${\mathcal{S}}^{\circ}(X)$ is defined as an open subscheme

$\displaystyle{\mathcal{S}}^{\circ}_{X}(\lambda)={\mathcal{S}}_{X}(\lambda)\backslash\cup_{\lambda^{\prime}\in{\mathrm{Conv}}(\lambda),\lambda^{\prime}\notin
W\lambda}{\mathcal{S}}_{X}(\lambda^{\prime})\subset{\mathcal{S}}_{X}(\lambda).$

Again, the base change
${\mathcal{S}}^{\circ}_{X^{0}}(\lambda)={\mathcal{S}}^{\circ}_{X}(\lambda)\times_{X}X^{0}$
is the constant family of open Schubert cells of
$\mathrm{Gr}_{\mathrm{GSp}_{4}}$ for $\lambda$ over $X^{0}$.

We have a map $L^{+}{\mathcal{G}}\rightarrow\mathrm{Gr}_{{\mathcal{G}},X}$
given by the orbit map $g\mapsto s_{\lambda}g$. Note that it factors through a
subscheme $\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq
h}$ with $d=\operatorname{\mathrm{sim}}(\lambda)$ and $h$ sufficiently large.
The stabilizer subgroup scheme $L^{+}{\mathcal{G}}_{\lambda}\subset
L^{+}{\mathcal{G}}$ of $s_{\lambda}$ is given by

$\displaystyle
L^{+}{\mathcal{G}}_{\lambda}(R)=L^{+}{\mathcal{G}}(R)\cap\mathrm{Ad}((v-t)^{-\lambda})(L^{+}{\mathcal{G}}(R))$

for $t:\operatorname{\mathrm{Spec}}R\rightarrow X$. Thus we have a
monomorphism

$\displaystyle
L^{+}{\mathcal{G}}_{\lambda}{\backslash}L^{+}{\mathcal{G}}\hookrightarrow\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq
h}$

whose scheme-theoretic image is ${\mathcal{S}}_{\lambda}(X)$. Over $X^{0}$, we
have an isomorphism
$(L^{+}{\mathcal{G}}_{\lambda}{\backslash}L^{+}{\mathcal{G}})\times_{X}X^{0}\simeq{\mathcal{S}}^{\circ}_{X^{0}}(\lambda)$.
Note that there is a map from $L^{+}{\mathcal{G}}_{\lambda}\times_{X}X^{0}$ to
$P_{\lambda}\times X^{0}$ sending $g\mapsto g\bmod(v-t)$ where
$P_{\lambda}\subset\mathrm{GSp}_{4}$ is a parabolic subgroup associated with
$\lambda$ and containing $B$. Thus, we have a natural map
$(L^{+}{\mathcal{G}}_{\lambda}{\backslash}L^{+}{\mathcal{G}})\times_{X}X^{0}\rightarrow
P_{\lambda}{\backslash}\mathrm{GSp}_{4}\times_{\mathbf{Z}}X^{0}$ given by
$g\mapsto g\bmod(v-t)$. When it is composed with the previous isomorphism, we
get a map

$\displaystyle\pi_{\lambda}:{\mathcal{S}}^{\circ}_{X^{0}}(\lambda)\xrightarrow{}(P_{\lambda}\backslash\mathrm{GSp}_{4})\times_{\mathbf{Z}}X^{0}.$

####  3.2.2. Universal local models

For convenience, we let
$\operatorname{{\mathrm{std}}}:\mathbb{A}^{3}\rightarrow\mathbb{A}^{4}$ be the
morphism sending $(a_{1},a_{2},a_{3})$ to
$(a_{1},a_{2},a_{3}-a_{2},a_{3}-a_{1})$. Note that this matches with the
description of $\operatorname{{\mathrm{std}}}:X_{*}(T^{\vee})\rightarrow
X_{*}(T^{\vee}_{4})$.

We define a subfunctor $L{\mathcal{G}}^{\nabla}$ of
$L{\mathcal{G}}\times_{\mathbf{Z}}\mathbb{A}^{3}$ given by

$\displaystyle L{\mathcal{G}}^{\nabla}(R)=\left\\{(g,{\mathbf{a}})\mid g\in
L{\mathcal{G}}(R),{\mathbf{a}}\in\mathbb{A}^{3},v\frac{d{\overline{g}}}{dv}\overline{g}^{-1}+\overline{g}\mathrm{Diag}(\operatorname{{\mathrm{std}}}({\mathbf{a}}))\overline{g}^{-1}\in\frac{1}{v-t}L^{+}{\mathcal{M}}(R)\right\\}.$

for $t:\operatorname{\mathrm{Spec}}R\rightarrow X$. Since
$L{\mathcal{G}}^{\nabla}$ is stable under left multiplication by
$L^{+}{\mathcal{G}}$, it defines a closed sub-ind-scheme
$\mathrm{Gr}^{\nabla}_{{\mathcal{G}},X}:=L^{+}{\mathcal{G}}{\backslash}L{\mathcal{G}}^{\nabla}\subset\mathrm{Gr}_{{\mathcal{G}},X}\times_{\mathbf{Z}}\mathbb{A}^{3}$
which is ind-proper over $X\times_{\mathbf{Z}}\mathbb{A}^{3}$.

######  Definition 3.2.3.

Let $\lambda\in X_{*}(T^{\vee})$ be a dominant cocharacter. We define the
_naive universal local model_
${\mathcal{M}}^{\mathrm{nv}}_{X}(\leq\lambda,\nabla)$ as
$\mathrm{Gr}^{\nabla}_{{\mathcal{G}},X}\cap({\mathcal{S}}_{X}(\lambda)\times_{\mathbf{Z}}\mathbb{A}^{3})$.

We write
${\mathcal{M}}^{\mathrm{nv}}_{X^{0}}(\leq\lambda,\nabla)={\mathcal{M}}^{\mathrm{nv}}_{X}(\leq\lambda,\nabla)\times_{X}X^{0}$.
It is a proper scheme over $X^{0}\times\mathbb{A}^{3}$.

######  Proposition 3.2.4.

Let $\lambda\in X_{*}(T^{\vee})$ be a dominant cocharacter. The map
$\pi_{\lambda}$ induces an isomorphism

$\displaystyle\left({\mathcal{M}}^{\mathrm{nv}}_{X^{0}}(\leq\lambda,\nabla)\cap({\mathcal{S}}^{\circ}_{X^{0}}(\lambda)\times_{\mathbf{Z}}\mathbb{A}^{3})\right)\left[\frac{1}{h_{\lambda}!}\right]\simeq(P_{\lambda}{\backslash}\mathrm{GSp}_{4})\times_{\mathbf{Z}}X^{0}\times_{\mathbf{Z}}\mathbb{A}^{3}\left[\frac{1}{h_{\lambda}!}\right].$

###### Proof.

Let $\overline{N}_{\lambda}$ be the unipotent radical of opposite parabolic to
$P_{\lambda}$. Recall the decomposition $\overline{N}_{\lambda}\simeq\prod
U_{\alpha^{\vee}}$ where the product runs over $\alpha^{\vee}\in\Phi^{\vee}$
satisfying $\langle\phi(\alpha^{\vee}),\lambda\rangle<0$. For any commutative
ring $R$ and $N\in\overline{N}_{\lambda}(R)$, we can write $N$ as a product of
$N_{\alpha^{\vee}}\in U_{\alpha^{\vee}}(R)$ over the same set of
$\alpha^{\vee}$.

Note that $\\{\overline{N}_{\lambda}w\\}_{w\in W}$ forms an affine open cover
of $P_{\lambda}{\backslash}\mathrm{GL}_{n}$. By pulling back along
$\pi_{\lambda}$, we get an affine open cover of
${\mathcal{S}}^{\circ}_{X^{0}}(\lambda)$. More precisely, we have
$\pi_{\lambda}^{-1}(\overline{N}_{\lambda}w)=\widetilde{N}_{\lambda}w$ where
$\widetilde{N}_{\lambda}$ is the affine scheme over $X^{0}$ whose $R$-points
for for $t:\operatorname{\mathrm{Spec}}R\rightarrow X$ are given by

$\displaystyle\widetilde{N}_{\lambda}(R)=\left\\{(v-t)^{\lambda}N\mid\text{$N\in\overline{N}_{\lambda}(R[v-t])$,
$\deg
N_{\alpha^{\vee}}\leq\langle\phi(\alpha^{\vee}),-\lambda\rangle-1$}\right\\}.$

Let us write
$N_{\alpha^{\vee}}=\sum_{j=0}^{\langle\phi(\alpha^{\vee}),-\lambda\rangle-1}X_{\alpha^{\vee},j}(v-t)^{j}$
with $X_{\alpha^{\vee},j}\in R$. From the proof of [LLHLMa, Proposition
3.3.4], we can deduce that taking the intersection
$\mathrm{Gr}^{\nabla}_{{\mathcal{G}},X}\cap(\widetilde{N}_{\lambda}w\times_{\mathbf{Z}}\mathbb{A}^{3})\left[\frac{1}{h_{\lambda}!}\right]$
imposes conditions that $X_{\alpha^{\vee},j}$ for $\alpha^{\vee}$ and $j>0$ is
determined by $X_{\alpha^{\prime\vee},0}$ for $\alpha^{\prime}\leq\alpha<0$.
Since the map

$\displaystyle\pi_{\lambda}:\mathrm{Gr}^{\nabla}_{{\mathcal{G}},X}\cap(\widetilde{N}_{\lambda}w\times_{\mathbf{Z}}\mathbb{A}^{3})\left[\frac{1}{h_{\lambda}!}\right]\rightarrow\overline{N}_{\lambda}w\times_{\mathbf{Z}}X^{0}\times_{\mathbf{Z}}\mathbb{A}^{3}\left[\frac{1}{h_{\lambda}!}\right]$

takes $(v-t)^{\lambda}Nw$ to $(N\bmod(v-t))w$, this proves that
$\pi_{\lambda}$ is an isomorphism. ∎

######  Remark 3.2.5.

The above Proposition implies that
${\mathcal{M}}^{\mathrm{nv}}_{X^{0}}(\leq\lambda,\nabla)\cap({\mathcal{S}}^{\circ}_{X^{0}}(\lambda)\times_{\mathbf{Z}}\mathbb{A}^{3})\left[\frac{1}{h_{\lambda}!}\right]$
is indeed proper over
$X^{0}\times_{\mathbf{Z}}\mathbb{A}^{3}\left[\frac{1}{h_{\lambda}!}\right]$
(by the properness of partial flag varieties). Thus the map

$\displaystyle{\mathcal{M}}^{\mathrm{nv}}_{X^{0}}(\leq\lambda,\nabla)\cap({\mathcal{S}}^{\circ}_{X^{0}}(\lambda)\times_{\mathbf{Z}}\mathbb{A}^{3})\left[\frac{1}{h_{\lambda}!}\right]\hookrightarrow{\mathcal{M}}^{\mathrm{nv}}_{X^{0}}(\leq\lambda,\nabla)\left[\frac{1}{h_{\lambda}!}\right]$

is a proper open immersion. Since proper morphism is universally closed, this
map is an inclusion of a connected component. Using this inductively, we
obtain an isomorphism (cf. [LLHLMa, Corollary 3.3.5])

$\displaystyle\left({\mathcal{M}}^{\mathrm{nv}}_{X^{0}}(\leq\lambda,\nabla)\left[\frac{1}{h_{\lambda}!}\right]\right)_{{\mathrm{red}}}\buildrel\sim\over{\rightarrow}\coprod_{\lambda^{\prime}\leq\lambda,\lambda^{\prime}\in
X_{*}^{+}(T^{\vee})}(P_{\lambda^{\prime}}{\backslash}\mathrm{GSp}_{4})\times_{\mathbf{Z}}X^{0}\times\mathbf{Z}\mathbb{A}^{3}\left[\frac{1}{h_{\lambda}!}\right].$

######  Definition 3.2.6.

Let $\lambda\in X_{*}(T^{\vee})$ be a dominant cocharacter. The _universal
local model_ ${\mathcal{M}}_{X}(\lambda,\nabla)$ is the closure of the
connected component
${\mathcal{M}}_{X^{0}}^{\mathrm{nv}}(\leq\lambda,\nabla)\cap({\mathcal{S}}^{\circ}_{X^{0}}(\lambda)\times_{\mathbf{Z}}\mathbb{A}^{3})$
of ${\mathcal{M}}_{X^{0}}^{\mathrm{nv}}(\leq\lambda,\nabla)$ inside
${\mathcal{M}}_{X}^{{\mathrm{nv}}}(\leq\lambda,\nabla)$. In particular, it is
$v$-flat.

######  Proposition 3.2.7.

Let $\lambda\in X_{*}(T^{\vee})$ be a dominant cocharacter. Then
${\mathcal{M}}_{X^{0}}^{\mathrm{nv}}(\leq\lambda,\nabla)\left[\frac{1}{(2h_{\lambda})!}\right]$
is smooth over
$X^{0}\times_{\mathbf{Z}}\mathbb{A}^{3}\left[\frac{1}{(2h_{\lambda})!}\right]$.

###### Proof.

Since the proof of [LLHLMa, Proposition 3.2.6] generalizes to our situation
quite straight forwardly, we only sketch the idea of the proof. We follow the
notation in _loc. cit._ and write
$Y={\mathcal{M}}_{X^{0}}^{\mathrm{nv}}(\leq\lambda,\nabla)\left[\frac{1}{(2h_{\lambda})!}\right]$,
$S=X^{0}\times_{\mathbf{Z}}\mathbb{A}^{3}\left[\frac{1}{(2h_{\lambda})!}\right]$,
and $pr$ for the natural projection map $Y\rightarrow S$.

There is a $T^{\vee}$-action on $Y$ induced by the right multiplication by
$T^{\vee}$ on ${\mathcal{G}}$. Note that this preserves the non-smooth locus
in $Y$. Since $\mathrm{pr}$ is proper and the non-smooth locus in $Y$ is
closed, the image of the non-smooth locus in $S$ is closed as well. If the
non-smooth locus is non-empty, it has a non-zero geometric fiber which is a
proper variety over a field with $T^{\vee}$-action. Thus, such fiber contains
a $T^{\vee}$-fixed point. Since it is contained in a geometric fiber of
${\mathcal{S}}_{X^{0}}(\lambda)$, it is known to be contained in the support
of a section $s_{\mu}:X^{0}\rightarrow{\mathcal{S}}_{X^{0}}(\lambda)$ for some
$\mu\in{\mathrm{Conv}}(\lambda)$ (see Lemma 3.3.7 in _loc. cit._).

Thus, it suffices to prove the smoothness at a closed point
$x\in\operatorname{\mathrm{Spec}}k\rightarrow Y$ lying in the support of
$s_{\mu}$. Let $s=\mathrm{pr}(x)$. We claim that $\dim
T_{x}Y/S\leq\dim_{x}Y_{s}$. If we have this bound, the completion
$\widehat{{\mathcal{O}}}_{Y,x}$ is generated over
$\widehat{{\mathcal{O}}}_{S,s}$ by $\dim_{x}Y_{s}$ many variables. Since $S$
is regular,
$\dim\widehat{{\mathcal{O}}}_{Y,x}\leq\dim\widehat{{\mathcal{O}}}_{S,s}+\dim_{x}Y_{s}$
where the equality holds if and only if $\widehat{{\mathcal{O}}}_{Y,x}$ is a
power series ring over $\widehat{{\mathcal{O}}}_{S,s}$ with $\dim_{x}Y_{s}$
many variables. By Remark 3.2.5 and the choice of $x$, we have

$\displaystyle\dim_{x}Y_{s}=\dim P_{\mu}{\backslash}\mathrm{GSp}_{4},\ \
\dim_{x}Y=\dim P_{\mu}{\backslash}\mathrm{GSp}_{4}+4$

Thus, the equality holds and $\mathrm{pr}$ is smooth at $x$ by [Sta18, Tag
07VH].

Finally, we prove the claimed inequality. Following the argument in [LLHLMa,
Proposition 3.3.6] and using Lemma 3.1.4, we can show that $T_{x}Y_{s}$ is a
subspace of the space of matrices of the form $(1+\epsilon X)(v-t)^{\mu}$
where $X\in\operatorname{\mathrm{Lie}}\mathrm{GSp}_{4}(k(\\!(v-t)\\!))$ whose
entries are polynomials with degree bounds, and $X$ satisfies certain equation
imposed by $\mathrm{Gr}_{{\mathcal{G}},X^{0}}^{\nabla}$. More precisely, the
computation in _loc. cit._ shows that all diagonal entries of $X$ are zero,
and for each root $\alpha$, the $\alpha$-th entry of $X$ is zero if
$\langle\mu,\alpha^{\vee}\rangle\geq 0$ and is determined by the coefficient
of the lowest degree term if $\langle\mu,\alpha^{\vee}\rangle<0$. Thus the
space of such $X$ has dimension at most $\dim
P_{\mu}{\backslash}\mathrm{GSp}_{4}$. This completes the proof. ∎

We finish this subsection with two lemmas regarding the normalization of the
universal local model. For an integral scheme $Y$, we let
$Y^{\mathrm{nm}}\rightarrow Y$ denote the normalization of $Y$.

Let $l>0$ be an integer. It will be useful to consider the base change

$\displaystyle{\mathcal{M}}_{X}(\lambda,\nabla)_{l}:={\mathcal{M}}_{X}(\lambda,\nabla)\times_{X,v\mapsto
v^{l}}X$

and its normalization $({\mathcal{M}}_{X}(\lambda,\nabla)_{l})^{\mathrm{nm}}$.
When $l=e$ and if we can take $\varpi=(-p)^{1/e}$ as the uniformizer in
${\mathcal{O}}$, this has the advantage that the map
$\operatorname{\mathrm{Spec}}{\mathcal{O}}\rightarrow X\times\mathbb{A}^{3}$
sending $v$ to $-p$ is the composite of the maps
$\operatorname{\mathrm{Spec}}{\mathcal{O}}\rightarrow X\times\mathbb{A}^{3}$
sending $v\mapsto\varpi$ and $X\times\mathbb{A}^{3}\rightarrow
X\times\mathbb{A}^{3}$ sending $v\mapsto v^{e}$. In particular, we have

$\displaystyle{\mathcal{M}}_{X}(\lambda,\nabla)\times_{X\times\mathbb{A}^{3},v\mapsto-p}\operatorname{\mathrm{Spec}}{\mathcal{O}}={\mathcal{M}}_{X}(\lambda,\nabla)_{e}\times_{X\times\mathbb{A}^{3},v\mapsto\varpi}\operatorname{\mathrm{Spec}}{\mathcal{O}}.$

######  Lemma 3.2.8.

There is an open subscheme $U\subset\mathbb{A}^{3}$ only depending on
$\lambda$ and $l$, such that
${\mathcal{M}}_{X}(\lambda,\nabla)_{l}\times_{\mathbb{A}^{3}}U\rightarrow
X\times_{\mathbf{Z}}U$ and
$({\mathcal{M}}_{X}(\lambda,\nabla)_{l})^{\mathrm{nm}}\times_{\mathbb{A}^{3}}U\rightarrow
X\times_{\mathbf{Z}}U$ are flat.

###### Proof.

This follows from [LLHLMa, Lemma 3.5.5, Remark 3.5.6]. ∎

######  Lemma 3.2.9.

There is an open subscheme $U\subset\mathbb{A}^{3}$ only depending on
$\lambda$ and $l$, such that if $R$ is complete DVR and
$f:\operatorname{\mathrm{Spec}}R\rightarrow X\times\mathbb{A}^{3}$ is a
morphism sending $v$ to a uniformizer of $R$ and factors through $X\times U$,
the base change

$\displaystyle({\mathcal{M}}_{X}(\lambda,\nabla)_{l})^{\mathrm{nm}}_{R}:=({\mathcal{M}}_{X}(\lambda,\nabla)_{l})^{\mathrm{nm}}\times_{X\times\mathbb{A}^{3}}\operatorname{\mathrm{Spec}}R$

is flat over $\operatorname{\mathrm{Spec}}R$, and
$({\mathcal{M}}_{X}(\lambda,\nabla)_{l})^{\mathrm{nm}}_{R}$ is normal.

###### Proof.

This follows from [LLHLMa, Proposition 3.5.2]. To satisfy Setup 3.5.1 of _loc.
cit._ , take $S=\mathbb{A}^{3}[\frac{1}{(2h_{\lambda})!}]$,
$M=({\mathcal{M}}_{X}(\lambda,\nabla)_{l})^{\mathrm{nm}}\times_{\mathbb{A}^{3}}S$,
and use Proposition 3.2.7. ∎

###  3.3. Local models in mixed characteristic

We specialize to objects over ${\mathcal{O}}$ by taking base change
$\operatorname{\mathrm{Spec}}{\mathcal{O}}\rightarrow X$ sending $v$ to $-p$.
For a fpqc sheaf $Y\rightarrow X$, we write
$Y_{\mathcal{O}}=Y\times_{X}\operatorname{\mathrm{Spec}}{\mathcal{O}}$. For
example, we have
$L{\mathcal{G}}_{\mathcal{O}}=L{\mathcal{G}}\times_{X}\operatorname{\mathrm{Spec}}{\mathcal{O}}$
and $L^{+}{\mathcal{G}}_{\mathcal{O}},L^{+}{\mathcal{M}}_{\mathcal{O}}$
similarly.

We have the global affine Grassmannian
$\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}=L^{+}{\mathcal{G}}_{\mathcal{O}}{\backslash}L{\mathcal{G}}_{\mathcal{O}}$.
Its generic fiber $\mathrm{Gr}_{{\mathcal{G}},E}$ is equal to the usual affine
Grassmannian associated to $\mathrm{GSp}_{4}$ over $E$. For $\lambda\in
X_{*}(T^{\vee})$ a dominant cocharacter, we write
$S_{E}^{\circ}(\lambda)\subset\mathrm{Gr}_{\mathrm{GSp}_{4},E}$ for the open
affine Schubert cell and $S_{E}(\lambda)$ for its reduced closure. The Zariski
clousre of $S_{E}(\lambda)$ in $\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}$ is
the Pappas–Zhu local model $M(\leq\lambda)$ associated to the group
$\mathrm{GSp}_{4}$, the conjugacy class of $\lambda$, and the Iwahori subgroup
${\mathcal{I}}$ ([PZ13]). It is known that $M(\leq\lambda)$ is projective over
$\operatorname{\mathrm{Spec}}{\mathcal{O}}$ (§7.1 in _loc. cit._).

Let ${\mathbf{a}}\in{\mathcal{O}}^{3}$. We define
$L{\mathcal{G}}_{\mathcal{O}}^{\nabla_{\mathbf{a}}}\subset
L{\mathcal{G}}_{\mathcal{O}}$ to be the subfunctor given by

$\displaystyle
L{\mathcal{G}}_{\mathcal{O}}^{\nabla_{\mathbf{a}}}(R):=\left\\{g\in
L{\mathcal{G}}_{\mathcal{O}}(R)\mid
v\frac{d{\overline{g}}}{dv}\overline{g}^{-1}+\overline{g}\mathrm{Diag}(\operatorname{{\mathrm{std}}}({\mathbf{a}}))\overline{g}^{-1}\in\frac{1}{v+p}L^{+}{\mathcal{M}}_{4}(R)\right\\}$

for an ${\mathcal{O}}$-algebra $R$. It is stable under left
$L^{+}{\mathcal{G}}_{{\mathcal{O}}}$ multiplication and induces a closed sub-
ind-scheme
$\mathrm{Gr}^{\nabla_{\mathbf{a}}}_{{\mathcal{G}},{\mathcal{O}}}\subset\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}$.

######  Definition 3.3.1.

We define the _naive local model_ as
$M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})=M(\leq\lambda)\cap\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}^{\nabla_{\mathbf{a}}}$.
The _(mixed characteristic) local model_ $M(\lambda,\nabla_{\mathbf{a}})$ is
defined to be the Zariski closure of
$S^{\circ}_{E}(\lambda)\cap\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}^{\nabla_{\mathbf{a}}}$
in $M(\leq\lambda)$. By its construction, $M(\lambda,\nabla_{{\mathbf{a}}})$
is projective and flat over $\operatorname{\mathrm{Spec}}{\mathcal{O}}$.

We define
$U(\widetilde{z}):={\mathcal{U}}(\widetilde{z})\times_{X}\operatorname{\mathrm{Spec}}{\mathcal{O}}$
and $U(\widetilde{z},\lambda,\nabla_{{\mathbf{a}}}):=U(\widetilde{z})\cap
M(\lambda,\nabla_{\mathbf{a}})$ where the intersection is taken inside
$\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}$. Note that the latter is equal to
$({\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq
h}\times_{X}\operatorname{\mathrm{Spec}}{\mathcal{O}})\cap
M(\lambda,\nabla_{\mathbf{a}})$ for large enough $h\geq 0$. Thus, it is an
open affine subscheme of $M(\lambda,\nabla_{\mathbf{a}})$ by Proposition
3.1.5.

Our main interest in the geometry of $M(\lambda,\nabla_{\mathbf{a}})$ is
whether a completed local ring
$\mathcal{O}_{M(\lambda,\nabla_{\mathbf{a}}),x}^{\wedge}$ is a domain for some
$x\in M(\lambda,\nabla_{\mathbf{a}})$ . To understand this property more
geometrically, we introduce the following definition.

######  Definition 3.3.2.

Let $Y$ be a scheme. A point $y\in Y$ is called _unibranch_ if the
normalization of the local ring $({\mathcal{O}}_{Y,y})_{\mathrm{red}}$ is
local.

######  Remark 3.3.3.

Suppose that $Y$ is Noetherian and excellent. The followings are equivalent:

  1. (1)

$y\in Y$ is unibranch;

  2. (2)

the fiber above $y$ of the normalization map $Y^{\mathrm{nm}}\rightarrow Y$ is
a single point ([Sta18, Tag 0C3B]);

  3. (3)

the completed local ring ${\mathcal{O}}_{Y,y}^{\wedge}$ is a domain ([Sta18,
Tag 0C2E]).

Let $\widetilde{z}=wt_{\nu}\in\widetilde{W}^{\vee}$. There is a constant
section
$\widetilde{z}:\operatorname{\mathrm{Spec}}\mathbf{Z}\hookrightarrow{\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq
h}\times_{X}X_{0}$ for $h$ large enough, given by
$wv^{\nu}\in\mathrm{GSp}_{4}(\mathbf{Z}(\\!(v)\\!))$. We denote its
composition with
${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq
h}\times_{X}X_{0}\hookrightarrow\mathrm{Gr}_{{\mathcal{G}},X}\times_{X}X_{0}$
again by $\widetilde{z}$.

Let ${\mathbf{a}}\in\mathbb{A}^{3}({\mathcal{O}})$. We write the induced
$\mathbf{F}$ point by ${\mathbf{a}}$ as well. We write
$\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ for the $\mathbf{F}$-point
$\operatorname{\mathrm{Spec}}\mathbf{F}\rightarrow(\mathrm{Gr}_{{\mathcal{G}},X}\times_{X}X_{0})\times_{\mathbf{Z}}\mathbb{A}^{3}$
given by $(\widetilde{z},{\mathbf{a}})$.

Recall that $e$ denotes the ramification index of the extension
${\mathcal{O}}/\mathbf{Z}_{p}$.

######  Theorem 3.3.4.

There exists a non-empty open subscheme $U\subset\mathbb{A}^{3}$ depending
only on $\lambda$ and $e$, such that if ${\mathbf{a}}\in U({\mathcal{O}})$,
then $M(\lambda,\nabla_{\mathbf{a}})$ is unibranch at any point
$\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ contained in the special fiber. In
addition,
${\mathcal{O}}(U(\widetilde{z},\lambda,\nabla_{\mathbf{a}}))^{\wedge_{p}}$ is
a domain.

To prove this Theorem, we introduce the local model in equal characteristic
and prove its unibranch property.

######  Definition 3.3.5.

Let ${\mathbf{a}}\in\mathbb{A}^{3}(\mathbf{F})$. We define
${\mathcal{M}}^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}}):={\mathcal{M}}_{X}^{\mathrm{nv}}(\leq\lambda,\nabla)\times_{\mathbb{A}^{3}}{\mathbf{a}}$.
We define the _equal characteristic local model_
${\mathcal{M}}(\lambda,\nabla_{\mathbf{a}})$ by the Zariski closure of

$\displaystyle\left({\mathcal{M}}^{\mathrm{nv}}_{X^{0}}(\leq\lambda,\nabla)\cap({\mathcal{S}}^{\circ}_{X^{0}}(\lambda)\times_{\mathbf{Z}}\mathbb{A}^{3})\right)\times_{\mathbb{A}^{3}}{\mathbf{a}}$

in ${\mathcal{M}}^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})$.

######  Remark 3.3.6.

Although mixed or equal characteristic local models are generally _not_ equal
to base changes of universal local models, their generic fibers can be
obtained by taking base changes of the generic fibers of universal local
models. Suppose that $h_{\lambda}<p$. By taking fiber product at $E$-point
$(-p,{\mathbf{a}})\in X\times\mathbb{A}^{3}$ (resp. at $\mathbf{F}$-point
${\mathbf{a}}$), we also have

$\displaystyle{\mathcal{M}}_{X^{0}}(\lambda,\nabla)\times_{X\times\mathbb{A}^{3}}\operatorname{\mathrm{Spec}}E$
$\displaystyle=M(\lambda,\nabla_{\mathbf{a}})\times_{\mathcal{O}}\operatorname{\mathrm{Spec}}E$
$\displaystyle{\mathcal{M}}_{X^{0}}(\lambda,\nabla)\times_{\mathbb{A}^{3}}\operatorname{\mathrm{Spec}}\mathbf{F}$
$\displaystyle={\mathcal{M}}(\lambda,\nabla_{\mathbf{a}})\times_{X}X^{0}.$

Moreover, $M(\lambda,\nabla_{\mathbf{a}})$ (resp.
${\mathcal{M}}(\lambda,\nabla_{\mathbf{a}})$) is characterized as $p$-flat
(resp. $v$-flat) closure of
${\mathcal{M}}_{X^{0}}(\lambda,\nabla)\times_{X\times\mathbb{A}^{3}}\operatorname{\mathrm{Spec}}E$
inside
${\mathcal{M}}_{X}(\lambda,\nabla)\times_{X\times\mathbb{A}^{3}}\operatorname{\mathrm{Spec}}{\mathcal{O}}$
(resp.
${\mathcal{M}}_{X^{0}}(\lambda,\nabla)\times_{X\times\mathbb{A}^{3}}\operatorname{\mathrm{Spec}}E$
inside
${\mathcal{M}}_{X}(\lambda,\nabla)\times_{\mathbb{A}^{3}}\operatorname{\mathrm{Spec}}\mathbf{F}$).

For $l\in\mathbf{Z}_{>0}$ and an $X$-scheme $Y$, we write $Y_{l}$ for the base
change $Y\times_{X,v\mapsto v^{l}}X$.

######  Proposition 3.3.7 (cf. Proposition 3.4.4 in [LLHLMa]).

Let $l>0$ be an integer. Then ${\mathcal{M}}(\lambda,\nabla_{\mathbf{a}})_{l}$
is unibranch at any $\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ contained in its
special fiber. Moreover, the preimage of ${\mathcal{U}}(\widetilde{z})$ in
$({\mathcal{M}}(\lambda,\nabla_{\mathbf{a}})_{l})^{\mathrm{nm}}\times_{X}X_{0}$
is connected.

###### Proof.

This mainly follows from [LLHLMa, Lemma 3.4.7, 3.4.8]. Indeed, we can take a
one-parameter subgroup in Lemma 3.4.7 in _loc. cit._ valued in
$T^{\vee}\times\mathbf{G}_{m}$ (instead of $T^{\vee}_{4}\times\mathbf{G}_{m}$)
whose induced action on
${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq h}$ (instead
of ${\mathcal{U}}_{4}(\widetilde{z})^{\det,\leq h}$) satisfies the conditions
stated in Lemma 3.4.7 in _loc. cit._. Using this, the proof of [LLHLMa,
Proposition 3.4.4] can be applied in our case too. ∎

###### Proof of Theorem 3.3.4.

The novel idea in [LLHLMa] is comparing the mixed characteristic and equal
characteristic local models inside the universal local model. Let $U$ be an
open subscheme $U\subset\mathbb{A}^{3}$ satisfying Lemma 3.2.8 and 3.2.9 for
$l=e$ and $(2h_{\lambda})!e!$ is invertible. Note that this implies
$U({\mathcal{O}})=\emptyset$ unless $p>(2h_{\lambda})!e!$. Thus we can assume
that ${\mathcal{O}}/\mathbf{Z}_{p}$ is tame and take $\varpi=(-p)^{1/e}$
(after enlarging the residue field if necessary). Let ${\mathbf{a}}\in
U({\mathcal{O}})$. The following diagram explains how to make such comparison.

${M(\lambda,\nabla_{\mathbf{a}})^{\mathrm{nm}}}$${({\mathcal{M}}_{X}(\lambda,\nabla)_{e})^{\mathrm{nm}}\times_{X\times
U}\operatorname{\mathrm{Spec}}{\mathcal{O}}}$${M(\lambda,\nabla_{\mathbf{a}})}$${\operatorname{\mathrm{Spec}}{\mathcal{O}}}$${({\mathcal{M}}_{X}(\lambda,\nabla)_{e})^{\mathrm{nm}}}$${{\mathcal{M}}_{X}(\lambda,\nabla)_{e}}$${X\times
U}$${({\mathcal{M}}_{X}(\lambda,\nabla)_{e})^{\mathrm{nm}}\times_{X\times
U}\mathbb{A}^{1}_{\mathbf{F}}}$${{\mathcal{M}}(\lambda,\nabla_{\mathbf{a}})_{e}}$${\mathbb{A}^{1}_{\mathbf{F}}}$${({\mathcal{M}}(\lambda,\nabla_{\mathbf{a}})_{e})^{\mathrm{nm}}}$$\scriptstyle{\sim}$$\scriptstyle{(v\mapsto\varpi,{\mathbf{a}})}$$\scriptstyle{(v\mapsto
v,{\mathbf{a}})}$

By Lemma 3.2.8, ${\mathcal{M}}_{X}(\lambda,\nabla)_{e}$ is flat over $X\times
U$. Thus, all rectangles are cartesian by Remark 3.3.6. Moreover, two base
changes of normalization map
$({\mathcal{M}}_{X}(\lambda,\nabla)_{e})^{\mathrm{nm}}\rightarrow{\mathcal{M}}_{X}(\lambda,\nabla)_{e}$
are finite and birational. Finiteness is obvious, and birationality is
preserved by base change because the dense open subscheme
${\mathcal{M}}_{X^{0}}(\lambda,\nabla)_{e}\subset{\mathcal{M}}_{X}(\lambda,\nabla)_{e}$,
which is already normal by Proposition 3.2.7, is still dense after each base
change. This induces two surjective dashed arrows by [Sta18, Tag 035Q], and
the top dashed arrow is an isomorphism by Lemma 3.2.9.

Note that $M(\lambda,\nabla_{\mathbf{a}})$ and
${\mathcal{M}}(\lambda,\nabla_{\mathbf{a}})$ share the same special fiber.
Suppose that $M(\lambda,\nabla_{\mathbf{a}})$ is _not_ unibranch at
$\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$. Then there are at least two points
in the preimage of $\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ in
$M(\lambda,\nabla_{\mathbf{a}})^{\mathrm{nm}}$. Therefore, the preimage of
$\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ in
$({\mathcal{M}}(\lambda,\nabla)_{e})^{\mathrm{nm}}$ contains at least two
points. In turn, this implies that the preimage of
$\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ in
$({\mathcal{M}}_{X}(\lambda,\nabla)_{e})^{\mathrm{nm}}\times_{X\times
U}\mathbb{A}_{\mathbf{F}}^{1}$ contains at least two points. By the
surjectivity of the bottom dashed map, this contradicts Proposition 3.3.7.
This proves that $M(\lambda,\nabla_{\mathbf{a}})$ is unibranch at
$\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$.

Note that
${\mathcal{O}}(U(\widetilde{z},\lambda,\nabla_{\mathbf{a}}))[1/\varpi]$ is
regular domain by Proposition 3.2.7. Also, the preimage of
$U(\widetilde{z},\lambda,\nabla_{\mathbf{a}})$ in the special fiber of
$M(\lambda,\nabla_{\mathbf{a}})^{\mathrm{nm}}$ is connected by Proposition
3.3.7 and the above diagram. Then
${\mathcal{O}}(U(\widetilde{z},\lambda,\nabla_{\mathbf{a}}))^{\wedge_{p}}$ is
a domain by [LLHLMa, Lemma 3.7.2]. ∎

###  3.4. Products of local models

We now generalize Theorem 3.3.4 to products of local models. Let
${\mathcal{J}}$ be a finite set. Let
$\lambda=(\lambda_{j})_{j\in{\mathcal{J}}}\in X_{*}(T^{\vee})^{\mathcal{J}}$
be a dominant cocharacter and
$\widetilde{z}=(\widetilde{z}_{j})_{j\in{\mathcal{J}}}\in\widetilde{W}^{\vee,{\mathcal{J}}}$.
We define

$\displaystyle{\mathcal{M}}_{X,{\mathcal{J}}}(\lambda,\nabla)=\prod_{{j\in{\mathcal{J}}}}{\mathcal{M}}_{X}(\lambda_{j},\nabla)\subset(\mathrm{Gr}_{{\mathcal{G}},X}\times_{\mathbf{Z}}\mathbb{A}^{3})^{\mathcal{J}}.$

Let
${\mathbf{a}}=({\mathbf{a}}_{j})_{j\in{\mathcal{J}}}\in(\mathbb{A}^{3})^{\mathcal{J}}({\mathcal{O}})$.
We also define
$M_{\mathcal{J}}(\lambda,\nabla_{\mathbf{a}}):=\prod_{{j\in{\mathcal{J}}}}M(\lambda_{j},\nabla_{{\mathbf{a}}_{j}})$
and
$U_{\mathcal{J}}(\widetilde{z},\lambda,\nabla_{\mathbf{a}})):=\prod_{j\in{\mathcal{J}}}U(\widetilde{z}_{j},\lambda_{j},\nabla_{{\mathbf{a}}_{j}})$.
The following is the main result of this section.

######  Theorem 3.4.1.

There exists a non-empty open subscheme
$U\subset(\mathbb{A}^{3})^{\mathcal{J}}$ depending only on $\lambda$ and $e$,
such that if ${\mathbf{a}}\in U({\mathcal{O}})$, then
$M_{\mathcal{J}}(\lambda,\nabla_{\mathbf{a}})$ is unibranch at any point
$\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ contained in its special fiber. In
addition,
${\mathcal{O}}(U_{\mathcal{J}}(\widetilde{z},\lambda,\nabla_{\mathbf{a}}))^{\wedge_{p}}$
is a domain.

###### Proof.

Let $M_{j}:={\mathcal{M}}_{X}(\lambda_{j},\nabla)\rightarrow
X\times\mathbb{A}^{3}$ and $\widetilde{z}_{j}\in\widetilde{W}^{\vee}$. The
special fiber $M_{j}\times_{X}X_{0}$ intersect with a section
$\widetilde{z}_{j}$ only if $\widetilde{z}_{j}=wt_{\nu}$ for some $w\in W$ and
$\nu\in{\mathrm{Conv}}(\lambda_{j})$. Note that there are finitely many
$\widetilde{z}_{j}$ satisfying these conditions.

Let $\eta$ be the generic point of $\mathbb{A}^{3}$. We define $Fix_{j}$ to be
the set of $\widetilde{z}_{j}\in\widetilde{W}^{\vee}$ which intersects with
$M_{j}\times_{X\times\mathbb{A}^{3}}(X_{0}\times\eta)$. For any
$\widetilde{z}_{j}$ such that $\widetilde{z}_{j}=wt_{\nu}$ for some $w\in W$
and $\nu\in{\mathrm{Conv}}(\lambda_{j})$, consider a map

$\displaystyle
f_{\widetilde{z}_{j}}:(M_{j}\times_{X}X_{0})\cap\widetilde{z}\rightarrow\mathbb{A}^{3}.$

(Here, the intersection is taken inside
${\mathrm{Gr}_{{\mathcal{G}},X}\times_{X}X_{0}\times\mathbb{A}^{3}}$.) The
image of $f_{\widetilde{z}_{j}}$ is constructible, and it contains $\eta$ if
and only if $\widetilde{z}_{j}\in Fix_{j}$. Note that any constructible set
containing the generic point of irreducible scheme contains an open
neighborhood of the generic point. Thus, there exists an open neighborhood
$V^{\prime}_{j}$ of $\eta$ which is contained in the image of
$f_{\widetilde{z}_{j}}$ for all $\widetilde{z}_{j}\in Fix_{j}$ and the
complement of the image of $f_{\widetilde{z}_{j}}$ for all
$\widetilde{z}_{j}\notin Fix_{j}$.

Let $V_{j}$ be the intersection of $V^{\prime}_{j}$ with the open subscheme of
$\mathbb{A}^{3}$ satisfying the conclusion of Lemma 3.2.8 for $\lambda_{j}$
and $e$. Then $M_{j}|_{V_{j}}\rightarrow X\times V_{j}$ and
$\widetilde{z}_{j}\in Fix_{j}$ satisfy the assumptions of [LLHLMa, Corollary
3.6.2]. As a result, there exists an integer $e^{\prime}$ and an open
subscheme $U^{\prime}_{j}\subset V_{j}$ satisfying the following: for any
$\widetilde{z}_{j}\in Fix_{j}$ and ${\mathbf{a}}_{j}\in U_{j}({\mathcal{O}})$,
there exists a finite DVR extension ${\mathcal{O}}^{\prime}$ of
${\mathcal{O}}$ of degree $\leq e^{\prime}$ and an
${\mathcal{O}}^{\prime}$-point of $M_{j}$ lifting
$(\widetilde{z}_{j},{\mathbf{a}}_{j})\in
M(\lambda_{j},\nabla_{{\mathbf{a}}_{j}})(\mathbf{F})$.

Let $U_{j}\subset U_{j}^{\prime}$ be the open subscheme in which Theorem 3.3.4
holds for $\lambda_{j}$ and $ee^{\prime}!$. Then we define
$U=\prod_{j\in{\mathcal{J}}}U_{j}$. Let ${\mathcal{O}}^{\prime}/{\mathcal{O}}$
be the extension obtained by adjoining $e^{\prime}!$-th root of uniformizer
and $e^{\prime}!$-th root of unity. Note that ${\mathcal{O}}^{\prime}$
contains any extension of ${\mathcal{O}}$ of degree $\leq e^{\prime}$. For
$\widetilde{z}\in\prod_{j\in{\mathcal{J}}}Fix_{j}$ and ${\mathbf{a}}\in
U({\mathcal{O}})$, $M_{\mathcal{J}}(\lambda,\nabla_{\mathbf{a}})$ is unibranch
at $\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ if and only if the completed
local ring of $M_{\mathcal{J}}(\lambda,\nabla_{\mathbf{a}})$ at
$\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ is a domain, which is a subring of
the completed local ring of
$M_{\mathcal{J}}(\lambda,\nabla_{\mathbf{a}})\times_{\mathcal{O}}\operatorname{\mathrm{Spec}}{\mathcal{O}}^{\prime}$
at $\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$. Similarly,
${\mathcal{O}}(U_{\mathcal{J}}(\widetilde{z},\lambda,\nabla_{{\mathbf{a}}}))^{\wedge_{p}}$
is a subring of
${\mathcal{O}}(U_{\mathcal{J}}(\widetilde{z},\lambda,\nabla_{{\mathbf{a}}}\times_{\mathcal{O}}\operatorname{\mathrm{Spec}}{\mathcal{O}}^{\prime}))^{\wedge_{p}}$.
Thus, it suffices to prove the claim after replacing ${\mathcal{O}}$ by
${\mathcal{O}}^{\prime}$. Note that both the completed local ring of
$M_{\mathcal{J}}(\lambda,\nabla_{\mathbf{a}})\times_{\mathcal{O}}\operatorname{\mathrm{Spec}}{\mathcal{O}}^{\prime}$
at $\widetilde{z}_{\mathbf{F},{\mathbf{a}}}$ and
${\mathcal{O}}(U_{\mathcal{J}}(\widetilde{z},\lambda,\nabla_{{\mathbf{a}}}\times_{\mathcal{O}}\operatorname{\mathrm{Spec}}{\mathcal{O}}^{\prime}))^{\wedge_{p}}$
are completed tensor products of the completed local ring of
$M(\lambda_{j},\nabla_{{\mathbf{a}}_{j}})\times_{\mathcal{O}}\operatorname{\mathrm{Spec}}{\mathcal{O}}^{\prime}$
at $\widetilde{z}_{j,\mathbf{F},{\mathbf{a}}_{j}}$ and
${\mathcal{O}}(U_{\mathcal{J}}(\widetilde{z}_{j},\lambda_{j},\nabla_{{\mathbf{a}}_{j}}\times_{\mathcal{O}}\operatorname{\mathrm{Spec}}{\mathcal{O}}^{\prime}))^{\wedge_{p}}$,
which are known to be domain by Theorem 3.3.4. By the choice of
${\mathcal{O}}^{\prime}$, each
$M_{j}\times_{{\mathcal{O}}}\operatorname{\mathrm{Spec}}{\mathcal{O}}^{\prime}$
has ${\mathcal{O}}^{\prime}$-point lifting
$(\widetilde{z}_{j},{\mathbf{a}}_{j})$ in its special fiber. Then the two
claims follow from [KW09b, Proposition 2.2], and [BLGGT14, Lemma A.1.1]
respectively (as explained in the last two paragraphs of the proof of [LLHLMa,
Theorem 3.7.1]). ∎

###  3.5. Special fiber of naive local models in mixed characteristic

In the remainder of this section, we study the special fiber of the naive
local models in mixed characteristic. Recall that the naive local models in
mixed characteristic are defined by imposing the monodromy condition on the
Pappas–Zhu local models inside
$\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}=L^{+}{\mathcal{G}}_{\mathcal{O}}{\backslash}L{\mathcal{G}}_{\mathcal{O}}$.
The special fiber
$\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}\times_{\mathcal{O}}\operatorname{\mathrm{Spec}}\mathbf{F}$
is the affine flag variety
$\mathrm{Fl}:={\mathcal{I}}_{\mathbf{F}}{\backslash}L(\mathrm{GSp}_{4})_{\mathbf{F}}$.
Given ${\mathbf{a}}\in{\mathcal{O}}^{3}$, we define
$\mathrm{Fl}^{\nabla_{\mathbf{a}}}:=\mathrm{Fl}\times_{\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}}\mathrm{Gr}_{{\mathcal{G}},{\mathcal{O}}}^{\nabla_{\mathbf{a}}}$.

It is known by [PZ13, Theorem 9.3] that the special fiber
$M(\leq\lambda)_{\mathbf{F}}$ is the reduced union of the affine Schubert
cells $S^{\circ}_{\mathbf{F}}(\widetilde{z})$ for
$\widetilde{z}\in{\mathrm{Adm}}^{\vee}(\lambda)$. Thus, the underlying reduced
closed subscheme of
$M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$ is equal to
the reduced union of
$S^{\circ}_{\mathbf{F}}(\widetilde{z})\cap\mathrm{Fl}^{\nabla_{\mathbf{a}}}$
for $\widetilde{z}\in{\mathrm{Adm}}^{\vee}(\lambda)$.

For $\alpha\in\Phi$, we define $H_{\alpha}^{(0,1)}=\\{x\in
X^{*}(T)\times_{\mathbf{Z}}\mathbf{R}\mid 0<\langle
x,\alpha^{\vee}\rangle<1\\}$. The following is the main result of this
subsection.

######  Theorem 3.5.1.

Let $h>0$ be an integer, $\widetilde{w}\in\widetilde{W}$, and
${\mathbf{a}}\in{\mathcal{O}}^{3}$. Suppose that $\widetilde{w}$ is $h$-small
and ${\mathbf{a}}\bmod\varpi\in\mathbf{F}^{3}$ is $h$-generic (Definition
2.1.8). Then
$S^{\circ}_{\mathbf{F}}(\widetilde{w}^{*})\cap\mathrm{Fl}^{\nabla_{\mathbf{a}}}$
is an affine space of dimension
$4-\\#\left\\{\alpha\in\Phi^{+}\mid\widetilde{w}(A_{0})\subset
H_{\alpha}^{(0,1)}\right\\}$.

As a Corollary, we get a classification of top($=4$)-dimensional irreducible
components of $M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$.
If $d\in\mathbf{Z}_{\geq 0}$ and $X$ is a reduced scheme, we let
$\mathrm{Irr}_{d}X$ denote the set of $d$-dimensional irreducible closed
subschemes of $X$.

######  Definition 3.5.2.

We say that $\widetilde{w}\in\widetilde{W}$ is _regular_ if
$\widetilde{w}^{*}(A_{0})\nsubseteq H^{(0,1)}_{\alpha}$ for any
$\alpha\in\Phi^{+}$. Let $\lambda\in X_{*}(T^{\vee})$. We define
${\mathrm{Adm}}_{\mathrm{reg}}^{\vee}(\lambda)\subset{\mathrm{Adm}}^{\vee}(\lambda)$
to be the subset of $\widetilde{z}\in{\mathrm{Adm}}^{\vee}(\lambda)$ such that
$\widetilde{z}^{*}$ is regular. We similarly define
${\mathrm{Adm}}_{\mathrm{reg}}(\nu)$ for $\nu\in X^{*}(T)$. Then $(-)^{*}$
induces a bijection between ${\mathrm{Adm}}_{\mathrm{reg}}(\nu)$ and
${\mathrm{Adm}}_{\mathrm{reg}}^{\vee}(\phi(\nu))$.

######  Corollary 3.5.3.

Let $\lambda\in X^{*}(T^{\vee})$ be a dominant cocharacter. Suppose that
${\mathbf{a}}\bmod\varpi\in\mathbf{F}^{n}$ is $h_{\lambda}$-generic. There is
a bijection

$\displaystyle{\mathrm{Adm}}_{\mathrm{reg}}^{\vee}(\lambda)$
$\displaystyle\xrightarrow{\sim}\mathrm{Irr}_{4}M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$
$\displaystyle\widetilde{z}$
$\displaystyle\mapsto\overline{(S^{\circ}_{\mathbf{F}}(\widetilde{z})\cap\mathrm{Fl}^{\nabla_{\mathbf{a}}})}.$

###### Proof.

This follows from [PZ13, Theorem 9.3] and Theorem 3.5.1. To apply Theorem
3.5.1, note that
$\widetilde{z}\in{\mathrm{Adm}}^{\vee}_{\mathrm{reg}}(\lambda)$ is
$h_{\lambda}$-small. ∎

To prove the Theorem 3.5.1, we describe
$S^{\circ}_{\mathbf{F}}(\widetilde{z})$ using explicit coordinates. A version
of Bruhat decomposition says that we have a double coset decomposition

$\displaystyle(L\mathrm{GSp}_{4})_{\mathbf{F}}=\cup_{\widetilde{z}\in\widetilde{W}^{\vee}}{\mathcal{I}}_{\mathbf{F}}\widetilde{z}{\mathcal{I}}_{\mathbf{F}}$

and the open Schubert cell $S^{\circ}_{\mathbf{F}}(\widetilde{z})$ can be
identified with
${\mathcal{I}}_{\mathbf{F}}{\backslash}{\mathcal{I}}_{\mathbf{F}}\widetilde{z}{\mathcal{I}}_{\mathbf{F}}$.

Let $L^{--}{\mathcal{G}}_{\mathbf{F}}\subset L{\mathcal{G}}_{\mathbf{F}}$ be
the subfunctor given by

$\displaystyle
L^{--}{\mathcal{G}}_{\mathbf{F}}(R)=\left\\{g\in\mathrm{GSp}_{4}\left(R\left[\frac{1}{v}\right]\right)\mid
g\bmod\frac{1}{v}\in\overline{B}(R)\right\\}$

for any $\mathbf{F}$-algebra $R$. We define
$N_{\widetilde{z}}:=\widetilde{z}^{-1}L^{--}{\mathcal{G}}_{\mathbf{F}}\widetilde{z}\cap{\mathcal{I}}_{\mathbf{F}}$.

Recall that the duality isomorphism $\phi$ identifies a coroot $\alpha^{\vee}$
of $G$ and a root $\phi(\alpha^{\vee})$ of $G^{\vee}$, and we write
$U_{\alpha^{\vee}}$ for the root subgroup of $G^{\vee}$ associated to
$\phi(\alpha^{\vee})$. Given
$(\alpha^{\vee},m)\in\Phi^{\vee}\times\mathbf{Z}$, we let
$U_{\alpha^{\vee},m}\subset LU_{\alpha^{\vee}}$ denote the subfunctor such
that for any $\mathbf{F}$-algebra $R$, $U_{\alpha^{\vee},m}(R)\subset
U_{\alpha^{\vee}}(R(\\!(v)\\!))$ is identified with $v^{m}R\subset
R(\\!(v)\\!)$ under the isomorphism $U_{\alpha^{\vee}}\simeq\mathbf{G}_{a}$.

In what follows, we fix $x\in A_{0}$. Note that $\lceil\langle
x,\alpha^{\vee}\rangle\rceil=\delta_{\alpha^{\vee}>0}$ and $\lfloor\langle
x,\alpha^{\vee}\rangle\rfloor=-\delta_{\alpha^{\vee}<0}$.

######  Proposition 3.5.4.

Let $\widetilde{z}\in\widetilde{W}^{\vee}$.

  1. (1)

We have the following decomposition

$\displaystyle
N_{\widetilde{z}}\simeq\prod_{(\alpha^{\vee},m)\in\Phi^{\vee}_{\widetilde{z}}}U_{-\alpha^{\vee},m}$

where the product runs over the set
$\Phi^{\vee}_{\widetilde{z}}=\left\\{(\alpha^{\vee},m)\in\Phi^{\vee}\times\mathbf{Z}\mid\langle
x,\alpha^{\vee}\rangle<m<\langle\widetilde{z}^{*}(x),\alpha^{\vee}\rangle\right\\}$.

  2. (2)

There is an isomorphism

$\displaystyle{\mathcal{I}}_{\mathbf{F}}\times_{\mathbf{F}}\left\\{\widetilde{z}\right\\}\times_{\mathbf{F}}N_{\widetilde{z}}\simeq{\mathcal{I}}_{\mathbf{F}}\widetilde{z}{\mathcal{I}}_{\mathbf{F}}$

given by multiplication. This induces an isomorphism
$\widetilde{z}N_{\widetilde{z}}\simeq S^{\circ}_{\mathbf{F}}(\widetilde{z})$.

###### Proof.

  1. (1)

We claim that $U_{-\alpha^{\vee},m}\subset N_{\widetilde{z}}$ if and only if
$(\alpha^{\vee},m)\in\Phi^{\vee}_{\widetilde{z}}$. It is easy to check that
$U_{-\alpha^{\vee},m}\subset{\mathcal{I}}_{\mathbf{F}}$ if and only if
$\delta_{\alpha^{\vee}>0}\leq m$. Let
$\widetilde{z}^{*}=st_{\nu}\in\widetilde{W}$. Direct computation shows that
$\widetilde{z}U_{-\alpha^{\vee},m}\widetilde{z}^{-1}=U_{-s^{-1}(\alpha^{\vee}),m-\langle\nu,\alpha^{\vee}\rangle}$.
Thus
$U_{-\alpha,m}\subset\widetilde{z}^{-1}L^{--}{\mathcal{G}}_{\mathbf{F}}\widetilde{z}$
if and only if $m-\langle\nu,\alpha^{\vee}\rangle<\langle
x,s^{-1}(\alpha^{\vee})\rangle$, or equivalently $m<\langle
s(x)+\nu,\alpha^{\vee}\rangle$. This proves the claim.

Observe that $U_{-\alpha^{\vee},m}\subset N_{\widetilde{z}}$ only if
$\langle\widetilde{z}^{*}(x),\alpha^{\vee}\rangle>0$. Let $w\in W$ be the
unique element such that $w\widetilde{z}^{*}\in\widetilde{W}^{+}$. If
$\langle\widetilde{z}^{*}(x),\alpha^{\vee}\rangle=\langle
w\widetilde{z}^{*}(x),w(\alpha^{\vee})\rangle>0$, then $w(\alpha^{\vee})>0$.
In other words, $\phi(w)N_{\widetilde{z}}\phi(w)^{-1}\subset L\overline{U}$.
Then the claimed decomposition follows from the decomposition of
$\overline{U}$.

  2. (2)

The injectivity of the multiplication map is obvious. The surjectivity follows
from
${\mathcal{I}}_{\mathbf{F}}=((\widetilde{z}^{-1}{\mathcal{I}}_{\mathbf{F}}\widetilde{z})\cap{\mathcal{I}}_{\mathbf{F}})N_{\widetilde{z}}$.

∎

Let
$d_{\alpha^{\vee},\widetilde{z}}=\lfloor\langle\widetilde{z}^{*}(x),\alpha^{\vee}\rangle\rfloor-\lceil\langle
x,\alpha^{\vee}\rangle\rceil$.

######  Corollary 3.5.5.

Let $R$ be a Noetherian $\mathbf{F}$-algebra. If $M\in N_{\widetilde{z}}(R)$,
we can write
$M=\prod_{\alpha^{\vee}\in\Phi^{\vee}}v^{\delta_{\alpha^{\vee}>0}}M_{\alpha^{\vee}}$
where $M_{\alpha^{\vee}}\in U_{-\alpha^{\vee}}(R[v])\simeq R[v]$ is a
polynomial of degree at most $d_{\alpha^{\vee},\widetilde{z}}$.

###### Proof of Theorem 3.5.1.

Let $\widetilde{z}=\widetilde{w}^{*}$ and $w\in W$ be the unique element such
that $w\widetilde{w}\in\widetilde{W}^{+}$. We follow the proof of [LLHLMa,
Theorem 4.2.4] using the Corollary 3.5.5 instead of Corollary 4.2.12 of _loc.
cit._. Indeed, _loc. cit._ shows that the monodromy condition on
$\widetilde{z}M$ for some $M\in N_{\widetilde{z}}(R)$ implies that
coefficients in the polynomial $M_{\alpha^{\vee}}$ are determined by its top
degree coefficient and coefficients of $M_{\alpha^{\prime\vee}}$ such that
$w(\alpha^{\prime\vee})<w(\alpha^{\vee})$. Inductively, this shows that
$S^{\circ}_{\mathbf{F}}(\widetilde{z})\cap\mathrm{Fl}^{\nabla_{\mathbf{a}}}$
is an affine space with coordinates given by the top degree coefficients of
$M_{\alpha^{\vee}}$. Thus, the dimension of
$S^{\circ}_{\mathbf{F}}(\widetilde{z})\cap\mathrm{Fl}^{\nabla_{\mathbf{a}}}$
is equal to

$\displaystyle\\#\left\\{\alpha^{\vee}\in\Phi^{\vee}\mid
U_{-\alpha^{\vee},m}\subset N_{\widetilde{z}}\text{ for some $m\geq
0$}\right\\}.$

This is equal to the number of $\alpha^{\vee}\in\Phi^{\vee}$ such that
$\langle\widetilde{z}^{*}(x),\alpha^{\vee}\rangle>0$ and
$\widetilde{z}^{*}(x)$ and $x$ does not lie in the same $\alpha$-strip. The
first condition says that $w(\alpha^{\vee})\in\Phi^{\vee,+}$, which has size
4. For such $\alpha$, the second condition holds unless
$\alpha^{\vee}\in\Phi^{\vee,+}$ and $\widetilde{z}^{*}(A_{0})\subset
H^{(0,1)}_{\alpha}$. This completes the proof. ∎

We now make comparison between irreducible components in
$M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$ for different
choices of $\lambda$ and ${\mathbf{a}}$. We define
$L\mathrm{GSp}_{4}^{\nabla_{0}}\subset L\mathrm{GSp}_{4}$ as the subsheaf
given by

$\displaystyle R\mapsto\left\\{A\in L\mathrm{GSp}_{4}(R)\mid
v\frac{d{A}}{dv}A^{-1}\in\frac{1}{v}L^{+}{\mathcal{M}}_{\mathbf{F}}(R)\right\\}$

for $\mathbf{F}$-algebra $R$. Let $\mathrm{Fl}^{\nabla_{0}}$ denote the fpqc-
quotient sheaf
${\mathcal{I}}_{\mathbf{F}}{\backslash}L\mathrm{GSp}_{4}^{\nabla_{0}}$.

######  Remark 3.5.6.

Let ${\mathbf{a}}\in{\mathcal{O}}^{3}$ and
$\widetilde{z}=s^{-1}t_{\mu}\in\widetilde{W}^{\vee}$ such that
${\mathbf{a}}\equiv s^{-1}(\mu)\bmod\varpi$. Choose a dominant cocharacter
$\lambda\in X_{*}(T^{\vee})$ and $\widetilde{w}\in\widetilde{W}$. A direct
computation shows that (cf. [LLHLMa, Proposition 4.3.1])

$\displaystyle
M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}\widetilde{z}$
$\displaystyle=M(\leq\lambda)_{\mathbf{F}}\widetilde{z}\cap\mathrm{Fl}^{\nabla_{0}}$
$\displaystyle(S^{\circ}_{\mathbf{F}}(\widetilde{w}^{*})\cap\mathrm{Fl}^{\nabla_{\mathbf{a}}})\widetilde{z}$
$\displaystyle=(S^{\circ}_{\mathbf{F}}(\widetilde{w}^{*})\widetilde{z})\cap\mathrm{Fl}^{{\nabla_{0}}}.$

This allows us to compare
$M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$ for different
$\lambda$ and ${\mathbf{a}}$ and its irreducible components inside
$\mathrm{Fl}^{\nabla_{0}}$.

######  Definition 3.5.7.

Let $\widetilde{s}\in\widetilde{W}$ and
$\widetilde{w}_{1},\widetilde{w}_{2}\in\widetilde{W}^{+}$. We define

$\displaystyle
S^{\circ}_{\mathbf{F}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{s})$
$\displaystyle:=S^{\circ}_{\mathbf{F}}((\widetilde{w}_{2}^{-1}w_{0}\widetilde{w}_{1})^{*})\widetilde{s}^{*}\subset\mathrm{Fl}$
$\displaystyle
S^{\circ}_{\mathbf{F}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{s})^{\nabla_{0}}$
$\displaystyle:=S^{\circ}_{\mathbf{F}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{s})\cap\mathrm{Fl}^{\nabla_{0}}\subset\mathrm{Fl}^{\nabla_{0}}$
$\displaystyle
S^{\nabla_{0}}_{\mathbf{F}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{s})$
$\displaystyle:=\overline{S^{\circ}_{\mathbf{F}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{s})^{\nabla_{0}}}$

where the closure is taken in $\mathrm{Fl}^{\nabla_{0}}$.

######  Lemma 3.5.8.

Let $\widetilde{s}=t_{\mu}s$, $\widetilde{w}_{1}$, and $\widetilde{w}_{2}$ be
as above. If further $\widetilde{w}_{2}^{-1}w_{0}\widetilde{w}_{1}$ is
$m$-small and $\widetilde{s}$ is $m$-generic for some integer $m$, then
$S^{\nabla_{0}}_{\mathbf{F}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{s})$
is irreducible closed subvariety of $\mathrm{Fl}^{\nabla_{0}}$ of dimension 4.

###### Proof.

By [LLHLMa, Proposition 2.1.5], $\widetilde{w}_{2}^{-1}w_{0}\widetilde{w}_{1}$
is regular, and
$S^{\nabla_{0}}_{\mathbf{F}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{s})$
is isomorphic to
$S^{\circ}_{\mathbf{F}}(\widetilde{w}^{*})\cap\mathrm{Fl}^{\nabla_{{\mathbf{a}}}}$
by Remark 3.5.6, where
$\widetilde{w}=\widetilde{w}_{2}^{-1}w_{0}\widetilde{w}_{1}$ and
${\mathbf{a}}\in\mathbf{Z}^{3}$ such that
${\mathbf{a}}\equiv\phi(s^{-1}(\mu))\bmod p$. Then the claim follows from
Theorem 3.5.1. ∎

In fact, many of
$S^{\nabla_{0}}_{\mathbf{F}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{s})$
for different $\widetilde{s},\widetilde{w}_{1},$ and $\widetilde{w}_{2}$ are
equal.

######  Proposition 3.5.9.

Let $\widetilde{s}\in\widetilde{W}$ and
$\widetilde{w}_{1},\widetilde{w}_{2}\in\widetilde{W}^{+}$.

  1. (1)

Suppose that, for $i=1,2$, $\widetilde{w}_{i}$ is $m_{i}$-small for some
integer $m_{i}$ and $\widetilde{s}$ is $(m_{1}+m_{2})$-generic. Then we have

$\displaystyle
S^{\nabla_{0}}_{\mathbf{F}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{s})=S^{\nabla_{0}}_{\mathbf{F}}(\widetilde{w}_{1},e,\widetilde{s}\widetilde{w}_{2}^{-1}).$

  2. (2)

Suppose that $\widetilde{w}_{1}$ is in $\widetilde{W}_{1}^{+}$ and
$\widetilde{s}$ is 3-generic. For all $w\in W$, we have

$\displaystyle
S^{\nabla_{0}}_{\mathbf{F}}(\widetilde{w}_{1},e,\widetilde{s})=S^{\nabla_{0}}_{\mathbf{F}}(\widetilde{w}_{1},e,\widetilde{s}w).$

###### Proof.

The first item is [LLHLMa, Proposition 4.3.5], and the second item is
Proposition 4.3.6 in _loc. cit._. Note that their proofs generalize to our
setup straightforwardly. ∎

######  Definition 3.5.10.

Let $(\widetilde{w}_{1},\omega)\in\widetilde{W}_{1}^{+}\times X^{*}(T)$ with
$t_{\omega}$ being 3-generic. We define
$C_{(\widetilde{w}_{1},\omega)}:=S^{\nabla_{0}}_{\mathbf{F}}(\widetilde{w}_{1},e,\widetilde{s})$
for any $\widetilde{s}\in\widetilde{W}$ such that $\widetilde{s}(0)=\omega$.

This is well-defined by Proposition 3.5.9. Since $\widetilde{w}_{1}$ is
3-small, $C_{(\widetilde{w}_{1},\omega)}$ is irreducible of dimension 4 by
Lemma 3.5.8.

Let $\lambda\in X_{*}(T^{\vee})$ be a regular dominant cocharacter. Recall
that the set of admissible pair

$\displaystyle\mathrm{AP}(\phi^{-1}(\lambda))=\left\\{(\widetilde{w}_{1},\widetilde{w}_{2})\in(\widetilde{W}_{1}^{+}\times\widetilde{W}^{+})/X^{0}(T)\mid\widetilde{w}_{1}\uparrow
t_{\phi^{-1}(\lambda)-\eta}\widetilde{w}_{h}^{-1}\widetilde{w}_{2}\right\\}.$

By [LLHLMa, Corollarly 2.1.7], there is a bijection

(3.5.11)
$\displaystyle\begin{split}\mathrm{AP}(\phi^{-1}(\lambda))&\xrightarrow{\sim}{\mathrm{Adm}}^{\vee}_{\mathrm{reg}}(\lambda)\\\
(\widetilde{w}_{1},\widetilde{w}_{2})&\mapsto(\widetilde{w}_{2}^{-1}w_{0}\widetilde{w}_{1})^{*}.\end{split}$

######  Theorem 3.5.12.

Let $\lambda\in X_{*}(T^{\vee})$ be a regular dominant cocharacter and
${\mathbf{a}}\in{\mathcal{O}}^{3}$. Let
$\widetilde{z}=s^{-1}t_{\mu}\in\widetilde{W}^{\vee}$ be
$(h_{\lambda}+3)$-generic such that
${\mathbf{a}}\equiv\phi(s^{-1}(\mu))\bmod\varpi$. We have a bijection

$\displaystyle\mathrm{AP}(\phi^{-1}(\lambda))$
$\displaystyle\xrightarrow{\sim}\mathrm{Irr}_{4}(M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}\widetilde{z})$
$\displaystyle(\widetilde{w}_{1},\widetilde{w}_{2})$ $\displaystyle\mapsto
C_{(\widetilde{w}_{1},\widetilde{z}^{*}\widetilde{w}_{2}^{-1}(0))}.$

###### Proof.

By Corollary 3.5.3, Remark 3.5.6, and (3.5.11), there is a bijection

$\displaystyle\mathrm{AP}(\phi^{-1}(\lambda))$
$\displaystyle\xrightarrow{\sim}\mathrm{Irr}_{4}(M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}\widetilde{z})$
$\displaystyle(\widetilde{w}_{1},\widetilde{w}_{2})$ $\displaystyle\mapsto
S_{\mathbf{F}}^{{\nabla_{0}}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{z}^{*}).$

Then the claim follows from the equlities

$\displaystyle
S_{\mathbf{F}}^{{\nabla_{0}}}(\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{z}^{*})=S_{\mathbf{F}}^{{\nabla_{0}}}(\widetilde{w}_{1},e,\widetilde{z}^{*}\widetilde{w}_{2}^{-1})=C_{(\widetilde{w}_{1},\widetilde{z}^{*}\widetilde{w}_{2}^{-1}(0))}$

by Proposition 3.5.9, the genericity assumption on $\widetilde{z}$, and noting
that $\widetilde{w}_{1}$ is 3-small and $\widetilde{w}_{2}$ is
$h_{\lambda}$-small. ∎

###  3.6. Matching irreducible components and Serre weights

Let ${\mathcal{I}}_{1,F}\subset{\mathcal{I}}_{\mathbf{F}}$ be the subfunctor
given by

$\displaystyle R\mapsto\left\\{A\in\mathrm{GSp}_{4}(R\llbracket
v\rrbracket)\mid A\bmod v\in U(R)\right\\}$

for $\mathbf{F}$-algebra $R$. We define $\widetilde{\mathrm{Fl}}$ as the fpqc-
quotient sheaf ${\mathcal{I}}_{1,\mathbf{F}}{\backslash}L\mathrm{GSp}_{4}$.
Then the natural quotient map
$\Psi:\widetilde{\mathrm{Fl}}\rightarrow\mathrm{Fl}$ is a
$T^{\vee}_{\mathbf{F}}$-torsor.

If $X\subset\mathrm{Fl}$ is a closed subvariety, we let $\widetilde{X}$ denote
the pullback $X\times_{\mathrm{Fl}}\widetilde{\mathrm{Fl}}$. Then
$\widetilde{X}\rightarrow X$ is again a $T^{\vee}_{\mathbf{F}}$-torsor. Let
$\lambda\in X_{*}(T^{\vee})$ be a dominant cocharacter and
${\mathbf{a}}\in{\mathcal{O}}^{3}$. Then we have the following
$T^{\vee}_{\mathbf{F}}$-torsors

$\displaystyle\widetilde{M}(\leq\lambda)_{\mathbf{F}}$
$\displaystyle\rightarrow M(\leq\lambda)_{\mathbf{F}}$
$\displaystyle\widetilde{M}^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$
$\displaystyle\rightarrow
M^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$
$\displaystyle\widetilde{M}(\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$
$\displaystyle\rightarrow M(\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}.$

For $(\widetilde{w},\omega)\in\widetilde{W}_{1}^{+}\times X^{*}(T)$ such that
$t_{\omega}$ is 3-generic, we also have a $T^{\vee}_{\mathbf{F}}$-torsor
$\widetilde{C}_{(\widetilde{w},\omega)}\rightarrow
C_{(\widetilde{w},\omega)}$.

Let ${\mathcal{J}}=\operatorname{\mathrm{Hom}}(k,\mathbf{F})$. For
${j\in{\mathcal{J}}}$, if $X_{j}\subset\mathrm{Fl}$ (resp.
$\widetilde{X}_{j}\subset\widetilde{\mathrm{Fl}}$) is a closed subvariety, we
let $X_{\mathcal{J}}:=\prod_{j\in{\mathcal{J}}}X_{j}$ (resp.
$\widetilde{X}_{\mathcal{J}}:=\prod_{j\in{\mathcal{J}}}\widetilde{X}_{j}$) to
be the closed subvariety in $\mathrm{Fl}^{\mathcal{J}}$ (resp.
$\widetilde{\mathrm{Fl}}^{\mathcal{J}}$). In particular, given a dominent
cocharacter $\lambda=(\lambda_{j})_{j\in{\mathcal{J}}}\in
X_{*}(\underline{T}^{\vee})$ and
${\mathbf{a}}=({\mathbf{a}}_{j})_{j\in{\mathcal{J}}}\in({\mathcal{O}}^{3})^{\mathcal{J}}$,
we have

$\displaystyle
M^{\mathrm{nv}}_{\mathcal{J}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$
$\displaystyle=\prod_{j\in{\mathcal{J}}}M(\leq\lambda_{j},\nabla_{{\mathbf{a}}_{j}})_{\mathbf{F}}$
$\displaystyle\widetilde{M}_{\mathcal{J}}^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}$
$\displaystyle=\prod_{j\in{\mathcal{J}}}\widetilde{M}(\leq\lambda_{j},\nabla_{{\mathbf{a}}_{j}})_{\mathbf{F}}.$

Let
$\widetilde{z}=(\widetilde{z}_{j})_{j\in{\mathcal{J}}}\in\underline{\widetilde{W}}^{\vee}$
such that $\widetilde{z}_{j}=s_{j}^{-1}t_{\mu_{j}}$ and
${\mathbf{a}}_{j}\equiv s_{j}^{-1}(\mu_{j})\bmod\varpi$. We have the following
cartesian diagram

${\widetilde{M}_{\mathcal{J}}^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}}$${\widetilde{\mathrm{Fl}}_{\mathcal{J}}^{\nabla_{0}}}$${M_{\mathcal{J}}^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}}$${\mathrm{Fl}_{\mathcal{J}}^{\nabla_{0}}}$$\scriptstyle{r_{\widetilde{z}}}$$\scriptstyle{r_{\widetilde{z}}}$

where $r_{\widetilde{z}}$ is right translation by $\widetilde{z}$. Let
$(\widetilde{w}_{1},\widetilde{w}_{2})\in\mathrm{AP}(\phi^{-1}(\lambda))=\prod_{j\in{\mathcal{J}}}\mathrm{AP}(\phi^{-1}(\lambda_{j}))$.
If $\widetilde{z}$ is $(h_{\lambda}+3)$-generic, by Theorem 3.5.12, we have a
$7f$-dimensional irreducible component of
$\widetilde{M}^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}}\widetilde{z}$
given by

$\displaystyle\widetilde{C}_{(\widetilde{w}_{1},\widetilde{z}^{*}\widetilde{w}_{2}^{-1}(0))}:=\prod_{j\in{\mathcal{J}}}\widetilde{C}_{(\widetilde{w}_{1,j},\widetilde{z}_{j}^{*}\widetilde{w}_{2,j}^{-1}(0))}.$

######  Definition 3.6.1.

Let $\sigma$ be a 3-deep Serre weight and $\zeta$ be an algebraic central
character (Definition 2.5.4) lifting central character of $\sigma$. Let
$(\widetilde{w}_{1},\omega)\in\underline{\widetilde{W}}_{1}^{+}\times
X^{*}(\underline{T})$ be the lowest alcove presentation of $\sigma$
corresponding to $\zeta$. We define
$C_{\sigma}^{\zeta}:=C_{(\widetilde{w}_{1},\omega)}$ and
$\widetilde{C}^{\zeta}_{\sigma}:=\widetilde{C}_{(\widetilde{w}_{1},\omega)}$.

######  Theorem 3.6.2.

Let $\lambda\in X_{*}(\underline{T}^{\vee})$ be a regular dominant
cocharacter. Let $R$ be a Deligne–Lusztig representation with
$\max\\{6,h_{\lambda}+3\\}$-generic lowest alcove presentation $(s,\mu)$ that
is $(\lambda-\eta)$-compatible with $\zeta\in X^{*}(\underline{Z})$. Let
${\mathbf{a}}\in({\mathcal{O}}^{3})^{\mathcal{J}}$ such that
${\mathbf{a}}\equiv\phi(s^{-1}(\mu+\eta))\bmod\varpi$. Then we have

$\displaystyle\mathrm{Irr}_{4f}\left((M_{\mathcal{J}}^{\mathrm{nv}}(\leq\lambda,\nabla_{\mathbf{a}})_{\mathbf{F}})\phi(s^{-1}t_{\mu+\eta})\right)=\left\\{C_{\sigma}^{\zeta}\mid\sigma\in\mathrm{JH}(\overline{R}\otimes
W(\phi^{-1}(\lambda)-\eta))\right\\}.$

###### Proof.

This follows from Proposition 2.5.6 and Theorem 3.5.12. ∎

###  3.7. Torus fixed points

The torus $T^{\vee,{\mathcal{J}}}$ acts on $\mathrm{Fl}^{\mathcal{J}}$ by
right translation and stablizes $\mathrm{Fl}_{\mathcal{J}}^{\nabla_{0}}$. The
natural inclusion
${\widetilde{W}}^{\vee,{\mathcal{J}}}\subset\mathrm{Fl}^{\mathcal{J}}$
(sending $wt_{\nu}\in\widetilde{W}^{\vee,{\mathcal{J}}}$ to
$wv^{\nu}\in\mathrm{Fl}^{\mathcal{J}}(\mathbf{F})$) identifies
${\widetilde{W}}^{\vee,{\mathcal{J}}}$ and the set of
$T^{\vee,{\mathcal{J}}}$-fixed points in $\mathrm{Fl}^{\mathcal{J}}$. Note
that $\widetilde{W}^{\vee,{\mathcal{J}}}$ is contained in
$\mathrm{Fl}_{\mathcal{J}}^{\nabla_{0}}$. In this subsection, we prove the
following result on the $T^{\vee,{\mathcal{J}}}$-fixed points of irreducible
components in $\mathrm{Fl}_{\mathcal{J}}^{\nabla_{0}}$.

######  Theorem 3.7.1.

Let $\sigma$ be a $3$-deep Serre weight with a lowest alcove presentation
$(\widetilde{w}_{1},\omega)$. Let
$C_{(\widetilde{w}_{1},\omega)}^{T^{\vee,{\mathcal{J}}}}$ be the set of
$T^{\vee,{\mathcal{J}}}$-fixed points in $C_{(\widetilde{w}_{1},\omega)}$.

  1. (1)

We have $\left\\{(t_{\omega}w\widetilde{w}_{1})^{*}\mid w\in
W^{\mathcal{J}}\right\\}\subset
C_{(\widetilde{w}_{1},\omega)}^{T^{\vee,{\mathcal{J}}}}\subset\\{(t_{\omega}\widetilde{w})^{*}\mid\widetilde{w}\in\widetilde{W}^{\mathcal{J}},\widetilde{w}\leq
w_{0}\widetilde{w}_{1}\\}$.

  2. (2)

For each ${j\in{\mathcal{J}}}$, there is a polynomial
$P_{\widetilde{w}_{1},j}\in\mathbf{Z}[X_{1},X_{2},X_{3}]$ depending only on
$\widetilde{w}_{1,j}\in\widetilde{W}_{1}^{+}$ such that if
$P_{\widetilde{w}_{1,j}}(\omega_{j})\not\equiv 0\bmod p$, then
$C_{(\widetilde{w}_{1},\omega)}^{T^{\vee,{\mathcal{J}}}}=\\{(t_{\omega}\widetilde{w})^{*}\mid\widetilde{w}\in\widetilde{W}^{\mathcal{J}},\widetilde{w}\leq
w_{0}\widetilde{w}_{1}\\}$

###### Proof.

It suffices to prove the claim for ${\mathcal{J}}=\\{*\\}$. By definition,
$C_{(\widetilde{w}_{1},\omega)}=S_{\mathbf{F}}^{\nabla_{0}}(\widetilde{w}_{1},e,t_{\omega})$.
By Proposition 3.5.9, for any $w\in W$, the latter is equal to
$S_{\mathbf{F}}^{\nabla_{0}}(\widetilde{w}_{1},e,t_{\omega}ww_{0})$ which
contains $(t_{\omega}w\widetilde{w}_{1})^{*}$. This proves the first inclusion
in item (1).

Since $C_{(\widetilde{w}_{1},\omega)}$ is contained in
$S_{\mathbf{F}}(\widetilde{w}_{1}^{*}w_{0}t_{\omega})$, the second inclusion
in the item (2) follows from the standard result on torus fixed points in
Schubert varieties.

Finally, we prove item (2) by following the proof of [LLHLMa, Proposition
4.7.3]. Since the proof of _loc. cit._ generalizes to our setup, we only
sketch the argument. Let $\widetilde{w}\in\widetilde{W}$ and
$\widetilde{w}\leq w_{0}\widetilde{w}_{1}$. By Lemma 4.7.1 in _loc. cit._ ,
$(t_{\omega}\widetilde{w})^{*}\in C_{(\widetilde{w}_{1},\omega)}^{T^{\vee}}$
if and only if
$S_{\mathbf{F}}^{\circ}(\widetilde{w}_{1}^{*}w_{0})\cap\mathrm{Fl}^{\nabla_{\phi(\omega)}}$
has nonempty intersection with the open neighborhood
$L^{--}{\mathcal{G}}\widetilde{w}^{*}$. Note that here we interpret
$\phi(\omega)$ as an element in $\mathbf{Z}^{3}$.

We claim that the intersection is nonempty if $\phi(\omega)\bmod
p\in\mathbf{F}^{3}$ is contained in some nonempty open subscheme
$U\subset\mathbb{A}^{3}$. Then we take
$P_{\widetilde{w}_{1}}(X_{1},X_{2},X_{3})$ to be a polynomial such that
$P_{\widetilde{w}_{1}}\circ\phi^{-1}$ vanishes on the complement of $U$ (here,
we view $\phi$ as a map $\mathbb{A}^{3}\rightarrow\mathbb{A}^{3}$). The idea
is that one can consider affine flag variety ${\mathcal{F}}l_{\mathbf{Z}}$ and
its open Schubert variety
${\mathcal{S}}^{\circ}_{\mathbf{Z}}(\widetilde{w}_{1}^{*}w_{0})$ defined over
$\mathbf{Z}$. Then we impose the monodromy condition with parameters
$(b,{\mathbf{a}})\in\mathbb{A}^{1}\times_{\mathbf{Z}}\mathbb{A}^{3}$ given by

$\displaystyle
bv\frac{dg}{dv}g^{-1}+g\mathrm{Diag}(\operatorname{{\mathrm{std}}}({\mathbf{a}}))g^{-1}\in\frac{1}{v}L^{+}{\mathcal{M}}$

which induces a closed subscheme ${\mathcal{Y}}^{\circ}$ in
${\mathcal{S}}^{\circ}_{\mathbf{Z}}(\widetilde{w}_{1}^{*}w_{0})\times\mathbb{A}^{1}\times_{\mathbf{Z}}\mathbb{A}^{3}$.
Its fiber at
$(1,\phi(\omega))\in\mathbb{A}^{1}\times_{\mathbf{Z}}\mathbb{A}^{3}(\mathbf{F})$
is equal to
$S_{\mathbf{F}}^{\circ}(\widetilde{w}_{1}^{*}w_{0})\cap\mathrm{Fl}^{\nabla_{\phi(\omega)}}$.
Meanwhile the reduced fiber at
$(0,{\mathbf{a}})\in\mathbb{A}^{1}\times_{\mathbf{Z}}\mathbb{A}^{3}(\mathbf{C})$
is an open dense subscheme of an irreducible component
$\mathrm{Fl}_{v{\mathbf{a}},\widetilde{w}_{1}}$ of a certain affine Springer
fiber considered in Appendix A. By Theorem A.3, this reduced fiber has
nonempty intersection with $L^{--}{\mathcal{G}}_{k}\widetilde{w}^{*}$. Thus
${\mathcal{Y}}^{\circ}\cap
L^{--}{\mathcal{G}}_{\mathbf{Z}}\widetilde{w}^{*}\subset{\mathcal{Y}}^{\circ}$
is a nonempty open subscheme. An easy generalization of Theorem 3.5.1 shows
that there is an open subscheme
$V\subset\mathbb{A}^{1}\times_{\mathbf{Z}}\mathbb{A}^{3}$ such that
${\mathcal{Y}}^{\circ}|_{V}\rightarrow V$ is a trivial vector bundle over $V$.
Thus the subspace of parameters $(b,{\mathbf{a}})\in V$ at which the fiber of
${\mathcal{Y}}^{\circ}$ has nonempty intersection with
$L^{--}{\mathcal{G}}_{\mathbf{Z}}\widetilde{w}^{*}$ is open. By observing that
such subspace is stable under scalar multiplication (acting on
$\mathbb{A}^{1}\times_{\mathbf{Z}}\mathbb{A}^{3}$ diagonally), the subspace of
parameters $(1,{\mathbf{a}})$ at which the fiber of ${\mathcal{Y}}^{\circ}$
(which is
$S_{\mathbf{F}}^{\circ}(\widetilde{w}_{1}^{*}w_{0})\cap\mathrm{Fl}^{\nabla_{{\mathbf{a}}}}$)
has nonempty intersection with
$L^{--}{\mathcal{G}}_{\mathbf{Z}}\widetilde{w}^{*}$ in $\mathbb{A}^{3}$ is
also open in $\mathbb{A}^{3}$. This proves our claim. ∎

Let $\overline{\rho}$ be a tame inertial $L$-parameter over $\mathbf{F}$ with
$6$-generic lowest alcove presentation $(s,\mu)$ compatible with $\zeta\in
X^{*}(\underline{Z})$. The following Theorem gives a geometric interpretation
of the set $W^{?}(\overline{\rho})$ under polynomial genericity assumptions.

######  Theorem 3.7.2.

Let $\overline{\rho}$ be as above. Let $W^{\zeta}_{g}(\overline{\rho})$ be a
set of 3-deep Serre weights $\sigma$ with lowest alcove presentation
compatible with $\zeta$ such that $\widetilde{w}^{*}(\overline{\rho})\in
C^{\zeta}_{\sigma}$.

  1. (1)

We have $W_{\mathrm{obv}}(\overline{\rho})\subset
W^{\zeta}_{g}(\overline{\rho})\subset W^{?}(\overline{\rho})$.

  2. (2)

Let $(\widetilde{w}_{1},\omega)$ be a lowest alcove presentation of $\sigma\in
W^{?}(\overline{\rho})$ compatible with $\zeta$. For each
${j\in{\mathcal{J}}}$, let $P_{\widetilde{w}_{1,j}}$ be the polynomial in
Theorem 3.7.1. If $P_{\widetilde{w}_{1,j}}(\omega_{j})\neq 0\bmod p$ for all
${j\in{\mathcal{J}}}$, then $\sigma\in W^{\zeta}_{g}(\overline{\rho})$

###### Proof.

Let $\sigma\in W^{?}(\overline{\rho})$. By Proposition 2.5.8, $\sigma$ is
$3$-deep with a lowest alcove presentation
$(\widetilde{w}_{1},\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{-1}(0))$
for some
$(\widetilde{w}_{1},\widetilde{w}_{2})\in(\underline{\widetilde{W}}_{1}^{+}\times\underline{\widetilde{W}}^{+})/X^{0}(\underline{T})$
satisfying $\widetilde{w}_{2}\uparrow\widetilde{w}_{1}$. We write
$\widetilde{w}(\overline{\rho})\widetilde{w}_{2}=t_{\omega}w$ so that
$\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{-1}(0)=\omega$. Also
$\sigma\in W_{\mathrm{obv}}(\overline{\rho})$ implies
$\widetilde{w}_{1}=\widetilde{w}_{2}$. Then both item (1) and (2) follow from
the corresponding items in Theorem 3.7.1 and noting that

$\displaystyle\\{(t_{\omega}\widetilde{w})^{*}\mid\widetilde{w}\in\widetilde{W}^{\mathcal{J}},\widetilde{w}\leq
w_{0}\widetilde{w}_{1}\\}=\\{(t_{\omega}w\widetilde{w})^{*}\mid w\in
W^{\mathcal{J}},\widetilde{w}\in\widetilde{W}^{+},\widetilde{w}\uparrow\widetilde{w}_{1}\\}.$

∎

We end this section by recording the following Proposition, which will be used
in §6.2.

######  Proposition 3.7.3.

$C_{(\widetilde{w}_{1},\omega)}$ is unibranch at its
$\underline{T}^{\vee}$-fixed points.

###### Proof.

This can be proven as the corresponding result for $\mathrm{GL}_{4}$ [LLHLMa,
Proposition 4.7.5] similarly to Proposition 3.3.7. ∎

##  4\. Moduli stacks in $p$-adic Hodge theory

In this section, we construct a symplectic variant of the moduli stacks of
étale $(\varphi,\Gamma)$-modules constructed in [EG23] and prove its
properties including the existence of closed substacks parameterizing
potentially crystalline representations. We also construct symplectic variants
for the $p$-adic formal algebraic stacks considered in [LLHLMa, §5], namely,
the moduli stacks of Breuil–Kisin modules, and étale $\phi$-modules. Then we
study the geometric properties of potentially crystalline substacks using
local models in §3.

###  4.1. Sympletic étale $(\varphi,\Gamma)$-modules

Only in this subsection, we allow $K/\mathbf{Q}_{p}$ to be ramified.

For a $p$-adically complete ${\mathcal{O}}$-algebra $R$, we write
${\mathcal{X}}_{n}(R)$ for the groupoid of projective étale
$(\varphi,\Gamma)$-modules of rank $n$ with $R$-coefficients, in the sense of
[EG23, Definition 2.7.2]. Then ${\mathcal{X}}_{n}$ is a Noetherian formal
algebraic stack over $\operatorname{\mathrm{Spf}}{\mathcal{O}}$ (Corollary
5.5.17 in _loc. cit._). When $R$ is a finite ${\mathcal{O}}$-algebra, there is
an equivalence of categories between the category of projective étale
$(\varphi,\Gamma)$-modules with $R$-coefficient and the category of continuous
representations of $G_{K}$ on finite projective $R$-modules. The goal of this
subsection is to construct a moduli stack of _symplectic_
$(\varphi,\Gamma)$-modules over ${\mathcal{O}}$ using moduli stacks of
Emerton–Gee.

We start by defining a symplectic variant of étale $(\varphi,\Gamma)$-modules.

######  Definition 4.1.1.

Let $R$ be a $p$-adically complete ${\mathcal{O}}$-algebra. A _symplectic_
projective étale $(\varphi,\Gamma)$-module (of rank 4) with $R$-coefficients
is a triple $(M,N,\alpha)$ where $M\in{\mathcal{X}}_{4}(R)$,
$N\in{\mathcal{X}}_{1}(R)$, and $\alpha:M\simeq M^{\vee}\otimes N$ is an
isomorphism between rank 4 étale $(\varphi,\Gamma)$-modules satisfying the
alternating condition $(\alpha^{\vee}\otimes N)^{-1}\circ\alpha=-1_{M}$. We
write ${\mathcal{X}}_{\mathrm{Sym}}(R)$ for the groupoid of projective
symplectic étale $(\varphi,\Gamma)$-modules with $R$-coefficients.

######  Lemma 4.1.2.

Let $R$ be a commutative ring with identity and $G$ be an abstract group.
There is an equivalence of groupoids

$\displaystyle\left\\{\rho:G\rightarrow\mathrm{GSp}_{4}(R)\right\\}$
$\displaystyle\simeq\left\\{(\rho^{\prime},\chi,\alpha)\mid\begin{array}[]{cc}\rho^{\prime}:G\rightarrow\mathrm{GL}_{4}(R),\chi:G\rightarrow\mathrm{GL}_{1}(R),\\\
\alpha:\rho^{\prime}\simeq(\rho^{\prime})^{\vee}\otimes\chi,(\alpha^{\vee}\otimes\chi)^{-1}\circ\alpha=-1\end{array}\right\\}$
$\displaystyle\rho$
$\displaystyle\mapsto(\operatorname{{\mathrm{std}}}(\rho),\operatorname{\mathrm{sim}}(\rho),\alpha_{\rho})$

where morphisms in the latter groupoid is given by isomorphisms of
$\rho^{\prime}$ and $\chi$ commuting with $\alpha$, and the isomorphism
$\alpha_{\rho}$ is given by the matrix $J$ with respect to the standard basis
of $\operatorname{{\mathrm{std}}}(\rho)$ and its the dual basis.

###### Proof.

Note that $\alpha$ defines a non-degenerate alternating $R$-bilinear pairing
$\langle-,-\rangle_{\alpha}:\rho^{\prime}\times\rho^{\prime}\rightarrow R$
given by $\langle x,y\rangle_{\alpha}=\alpha(y)(x)$. Then the essential
surjectivity follows from the presentation of standard symplectic space. More
precisely, one can find two vectors $v_{1},v_{4}\in\rho^{\prime}$ such that
$\langle v_{1},v_{4}\rangle_{\alpha}=1$, and two vectors $v_{2},v_{3}$ such
that $\langle v_{2},v_{3}\rangle_{\alpha}=1$ in the complement of the span of
$v_{1},v_{4}$. Then the basis $\\{v_{1},v_{2},v_{3},v_{4}\\}$ gives the
standard symplectic space. The proof of fully faithfulness is elementary. ∎

######  Corollary 4.1.3.

Let $R$ be a finite local ${\mathcal{O}}$-algebra. There is an equivalence of
groupoids

$\displaystyle\left\\{\rho:G_{K}\rightarrow\mathrm{GSp}_{4}(R)\right\\}\simeq{\mathcal{X}}_{\mathrm{Sym}}(A).$

###### Proof.

This follows from Lemma 4.1.2 and Fontaine’s equivalence between Galois
representations and étale $(\varphi,\Gamma)$-modules. ∎

We recall some definitions from [Eme]. A formal algebraic stack is quasi-
compact if it admits a morphism from a quasi-compact formal algebraic space
which is representable by algebraic spaces, smooth, and surjective. A morphism
${\mathcal{X}}\rightarrow{\mathcal{Y}}$ of formal algebraic stacks is quasi-
compact if for every morphism $Z\rightarrow{\mathcal{Y}}$ whose source is an
affine scheme, the fiber product ${\mathcal{X}}\times_{{\mathcal{Y}}}Z$ is a
quasi-compact formal algebraic stack. A formal algebraic stack ${\mathcal{X}}$
is quasi-separated if its diagonal morphism (which is representable by
algebraic spaces) is quasi-compact and quasi-separated. We have the following
obvious generalization.

######  Definition 4.1.4.

A morphism $f:{\mathcal{X}}\rightarrow{\mathcal{Y}}$ of formal algebraic
stacks is _quasi-separated_ if the relative diagonal
$\Delta_{f}:{\mathcal{X}}\rightarrow{\mathcal{X}}\times_{\mathcal{Y}}{\mathcal{X}}$
is quasi-compact and quasi-separated.

It is easy to see that if $f:{\mathcal{X}}\rightarrow{\mathcal{Y}}$ is a
quasi-separated morphism of formal algebraic stacks whose target is quasi-
separated, then ${\mathcal{X}}$ is quasi-separated.

A formal algebraic stack ${\mathcal{X}}$ is _locally Noetherian_ if it admits
a morphism from a disjoint union of affine formal algebraic spaces which is
representable by algebraic spaces, smooth, and surjective. Moreover,
${\mathcal{X}}$ is _Noetherian_ if it is locally Noetherian, quasi-compact,
and quasi-separated.

We denote by
$\operatorname{{\mathrm{std}}}:{\mathcal{X}}_{\mathrm{Sym}}\rightarrow{\mathcal{X}}_{4}$
and
$\operatorname{\mathrm{sim}}:{\mathcal{X}}_{\mathrm{Sym}}\rightarrow{\mathcal{X}}_{1}$
morphisms sending $(M,N,\alpha)$ to $M$ and $N$ respectively.

######  Theorem 4.1.5.

The morphism
$\operatorname{{\mathrm{std}}}\times\operatorname{\mathrm{sim}}:{\mathcal{X}}_{{\mathrm{Sym}}}\rightarrow{\mathcal{X}}_{4}\times_{\operatorname{\mathrm{Spf}}{\mathcal{O}}}{\mathcal{X}}_{1}$
is representable by algebraic spaces, quasi-compact, and quasi-separated. In
particular, the stack ${\mathcal{X}}_{\mathrm{Sym}}$ is a Noetherian formal
algebraic stack over $\operatorname{\mathrm{Spf}}{\mathcal{O}}$.

###### Proof.

We first prove that
$\operatorname{{\mathrm{std}}}\times\operatorname{\mathrm{sim}}$ is
representable by algebraic spaces and quasi-compact. Let
$S\rightarrow\operatorname{\mathrm{Spf}}{\mathcal{O}}$ be a test scheme and
$S\rightarrow{\mathcal{X}}_{4}\times_{\operatorname{\mathrm{Spf}}{\mathcal{O}}}{\mathcal{X}}_{1}$
be a morphism corresponding to étale $(\varphi,\Gamma)$-modules $M_{S}$ and
$N_{S}$ (of rank 4 and 1, respectively). If $S^{\prime}$ is a $S$-scheme, we
write $(M_{S^{\prime}},N_{S^{\prime}})$ for the pullback of $(M_{S},N_{S})$ to
$S^{\prime}$. The fiber product

$\displaystyle\underline{\alpha}_{S}:={\mathcal{X}}_{\mathrm{Sym}}\times_{{\mathcal{X}}_{4}\times{\mathcal{X}}_{1}}S$

is given by the following subsheaf of
$\underline{Isom}(M_{S},M_{S}^{\vee}\otimes N_{S})$

$\displaystyle\underline{\alpha}_{S}:S^{\prime}\mapsto\left\\{\alpha^{\prime}\in
Isom(M_{S^{\prime}},M_{S^{\prime}}^{\vee}\otimes
N_{S^{\prime}})\mid((\alpha^{\prime})^{\vee}\otimes
N_{S^{\prime}})\circ\alpha^{\prime}=-1_{M_{S}}\right\\}.$

We have the following cartesian diagram

$\leavevmode\hbox to322.52pt{\vbox
to52.37pt{\pgfpicture\makeatletter\hbox{\hskip
161.25862pt\lower-26.23703pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-161.25862pt}{-26.13719pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
8.68387pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.37833pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\underline{\alpha}_{S}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.68387pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 31.65967pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${S}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
7.6597pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
38.90576pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-34.60022pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\underline{Isom}(M_{S},M_{S}^{\vee}\otimes N_{S})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
38.90576pt\hfil&\hfil\hskip
104.38426pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-76.07875pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\underline{Isom}(M_{S},M_{S}^{\vee}\otimes
N_{S})\times_{S}\underline{Isom}(M_{S}^{\vee}\otimes N_{S},M_{S})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
80.3843pt\hfil&\hfil\hskip 41.96857pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.66306pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\underline{Aut}(M_{S})}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
17.9686pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-113.469pt}{11.98447pt}\pgfsys@lineto{135.03036pt}{11.98447pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{135.23035pt}{11.98447pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-122.35286pt}{3.71143pt}\pgfsys@lineto{-122.35286pt}{-13.48866pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-122.35286pt}{-13.68864pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{143.29002pt}{5.62476pt}\pgfsys@lineto{143.29002pt}{-14.37752pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{143.29002pt}{-14.5775pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{145.64279pt}{-5.60191pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{-1_{M_{S}}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-83.2471pt}{-23.63719pt}\pgfsys@lineto{-60.04709pt}{-23.63719pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.8471pt}{-23.63719pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{101.52145pt}{-23.63719pt}\pgfsys@lineto{124.72147pt}{-23.63719pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{124.92145pt}{-23.63719pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{111.80681pt}{-21.28442pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{c}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where the right vertical map is a constant section given by
$-1_{M_{S}}\in\underline{Aut}(M_{S})$, the left bottom horizontal map sends
$\alpha$ to $(\alpha,(\alpha^{\vee}\otimes N_{S})^{-1})$, and $c$ is the
composition. Since ${\mathcal{X}}_{4}$ is quasi-separated,
$\underline{Aut}(M_{S})\rightarrow S$ is quasi-compact and quasi-separated.
This implies that the section $-1_{M_{S}}$ is quasi-compact, and so is
$\underline{\alpha}_{S}\rightarrow\underline{Isom}(M_{S},M_{S}^{\vee}\otimes
N_{S})$. In particular, $\underline{\alpha}_{S}\rightarrow S$ is quasi-
compact. This proves that
$\operatorname{{\mathrm{std}}}\times\operatorname{\mathrm{sim}}$ is
representable by algebraic spaces and quasi-compact.

We now prove that
$\operatorname{{\mathrm{std}}}\times\operatorname{\mathrm{sim}}$ is quasi-
separated. By the following cartesian diagram

$\leavevmode\hbox to333.54pt{\vbox
to52.57pt{\pgfpicture\makeatletter\hbox{\hskip
166.76912pt\lower-26.33704pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-166.76912pt}{-26.2372pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
64.29694pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.9914pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\underline{Isom}((M_{1},N_{1},\alpha_{1}),(M_{2},N_{2},\alpha_{2}))}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
64.29694pt\hfil&\hfil\hskip 76.04436pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-47.73885pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\underline{Isom}(M_{1},M_{2})\times\underline{Isom}(N_{1},N_{2})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
52.04439pt\hfil&\hfil\hskip 31.65967pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${S}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
7.6597pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
13.42223pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.11668pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${{\mathcal{X}}_{\mathrm{Sym}}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
13.42223pt\hfil&\hfil\hskip 58.40558pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-30.10007pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${{\mathcal{X}}_{\mathrm{Sym}}\times_{{\mathcal{X}}_{4}\times{\mathcal{X}}_{1}}{\mathcal{X}}_{\mathrm{Sym}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
34.40561pt\hfil&\hfil\hskip 50.4278pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.12228pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${{\mathcal{X}}_{\mathrm{Sym}}\times{\mathcal{X}}_{\mathrm{Sym}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
26.42783pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-37.97525pt}{10.91559pt}\pgfsys@lineto{-14.77524pt}{10.91559pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-14.57526pt}{10.91559pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-102.47218pt}{2.05588pt}\pgfsys@lineto{-102.47218pt}{-15.14421pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-102.47218pt}{-15.3442pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{90.1135pt}{10.91559pt}\pgfsys@lineto{132.08163pt}{10.91559pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{132.28162pt}{10.91559pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{37.86911pt}{2.05588pt}\pgfsys@lineto{37.86911pt}{-15.14421pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{37.86911pt}{-15.3442pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{140.3413pt}{4.55588pt}\pgfsys@lineto{140.3413pt}{-15.14421pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{140.3413pt}{-15.3442pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-88.84996pt}{-23.7372pt}\pgfsys@lineto{2.86354pt}{-23.7372pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.06352pt}{-23.7372pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{72.47472pt}{-23.7372pt}\pgfsys@lineto{113.3135pt}{-23.7372pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{113.51349pt}{-23.7372pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

it is suffice to show that

$\displaystyle\underline{Isom}((M_{1},N_{1},\alpha_{1}),(M_{2},N_{2},\alpha_{2}))\rightarrow\underline{Isom}(M_{1},M_{2})\times_{S}\underline{Isom}(N_{1},N_{2})$

is quasi-compact and quasi-separated. It is clearly a monomorphism, and thus
is quasi-separated. Note that there is a morphism

$\displaystyle
d:\underline{Isom}(M_{1},M_{2})\times_{S}\underline{Isom}(N_{1},N_{2})$
$\displaystyle\rightarrow\underline{Isom}(M_{1},M_{2})\times_{S}\underline{Isom}(N_{1},N_{2})$
$\displaystyle(f,g)$
$\displaystyle\mapsto(\alpha_{2}^{-1}\circ((f^{\vee})^{-1}\otimes
g)\circ\alpha_{1},g)$

which makes the following diagram cartesian

$\leavevmode\hbox to271.02pt{\vbox
to54.54pt{\pgfpicture\makeatletter\hbox{\hskip
135.50853pt\lower-27.3215pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-135.50853pt}{-27.22166pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
64.29694pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.9914pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\underline{Isom}((M_{1},N_{1},\alpha_{1}),(M_{2},N_{2},\alpha_{2}))}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
64.29694pt\hfil&\hfil\hskip 77.92268pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-49.61717pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\underline{Isom}(M_{1},M_{2})\times_{S}\underline{Isom}(N_{1},N_{2})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
53.92271pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
53.92271pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-49.61717pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\underline{Isom}(M_{1},M_{2})\times_{S}\underline{Isom}(N_{1},N_{2})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
53.92271pt\hfil&\hfil\hskip 83.21158pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.90607pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${(\underline{Isom}(M_{1},M_{2})\times_{S}\underline{Isom}(N_{1},N_{2}))^{2}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
59.21161pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-6.71466pt}{12.40225pt}\pgfsys@lineto{21.77425pt}{12.40225pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.97423pt}{12.40225pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-71.2116pt}{3.54254pt}\pgfsys@lineto{-71.2116pt}{-15.46199pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-71.2116pt}{-15.66197pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{76.29692pt}{3.54254pt}\pgfsys@lineto{76.29692pt}{-13.65755pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{76.29692pt}{-13.85753pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{78.64969pt}{-7.00748pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\left(\mathrm{id},d\right)}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-17.08888pt}{-24.72166pt}\pgfsys@lineto{16.48535pt}{-24.72166pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.68533pt}{-24.72166pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.01846pt}{-22.3689pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\Delta}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
Then the top horizontal arrow is quasi-compact because
$\underline{Isom}(M_{1},M_{2})\times_{S}\underline{Isom}(N_{1},N_{2})$ is
quasi-separated.

Finally, ${\mathcal{X}}_{\mathrm{Sym}}$ is a formal algebraic stack over
$\operatorname{\mathrm{Spf}}{\mathcal{O}}$ by [Eme, Lemma 5.19]. It is quasi-
compact and quasi-separated because
$\operatorname{{\mathrm{std}}}\times\operatorname{\mathrm{sim}}$ and
${\mathcal{X}}_{4}\times_{\operatorname{\mathrm{Spf}}{\mathcal{O}}}{\mathcal{X}}_{1}$
are. It is also locally Noetherian because
${\mathcal{X}}_{4}\times_{\operatorname{\mathrm{Spf}}{\mathcal{O}}}{\mathcal{X}}_{1}$
is, and $\operatorname{{\mathrm{std}}}\times\operatorname{\mathrm{sim}}$ is
quasi-compact. ∎

Let $\tau:I_{K}\rightarrow T^{\vee}(E)$ be an inertial $K$-type and
$\lambda\in X_{*}(\underline{T}^{\vee})$ be a dominant cocharacter. Recall the
potentially crystalline (resp. semistable) substack
${\mathcal{X}}_{4}^{\lambda^{\prime},\tau^{\prime}}$ (resp.
${\mathcal{X}}_{4}^{{\mathrm{ss}},\lambda^{\prime},\tau^{\prime}}$) of
${\mathcal{X}}_{4}$ with Hodge type
$\lambda^{\prime}:=\operatorname{{\mathrm{std}}}(\lambda)$ and descent type
$\tau^{\prime}:=\operatorname{{\mathrm{std}}}(\tau)$ ([EG23, Theorem 4.8.12]).
It is a ${\mathcal{O}}$-flat $p$-adic formal algebraic substack. For finite
flat ${\mathcal{O}}$-algebra $A$,
${\mathcal{X}}_{4}^{\lambda^{\prime},\tau^{\prime}}(A)$ (resp.
${\mathcal{X}}_{4}^{{\mathrm{ss}},\lambda^{\prime},\tau^{\prime}}(A)$) is the
full subgroupoid of ${\mathcal{X}}_{4}(A)$ consisting of lattices in
potentially crystalline (resp. semistable) $G_{K}$-representations of Hodge
type $\lambda^{\prime}$ and inertia type $\tau^{\prime}$. For convenience, we
write ${\mathcal{X}}_{4}^{\lambda,\tau}$ and
${\mathcal{X}}_{4}^{{\mathrm{ss}},\lambda,\tau}$ instead of
${\mathcal{X}}_{4}^{\lambda^{\prime},\tau^{\prime}}$ and
${\mathcal{X}}_{4}^{{\mathrm{ss}},\lambda^{\prime},\tau^{\prime}}$. We define
${\mathcal{X}}_{{\mathrm{Sym}}}^{\lambda,\tau}$ (resp.
${\mathcal{X}}_{{\mathrm{Sym}}}^{{\mathrm{ss}},\lambda,\tau}$) to be the
${\mathcal{O}}$-flat part (see [Eme, Example 9.11]) of
${\mathcal{X}}_{\mathrm{Sym}}\times_{{\mathcal{X}}_{4}}{\mathcal{X}}_{4}^{\lambda^{\prime},\tau^{\prime}}$
(resp.
${\mathcal{X}}_{\mathrm{Sym}}\times_{{\mathcal{X}}_{4}}{\mathcal{X}}_{4}^{{\mathrm{ss}},\lambda^{\prime},\tau^{\prime}}$).

######  Proposition 4.1.6.

Let $\lambda,\tau$ be as above.

  1. (1)

The stack ${\mathcal{X}}_{\mathrm{Sym}}^{\lambda,\tau}$ (resp.
${\mathcal{X}}_{{\mathrm{Sym}}}^{{\mathrm{ss}},\lambda,\tau}$) is a $p$-adic
formal algebraic stack and uniquely determined as ${\mathcal{O}}$-flat closed
substack of ${\mathcal{X}}_{\mathrm{Sym}}$ such that for finite flat
${\mathcal{O}}$-algebra $A$,
${\mathcal{X}}_{\mathrm{Sym}}^{\lambda,\tau}(A)\subset{\mathcal{X}}_{\mathrm{Sym}}(A)$
(resp.
${\mathcal{X}}_{\mathrm{Sym}}^{{\mathrm{ss}},\lambda,\tau}(A)\subset{\mathcal{X}}_{\mathrm{Sym}}(A)$)
is precisely the subgroupoid consisting of lattices in potentially crystalline
(resp. semistable) $G_{K}$-representations of Hodge type $\lambda$ and inertia
type $\tau$.

  2. (2)

The algebraic stacks
${\mathcal{X}}_{\mathrm{Sym}}^{\lambda,\tau}\times_{\operatorname{\mathrm{Spf}}{\mathcal{O}}}\operatorname{\mathrm{Spec}}\mathbf{F}$
(resp.
${\mathcal{X}}_{{\mathrm{Sym}}}^{{\mathrm{ss}},\lambda,\tau}\times_{\operatorname{\mathrm{Spf}}{\mathcal{O}}}\operatorname{\mathrm{Spec}}\mathbf{F}$)
are equidimensional of dimension $d_{\lambda}:=\sum_{j\in{\mathcal{J}}}\dim
P_{\lambda_{j}}{\backslash}\mathrm{GSp}_{4}$.

###### Proof.

The first claim follows from the construction. For the second claim, the proof
is identical to that of [EG23, Theorem 4.8.14], using $\mathrm{GSp}_{4}$
instead of $\mathrm{GL}_{4}$ and the dimension formula for the potentially
crystalline (or semistable) symplectic deformation ring ([BG19, Theorem A]). ∎

####  4.1.7. Irreducible components in
${\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}}}$

Let ${\mathcal{X}}_{4,{\mathrm{red}}}$ be the underlying reduced substack of
${\mathcal{X}}_{4}$. It is an algebraic stack of finite presentation over
$\mathbf{F}$ and equidimensional of dimension $6[K:\mathbf{Q}_{p}]$ ([EG23,
Theorem 5.5.11 and 6.5.1]). Moreover, its irreducible components are labelled
by Serre weights (of $\mathrm{GL}_{4}(k)$). Our goal is to prove analogous
results for the underlying reduced substack
${\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}}}$ of
${\mathcal{X}}_{{\mathrm{Sym}}}$.

Let $\sigma$ be a Serre weight (of $\mathrm{GSp}_{4}(k)$). Then there exists
$\lambda\in X^{*}_{1}(\underline{T})$ such that $\sigma\simeq F(\lambda)$. For
each ${j\in{\mathcal{J}}}$, we identify $\lambda_{j}$ with a triple of
integers $(\lambda_{j,1},\lambda_{j,2};\lambda_{j,3})$ such that
$0\leq\lambda_{j,1}-\lambda_{j,2},\lambda_{j,2}\leq p-1$ as explained in §2.1.

######  Definition 4.1.8.

We say that $\overline{\rho}:G_{K}\rightarrow\mathrm{GSp}_{4}(\mathbf{F})$ is
_maximally non-split of niveau 1 and of weight $\sigma$_ if

(4.1.9) $\displaystyle\overline{\rho}={\begin{pmatrix}\chi_{1}&*&*&*\\\
0&\chi_{2}&*&*\\\ 0&0&\chi_{3}&*\\\ 0&0&0&\chi_{4}\end{pmatrix}}$

where

  * •

$\overline{\rho}$ is maximally non-split of niveau 1, i.e. it has a unique
$G_{K}$-stable complete flag;

  * •

$\bigoplus_{i=1}^{4}\chi_{i}|_{I_{K}}=\prod_{{j\in{\mathcal{J}}}}\overline{\omega}_{K,\sigma_{j}}^{\phi(\lambda_{j}+\eta_{j})}$;

  * •

If $\chi_{1}\chi_{2}^{-1}|_{I_{K}}=\overline{\varepsilon}^{-1}$ (resp.
$\chi_{2}\chi_{3}^{-1}|_{I_{K}}=\overline{\varepsilon}^{-1}$), then
$\lambda_{j,2}=p-1$ (resp. $\lambda_{j,1}-\lambda_{j,2}=p-1$) for all
${j\in{\mathcal{J}}}$ if and only
$\chi_{1}\chi_{2}^{-1}=\overline{\varepsilon}^{-1}$ (resp.
$\chi_{2}\chi_{3}^{-1}=\overline{\varepsilon}^{-1}$) and the element of
$\operatorname{\mathrm{Ext}}^{1}_{G_{K}}(\chi_{1},\chi_{2})$ (resp.
$\operatorname{\mathrm{Ext}}^{1}_{G_{K}}(\chi_{2},\chi_{3})$) determined by
$\overline{\rho}$ is très ramifiée. Otherwise, $\lambda_{j,1}-\lambda_{j,2}=0$
(resp. $\lambda_{j,2}-\lambda_{j,3}=0$) for all ${j\in{\mathcal{J}}}$. (Note
that $\chi_{1}\chi_{2}^{-1}=\chi_{3}\chi_{4}^{-1}$.)

The following is the main Theorem of this subsection.

######  Theorem 4.1.10.

The stack ${\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}}}$ is an algebraic
stack over $\operatorname{\mathrm{Spec}}\mathbf{F}$ of finite presentation and
equidimensional of dimension $4[K:\mathbf{Q}_{p}]$. For each (isomorphism
class of) Serre weight $\sigma$, there exists an irreducible component
${\mathcal{C}}_{\sigma}\subset{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}}}$
containing a dense locus of $\overline{\rho}$ maximally non-split of niveau 1
and of weight $\sigma$. This induces a bijection between the set of
isomorphism classes of Serre weights and the set of irreducible components of
${\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}}}$.

Before we proceed, we define _potentially diagonalizable representations_. Let
$\operatorname{{\mathrm{std}}}^{\prime}:\mathrm{GSp}_{4}\rightarrow\mathrm{GL}_{5}$
denote the composite of the projection
$\mathrm{GSp}_{4}\twoheadrightarrow\mathrm{SO}_{5}$ and the standard
representation $\mathrm{SO}_{5}\rightarrow\mathrm{GL}_{5}$.

######  Definition 4.1.11.

A continuous representation
$\rho:G_{K}\rightarrow\mathrm{GSp}_{4}({\mathcal{O}})$ is _potentially
diagonalizable_ if $\rho\otimes_{\mathcal{O}}E$ is potentially crystalline and
$\operatorname{{\mathrm{std}}}(\rho)$ is potentially diagonalizable in the
sense of [BLGGT14, §1.4].

######  Example 4.1.12.

Suppose that $\rho:G_{K}\rightarrow\mathrm{GSp}_{4}({\mathcal{O}})$ is
potentially crystalline and ordinary (in the sense of [BLGGT14, §1.4]). Then
$\rho$ is potentially diagonalizable by Lemma 1.4.3 of _loc. cit._.

The following two results will be proven in §4.1.15 and §4.1.19 respectively.
We write
${\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$ for
${\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}}}\times_{\mathbf{F}}\operatorname{\mathrm{Spec}}\overline{\mathbf{F}}_{p}$.

######  Proposition 4.1.13.

For each Serre weight $\sigma$, there exists an algebraic stack
${\mathcal{C}}_{\sigma,\overline{\mathbf{F}}_{p}}\subset{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$
irreducible of dimension $4[K:\mathbf{Q}_{p}]$ containing a dense locus of
$\overline{\rho}$ maximally non-split of niveau 1 and of weight $\sigma$.
Furthermore, there exists a closed substack
${\mathcal{C}}_{small}\subset{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$
of dimension strictly less than $4[K:\mathbf{Q}_{p}]$ such that

$\displaystyle{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}=\bigcup_{\sigma}{\mathcal{C}}_{\sigma,\overline{\mathbf{F}}_{p}}\cup{\mathcal{C}}_{small}.$

######  Theorem 4.1.14.

Any continuous representation
$\overline{\rho}:G_{K}\rightarrow\mathrm{GSp}_{4}(\overline{\mathbf{F}}_{p})$
admits a lift
$\rho:G_{K}\rightarrow\mathrm{GSp}_{4}(\overline{\mathbf{Z}}_{p})$ such that
$\rho\otimes_{\overline{\mathbf{Z}}_{p}}\overline{\mathbf{Q}}_{p}$ is
crystalline with regular Hodge–Tate weights. Furthermore, $\rho$ can be taken
to be potentially diagonalizable.

Granting the Proposition 4.1.13 and Theorem 4.1.14, we prove:

###### Proof of Theorem 4.1.10.

If a closed immersion of reduced algebraic stacks that are locally of finite
type over $\operatorname{\mathrm{Spec}}\overline{\mathbf{F}}_{p}$ is
surjective on finite type points (see [EG23, §6.6]), then it is an
isomorphism. Thus, if we prove that any
$\overline{\rho}:G_{K}\rightarrow\mathrm{GSp}_{4}(\overline{\mathbf{F}}_{p})$
is contained in some ${\mathcal{C}}_{\sigma,\overline{\mathbf{F}}_{p}}$, then
the closed immersion
$\cup_{\sigma}{\mathcal{C}}_{\sigma,\overline{\mathbf{F}}_{p}}\hookrightarrow{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$
from Proposition 4.1.13 is an isomorphism. By Theorem 4.1.14,
$\overline{\rho}$ is contained in the reduction of a crystalline stack
${\mathcal{X}}_{\mathrm{Sym}}^{\lambda}\times_{\operatorname{\mathrm{Spf}}{\mathcal{O}}}\operatorname{\mathrm{Spec}}\overline{\mathbf{F}}_{p}$
for some regular cocharacter $\lambda$. By Proposition 4.1.6, the algebraic
stack
${\mathcal{X}}_{\mathrm{Sym}}^{\lambda}\times_{\operatorname{\mathrm{Spf}}{\mathcal{O}}}\operatorname{\mathrm{Spec}}\overline{\mathbf{F}}_{p}$
is equidimensional of dimension $4[K:\mathbf{Q}_{p}]$. So its underlying space
is a union of ${\mathcal{C}}_{\sigma,\overline{\mathbf{F}}_{p}}$, and one of
them contains $\overline{\rho}$.

It remains to show that ${\mathcal{C}}_{\sigma,\overline{\mathbf{F}}_{p}}$
descends to ${\mathcal{C}}_{\sigma}$ over
$\operatorname{\mathrm{Spec}}\mathbf{F}$. We need to show that
$\operatorname{\mathrm{Gal}}(\overline{\mathbf{F}}_{p}/\mathbf{F})$ stabilizes
each component
${\mathcal{C}}_{\sigma,\overline{\mathbf{F}}_{p}}\subset{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$.
This follows because
$\operatorname{\mathrm{Gal}}(\overline{\mathbf{F}}_{p}/\mathbf{F})$-action
preserves the property of being maximally non-split of niveau 1 and of weight
$\sigma$. ∎

####  4.1.15. Families of extensions

We prove the Proposition 4.1.13. The essential ingredient is generalizations
of the Proposition 5.4.4 in [EG23] which computes the dimension of families of
extensions inside
${\mathcal{X}}_{n,{\mathrm{red}},\overline{\mathbf{F}}_{p}}$. We start by
introducing several notations.

Recall that $S$ (resp. $Q$) denotes the Siegel (resp. Klingen) parabolic
subgroup of $\mathrm{GSp}_{4}$. We write $U_{S}$ (resp. $U_{Q}$) for its
unipotent radical and $L_{S}$ (resp. $L_{Q}$) for its Levi component. Then
$U_{S}\simeq\mathbf{G}_{a}^{\oplus 3}$ and $U_{Q}$ is an extension of
$\mathbf{G}_{a}^{\oplus 2}$ by $\mathbf{G}_{a}$.

We also need the following auxiliary groups. Let $Q_{4}$ be the minimal
parabolic of $\mathrm{GL}_{4}$ containing $\operatorname{{\mathrm{std}}}(Q)$
and $U_{Q_{4}}$ be its unipotent radical. We define the following subgroups of
$U_{Q_{4}}$;

$\displaystyle U_{Q_{3}}=\left\\{{\begin{pmatrix}1&*&*&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{pmatrix}}\right\\},\
U_{Q_{3}^{\prime}}=\left\\{{\begin{pmatrix}1&0&0&0\\\ 0&1&0&*\\\ 0&0&1&*\\\
0&0&0&1\end{pmatrix}}\right\\},\ U_{P_{4}}=\left\\{{\begin{pmatrix}1&0&0&*\\\
0&1&0&*\\\ 0&0&1&*\\\ 0&0&0&1\end{pmatrix}}\right\\}.$

Let $\iota:\mathrm{GL}_{4}\rightarrow\mathrm{GL}_{4}$ be an involution given
by $\iota(A)=J^{-1}A^{-\top}J$. Then $(U_{Q_{4}})^{\iota}=U_{Q}$ and
$\iota:U_{Q_{3}}\buildrel\sim\over{\rightarrow}U_{Q^{\prime}_{3}}$.

Let $R$ be a finite $\overline{\mathbf{Z}}_{p}$-algebra. Let
${\theta}:G_{K}\rightarrow\mathrm{GL}_{2}(R)$ be a continuous representation
and $\chi:G_{K}\rightarrow R^{\times}$ be a continuous character. Then
${\theta}\oplus({\theta}^{\vee}\otimes\chi)$ is $L_{S}$-valued representation
of $G_{K}$ and $\chi\oplus{\theta}\oplus\det({\theta})\chi^{-1}$ is
$L_{Q}$-valued representation of $G_{K}$. We write

$\displaystyle\mathrm{Ad}_{S}({\theta},\chi):G_{K}$
$\displaystyle\xrightarrow{{\theta}\oplus({\theta}^{\vee}\otimes\chi)}L_{S}(R)\xrightarrow{\mathrm{Ad}}\operatorname{\mathrm{Aut}}(U_{S}(R))$
$\displaystyle\mathrm{Ad}_{Q}(\chi,{\theta}):G_{K}$
$\displaystyle\xrightarrow{\chi\oplus{\theta}\oplus\det({\theta})\chi^{-1}}L_{Q}(R)\xrightarrow{\mathrm{Ad}}\operatorname{\mathrm{Aut}}(U_{Q}(R))$
$\displaystyle\mathrm{Ad}_{Q_{4}}(\chi,\theta):G_{K}$
$\displaystyle\xrightarrow{\chi\oplus{\theta}\oplus\det({\theta})\chi^{-1}}L_{Q_{4}}(R)\xrightarrow{\mathrm{Ad}}\operatorname{\mathrm{Aut}}(U_{Q_{4}}(R)).$

Note that $G_{K}$-action on $U_{Q_{3}}$ and $U_{Q^{\prime}_{3}}$ induced by
$\mathrm{Ad}_{Q_{4}}(\chi,\theta)$ define subrepresentations of
$\mathrm{Ad}_{Q_{4}}(\chi,\theta)$, which we denote by
$\mathrm{Ad}_{Q_{3}}(\chi,\theta)$ and
$\mathrm{Ad}_{Q^{\prime}_{3}}(\chi,\theta)$ respectively. The isomorphism
$\iota:U_{Q_{3}}\buildrel\sim\over{\rightarrow}U_{Q^{\prime}_{3}}$ induces an
isomorphism of $G_{K}$-modules

$\displaystyle\mathrm{Ad}_{Q_{3}}(\chi,\theta)\buildrel\sim\over{\rightarrow}\mathrm{Ad}_{Q^{\prime}_{3}}(\chi,\theta).$

There is an obvious quotient map $U_{P_{4}}\twoheadrightarrow
U_{Q^{\prime}_{3}}.$ We can view $U_{P_{4}}$ as the unipotent of standard
parabolic subgroup of $\mathrm{GL}_{4}$ whose Levi factor is
$\mathrm{GL}_{3}\times\mathrm{GL}_{1}$. If
$\psi:G_{K}\rightarrow\mathrm{GL}_{3}(R)$ is a continuous representation, we
write

$\displaystyle\mathrm{Ad}_{P_{4}}(\psi,\chi):G_{K}\xrightarrow{\psi\oplus\chi}\mathrm{GL}_{3}(R)\times\mathrm{GL}_{1}(R)\xrightarrow{\mathrm{Ad}}\operatorname{\mathrm{Aut}}(U_{P_{4}}(R)).$

Finally, we remark that there is an auxiliary embedding

$\displaystyle\iota_{\mathrm{aux}}:\\{(g,h)\in\mathrm{GL}_{2}\times\mathrm{GL}_{2}\mid\det(g)=\det(h)\\}$
$\displaystyle\rightarrow\mathrm{GSp}_{4}$
$\displaystyle\left({\begin{pmatrix}a&b\\\
c&d\end{pmatrix}},{\begin{pmatrix}x&y\\\ z&x\end{pmatrix}}\right)$
$\displaystyle\mapsto{\begin{pmatrix}a&&&b\\\ &x&y&\\\ &z&w&\\\
c&&&d\end{pmatrix}}.$

We now consider families of extensions valued in the Siegel parabolic
subgroup. Let $T$ be a reduced finite type $\overline{\mathbf{F}}_{p}$-scheme
with a morphism
$T\rightarrow{\mathcal{X}}_{2,{\mathrm{red}},\overline{\mathbf{F}}_{p}}$ whose
scheme-theoretic image is of pure dimension $d$. We let
$\chi:G_{K}\rightarrow\overline{\mathbf{F}}_{p}^{\times}$ be a continuous
character. We write $\overline{\theta}_{T}$ for the family of
$G_{K}$-representations corresponding to
$T\rightarrow{\mathcal{X}}_{2,{\mathrm{red}}}$. Following the convention of
[EG23], we write $H^{2}(G_{K},\mathrm{Ad}_{S}(\overline{\theta}_{T},\chi))$
for the coherent sheaf on $T$ defined as the cohomology group of the _Herr
complex_ $H^{2}({\mathcal{C}}^{\bullet}(M))$ on $T$ where $M$ is the rank 3
étale $(\varphi,\Gamma)$-module corresponding to the family
$\mathrm{Ad}_{S}(\overline{\theta}_{T},\chi)$ (see Remark 5.1.30 of _loc.
cit._).

We suppose that $H^{2}(G_{K},\mathrm{Ad}_{S}(\overline{\theta}_{T},\chi))$ is
locally free of constant rank $r$. Following the discussion before Proposition
5.4.4 in _loc. cit._ , we find a complex of finite rank locally free
${\mathcal{O}}_{T}$-modules

$\displaystyle C^{0}_{T}\rightarrow Z^{1}_{T}$

such that $\operatorname{\mathrm{coker}}(C^{0}_{T}\rightarrow Z^{1}_{T})\simeq
H^{1}(G_{K},\mathrm{Ad}_{S}(\overline{\theta}_{\mathcal{T}},\chi))$. Let
$V=\underline{Spec}({\mathrm{Sym}}((Z_{T}^{1})^{\vee}))$. Then $V$
parameterizes a universal family of extension

$\displaystyle
0\rightarrow\overline{\theta}\rightarrow\overline{\rho}_{V}\rightarrow\overline{\theta}^{\vee}\otimes\chi\rightarrow
0$

such that $\overline{\rho}_{V}$ is valued in $\mathrm{GSp}_{4}$. This induces
a morphism
$V\rightarrow{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$.

######  Lemma 4.1.16.

Let
$T\rightarrow{\mathcal{X}}_{2,{\mathrm{red}},\overline{\mathbf{F}}_{p}},\chi$,
and $V$ be as above. In particular, the scheme-theoretic image of
$T\rightarrow{\mathcal{X}}_{2,{\mathrm{red}},\overline{\mathbf{F}}_{p}}$ is of
pure dimension $d$, and
$H^{2}(G_{K},\mathrm{Ad}_{S}(\overline{\theta}_{T},\chi))$ is locally free of
constant rank $r$. Then the scheme-theoretic image of
$V\rightarrow{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$
is of dimension $\leq d+3[K:\mathbf{Q}_{p}]+r-1$. Furthermore, if $T$ is
generically maximally non-split of niveau 1, then the equality holds.

###### Proof.

This follows from the proof of [EG23, Proposition 5.4.4]. ∎

We now consider the Klingen parabolic case. Let
$\overline{\theta}:G_{K}\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{F}}_{p})$
be a continuous irreducible representation. Let $T$ be a reduced finite type
$\overline{\mathbf{F}}_{p}$-scheme with a morphism
$T\rightarrow{\mathcal{X}}_{1,{\mathrm{red}},\overline{\mathbf{F}}_{p}}$ whose
scheme-theoretic image is of pure dimension $d$. For each $t\in
T(\overline{\mathbf{F}}_{p})$,
$H^{1}(G_{K},\mathrm{Ad}_{Q}(\chi_{t},\overline{\theta}))$ parameterizes
$\overline{\rho}:G_{K}\rightarrow Q(\overline{\mathbf{F}}_{p})$ whose
projection onto $L_{Q}(\overline{\mathbf{F}}_{p})$ is
$\chi_{t}\oplus\overline{\theta}\oplus\det(\overline{\theta})\chi^{-1}$.
However, the unipotent radical $U_{Q}$ is non-abelian. To avoid using non-
abelian cohomology, we construct a family of extensions in two steps.

Since $\theta$ is irreducible, we have
$H^{2}(G_{K},\mathrm{Ad}_{Q_{3}}(\chi_{T},\overline{\theta}))=0$. Similar to
Siegel parabolic case, there exists a locally free ${\mathcal{O}}_{T}$-module
$Z^{1}_{T}$ of constant rank with a surjection onto
$H^{1}(G_{K},\mathrm{Ad}_{Q_{3}}(\chi_{T},\overline{\theta}))$. The vector
bundle $V:=\underline{Spec}({\mathrm{Sym}}((Z_{T}^{1})^{\vee}))$ parameterizes
a universal family of extension

$\displaystyle
0\rightarrow\chi_{T}\rightarrow\overline{\psi}_{V}\rightarrow\overline{\theta}\rightarrow
0.$

By [EG23, Proposition 5.4.4], the scheme-theoretic image of
$V\rightarrow{\mathcal{X}}_{3,{\mathrm{red}},\overline{\mathbf{F}}_{p}}$ is of
dimension $\leq d+2[K:\mathbf{Q}_{p}]-1$.

We can and do replace $V$ by its open dense locus of non-split extensions. Let
$\delta=\det(\overline{\theta})$ and $\chi_{V}$ be the pullback of $\chi_{T}$
to $V$. By local Tate duality, we have
$H^{2}(G_{K},\mathrm{Ad}_{P_{4}}(\overline{\psi}_{V},\chi_{V}^{-1}\delta))=0$.
Thus, there is a locally free ${\mathcal{O}}_{V}$-module $Z^{1}_{V}$ of
constant rank with a surjection onto
$H^{1}(G_{K},\mathrm{Ad}_{P_{4}}(\overline{\psi}_{V},\chi_{V}^{-1}\delta))$,
and the vector bundle
$W:=\underline{Spec}({\mathrm{Sym}}((Z^{1}_{V})^{\vee}))$ parameterizes a
universal family of extensions

$\displaystyle
0\rightarrow\overline{\psi}_{V}\rightarrow\overline{\rho}_{W}\rightarrow\det(\overline{\rho})\chi_{T}\rightarrow
0.$

This induces a morphism
$f:W\rightarrow{\mathcal{X}}_{4,{\mathrm{red}},\overline{\mathbf{F}}_{p}}$
whose scheme-theoretic image is of dimension $\leq d+5[K:\mathbf{Q}_{p}]-2$ by
_loc. cit._. We replace $W$ by its open dense locus of non-split extensions.
Then
$\widetilde{W}=W\times_{{\mathcal{X}}_{4,{\mathrm{red}},\overline{\mathbf{F}}_{p}}}{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$
is again a scheme. We write
$\widetilde{f}:\widetilde{W}\rightarrow{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$.

######  Lemma 4.1.17.

Let
$\overline{\theta},T\rightarrow{\mathcal{X}}_{1,{\mathrm{red}},\overline{\mathbf{F}}_{p}}$,
and
$\widetilde{f}:\widetilde{W}\rightarrow{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$
be as above. In particular, the scheme-theoretic image of
$T\rightarrow{\mathcal{X}}_{1,{\mathrm{red}},\overline{\mathbf{F}}_{p}}$ is of
pure dimension $d$. We further assume that
$H^{2}(G_{K},\chi_{T}^{2}\delta^{-1})$ is locally free of constant rank $r$.
Then the scheme-theoretic image of $\widetilde{f}$ is of dimension $\leq
d+3[K:\mathbf{Q}_{p}]+r-1$.

###### Proof.

For
$w:\operatorname{\mathrm{Spec}}\mathbf{F}\rightarrow{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$,
we write $\widetilde{W}_{\widetilde{f}(w)}$ (resp. $W_{f(w)}$) for the fiber
of $\widetilde{f}$ at $w$ (resp. of $f$ at
$\operatorname{{\mathrm{std}}}(w)$). We have the following commutative diagram

${\widetilde{W}_{\widetilde{f}(w)}}$${\operatorname{\mathrm{Spec}}\overline{\mathbf{F}}_{p}}$${W_{f(w)}}$${\widetilde{W}}$${{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}}$${W}$${{\mathcal{X}}_{4,{\mathrm{red}},\overline{\mathbf{F}}_{p}}.}$$\scriptstyle{w}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{\operatorname{{\mathrm{std}}}}$$\scriptstyle{f}$

By [Sta18, Tag 0DS4], we have

$\displaystyle\dim W-\dim W_{f(w)}\leq d+5[K:\mathbf{Q}_{p}]-2.$

By the same Lemma, it is suffice to show $\dim\widetilde{W}\leq\dim
W-2[K:\mathbf{Q}_{p}]+r+1$ in order to get the bound on the dimension of
scheme-theoretic image of $\widetilde{f}$. Recall that $W$ is a dense open
subscheme of a vector bundle over $V$. For
$v:\operatorname{\mathrm{Spec}}\overline{\mathbf{F}}_{p}\rightarrow V$, we
define
$W_{v}:=W\times_{V,v}\operatorname{\mathrm{Spec}}\overline{\mathbf{F}}_{p}$
and
$\widetilde{W}_{v}:=\widetilde{W}\times_{V,v}\operatorname{\mathrm{Spec}}\overline{\mathbf{F}}_{p}$.
To prove the claimed inequality, we prove

(4.1.18) $\displaystyle\dim\widetilde{W}_{v}\leq\dim
W_{v}-2[K:\mathbf{Q}_{p}]+r+1.$

We first claim that the image of $\widetilde{W}_{v}$ in $W_{v}$ is contained
in a codimension $2[K:\mathbf{Q}_{p}]-r-1$ closed subscheme of $W_{v}$. By our
construction, $v$ correspond to a non-split extension $\overline{\psi}_{v}$ of
$\overline{\theta}$ by $\chi_{t}$, where $t$ is the image of $v$ under
$V\rightarrow T$. Then $\overline{\psi}_{v}$ determines a class $[c]\in
H^{1}(G_{K},\mathrm{Ad}_{Q_{3}}(\chi_{t},\overline{\theta}))$. By the
isomorphism
$\iota:\mathrm{Ad}_{Q_{3}}(\chi_{t},\overline{\theta})\buildrel\sim\over{\rightarrow}\mathrm{Ad}_{Q_{3}^{\prime}}(\chi_{t},\overline{\theta})$,
we get $[c^{\prime}]:=\iota([c])$. Any $w\in W_{v}(\overline{\mathbf{F}}_{p})$
determines a class $[b]\in
H^{1}(G_{K},\mathrm{Ad}_{P_{4}}(\overline{\psi}_{v},\chi_{t}^{-1}\delta))$.
Then $w$ is in the image of $\widetilde{W}_{v}$ if and only if $[b]$ is mapped
to a line $\overline{\mathbf{F}}_{p}[c^{\prime}]$ under the map

$\displaystyle
H^{1}(G_{K},\mathrm{Ad}_{P_{4}}(\overline{\psi}_{v},\chi_{t}^{-1}\delta))\rightarrow
H^{1}(G_{K},\mathrm{Ad}_{Q_{3}^{\prime}}(\chi_{t},\overline{\theta})).$

We take the long exact sequence of Galois cohomology to the following short
exact sequence

$\displaystyle
0\rightarrow\chi_{t}^{2}\delta^{-1}\rightarrow\mathrm{Ad}_{P_{4}}(\overline{\psi}_{v},\chi_{t}^{-1}\delta)\rightarrow\mathrm{Ad}_{Q_{3}^{\prime}}(\chi_{t},\overline{\theta})\rightarrow
0.$

Since
$\dim_{\overline{\mathbf{F}}_{p}}H^{2}(G_{K},\chi_{t}^{2}\delta^{-1})=r$, it
shows that the set of $[b]$ mapped to $\overline{\mathbf{F}}_{p}[c^{\prime}]$
is a subspace $H^{\prime}\subset
H^{1}(G_{K},\mathrm{Ad}_{P_{4}}(\overline{\psi}_{v},\chi_{t}^{-1}\delta))$ of
codimension $2[K:\mathbf{Q}_{p}]-r-1$. Recall that
$W=\underline{Spec}((Z^{1}_{V})^{\vee})$ where $Z^{1}_{V}$ is locally free
${\mathcal{O}}_{V}$-module with a surjection onto
$H^{1}(G_{K},\mathrm{Ad}_{P_{4}}(\overline{\psi}_{V},\chi_{V}^{-1}\delta))$.
Then $W_{v}=\underline{Spec}((Z^{1}_{v})^{\vee})$ where
$Z^{1}_{v}:=v^{*}(Z^{1}_{V})$, and the image of $\widetilde{W}_{v}$ in $W_{v}$
is given by the inverse image of $H^{\prime}$ in $Z^{1}_{v}$. This proves the
claim.

Take $w\in W_{v}(\overline{\mathbf{F}}_{p})$ in the image of
$\widetilde{W}_{v}$. We compute the dimension of the fiber

$\displaystyle\widetilde{W}_{w}:=\widetilde{W}_{v}\times_{W_{v},w}\operatorname{\mathrm{Spec}}\overline{\mathbf{F}}_{p}.$

If the dimension of $\widetilde{W}_{w}$ is zero, we get the inequality
(4.1.18). Let
$\overline{\rho}_{w}:G_{K}\rightarrow\mathrm{GL}_{4}(\overline{\mathbf{F}}_{p})$
be the continuous representation corresponding to $w$. Then
$\widetilde{W}_{w}$ parameterizes triples $(\overline{\rho}_{w},\zeta,\alpha)$
satisfying the conditions in Lemma 4.1.2. One can easily see that
$\zeta\simeq\det(\overline{\theta})$. Suppose that
$(\overline{\rho}_{w},\det(\overline{\theta}),\alpha_{1})$ and
$(\overline{\rho}_{w},\det(\overline{\theta}),\alpha_{2})$ are such triples.
Since
$\operatorname{\mathrm{Aut}}(\overline{\rho}_{w})=\overline{\mathbf{F}}_{p}^{\times}$,
$\alpha_{2}^{-1}\alpha_{1}=c$ for some
$c\in\overline{\mathbf{F}}_{p}^{\times}$. Then map
$(\overline{\rho}_{w},\det(\overline{\theta}),\alpha_{1})\rightarrow(\overline{\rho}_{w},\det(\overline{\theta}),\alpha_{2})$
identity on $\overline{\rho}_{w}$ and multiplication by $c^{-1}$ on
$\det(\overline{\theta})$ is then an isomorphism. Thus, there is the unique up
to isomorphism triple $(\overline{\rho},\det(\overline{\theta}),\alpha)$
parameterized by $\widetilde{W}_{w}$. This shows that $\widetilde{W}_{w}$ is
of dimension 0. ∎

###### Proof of Proposition 4.1.13.

Let $\sigma$ be a Serre weight. Then $\sigma\simeq F(\lambda)$ for $\lambda\in
X^{*}_{1}(\underline{T})$ well-defined up to $(p-\pi)X^{0}(\underline{T})$. We
identify $\lambda_{j}$ to a triple
$(\lambda_{j,1},\lambda_{j,2};\lambda_{j,3})$ as in §2.1. Then
$\lambda^{\prime}:=(\lambda_{j,1}+\lambda_{j,2}+\lambda_{j,3},\lambda_{j,1}+\lambda_{j,3})$
defines an element in
$X^{*}_{1}(\underline{T}_{2})/(p-\pi)X^{0}(\underline{T}_{2})$, and thus
$\sigma^{\prime}:=F(\lambda^{\prime})$ is a well-defined Serre weight of
$\mathrm{GL}_{2}(k)$.

There exists an irreducible component
${\mathcal{C}}_{\sigma^{\prime},\overline{\mathbf{F}}_{p}}\subset{\mathcal{X}}_{2,{\mathrm{red}},\overline{\mathbf{F}}_{p}}$
of dimension $[K:\mathbf{Q}_{p}]$ characterized as a closure of the locus of
$G_{K}$-representations of the form

$\displaystyle\overline{\theta}={\begin{pmatrix}\chi_{1}&*\\\
0&\chi_{2}\end{pmatrix}}$

such that
$\chi_{1}\mathrm{ur}_{t_{1}^{-1}}\oplus\chi_{2}\mathrm{ur}_{t_{2}^{-1}}=\prod_{j\in{\mathcal{J}}}\overline{\varpi}_{K,\sigma_{j}}^{\lambda_{j}^{\prime}+(1,0)}$
for some $t_{1},t_{2}\in\overline{\mathbf{F}}_{p}^{\times}$, and moreover, if
$\chi_{1}\chi_{2}^{-1}|_{I_{K}}=\overline{\varepsilon}^{-1}$, then
$\lambda_{j,2}=p-1$ for all ${j\in{\mathcal{J}}}$ if and only if
$\chi_{1}\chi_{2}^{-1}=\overline{\varepsilon}^{-1}$ and the extension class is
très ramifiée (see [LLHLMa, §7.4]). We call such nonsplit $\overline{\theta}$
as of weight $\sigma^{\prime}$. We recall some of its properties from the
proof of [EG23, Theorem 5.5.12] (corrected in [EG]). There exists an open
substack
${\mathcal{U}}_{\sigma^{\prime}}\subset{\mathcal{C}}_{\sigma^{\prime},\overline{\mathbf{F}}_{p}}$
consisting of nonsplit $\overline{\theta}$ of weight $\sigma^{\prime}$. Also,
there exists a codimension 1 closed substack
${\mathcal{C}}_{\sigma^{\prime}}^{\mathrm{fixed}}\subset{\mathcal{C}}_{\sigma^{\prime},\overline{\mathbf{F}}_{p}}$
containing dense locus of nonsplit $\overline{\theta}$ of weight
$\sigma^{\prime}$ as above with $t_{2}=1$. We let
${\mathcal{U}}_{\sigma^{\prime}}^{\mathrm{fixed}}={\mathcal{C}}_{\sigma^{\prime}}^{\mathrm{fixed}}\cap{\mathcal{U}}_{\sigma^{\prime}}$.
We can and do assume that
${\mathcal{C}}_{\sigma^{\prime},\overline{\mathbf{F}}_{p}}$ (resp.
${\mathcal{U}}_{\sigma^{\prime}}$) is obtained from
${\mathcal{C}}_{\sigma^{\prime}}^{\mathrm{fixed}}$ (resp.
${\mathcal{U}}_{\sigma^{\prime}}^{\mathrm{fixed}}$) by taking unramified
twists.

We write $\chi_{\sigma}$ for the continuous character
$\prod_{{j\in{\mathcal{J}}}}\overline{\varpi}^{\operatorname{\mathrm{sim}}(\phi(\lambda+\eta))}_{K,\sigma_{j}}:G_{K}\rightarrow\overline{\mathbf{F}}_{p}^{\times}$.
For nonsplit $\overline{\theta}$ of weight $\sigma^{\prime}$ as above, a
direct computation shows that the dimension of
$H^{2}(G_{K},\mathrm{Ad}_{S}(\overline{\theta},\chi_{\sigma}))$ is equal to 1
if $\chi_{\sigma}\chi_{2}^{-2}=\overline{\varepsilon}^{-1}$ and $0$ otherwise.
Note that $\chi_{\sigma}\chi_{2}^{-2}=\overline{\varepsilon}^{-1}$ if and only
if $\lambda_{j,1}-\lambda_{j,2}$ is equal to 0 for all ${j\in{\mathcal{J}}}$
or equal to $p-1$ for all ${j\in{\mathcal{J}}}$, and $t_{2}^{2}=1$.

Suppose that $\lambda_{j,1}-\lambda_{j,2}=p-1$ for all ${j\in{\mathcal{J}}}$,
which we call très ramifiée case. We let $T$ be a
$\overline{\mathbf{F}}_{p}$-scheme smoothly covering
${\mathcal{U}}_{\sigma^{\prime}}^{\mathrm{fixed}}$. Then
$H^{2}(G_{K},\mathrm{Ad}_{S}(\overline{\theta}_{T},\chi_{\sigma}))$ is locally
free of constant rank 1. Otherwise, we replace
${\mathcal{U}}_{\sigma^{\prime}}$ by its open substack so that $t_{2}^{2}\neq
1$, and let $T$ be a $\overline{\mathbf{F}}_{p}$-scheme smoothly covering
${\mathcal{U}}_{\sigma^{\prime}}$. Then
$H^{2}(G_{K},\mathrm{Ad}_{S}(\overline{\theta}_{T},\chi_{\sigma}))$ is locally
free of constant rank 0. In both cases, $T$ is necessarily irreducible,
reduced, and of finite type over $\overline{\mathbf{F}}_{p}$. By applying
Lemma 4.1.16 and the preceding discussion, we obtain a vector bundle $V$ over
$T$ parameterizing extensions $\overline{\rho}_{V}$ of
$\overline{\theta}_{T}^{\vee}\otimes\chi_{\sigma}$ by $\overline{\theta}_{T}$,
and the induced morphism
$V\rightarrow{\mathcal{X}}_{{\mathrm{Sym}},{\mathrm{red}},\overline{\mathbf{F}}_{p}}$