr{S}_{\mathsf{K}_{1}^{p}}^{\mathfrak{d}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathcal{O}_{E}}\times\mathscr{S}_{{\mathsf{M}^{\prime}_{1}}^{p}}(\mathbf{T}^{\prime}_{1},\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}}}}$${{\mathscr{S}_{\mathsf{K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})}}$${{\mathscr{S}_{\mathsf{K}^{p}_{1}\times{\mathsf{M}^{\prime}_{1}}^{p}}^{\mathfrak{d}}(\mathbf{G}_{1}\times\mathbf{T}^{\prime}_{1},\mathbf{X}_{1}\times\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}},}}$$\scriptstyle{e}$$\scriptstyle{(a,c)}$$\scriptstyle{\wr}$

from (5.2) (excluding some terms) and Proposition 3.28.

Let us now fix a point $w$ of
$\mathscr{S}^{\mathfrak{d}}_{{\mathsf{L}^{\prime}_{1}}^{p}}(\mathbf{B}_{1}^{\prime},\mathbf{Y}_{1}^{\prime})(\overline{k}_{E})$.
We set

$x=e(w)\in{\mathscr{S}_{\mathsf{K}^{p}}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})}(\overline{k}_{E}),\qquad
y=a(w)\in\mathscr{S}_{\mathsf{K}_{1}^{p}}^{\mathfrak{d}}(\mathbf{G}_{1},\mathbf{X}_{1})_{\mathcal{O}_{E}}(\overline{k}_{E}),$

and

$z=c(w)\in\mathscr{S}_{{\mathsf{M}^{\prime}_{1}}^{p}}(\mathbf{T}^{\prime}_{1},\\{h_{\mathbf{T}^{\prime}_{1}}\\})_{\mathcal{O}_{E}}(\overline{k}_{E}).$

Since the maps

$(\mathbf{B}^{\prime}_{1})^{\textup{der}}\to\mathbf{G}^{\textup{der}}\ \text{
and }\
(\mathbf{B}^{\prime}_{1})^{\textup{der}}\to(\mathbf{G}_{1}\times\mathbf{T}^{\prime}_{1})^{\textup{der}}$

are isogenies, the maps

$\mathbf{B}^{\prime}_{1}\to\mathbf{G}\ \text{ and }\
\mathbf{B}^{\prime}_{1}\to\mathbf{G}_{1}\times\mathbf{T}^{\prime}_{1}$

are ad-isomorphisms by Proposition 2.4. Thus, the maps $e$ and $(a,c)$ are
finite étale by Proposition 3.24, and we obtain induced isomorphisms

$e\colon\widehat{\mathcal{O}}_{x}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}},\widehat{\mathcal{O}}_{w}\qquad(a,c)\colon\widehat{\mathcal{O}}_{y}\otimes_{\mathcal{O}_{\breve{E}}}\widehat{\mathcal{O}}_{z}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\widehat{\mathcal{O}}_{w}.$

Here we use the obvious notation for the complete local rings of these
Kisin–Pappas–Zhou models at these $\overline{k}_{E}$-points. We also use the
fact that $\widehat{\mathcal{O}}_{z}$ is isomorphic to
$\mathcal{O}_{\breve{E}}$ (see [7, Lemma 4.2]) to relate the local ring
$(y,z)$ to the tensor product of local rings.

Now, by Theorem 4.6 and Theorem 4.8 we may choose isomorphisms

$\Theta_{y}\colon\Big{(}\mathcal{M}^{\mathrm{int}}_{\mathcal{G}_{1},b_{y},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{y}}}\Big{)}^{\wedge}_{/y_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\textup{Spf
}(\widehat{\mathcal{O}}_{y})^{\lozenge}\ \text{ and }\
\Theta_{z}\colon\Big{(}\mathcal{M}^{\mathrm{int}}_{{\mathcal{T}^{\prime}_{1}}^{c},b_{z},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{z}}}\Big{)}^{\wedge}_{/z_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\textup{Spf
}(\widehat{\mathcal{O}}_{z})^{\lozenge},$

such that
$\Theta_{y}^{\ast}(\mathscr{P}_{\mathsf{K}^{p}_{1}})\simeq\mathscr{P}_{y}^{\mathrm{univ}}$
and
$\Theta_{z}^{\ast}(\mathscr{P}_{\mathsf{M}^{{}^{\prime}p}_{1}})\simeq\mathscr{P}_{z}^{\mathrm{univ}}$,
where have used the obvious shortenings for the local cocharacters and
universal shtukas. From these, we obtain an isomorphism

$\Theta_{y}\times\Theta_{z}\colon\Big{(}\mathcal{M}^{\mathrm{int}}_{\mathcal{G}_{1},b_{y},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{y}}}\Big{)}^{\wedge}_{/y_{0}}\times\Big{(}\mathcal{M}^{\mathrm{int}}_{{\mathcal{T}^{\prime}_{1}}^{c},b_{z},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{z}}}\Big{)}^{\wedge}_{/z_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\textup{Spf
}(\widehat{\mathcal{O}}_{y})^{\lozenge}\times\textup{Spf
}(\widehat{\mathcal{O}}_{z})^{\lozenge}.$

On the other hand, since ${B}_{1}^{\prime
c}\to{G}_{1}\times{{T}^{\prime}_{1}}^{c}$ is an ad-isomorphism, we see from
Proposition 5.12 that we have an induced isomorphism

$\Big{(}\mathcal{M}^{\mathrm{int}}_{\mathcal{B}_{1}^{\prime
c},b_{w},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{w}}}\Big{)}^{\wedge}_{/w_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\Big{(}\mathcal{M}^{\mathrm{int}}_{\mathcal{G}_{1}\times{\mathcal{T}^{\prime}_{1}}^{c},b_{(y,z)},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{(y,z)}}}\Big{)}^{\wedge}_{/(y_{0},z_{0})}.$

Using the fact that the formation of integral local Shimura varieties commutes
with products in the obvious way, we obtain an induced isomorphism

$(\alpha,\gamma)\colon\Big{(}\mathcal{M}^{\mathrm{int}}_{\mathcal{B}_{1}^{\prime
c},b_{w},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{w}}}\Big{)}^{\wedge}_{/w_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\Big{(}\mathcal{M}^{\mathrm{int}}_{\mathcal{G}_{1},b_{y},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{y}}}\Big{)}^{\wedge}_{/y_{0}}\times\Big{(}\mathcal{M}^{\mathrm{int}}_{{\mathcal{T}^{\prime}_{1}}^{c},b_{z},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{z}}}\Big{)}^{\wedge}_{/z_{0}},$

where we are again using the fact that
$(\mathcal{M}^{\mathrm{int}}_{{\mathcal{T}^{\prime}_{1}}^{c},b_{z},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{z}}})^{\wedge}_{/z_{0}}$ is a point to pass to the
completions.

We may then form the map

$\Theta_{w}\vcentcolon=(a,c)^{-1}\circ(\Theta_{y}\times\Theta_{z})\circ(\alpha,\gamma)\colon\Big{(}\mathcal{M}^{\mathrm{int}}_{\mathcal{B}_{1}^{\prime
c},b_{w},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{w}}}\Big{)}^{\wedge}_{/w_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\textup{Spf
}(\widehat{\mathcal{O}}_{w})^{\lozenge}.$

Let us observe that by design

$\Theta_{w}^{\ast}(\mathscr{P}_{{\mathsf{L}^{\prime}_{1}}^{p}})\times^{\mathcal{B}_{1}^{\prime
c}}(\mathcal{G}_{1}\times{\mathcal{T}^{\prime}_{1}}^{c})\simeq\alpha^{\ast}\mathscr{P}_{y}^{\mathrm{univ}}\times\gamma^{\ast}\mathscr{P}_{z}^{\mathrm{univ}}\simeq\mathscr{P}_{w}^{\mathrm{univ}}\times^{\mathcal{B}_{1}^{\prime
c}}(\mathcal{G}_{1}\times{\mathcal{T}^{\prime}_{1}}^{c}).$ (5.9)

But, since

$\mathcal{B}_{1}^{\prime
c}(\mathbb{Z}_{p}^{\mathrm{ur}})=(\mathcal{G}_{1}(\mathbb{Z}_{p}^{\mathrm{ur}})\times{\mathcal{T}^{\prime}_{1}}^{c}(\mathbb{Z}_{p}^{\mathrm{ur}}))\cap
B^{{}^{\prime}c}_{1}(\mathbb{Q}_{p}^{\mathrm{ur}})$

(as follows from Proposition 5.4), [43, Proposition 5.1.3] implies that the
map of integral local Shimura varieties

$\mathcal{M}^{\mathrm{int}}_{\mathcal{B}_{1}^{\prime
c},b_{w},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{w}}}\to\mathcal{M}^{\mathrm{int}}_{\mathcal{G}_{1}\times{\mathcal{T}^{\prime}_{1}}^{c},b_{(y,z)},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{(y,z)}}}$ (5.10)

is a closed immersion. Because the map (5.10) is given at the level of objects
by the push forward of shtukas along $\mathcal{B}_{1}^{\prime
c}\to\mathcal{G}_{1}\times{\mathcal{T}^{\prime}_{1}}^{c}$, it follows from
(5.9) that $\Theta_{w}^{\ast}(\mathscr{P}_{{\mathsf{L}^{\prime}_{1}}^{p}})$ is
isomorphic to $\mathscr{P}_{w}^{\mathrm{univ}}$.

On the other hand, since ${B}_{1}^{\prime c}\to{G}^{c}$ is an ad-isomorphism,
we deduce from Proposition 5.12 an isomorphism

$\varepsilon\colon\Big{(}\mathcal{M}^{\mathrm{int}}_{\mathcal{B}_{1}^{\prime
c},b_{w},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{w}}}\Big{)}^{\wedge}_{/w_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\Big{(}\mathcal{M}^{\mathrm{int}}_{\mathcal{G},b_{x},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{x}}}\Big{)}^{\wedge}_{/x_{0}}.$

We may then define

$\Theta_{x}\vcentcolon=e\circ\Theta_{w}\circ\varepsilon^{-1}\colon\Big{(}\mathcal{M}_{\mathcal{G},b_{x},{\scaleobj{.65}{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}_{x}}}\Big{)}^{\wedge}_{/x_{0}}\xrightarrow{\raisebox{-1.99997pt}{\smash{$\sim$}}}\textup{Spf
}(\widehat{\mathcal{O}}_{x})^{\lozenge}.$

That $\Theta_{x}(\mathscr{P}_{\mathsf{K}^{p}})$ is isomorphic to
$\mathscr{P}^{\mathrm{univ}}_{x}$ is easy as

$\displaystyle\Theta_{x}(\mathscr{P}_{\mathsf{K}^{p}})$
$\displaystyle\simeq(\varepsilon^{-1})^{\ast}(\Theta_{w}^{\ast}(e^{\ast}\mathscr{P}_{\mathsf{K}^{p}})))$
$\displaystyle\simeq(\varepsilon^{-1})^{\ast}(\Theta_{w}^{\ast}(\mathscr{P}_{{\mathsf{L}^{\prime}_{1}}^{p}}\times^{\mathcal{B}_{1}^{\prime
c}}\mathcal{G}^{c}))$
$\displaystyle\simeq(\varepsilon^{-1})^{\ast}(\Theta_{w}^{\ast}(\mathscr{P}_{{\mathsf{L}^{\prime}_{1}}^{p}})\times^{\mathcal{B}_{1}^{\prime
c}}\mathcal{G}^{c})$
$\displaystyle\simeq(\varepsilon^{-1})^{\ast}(\mathscr{P}_{w}^{\mathrm{univ}}\times^{\mathcal{B}_{1}^{\prime
c}}\mathcal{G}^{c})$ $\displaystyle\simeq\mathscr{P}^{\mathrm{univ}}_{x}.$

Here the first isomorphism holds by definition, the second holds by our
construction of $\mathscr{P}_{\mathsf{K}^{p}}$ as in Step 3 of §5.2, the third
holds from abstract nonsense, the fourth holds by our construction of
$\Theta_{w}$, and the final isomorphism holds from the properties of
$\varepsilon$ stated in Proposition 5.12. ∎

The proof of Theorem 4.10 now follows by combining Theorem 3.12 with Theorem
5.3 and Proposition 5.11.

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