$\mathcal{A}_{\chi}=\bigoplus_{\mu}\mathcal{A}_{\chi,\mu}$ and

$\mathcal{A}_{\chi,\mu}=\bigoplus_{\pi}m_{\pi}\cdot\pi$

for some finite multiplicity $m_{\pi}\geq 1$, the sum running over automorphic
representations $\pi$ of $\mathcal{G}$ of central character $\chi$ such that
$\pi_{\infty}\cong\xi_{\mu}$.

A multiplicity-preserving Jacquet-Langlands transfer from automorphic
representations of $\mathcal{G}$ to $\operatorname{GSp}_{4}$ was proven by
Sorensen in [Sor09, Theorem B] for stable and tempered representations. (See
[Sor09, §1] for the definition of stable and tempered.)

Arthur described a classification of the discrete spectrum of
$\operatorname{GSp}_{4}$ with central character $\chi$ in [Art04]. This
classification was proven in [GT19], conditional on unpublished results of
Arthur, Moeglin and Waldspurger. We freely use consequences of the
classification below.

######  Corollary 4.1.1.

Stable and tempered automorphic representations of $\mathcal{G}$ occur with
multiplicity one in the discrete spectrum.

###### Proof.

Automorphic representations of $\operatorname{GSp}_{4}$ of general type
(meaning its Arthur parameter is a $\chi$-self dual cuspidal automorphic
representation of $\operatorname{GL}_{4}$ of symplectic type - equivalently
$\Pi$ is non-CAP and non-endoscopic) occur with multiplicity one in the
discrete spectrum by Arthur’s multiplicity theorem (cf. [GT19] or [BCGP21,
Theorem 2.9.3]). It follows from Arthur’s classification that an automorphic
representation of $\operatorname{GSp}_{4}$ being of general type is equivalent
to being stable and tempered. Now the claim follows from [Sor09, Theorem B]. ∎

From now on we assume that there exists $w\in\mathbb{Z}$ such that

$\textrm{$c_{v}=w$ for each infinite place $v$}.$ (4.1.2)

This assumption is necessary because any algebraic Hecke character over
totally real field has a constant weight.

######  Definition 4.1.3 (cf. Definition 7.6.2 of [BCGP21]).

If $r:G_{F}\rightarrow\operatorname{GSp}_{4}(\overline{\mathbb{Q}}_{\ell})$ is
a continuous homomorphism, we say that $r$ is _odd_ if for each infinite place
$v$ and corresponding choice of complex conjugation $c_{v}\in G_{F}$ we have
$\nu_{\mathrm{sim}}(r(c_{v}))=-1$.

######  Theorem 4.1.4.

Assume that $\chi$ satisfies (4.1.2), and let $\pi$ be a stable and tempered
automorphic representation of $\mathcal{G}$ that contributes to
$\mathcal{A}_{\chi,\mu}$. For any prime $\ell$ and choice of isomorphism
$\iota:\overline{\mathbb{Q}}_{\ell}\xrightarrow{\sim}\mathbb{C}$ there exists
a (unique up to isomorphism) continuous semisimple representation

$r_{\pi,\ell,\iota}:G_{F}\rightarrow\operatorname{GSp}_{4}(\overline{\mathbb{Q}}_{\ell})$

satisfying:

  * (i)

$r_{\pi,\ell,\iota}$ is de Rham at places dividing $\ell$.

  * (ii)

$r_{\pi,\ell,\iota}$ obeys local-global compatibility

$\iota\circ\operatorname{WD}(r_{\pi,\ell,\iota}|_{G_{F_{v}}})^{\mathrm{F-ss}}\cong\mathrm{rec}_{\mathrm{GT}}(\pi_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})$

for each finite place $v$ of $F$. If $v|\ell$, the Hodge-Tate weights of
$r_{\pi,\ell,\iota}|_{G_{F_{v}}}$ are given by $\bar{\mu}_{v}+(3,2,1,0)$.

  * (iii)

$r_{\pi,\ell,\iota}$ has similitude character $\chi_{\ell,\iota}\epsilon^{-3}$
(where $\chi_{\ell,\iota}$ is the $\ell$-adic realization of $\chi$ through
$\iota$).

###### Proof.

The existence of $r_{\pi,\ell,\iota}$ satisfying local-global compatibility
with the stated Hodge-Tate weights follows from [Sor09, Theorem B] together
with Theorem 3.5 of [Mok14]. The symplecticity and similitude character in (i)
follow from Corollary 1.3 of [BC11] (cf. [Mok14, Remark 3.3(3)]). ∎

For $\pi$ a regular cuspidal automorphic representation of
$\operatorname{GSp}_{4}(\mathbb{A}_{F})$ with central character $\chi$, we
also write $r_{\pi,l,\iota}$ for a continuous semisimple representation

$\displaystyle
r_{\pi,l,\iota}:G_{F}\rightarrow\operatorname{GSp}_{4}(\overline{\mathbb{Q}}_{p})$

satisfying items (i)-(iii) above constructed by [Mok14, Theorem 3.5].

######  Remark 4.1.5.

(i) One expects $r_{\pi,\ell,\iota}$ to always be irreducible if $\pi$ is of
general type. This is known for $F=\mathbb{Q}$ and $l\geq 5$ ([Weib, Theorem
1.1]) and $F$ totally real and sufficiently large $l$ ([Weia, Theorem A]).

(ii) The existence of these Galois representations is compatible with the
conjectures of [BG14]: the automorphic representation of
$\operatorname{GSp}_{4}$ corresponding to $\pi$ is $C$-algebraic and one
twists this into an $L$-algebraic automorphic representation (see [BG14, §5]).

(iii) It follows immediately from the theorem that
$r_{\pi,\ell,\iota}:G_{F}\rightarrow\operatorname{GSp}_{4}(\overline{\mathbb{Q}}_{\ell})$
is odd in the sense of Definition 4.1.3.

We finish this subsection by proving that all automorphic representations of
$\mathcal{G}(\mathbb{A}_{F})$ of our interest are stable and tempered.

Let $\pi$ be a regular algebraic cuspidal automorphic representation of
$\operatorname{GSp}_{4}(\mathbb{A}_{F})$ or $\mathcal{G}(\mathbb{A}_{F})$. We
say that a continuous representation
$\bar{r}:G_{F}\rightarrow\operatorname{GSp}_{4}(\mathbb{F})$ is _attached to
$\pi$_ if for almost all finite places $v$ at which $\pi$ and $\bar{r}$ are
unramified, we have

$\displaystyle\iota\circ\operatorname{WD}(\bar{r}|_{G_{F_{v}}})\simeq\mathrm{rec}_{\mathrm{GT}}(\pi_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})\mod\varpi.$

######  Lemma 4.1.6.

Let $\pi^{\prime}$ be a regular algebraic cuspidal automorphic representation
of $\mathcal{G}(\mathbb{A}_{F})$. Suppose that there exists a continuous
semisimple representation
$\bar{r}:G_{F}\rightarrow\operatorname{GSp}_{4}(\mathbb{F})$ attached to
$\pi^{\prime}$. Then there exists a regular algebraic cuspidal automorphic
representation $\pi$ of $\operatorname{GSp}_{4}(\mathbb{A}_{F})$ which is a
weak Jacquet–Langlands transfer of $\pi^{\prime}$. In other words, we have
$\pi_{v}\simeq\pi^{\prime}_{v}$ for almost all finite places $v$ of $F$.

###### Proof.

Let $\mathcal{H}:=\mathcal{G}^{\mathrm{der}}$ be the derived subgroup of
$\mathcal{G}$. Then $\mathcal{H}$ is an inner form of $\operatorname{Sp}_{4}$
which splits at all finite places. By a Theorem of Hiraga–Saito ([GT19,
Theorem 5.1.2]), $\pi^{\prime}|_{\mathcal{H}(\mathbb{A}_{F})}$ contains a
regular algebraic cuspidal automorphic representation of
$\mathcal{H}(\mathbb{A}_{F})$ which we denote by $\tilde{\pi}^{\prime}$. By
[Taï19, Theorem 4.0.1], there exists a self-dual regular algebraic discrete
automorphic representation $\Pi$ of $\operatorname{GL}_{5}(\mathbb{A}_{F})$
which is a transfer of $\tilde{\pi}^{\prime}$. It is moreover cuspidal by
[Ram08, Theorem A]. Then by [Art13, Theorem 1.4.1], there exists a regular
algebraic discrete automorphic representation $\tilde{\pi}$ of
$\operatorname{Sp}_{4}(\mathbb{A}_{F})$ whose transfer to
$\operatorname{GL}_{5}(\mathbb{A}_{F})$ is $\Pi$. Since $\Pi$ is algebraic,
$\Pi_{\infty}$ is essentially tempered by [Clo90, Lemma 4.9], and
$\pi_{\infty}$ is also essentially tempered, so that $\pi$ is cuspidal by
[Wal84, Theorem 4.3]. Note that at all finite places $v$,
$\tilde{\pi}^{\prime}_{v}$ and $\tilde{\pi}_{v}$ have the same $L$-parameter
(these $L$-parameters are constructed in [Art13]).

By [Pat19, Proposition 3.1.4], there exists a cuspidal automorphic
representation $\pi$ of $\operatorname{GSp}_{4}(\mathbb{A}_{F})$ lifting
$\tilde{\pi}$ and whose central character is equal to the central character of
$\pi^{\prime}$. For any finite place $v$ of $F$,
${\mathrm{std}}^{\prime}(\bar{r})_{v}$ and
${\mathrm{std}}^{\prime}(\bar{r}_{\pi,p,\iota})_{v}$ are isomorphic, as they
are mod $p$ reduction of $L$-parameters attached to $\tilde{\pi}^{\prime}_{v}$
and $\tilde{\pi}_{v}$ respectively (here, we use the compatibility between the
local Langlands correspondence for $\operatorname{GSp}_{4}$ and
$\operatorname{Sp}_{4}$ upon restriction; see [GT10, Main Theorem (v)]). By
Brauer–Nesbitt Theorem, we have
${\mathrm{std}}^{\prime}(\bar{r})\simeq{\mathrm{std}}^{\prime}(\bar{r}_{\pi,p,\iota})$.
Since $\bar{r}_{\pi,p,\iota}$ and $\bar{r}$ also have the same similitude
character, we have $\bar{r}\simeq\bar{r}_{\pi,p,\iota}\otimes\theta$ for some
quadratic character
$\theta:G_{F}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$. By twisting $\pi$
by $\theta^{{-1}}$, we can ensure that $\bar{r}_{\pi,p,\iota}\simeq\bar{r}$.

Let $v$ be a finite place of $F$ at which $\pi$ and $\pi^{\prime}$ are
unramified. Then
$\mathrm{rec}_{\mathrm{GT}}(\pi_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})$ and
$\mathrm{rec}_{\mathrm{GT}}(\pi^{\prime}_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})$
have the same similitude character and are isomorphic after composed with
${\mathrm{std}}^{\prime}$ (again, by using the local Langlands for
$\operatorname{Sp}_{4}$). Thus,
$\mathrm{rec}_{\mathrm{GT}}(\pi_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})\simeq\mathrm{rec}_{\mathrm{GT}}(\pi^{\prime}_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})\otimes\theta_{v}$
for some quadratic character
$\theta_{v}:W_{F_{v}}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$. However,
by the previous paragraph, we have
$\mathrm{rec}_{\mathrm{GT}}(\pi_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})\simeq\mathrm{rec}_{\mathrm{GT}}(\pi^{\prime}_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})\mod\varpi$,
which means $\theta_{v}$ is trivial modulo $\varpi$. Since $\theta_{v}$ has
order at most 2 and $p>2$, this implies that $\theta_{v}$ is trivial and
$\mathrm{rec}_{\mathrm{GT}}(\pi_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})\simeq\mathrm{rec}_{\mathrm{GT}}(\pi^{\prime}_{v}\otimes|\nu_{\mathrm{sim}}|^{-3/2})$.
Since the $L$-packet of an unramified $L$-parameter valued in
$\operatorname{GSp}_{4}$ is a singleton, we have
$\pi_{v}\simeq\pi^{\prime}_{v}$. ∎

######  Lemma 4.1.7.

Let $\pi$ be a regular cuspidal automorphic representation of
$G(\mathbb{A}_{F})$ where $G=\operatorname{GSp}_{4}$ or $G=\mathcal{G}$.
Suppose that there exists a continuous irreducible representation
$\bar{r}:G_{F}\rightarrow\operatorname{GSp}_{4}(\mathbb{A}_{F})$ attached to
$\pi$, then $\pi$ is stable and tempered.

###### Proof.

If $G=\mathcal{G}$, then we can apply Lemma 4.1.6 to replace $\pi$ by a
regular cuspidal automorphic representation of
$\operatorname{GSp}_{4}(\mathbb{A}_{F})$ to which $\bar{r}$ is attached. Since
being stable and tempered is characterized by components at almost all finite
places, it suffices to prove the claim for $G=\operatorname{GSp}_{4}$, and
$\bar{r}\simeq\bar{r}_{\pi,p,\iota}$ in this case. By [BCGP21, Theorem 2.9.3],
the irreducibility of $\bar{r}$ implies that $\pi$ is of general type, which
implies that $\pi$ is stable and tempered by Arthur’s classification (see item
(a)-(f) at the end of [Art04]).

To be more precise, suppose that $\pi$ is not stable, i.e. $\pi$ is either CAP
or endoscopic (again, in the sense of [Sor09]). If $\pi$ is CAP, then by the
results of Piatetski-Shapiro and Soudry [PS83, Sou88], $\pi$ can be realized
as a theta lift (such $\pi$ corresponds to either Soudry type or
Saito–Kurokawa type in [Art04]). Using this, one can easily see that $\bar{r}$
has to be reducible. If $\pi$ is endoscopic, it corresponds to Yoshida type in
_loc. cit._ , and $\bar{r}$ in this case has to be reducible as well.

To show that $\pi$ is tempered, note that the transfer $\Pi$ of $\pi$ to
$\operatorname{GL}_{4}$ is cuspidal. By Ramanujan conjecture (proven in
[Shi11] in this case), $\Pi_{v}$ is tempered for all places $v$. This implies
that the $L$-parameter attached to $\Pi_{v}$ has bounded image. By [GT11, Main
Theorem (vii)], the $L$-packet containing $\pi_{v}$ has a unique generic
tempered element. Then for any finite place $v$ at which $\pi$ is unramified,
$\pi_{v}$ is generic tempered because the $L$-packet containing it is a
singleton. ∎

###  4.2 $p$-adic automorphic forms

We now define integral models of spaces of automorphic forms on $\mathcal{G}$
over $\mathcal{O}$. Fix a choice of isomorphism
$\iota:\overline{\mathbb{Q}}_{p}\rightarrow\mathbb{C}$, and let
$\psi:=\chi_{p,\iota}\epsilon^{-3}$. All the constructions in this section
depend on $\iota$, so we omit it from the notation. If $V$ is an
$\mathcal{O}$-module with a linear action of
$\mathcal{G}(\mathcal{O}_{F_{p}})$ and
$U\leq\mathcal{G}(\mathbb{A}_{F}^{\infty,p})\times\mathcal{G}(\mathcal{O}_{F_{p}})$
is a compact open subgroup, we define $S_{\chi}(U,V)$ to be the
$\mathcal{O}$-module of functions
$f:\mathcal{G}(F)\backslash\mathcal{G}(\mathbb{A}_{F}^{\infty})\rightarrow V$
such that $f(zg)=\iota^{-1}(\chi(z))f(g)$ and $f(gu)=u_{p}^{-1}f(g)$ for $z\in
Z(\mathbb{A}_{F}^{\infty})$, $g\in\mathcal{G}(\mathbb{A}_{F}^{\infty})$ and
$u\in U$. Note this space is zero unless $V$ has central character equal to
$\iota^{-1}\circ\chi^{-1}|_{\mathcal{O}_{F_{p}}^{\times}}$.

######  Definition 4.2.1.

We say that a compact open subgroup $U$ as above is _sufficiently small_ if
its projection to $\mathcal{G}(F_{v})$ contains no element of exact order $p$
for some finite place $v$.

We say that $U$ is _unramified_ at a finite place $v$ if $U=U_{v}U^{v}$ where
$U_{v}=\mathcal{G}(\mathcal{O}_{F_{v}})$.

By finiteness of class numbers, $S_{\chi}(U,V)$ is finitely generated (resp.
free) over $\mathcal{O}$ whenever $V$ is, and if $U$ is sufficiently small
then $S_{\chi}(U,V\otimes_{\mathcal{O}}A)=S_{\chi}(U,V)\otimes_{\mathcal{O}}A$
for any $\mathcal{O}$-module $A$ with trivial
$\mathcal{G}(\mathcal{O}_{F_{p}})$-action.

Let $P_{U}$ denote the finite set of finite places at which $U$ is ramified.
Let $S_{p}$ denote the set of places of $F$ dividing $p$. If $P\supseteq
P_{U}\cup S_{p}$ is any finite set of finite places we let
$\mathbb{T}^{P,{\operatorname{univ}}}$ denote the polynomial
$\mathcal{O}$-algebra generated by variables $S_{v},T_{v,1},T_{v,2}$ for each
$v\notin P$. Then $\mathbb{T}^{P,{\operatorname{univ}}}$ has a natural action
on $S_{\chi}(U,V)$, where the elements above act through the double coset
operators of
$\beta_{0}(\varpi_{v}),\beta_{1}(\varpi_{v}),\beta_{2}(\varpi_{v})$ in the
spherical Hecke algebra
$\mathcal{O}[\operatorname{GSp}_{4}(F_{v})//\operatorname{GSp}_{4}(\mathcal{O}_{F_{v}})]$
respectively.

For simplicity we now assume that $p$ splits completely in $F$. Then we can
identify places above $p$ with $\mathbb{Q}$-embeddings $F\hookrightarrow E$
and $\iota$ induces a bijections $v\mapsto\iota(v):S_{p}\rightarrow
S_{\infty}$. For $\lambda\in X^{*}(T)^{+}$, let $V_{\lambda}$ denote the Weyl
module of $\operatorname{GSp}_{4}$ over $\mathcal{O}_{F}$ of highest weight
$\lambda$. So $V_{\lambda}$ is a finite free $\mathcal{O}_{F}$-module with an
action of $\operatorname{GSp}_{4}(\mathcal{O}_{F})$. Now if
$\mu=(\mu_{v})_{v|p}\in(X^{*}(T)^{+})^{\operatorname{Hom}_{\mathbb{Q}}(F,\overline{\mathbb{Q}}_{p})}$
we let $V(\mu)$ denote the
$\mathcal{O}[\operatorname{GSp}_{4}(\mathcal{O}_{F,p})]$-module

$V(\mu)=\bigotimes_{v\in
S_{p}}V_{\mu_{v}}\otimes_{\mathcal{O}_{F_{v}}}\mathcal{O}$

which we regard a $\mathcal{O}[\mathcal{G}(\mathcal{O}_{F,p})]$-module. Then
$V(\mu)\otimes_{\mathcal{O},\iota}\mathbb{C}$ has a natural action of
$\mathcal{G}(F_{\infty})$ making it isomorphic to $\xi_{\iota(\mu)}$.
Moreover, $\iota$ induces an isomorphism of
$\overline{\mathbb{Q}}_{p}[\mathcal{G}(\mathbb{A}_{F}^{\infty})]$-modules

$\varinjlim_{U}S_{\chi}(U,V(\mu))\cong\mathcal{A}_{\chi,\iota(\mu)}\otimes_{\mathbb{C},\iota^{-1}}\overline{\mathbb{Q}}_{p},$
(4.2.2)

where $U$ runs over all compact open subgroups of
$\mathcal{G}(\mathbb{A}_{F}^{\infty,p})\times\mathcal{G}(\mathcal{O}_{F,p})$.
See for example the proof of Proposition 3.3.2 in [CHT08]. We define

$\displaystyle S_{\chi,\mu}(U,W):=S_{\chi}(U,V(\mu)\otimes_{\mathcal{O}}W)$

for any $\mathcal{O}[\mathcal{G}(\mathcal{O}_{F_{p}})]$-module $W$ and level
$U\leq\mathcal{G}(\mathbb{A}_{F}^{\infty,p})\times\mathcal{G}(\mathcal{O}_{F,p})$.

Suppose that $\bar{r}:G_{F}\rightarrow\operatorname{GSp}_{4}(\mathbb{F})$ is
an absolutely irreducible continuous representation. Let $P_{\bar{r}}$ denote
the set of finite places either dividing $p$ or at which $\bar{r}$ is
ramified. For any finite set of finite places $P\supseteq P_{\bar{r}}$ we
define a maximal ideal
$\mathfrak{m}_{\bar{r},\chi}^{P}\leq\mathbb{T}^{P,{\operatorname{univ}}}$ with
residue field $\mathbb{F}$ by demanding that for each $v\notin P$,

$S_{v}\mod\mathfrak{m}_{\bar{r},\chi}^{P}=\bar{\psi}(\operatorname{Frob}_{v}),$

and the characteristic polynomial $\bar{r}(\operatorname{Frob}_{v})$ in
$\mathbb{F}[X]$ is given by

$X^{4}-T_{v,1}X^{3}+(q_{v}T_{v,2}+(q_{v}^{3}+q_{v})S_{v})X^{2}-q_{v}^{3}T_{v,1}S_{v}X+q_{v}^{6}S_{v}^{2}\mod\mathfrak{m}_{\bar{r},\chi}^{P}$

(cf. [BCGP21, §2.4.7]; note $\mathfrak{m}_{\bar{r},\chi}^{P}$ is well-defined
by symplecticity of $\bar{r}$).

######  Definition 4.2.3.

We say that a pair $(\bar{r},\chi)$ as above is _modular of weight $\mu$
(resp. weight $F(\mu)$) and level $U$_ if there exists a weight
$\mu\in(X^{*}(T)^{+})^{S_{p}}$ (resp. $\mu\in(X_{1}(T))^{S_{p}}$ and
$F(\mu):=\otimes_{v\in S_{p}}F(\mu_{v})$), a level
$U\leq\mathcal{G}(\mathbb{A}_{F}^{\infty,p})\times\mathcal{G}(\mathcal{O}_{F_{p}})$,
and a finite set of finite places $P$ containing $P_{U}\cup P_{\bar{r}}$ such
that $S_{\chi,\mu}(U,\mathcal{O})_{\mathfrak{m}_{\bar{r},\chi}^{P}}\neq 0$
(resp. $S_{\chi}(U,F(\mu))_{\mathfrak{m}_{\bar{r},\chi}^{P}}\neq 0$).

######  Remark 4.2.4.

A pair $(\bar{r},\chi)$ being modular is equivalent to asking that there
exists a regular cuspidal automorphic representation $\pi^{\prime}$ of
$\mathcal{G}(\mathbb{A}_{F})$ to which $\bar{r}$ is attached such that
$\pi^{U}\neq 0$ and $\iota^{-1}(\pi)$ contributes to (4.2.2). As such we are
free to shrink $U$ and enlarge $P$. Moreover, such $\pi$ is necessarily stable
and tempered by Lemma 4.1.7, and we can apply Theorem 4.1.4 to obtain

$\displaystyle
r_{\pi,p,\iota}:G_{F}\rightarrow\operatorname{GSp}_{4}(\overline{\mathbb{Q}}_{p})$

such that $\nu_{\mathrm{sim}}(r_{\pi,p,\iota})=\chi_{p,\iota}\epsilon^{-3}$
and the reduction mod $p$ of $r_{\pi,p,\iota}$ is
$\operatorname{GL}_{4}(\overline{\mathbb{F}}_{p})$-conjugate to $\bar{r}$.

Also, by applying [Sor09, Theorem B], $(\bar{r},\chi)$ being modular is in
turn equivalent to asking that there exists a regular algebraic cuspidal
automorphic representation $\pi$ of $\operatorname{GSp}_{4}(\mathbb{A}_{F})$
with central character $\chi$ such that $\bar{r}_{\pi,p,\iota}\simeq\bar{r}$.

######  Definition 4.2.5.

We say that a pair $(\bar{r},\chi)$ is _potentially diagonalizably
automorphic_ if it is modular and $\pi$ as above can be taken so that
${\mathrm{std}}(r_{\pi,p,\iota})|_{G_{F_{v}}}$ is potentially diagonalizable
(in the sense of [BLGGT14, §1.4]) for all $v|p$.

For any finite set of finite places $P\supseteq P_{U}$ let
$\mathbb{T}_{\chi,\mu}^{P}(U)$ denote the image of
$\mathbb{T}^{P,{\operatorname{univ}}}$ inside
$\operatorname{End}_{\mathcal{O}}(S_{\chi,\mu}(U,\mathcal{O}))$. Observe that
$\mathbb{T}_{\chi,\mu}^{P}(U,\mathcal{O})_{\mathfrak{m}_{\bar{r}}^{P}}$ is an
object of $\widehat{\mathcal{C}}_{\mathcal{O}}$.

######  Lemma 4.2.6.

The ring $\mathbb{T}_{\chi,\mu}^{P}(U)_{\mathfrak{m}_{\bar{r}}^{P}}$ is
reduced.

###### Proof.

By commutative algebra the semisimplicity of
$S_{\chi,\mu}(U,\mathcal{O})\otimes_{\mathcal{O}}\overline{\mathbb{Q}}_{p}$,
which follows from (4.2.2), shows that $\mathbb{T}_{\chi,\mu}^{P}(U)$ is
reduced. The claim follows. ∎

######  Proposition 4.2.7.

Assume that $\bar{r}:G_{F}\rightarrow\operatorname{GSp}_{4}(\mathbb{F})$ is
absolutely irreducible and $(\bar{r},\chi)$ is modular of weight $\mu$ and
level $U$. Then for any finite set $P\supset P_{U}$ there exists a unique
continuous homomorphism

$r_{\chi,\mu}^{P}(U):G_{F}\rightarrow\operatorname{GSp}_{4}(\mathbb{T}_{\chi,\mu}^{P}(U)_{\mathfrak{m}_{\bar{r},\chi}^{P}})$

lifting $\bar{r}$ such that

  * (i)

$\nu_{\mathrm{sim}}\circ r_{\chi,\mu}^{P}(U)=\psi$,

  * (ii)

if $v\notin P$ then $r_{\chi,\mu}^{P}(U)$ is unramified and the characteristic
polynomial of $r_{\chi,\mu}^{P}(U)(\operatorname{Frob}_{v})$ in
$\mathbb{T}^{P}_{\chi,\mu}(U)_{\mathfrak{m}^{P}_{\bar{r},\chi}}[X]$ is equal
to

$X^{4}-T_{v,1}X^{3}+(q_{v}T_{v,2}+(q_{v}^{3}+q_{v})\chi_{v}(\varpi_{v}))X^{2}-q_{v}^{3}T_{v,1}\chi_{v}(\varpi_{v})X+q_{v}^{6}\chi_{v}(\varpi_{v})^{2},$

  * (iii)

and for every $\mathcal{O}$-algebra homomorphism
$\zeta:\mathbb{T}_{\chi,\mu}^{P}(U)_{\mathfrak{m}_{\bar{r}}^{P}}\rightarrow
E^{\prime}$ where $E^{\prime}$ is a finite extension of $E$, the
representation $\zeta\circ r_{\chi,\mu}^{P}(U)|_{G_{F_{v}}}$ is de Rham of
Hodge-Tate weights $\bar{\mu}_{v}+(3,2,1,0)$ for all $v\in S_{p}$.

###### Proof.

By Lemma 4.1.7 and the irreducibility of $\bar{r}$, we know that the
automorphic representation associated with Hecke eigensystem
$\mathbb{T}^{P}_{\chi,\mu}(U)_{\mathfrak{m}_{\bar{r}}^{P}}\xrightarrow{}E^{\prime}$
is stable and tempered. Then the Proposition can be shown by using Theorem
4.1.4 and a standard argument using Lemma 4.2.6; see for example the proof of
[CHT08, Proposition 3.4.4]. In order to ensure the resulting representation is
symplectic, one applies Lemma 1.2.1. Item (ii) follows from [BCGP21, §2.4.7].
∎

From now on, we fix a preferred place $w|p$. If $W$ is any
$\mathcal{O}[\mathcal{G}(\mathcal{O}_{F_{w}})]$-module and $\sigma$ is a
finite $\mathcal{O}$-module with smooth
$\operatorname{GSp}_{4}(\mathbb{Z}_{p})$-action, we define

$S_{\chi,\mu,\sigma}(U^{w},W):=\varinjlim_{U_{w}\leq\mathcal{G}(\mathcal{O}_{F_{w}})}S_{\chi}(U^{w}U_{w},V(\mu^{\prime})\otimes_{\mathcal{O}}\sigma^{S_{p}\backslash\\{w\\}}\otimes_{\mathcal{O}}W)$

where $\mu^{\prime}$ is equal to $\mu$ but with $\mu_{w}$ replaced by
$(0,0;0)$, and $\sigma^{S_{p}\backslash\\{w\\}}$ is viewed as smooth
$\mathcal{O}[\prod_{v\in
S_{p}\backslash\\{w\\}}\mathcal{G}(\mathcal{O}_{F_{v}})]$-module by
identifying
$\mathcal{G}(\mathcal{O}_{F_{v}})\simeq\operatorname{GSp}_{4}(\mathbb{Z}_{p})$.
Note that the functor $W\mapsto S_{\chi,\mu,\sigma}(U^{w},W)$ is exact if
$U^{w}$ is sufficiently small.

By definition $S_{\chi,\mu,\sigma}(U^{w},\mathbb{F})$ is an admissible
representation of
$\mathcal{G}(F_{w})\cong\operatorname{GSp}_{4}(\mathbb{Q}_{p})$ over
$\mathbb{F}$ with central character $\iota^{-1}\circ\chi_{w}$. Suppose that
$\overline{r}:G_{F}\rightarrow\operatorname{GSp}_{4}(\mathbb{F})$ is modular
of some weight and level. We will show that
$S_{\chi,\mu,\sigma}(U^{w},\mathbb{F})[\mathfrak{m}_{\bar{r},\chi}^{P}]$ is
nonzero for certain $\mu$ and $\sigma$. We are interested in studying the
relationship between the $\mathbb{F}[\mathcal{G}(F_{w})]$-module
$S_{\chi,\mu,\sigma}(U^{w},\mathbb{F})[\mathfrak{m}_{\bar{r},\chi}^{P}]$ and
the local Galois representation $\bar{r}|_{G_{F_{w}}}$. The main tool for
doing this is the patching method described in the next two sections.

###  4.3 A patching lemma

In order to clarify the patching method used in the next section, we present
here a formalized version. In this section $G$ denotes a locally profinite
group having a countable fundamental system of open neighbourhoods at the
identity.

######  Definition 4.3.1.

Let $A$ be a commutative ring and
$\mathfrak{j}=\\{\mathfrak{j}_{1}\supseteq\mathfrak{j}_{2}\supseteq\cdots\\}$
a system of ideals. A _$(G,A,\mathfrak{j})$ -inverse system_ is an inverse
system $\mathfrak{M}=\\{M_{r}(H)\\}_{r\geq 1,H\leq_{\mathrm{c.o.}}G}$ of
finite $A$-modules together with a collection of morphisms of inverse systems
$g_{*}:M_{r}(H)\rightarrow M_{r}(gHg^{-1})$ for each $g\in G$ satisfying
$(gh)_{*}=g_{*}h_{*}$ and $\mathrm{id}_{*}=\mathrm{id}$. We assume that the
transition maps have the property that they induce isomorphisms

$M_{r+1}(H)/\mathfrak{j}_{r}\xrightarrow{\sim}M_{r}(H)$

and

$M_{r}(H^{\prime})_{H}\xrightarrow{\sim}M_{r}(H)$

for all $r\geq 1,H\leq_{\mathrm{c.o.}}G$, and any open normal subgroup
$H^{\prime}\subseteq H$. (In particular, all transition maps are surjective.)
These objects form a category in an obvious way.

We say that $\mathfrak{M}$ is _projective_ if $M_{r}(H^{\prime})$ is a
projective $(A/\mathfrak{j}_{r})[H/H^{\prime}]$-module whenever
$H^{\prime}\subseteq H$ is an open normal subgroup.

######  Remark 4.3.2.

If
$\mathfrak{b}=\\{\mathfrak{b}_{1}\supseteq\mathfrak{b}_{2}\supseteq\cdots\\}$
is another system of ideals of $A$ such that
$\mathfrak{b}_{r}\supseteq\mathfrak{j}_{r}$ then
$(M_{r}(K)/\mathfrak{b}_{r})_{r,K}$ is a (projective)
$(G,A,\mathfrak{b})$-inverse system.

######  Definition 4.3.3 (cf. [Eme10]).

Let $A$ be a complete local noetherian $\mathcal{O}$-algebra with finite
residue field. We let $\operatorname{Mod}^{\textrm{pro aug}}_{G}(A)$ denote
the abelian category of left $A[G]$-modules $M$ equipped with a profinite
topology such that for any compact open $H\subseteq G$, the $A[H]$-action
extends to an action of $A[[H]]$ making $M$ into a linear topological
$A[[H]]$-module for the profinite topology on $A[[H]]$.

######  Lemma 4.3.4.

Let $A$ be a complete local noetherian $\mathcal{O}$-algebra with finite
residue field, and suppose that the topology determined by $\mathfrak{j}$ is
the max-adic one. The functor

$\mathfrak{M}\mapsto\mathfrak{M}_{\infty}:=\varprojlim_{r,H}M_{r}(H)$

is an equivalence of categories between the category of
$(G,A,\mathfrak{j})$-inverse systems and
$\operatorname{Mod}^{\textrm{\emph{{pro aug}}}}_{G}(A)$. Moreover, if
$\mathfrak{M}$ is projective then $\mathfrak{M}_{\infty}$ is a finitely
generated projective $A[[H]]$-module for any compact open $H\subseteq G$.

###### Proof.

The equivalence of categories follows from [Eme10, Lemma 2.2.6]. In
particular, for any open subgroup $H\subseteq G$ we have
$(\mathfrak{M}_{\infty})_{H}/\mathfrak{j}_{r}=M_{r}(H)$. Using this, the
topological Nakayama lemma for $A[[H]]$ shows that $\mathfrak{M}_{\infty}$ is
finitely generated over $A[[H]]$. The argument in the proof of [CEG+16,
Proposition 2.10] now shows that $\mathfrak{M}_{\infty}$ is a projective
$A[[H]]$-module if $\mathfrak{M}$ is projective. ∎

######  Definition 4.3.5.

A _$G$ -patching datum over $\mathcal{O}$_ is a tuple

$(S_{\infty},R_{\infty},(R_{n},\varphi_{n},\mathfrak{M}_{n},\alpha_{n})_{n\geq
1},\mathfrak{M}_{0})$

where

  * •

$S_{\infty}$ is a power series ring over $\mathcal{O}$ with augmentation ideal
$\mathfrak{a}_{\infty}=\ker(S_{\infty}\rightarrow\mathcal{O})$,

  * •

$R_{\infty}$ is a complete local $\mathcal{O}$-algebra with finite residue
field,

  * •

$R_{n}$ is a local $S_{\infty}$-algebra for $n\geq 1$,

  * •

$\varphi_{n}:R_{\infty}\twoheadrightarrow R_{n}$ is a local
$\mathcal{O}$-algebra surjection for $n\geq 1$,

  * •

$\mathfrak{M}_{n}=\\{M_{r}(H)_{n}\\}_{r\geq 1,H\leq_{\mathrm{c.o.}}G}$ is a
$(G,R_{n},(\varpi^{r})_{r\geq 1}$-inverse system for $n\geq 0$ with
$\mathfrak{M}_{0}$ projective (where we define $R_{0}:=\mathcal{O}$),

  * •

and
$\alpha_{n}:\mathfrak{M}_{n}/\mathfrak{a}_{\infty}\rightarrow\mathfrak{M}_{0}$
is an isomorphism of $(G,S_{\infty},(\varpi^{r})_{r\geq 1})$-inverse systems
for $n\geq 1$.

This data is assumed to satisfying the following axioms:

  1. (1)

For all $r,n\geq 1$ the $S_{\infty}$-module $M_{r}(H)_{n}$ is free over a
quotient $S_{\infty}/I_{r,n}$ (independent of $H$) such that
$I_{r,n}\subseteq(\varpi^{r},\mathfrak{a}_{\infty})$ and

for any open ideal $\mathfrak{b}\subseteq S_{\infty}$,
$I_{r,n}\subseteq(\varpi^{r},\mathfrak{b})$ for almost all $n\geq 1$. (4.3.6)

  2. (2)

Whenever $H^{\prime}\subseteq H$ is an open normal subgroup,
$M_{r}(H^{\prime})_{n}$ is a projective
$S_{\infty}/I_{r,n}[H/H^{\prime}]$-module for all $r,n\geq 1$. Also
$M_{r}(H^{\prime})_{0}$ is a projective
$(\mathcal{O}/\varpi^{r})[H/H^{\prime}]$-module.

One can visualize a $G$-patching datum via the commutative diagram

$\textstyle{R_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{n}}$$\textstyle{R_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{n}/\varpi^{r}}$$\textstyle{\curvearrowright
M_{r}(H)_{n}.}$$\textstyle{S_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(4.3.7)

The dotted arrow exists for each $n\geq 1$ because $S_{\infty}$ is a power
series ring.

######  Definition 4.3.8 (cf. [Man]).

Let $\mathcal{F}\subset 2^{\mathbb{N}}$ be a nonprincipal ultrafilter. If $S$
is a ring and $\mathcal{Q}=(Q_{n})_{n\geq 1}$ is a sequence of left
$S$-modules we let $\mathcal{U}_{\mathcal{F}}(\mathcal{Q})$ denote their
ultraproduct over $\mathcal{F}$. The properties of this construction that we
will use are

  1. 1.

$\mathcal{U}_{\mathcal{F}}(-)$ is an exact functor from the category of
sequences of $S$-modules with $\mathcal{F}$-morphisms, and

  2. 2.

if $S$ is finite and the cardinalities of the $Q_{n}$ are uniformly bounded,
then $\mathcal{U}_{\mathcal{F}}(\mathcal{Q})\cong Q_{i}$ for
$\mathcal{F}$-many $i\geq 1$.

######  Lemma/Definition 4.3.9.

Fix a nonprincipal ultrafilter $\mathcal{F}$. Given a $G$-patching datum as in
Definition 4.3.5 we define the associated _patched module_ to be

$M_{\infty}:=\varprojlim_{r,H}\,\,\mathcal{U}_{\mathcal{F}}(\\{(M_{r}(H)_{n}\otimes_{S_{\infty}}S_{\infty}/\mathfrak{m}_{S_{\infty}}^{r}\\}_{n\geq
1}).$

It is a finitely generated projective object in
$\operatorname{Mod}^{\mathrm{pro\,aug}}_{G}(S_{\infty})$ such that
$M_{\infty}/\mathfrak{a}_{\infty}\cong(\mathfrak{M}_{0})_{\infty}$. It also
has a compatible $R_{\infty}[G]$-module structure via a local
$\mathcal{O}$-algebra map $S_{\infty}\rightarrow R_{\infty}$.

###### Proof.

For each $r,H$, the isomorphisms $\alpha_{n}$ imply that
$M_{r}(H)_{n}\otimes_{S_{\infty}}S_{\infty}/\mathfrak{m}_{S_{\infty}}^{r}$ is
of uniformly bounded cardinality, so the ultraproduct is well behaved. Axioms
(1) and (2) together with the properties of $\mathcal{U}_{\mathcal{F}}$ above
imply that
$(\mathcal{U}_{\mathcal{F}}(\\{(M_{r}(H)_{n}\otimes_{S_{\infty}}S_{\infty}/\mathfrak{m}_{S_{\infty}}^{r}\\}_{n\geq
1})_{r\geq 1,H\leq_{\mathrm{c.o.}}G}$ is a projective
$(G,S_{\infty},(\mathfrak{m}_{S_{\infty}}^{r})_{r\geq 1})$-inverse system
whose reduction mod $\mathfrak{a}_{\infty}$ is isomorphic to
$\mathfrak{M}_{0}$. The first claim now follows from Lemma 4.3.4.

For the second claim, at each finite level $n\geq 1$ the $S_{\infty}$ action
on $\mathfrak{M}_{n}$ factors through the local map $S_{\infty}\rightarrow
R_{\infty}$ in (4.3.7). Since this map is independent of $r$ and $H$ the same
is true of $M_{\infty}$. ∎

###  4.4 Patching

We give the precise conditions of the global Galois representations to which
our main result applies.

######  Definition 4.4.1.

A triple $(F,\overline{r},\chi)$ of a totally real field $F$, a continuous
representation
$\overline{r}:G_{F}\rightarrow\operatorname{GSp}_{4}(\mathbb{F})$, and a Hecke
character
$\chi:\mathbb{A}_{F}^{\times}/F^{\times}\rightarrow\mathbb{C}^{\times}$ is
called _suitable of weight $(a_{3},a_{2},a_{1},a_{0})$_ if it satisfies the
followings:

  * (A1)

$F$ has even degree over $\mathbb{Q}$,

  * (A2)

$\overline{r}$ is odd,

  * (A3)

$\overline{r}$ is _vast_ and _tidy_ in the sense of [BCGP21, §7.5] (this
implies that $\bar{r}$ is absolutely irreducible),

  * (A4)

$(\overline{r},\chi)$ is potentially diagonaliazably automorphic of some
weight $\xi^{\prime}$ and level unramified away from $p$,

  * (A5)

$\overline{r}|_{G_{F_{v}}}$ is strongly 7-generic of weight
$(a_{3},a_{2},a_{1},a_{0})$ as in Definition 2.3.7(ii) for each $v|p$.

Let
$\bar{\rho}:G_{\mathbb{Q}_{p}}\rightarrow\operatorname{GSp}_{4}(\mathbb{F})$
be strongly 7-generic of weight $(a_{3},a_{2},a_{1},a_{0})$. We say that a
triple $(F,\overline{r},\chi)$ is a _suitable globalization of $\bar{\rho}$_
if it is suitable of weight $(a_{3},a_{2},a_{1},a_{0})$, and there is a place
$w|p$ of $F$ such that $\overline{r}|_{G_{F_{w}}}\simeq\bar{\rho}$.

######  Remark 4.4.2.

In (A4), we assume $(\bar{r},\chi)$ to be potentially diagonalizably
automorphic as opposed to being modular. This is used to show that
$(\bar{r},\chi)$ is modular of the “obvious” Serre weight $F(\mu)$ in
Corollary 4.4.7 by applying the automorphy lifting result for potentially
diagonalizable lifts ([BLGGT14, 4.2.1]). The unramifiedness in (A4) is assumed
for simplicity in the construction of patched modules. Assumption (A5) is not
important for the construction of patching modules.

For the following series of Lemmas, we let $K/\mathbb{Q}_{p}$ be a finite
unramified extension. We only need the case $K=\mathbb{Q}_{p}$ for this
article.

The first Lemma provides a lift of global residual character with prescribed
local lifts. This must be well-known but we give the proof as we could not
find it in the literature.

######  Lemma 4.4.3.

Let $F$ be a totally real field and $S$ be a finite set of places of $F$
containing all places dividing $p$. Suppose we have a continuous character
$\overline{\psi}:G_{F}\xrightarrow{}\mathbb{F}^{\times}$ and
$\psi_{v}:G_{F_{v}}\xrightarrow{}\mathcal{O}^{\times}$ lifting
$\overline{\psi}|_{G_{F_{v}}}$ for all $v\in S$. We further assume that for
all $v|p$, $\psi_{v}$ is de Rham with fixed Hodge–Tate weight $c$ and
$\psi_{v}\epsilon^{-c}$ has finite image. Then there is a continous character
$\psi:G_{F}\xrightarrow{}\mathcal{O}^{\times}$ lifting $\overline{\theta}$ and
$\psi|_{G_{F_{v}}}=\psi_{v}$ for all $v\in S$.

###### Proof.

Twisting by a power of cyclotomic character, we can assume $c=0$. Also, we use
class field theory to consider $\overline{\psi}$ and $\psi_{v}$ as characters
of $\mathbb{A}_{F}^{\times}/F^{\times}$ and $F_{v}^{\times}$ respectively.
Following the argument in [CHT08, Lemma 4.1.1], we let
$U\subset(\mathbb{A}_{F}^{S})^{\times}$ be the open compact subgroup such that
$\overline{\psi}$ is trivial on $U$, and $\psi_{v}$ is trivial on $U\cap
F^{\times}$. We define $\psi:U\prod_{v\in
S}F_{v}^{\times}/(U\cap{F}^{\times})$ by setting trivial on $U$ and $\psi_{v}$
on $F_{v}$. Since the quotient
$(\mathbb{A}_{F}^{\times}/F^{\times})/(U\prod_{v\in
S}F_{v}^{\times}/(U\cap{F}^{\times}))$ is finite abelian group, we can choose
a sequence of subgroups

$\displaystyle U\prod_{v\in
S}F_{v}^{\times}/(U\cap{F}^{\times})=U_{0}\subsetneq
U_{1}\subsetneq\cdots\subsetneq U_{r}=\mathbb{A}_{F}^{\times}/F^{\times}$

where each quotient $U_{i}/U_{i-1}$ is a finite cyclic group. We can
inductively extend $\psi$ from $U_{i-1}$ to $U_{i}$ while ensuring that $\psi$
lifts $\overline{\psi}|_{U_{i}}$ at each step. ∎

The following Lemma shows that the modularity lifting for potentially
diagonalizable representations for $\operatorname{GSp}_{4}$ follows from that
for $\operatorname{GL}_{4}$.

######  Lemma 4.4.4.

Let $r:G_{F}\rightarrow\operatorname{GSp}_{4}(E)$ be a continuous
representation. Let
$\chi:\mathbb{A}_{F}^{\times}/F^{\times}\rightarrow\mathbb{C}^{\times}$ be a
Hecke character such that $\chi_{p,\iota}\epsilon^{-3}=\nu_{\mathrm{sim}}(r)$.
Suppose that

  1. 1.

for all $v|p$, ${\mathrm{std}}(r)|_{G_{F_{v}}}$ is potentially diagonalizable
with distinct Hodge–Tate weights;

  2. 2.

$\bar{r}$ is vast;

  3. 3.

$(\bar{r},\chi)$ is potentially diagonalizably automorphic for some
$\chi:\mathbb{A}_{F}^{\times}/F^{\times}\rightarrow\mathbb{C}^{\times}$.

Then there exists a regular algebraic cuspidal automorphic representation
$\pi$ of $\operatorname{GSp}_{4}(\mathbb{A}_{F})$ such that $r\simeq
r_{\pi,p,\iota}$.

###### Proof.

By [GT19, Theorem 7.4.1], ${\mathrm{std}}(\bar{r})$ is potentially
diagonalizably automorphic in the sense of [BLGGT14]. Thus, Theorem 4.2.1 in
_loc. cit._ implies that there exists a regular algebraic cuspidal automorphic
representation $\Pi$ of $\operatorname{GL}_{4}(\mathbb{A}_{F})$ such that a
continuous representation $r_{\Pi,p,\iota}$ (see Theorem 2.1.1 in _loc. cit._)
is isomorphic to ${\mathrm{std}}(r)$. Such $\Pi$ is then $\chi$-self dual.
Again by applying [GT19, Theorem 7.4.1] (also see Theorem 2.9.3 in [BCGP21]),
this proves the existence of $\pi$ such that $r\simeq r_{\pi,p,\iota}$. ∎

We now prove the existence of a suitable globalization of $\bar{\rho}$
following the strategy of [EG14, Appendix A].

######  Lemma 4.4.5.

Assume that $p>7$. Let
$\bar{\rho}:G_{K}\xrightarrow{}\operatorname{GSp}_{4}(\mathbb{F})$ be a
continuous representation. Suppose there is a lift
$\rho:G_{K}\xrightarrow{}\operatorname{GSp}_{4}(E)$ such that
${\mathrm{std}}(\rho)$ is potentially diagonalizable. Then there is a totally
real field $F$ of even degree over $\mathbb{Q}$ such that $F_{v}\simeq K$ for
all $v|p$ and a continuous representation
$r:G_{F}\xrightarrow{}\operatorname{GSp}_{4}(E)$, with its reduction modulo
$\varpi$ denoted by $\bar{r}$, satisfying that

  1. 1.

$\overline{r}$ is odd,

  2. 2.

$\overline{r}$ is vast and tidy,

  3. 3.

$\overline{r}|_{G_{F_{v}}}\simeq\bar{\rho}$ for all $v|p$,

  4. 4.

$\overline{r}$ is unramified at all finite places not dividing $p$,

  5. 5.

$r|_{G_{F_{v}}}$ and $\rho$ correspond to points contained in the same
component of a potentially crystalline deformation ring for all $v|p$ (in
particular, ${\mathrm{std}}(r|_{G_{F_{v}}})$ is potentially diagonalizable),
and

  6. 6.

there exists a regular algebraic cuspidal automorphic representation $\pi$ of
$\operatorname{GSp}_{4}(\mathbb{A}_{F})$ such that $r\simeq r_{\pi,p,\iota}$.

###### Proof.

In this proof, we let $S_{p,L}$ (resp. $S_{\infty,L}$) be the set of places of
a number field $L$ dividing $p$ (resp. $\infty$).

Let $L$ be a totally real field of even degree over $\mathbb{Q}$ satisfying
$L_{v}\simeq K$ for all $v|p$. We apply [Cal12, Proposition 3.2]. In the
notation of _loc. cit._ , we set $G:=\operatorname{GSp}_{4}(\mathbb{F})$,
$E:=L$, $S:=S_{p,L}\cup S_{\infty,L}$, $F:=L(\zeta_{p})$,
$H_{v}:=\overline{L_{v}}^{\ker\bar{\rho}}$ and $\phi_{v}$ given by
$\bar{\rho}$ with image $D_{v}$ and the isomorphism $L_{v}\simeq K$ for $v\in
S_{p,L}$, and $c_{v}$ to be a Chevalley involution for all $v\in
S_{\infty,L}$. Then there exists extensions of totally real fields
$L^{\prime\prime}/L^{\prime}/L$ such that

  * •

all places in $S$ split completely in $L^{\prime}$,

  * •

$L^{\prime\prime}/L$ is linearly disjoint from $L(\zeta_{p})/L$,

  * •

$L^{\prime\prime}/L$ is Galois and
$\operatorname{Gal}(L^{\prime\prime}/L)\simeq\operatorname{GSp}_{4}(\mathbb{F})$,

  * •

for all places $w$ of $L^{\prime}$ above $v\in S_{p,L}$, the isomorphism
$\operatorname{Gal}(L^{\prime\prime}/L)\simeq\operatorname{GSp}_{4}(\mathbb{F})$
induces $\operatorname{Gal}(L^{\prime\prime}_{w}/L^{\prime}_{w})\simeq D_{v}$.

We write
$\bar{r}^{\prime}:G_{L^{\prime}}\twoheadrightarrow\operatorname{Gal}(L^{\prime\prime}/L)\xrightarrow{\sim}\operatorname{GSp}_{4}(\mathbb{F})$.
Then $\bar{r}^{\prime}$ satisfies conditions (1) and (3) above. Moreover, such
$\bar{r}^{\prime}$ is tidy by [BCGP21, Lemma 7.5.12]. By Lemma 7.5.15 of _loc.
cit._ , $\bar{r}^{\prime}$ is vast if

$\displaystyle\operatorname{Sp}_{4}(\mathbb{F}_{p})\subset\overline{r}^{\prime}(G_{L^{\prime}(\zeta_{p}^{N})})$

for all $N\geq 1$. For $N=1$, this is true because $L^{\prime\prime}/L$ is
linearly disjoint from $L(\zeta_{p})/L$. Then this holds for all $N\geq 1$
because $\operatorname{Sp}_{4}(\mathbb{F}_{p})$ has no $p$-power order
quotient. Moreover, using the argument explained in the last two paragraph of
the proof of [EG14, Proposition A.2], we can and do replace $L^{\prime}$ by
its totally real extension such that $\overline{r}^{\prime}$ further satisfies
(4).

Using Lemma 4.4.3, choose an odd continuous character
$\psi:G_{L^{\prime}}\xrightarrow{}\mathcal{O}^{\times}$ lifting
$\nu_{\operatorname{sim}}(\overline{r})$ and
$\psi|_{G_{L^{\prime}_{v}}}\simeq\nu_{\operatorname{sim}}(\rho)$ for all $v\in
S_{p,L^{\prime}}$. Then [PT21, Theorem 3.4] provides a lift
$r^{\prime}:G_{L^{\prime}}\xrightarrow{}\operatorname{GSp}_{4}(E)$ of
$\overline{r}^{\prime}$ such that $r^{\prime}|_{G_{L^{\prime}_{v}}}$ and
$\rho_{v}$ lie on the same component of potentially crystalline deformation
ring of $\bar{r}|_{G_{L^{\prime}_{v}}}$ for all $v\in S_{p,L^{\prime}}$. By
[EG14, Proposition A.6] (which is based on [BLGGT14, Theorem 4.5.1]), there
exists a finite extension $F/L^{\prime}$ of totally real fields in which $p$
splits completely and a regular algebraic cuspidal automorphic representation
$\Pi$ of $\operatorname{GL}_{4}(\mathbb{A}_{F})$ such that
$r_{\Pi,p,\iota}\simeq r^{\prime}|_{G_{F}}$. Let
$\chi:\mathbb{A}_{F}^{\times}/F^{\times}$ be a Hecke character such that
$\chi_{p,\iota}\epsilon^{-3}\simeq\nu_{\mathrm{sim}}(r^{\prime})|_{G_{F}}$.
Then $\Pi$ is $\chi$-self dual, and it descends to a regular algebraic
cuspidal automorphic representation $\pi$ of
$\operatorname{GSp}_{4}(\mathbb{A}_{F})$. Thus, the representation
$r:=r^{\prime}|_{G_{F}}$ satisfies all desired properties. ∎

######  Corollary 4.4.6.

Suppose
$\bar{\rho}:G_{\mathbb{Q}_{p}}\xrightarrow{}\operatorname{GSp}_{4}(\mathbb{F})$
is $7$-generic of weight $(a_{3},a_{2},a_{2},a_{0})$. Then there is a suitable
globalization $(F,\overline{r},\chi_{\mathrm{cris}})$ of $\bar{\rho}$ such
that
$(\chi_{\mathrm{cris},p,\iota}\otimes\epsilon^{-3})|_{G_{F_{v}}}=\epsilon^{b}\mathrm{nr}_{\xi_{v}}$.

###### Proof.

By [GG12, Lemma 7.6.7], $\bar{\rho}$ has an ordinary crystalline symplectic
lift $\rho$ of weight $\overline{\mu}_{\bar{\rho},v}+\eta$. Twisting by
unramified character, we can assume that
$\nu_{\operatorname{sim}}(\rho)=\epsilon^{b+3}\mathrm{nr}_{\xi}$ with finite
order $\mathrm{nr}_{\xi}$. Then ${\mathrm{std}}(\rho)$ is potentially
diagonalizable by [BLGGT14, Lemma 1.4.3]. The claim follows from Lemma 4.4.5
and Remark 4.2.4 with
$\chi_{\mathrm{cris},p,\iota}=\nu_{\operatorname{sim}}(r)\otimes\epsilon^{3}$.
∎

######  Corollary 4.4.7.

For any $(F,\overline{r},\chi)$ suitable of weight
$(a_{3},a_{2},a_{1},a_{0})$, there is a Hecke character $\chi_{\mathrm{cris}}$
such that
$(\chi_{\mathrm{cris}}\otimes\epsilon^{-3})|_{G_{F_{v}}}=\epsilon^{b}\mathrm{nr}_{\xi_{v}}$
for $v|p$, and $(\overline{r},\chi_{\mathrm{cris}})$ is modular of weight
$F(\mu)$ and level unramified outside $p$ where $\mu_{v}=\mu_{\bar{\rho}}$ for
$v|p$.

###### Proof.

As in the proof of Lemma 4.4.5, we can apply [PT21, Theorem 3.4] to construct
a lift $r$ of $\overline{r}$ such that $r|_{G_{F_{v}}}$ is crystalline of
weight $\overline{\mu}_{v}+\eta$ for each $v|p$ and unramified outside $p$,
and $\chi_{\mathrm{cris}}=\nu_{\operatorname{sim}}(r)\otimes\epsilon^{3}$. By
Lemma 4.4.4, 4.1.7, and [Sor09, Theorem B], there exists an automorphic
representation $\pi$ of $\mathcal{G}(\mathbb{A}_{F})$ such that $r\simeq
r_{\pi,p,\iota}$. Moreover, such $\pi$ arise from
$\mathcal{A}_{\chi_{\mathrm{cris}},\iota(\mu)}$. Since ${\mu}_{v}$ is a lowest
alcove weight the claim follows from Lemma 3.2.7. ∎

From now on, we let $(F,\overline{r},\chi_{\mathrm{cris}})$ be suitable of
weight $(a_{3},a_{2},a_{1},a_{0})$ as in Corollary 4.4.7 and $\mathcal{G}$ be
as in §4.1. We write
$\psi_{\mathrm{cris}}=\chi_{\mathrm{cris}}\epsilon^{-3}:G_{F}\xrightarrow{}\mathcal{O}^{\times}$
and
$\psi_{v,\mathrm{cris}}:=\psi|_{G_{F_{v}}}=\epsilon^{b}\mathrm{nr}_{\xi_{v}}$
for all $v|p$. We also write
$\psi_{v,{\mathrm{p}\text{-}\mathrm{cris}}}=\widetilde{\omega}^{b-3}\epsilon^{3}\mathrm{nr}_{\xi_{v}}$
for $v|p$. By Lemma 4.4.3, there exists
$\psi_{\mathrm{p}\text{-}\mathrm{cris}}:G_{F}\xrightarrow{}\mathcal{O}^{\times}$
lifting $\nu_{\operatorname{sim}}(\bar{r})$ such that
$(\psi_{\mathrm{p}\text{-}\mathrm{cris}})|_{G_{F_{v}}}\simeq\psi_{v,{\mathrm{p}\text{-}\mathrm{cris}}}$
for all $v|p$. Finally, we set
$\chi=\chi_{\mathrm{p}\text{-}\mathrm{cris}}:=\psi_{\mathrm{p}\text{-}\mathrm{cris}}\otimes\epsilon^{3}$.
Let $S_{p}$ be the set of places of $F$ dividing $p$. Fix a choice of
preferred place $w|p$. Since $\overline{r}$ is vast and tidy, we can find a
finite place $v_{0}\notin S_{p}$ such that $q_{v_{0}}\not\equiv 1\mod p$, no
two eigenvalues of $\bar{r}(\operatorname{Frob}_{v_{0}})$ have ratio
$q_{v_{0}}$, and the residue characteristic of $v_{0}$ is $>5$ ([BCGP21,
§7.7]).

Let $S$ be a finite set of finite places of $F$ containing
$S_{p}\cup\\{v_{0}\\}$. For $v\neq v_{0}$, we set
$U_{v}:=\mathcal{G}(\mathcal{O}_{F_{v}})$, and we also define
$U_{v_{0}}=\mathrm{Iw}_{1}(v_{0})$. Then set $U^{p}:=\prod_{v\nmid p}U_{v}$
and $U_{p}:=\prod_{v|p}U_{v}$. Corollary 4.4.7 ensures that
$S_{\chi_{\mathrm{cris}}}(U^{p}U_{p},F(\mu))[\mathfrak{m}_{\bar{r},{\chi_{\mathrm{cris}}}}^{S}]\neq
0$. The choice of $v_{0}$ ensures that $U^{p}U_{p}$ is sufficiently small.

Let $T$ be a subset of $S$. For each $v\in S$, if $\mathcal{D}_{v}$ (resp.
$\mathcal{D}_{v}^{\mathrm{cris}}$) denotes a deformation problem on symplectic
lifts of $\bar{r}|_{G_{F_{v}}}$ and $R_{v}$ (resp. $R_{v}^{\mathrm{cris}}$) is
the corresponding quotient of the universal symplectic lifting ring, the
tuples

$\mathcal{S}=(S,T,\\{\mathcal{D}_{v}\\}_{v\in
S},\psi_{\mathrm{p}\text{-}\mathrm{cris}}),\
\mathcal{S}_{\mathrm{cris}}=(S,T,\\{\mathcal{D}_{v}^{\mathrm{cris}}\\}_{v\in
S},\psi_{\mathrm{cris}})$

are global deformation problems and we write $\mathcal{D}_{\mathcal{S}}^{T}$
(resp. $\mathcal{D}_{\mathcal{S}_{\mathrm{cris}}}^{T}$) for the the functor of
$T$-framed symplectic deformations of type $\mathcal{S}$ (resp
$\mathcal{S}_{\mathrm{cris}})$. To simplify the notations, we write
$\star\in\\{\emptyset,\mathrm{cris}\\}$. Since $\bar{r}$ is absolutely
irreducible, $\mathcal{D}_{\mathcal{S}_{\star}}^{T}$ is representable by a
ring $R_{\mathcal{S}_{\star}}^{T}$. When $T=\emptyset$, we omit the
superscript $T$. The natural map $R_{\mathcal{S}_{\star}}\rightarrow
R_{\mathcal{S}_{\star}}^{T}$ is formally smooth of relative dimension
$11|T|-1$. Write $\mathcal{T}:=\mathcal{O}[[y_{1},\ldots,y_{11|T|-1}]]$. The
choice of a universal lift
$r_{\mathcal{S}_{\star}}:G_{F}\rightarrow\operatorname{GSp}_{4}(R_{\mathcal{S}_{\star}})$
determines a canonical isomorphism $R_{\mathcal{S}_{\star}}^{T}\cong
R_{\mathcal{S}_{\star}}\hat{\otimes}_{\mathcal{O}}\mathcal{T}$. Define
$R^{T,\mathrm{loc}}_{\mathcal{S}_{\star}}=\widehat{\bigotimes}_{v\in
T}R_{v}^{\star}$. There is a natural map
$R^{T,\mathrm{loc}}_{\mathcal{S}_{\star}}\rightarrow
R_{\mathcal{S}_{\star}}^{T}$.

######  Definition 4.4.8.

A _Taylor-Wiles datum_ is a tuple
$(Q,(\bar{\alpha}_{v,1},\ldots,\bar{\alpha}_{v,4})_{v\in Q})$ where $Q$ is a
finite set of finite places disjoint from $S$ such that $q_{v}\equiv 1\mod p$
for all $v\in Q$ and an ordering of the eigenvalues of
$\bar{r}(\operatorname{Frob}_{v})$ which are assumed to be
$\mathbb{F}$-rational and pairwise distinct:
$\bar{\alpha}_{v,1},\bar{\alpha}_{v,2}$ and
$\bar{\alpha}_{v,3}=\nu_{\operatorname{sim}}(\bar{r})(\operatorname{Frob}_{v})\bar{\alpha}_{v,2}^{-1}$
and
$\bar{\alpha}_{v,4}=\nu_{\operatorname{sim}}(\bar{r})(\operatorname{Frob}_{v})\bar{\alpha}_{v,1}^{-1}$.

Given a Taylor-Wiles datum we define the augmented deformation problem
$\mathcal{S}_{Q}=(S\cup Q,\\{\mathcal{D}_{v}\\}_{v\in
S}\cup\\{\mathcal{D}_{v}^{\square}\\}_{v\in
Q},\psi_{\mathrm{p}\text{-}\mathrm{cris}})$ and
$\mathcal{S}_{\mathrm{cris},Q}$ similarly. For $v\in Q$ let
$\Delta_{v}=k_{v}^{\times}(p)^{2}$ where $k_{v}^{\times}(p)$ is the maximal
$p$-power quotient of $k_{v}^{\times}$. Set $\Delta_{Q}:=\prod_{v\in
Q}\Delta_{v}$. There is a canonical map $\mathcal{O}[\Delta_{Q}]\rightarrow
R_{\mathcal{S}_{\star,Q}}^{T}$ for any subset $T\subseteq S$ such that the
natural surjection $R_{\mathcal{S}_{\star,Q}}^{T}\rightarrow
R_{\mathcal{S}_{\star}}^{T}$ has kernel
$\mathfrak{a}_{Q}R_{\mathcal{S}_{\star,Q}}^{T}$. Here
$\mathfrak{a}_{Q}\leq\mathcal{O}[\Delta_{Q}]$ denotes the augmentation ideal.

######  Proposition 4.4.9.

Let $q\geq h^{1}(F_{S}/F,\mathrm{ad}(\bar{r})(1))$. For every $n\geq 1$ there
exists a choice of Taylor-Wiles datum $Q_{n}$ such that

  1. 1.

$|Q_{n}|=q$;

  2. 2.

$q_{v}\equiv 1\mod p^{n}$ for each $v\in Q_{n}$;

  3. 3.

$R_{\mathcal{S}_{\star,Q_{n}}}^{S}$ is a quotient of a power series ring over
$R_{\mathcal{S}_{\star}}^{S,\mathrm{loc}}=R_{\mathcal{S}_{\star,Q_{n}}}^{S,\mathrm{loc}}$
in

$g:=2q-4|F:\mathbb{Q}|+|S|-1$

variables.

###### Proof.

This follows immediately from Corollary 7.6.3 of [BCGP21] because $\bar{r}$ is
odd with large image. The assumption that $\psi=\epsilon^{-1}$ in this
reference is not necessary for the proof. ∎

######  Remark 4.4.10.

Note that we can and do choose the same Taylor–Wiles datum $Q_{n}$ for both
$\star=\emptyset$ and $\star=\mathrm{cris}$.

From now on, we take $S=S_{p}\cup\\{v_{0}\\}$,
$\mathcal{D}_{v}=\mathcal{D}_{v}^{\square,\psi_{v,{\mathrm{p}\text{-}\mathrm{cris}}}}$
(resp.
$\mathcal{D}_{v}^{\mathrm{cris}}=\mathcal{D}_{v}^{\square,\psi_{v,\mathrm{cris}}}$)
for $v\in S$. Write
$R_{n}^{\star}:=R_{\mathcal{S}_{\star,\mathcal{Q}_{n}}}^{S}$. Let
$R_{\infty}^{\star}:=R_{\mathcal{S}_{\star}}^{S,\mathrm{loc}}\hat{\otimes}[[x_{1},\ldots,x_{g}]]$.
By Proposition 4.4.9(3) we may choose surjections
$\varphi_{n}^{\star}:R^{\star}_{\infty}\twoheadrightarrow R^{\star}_{n}$ for
each $n\geq 1$. Define
$S_{\infty}:=\mathcal{O}[[\mathbb{Z}_{p}^{2q}]]\hat{\otimes}_{\mathcal{O}}\mathcal{T}$.
For each $n\geq 1$, choose a surjection
$S_{\infty}\twoheadrightarrow\mathcal{O}[\Delta_{Q_{n}}]\hat{\otimes}_{\mathcal{O}}\mathcal{T}$.
This makes $R_{n}^{\star}$ into an $S_{\infty}$ algebra. Note that if we let
$I_{r,n}\leq S_{\infty}$ be the ideal generated by the kernel of this map
together with $\varpi^{r}$ then the collection $\\{I_{r,n}\\}$ obeys (4.3.6)
by Proposition 4.4.9(2).

######  Lemma 4.4.11.

Suppose that $\sigma=\sigma(\tau)$ for a regular principal series $\tau$ and
$\sigma^{\mathrm{cris}}=V(\mu_{\bar{\rho}})_{\mathbb{Q}_{p}}\otimes E|_{K}$,
then we have $\dim(S_{\infty})=\dim(R^{\star}_{\infty}(\sigma^{\star}))$ for
$\star\in\\{\emptyset,\mathrm{cris}\\}$.

###### Proof.

This follows from Lemma 2.1.1 and Proposition 2.4.2. ∎

For each $n\geq 1$ we now define $U_{1}^{p}(Q_{n})$ to be $\prod_{v\nmid
p}U_{1}(Q_{n})_{v}$, where $U_{1}(Q_{n})_{v}=U_{v}$ if $v\notin Q_{n}\cup
S_{p}$ and $U_{1}(Q_{n})_{v}=\mathrm{Iw}_{1}(v)$ if $v\in Q_{n}$. Let
$G:=\mathcal{G}(F_{w})$, $G_{p}:=\prod_{v|p}\mathcal{G}(F_{v})$, and
$K_{p}=\prod_{v|p}\mathcal{G}(\mathcal{O}_{F_{v}})$. For $r,n\geq 1$ and
$H\leq_{\mathrm{c.o.}}G_{p}$, we define

$M_{r}^{\star}(H)_{n}=[S_{\chi_{\star}}(H\cdot
U_{1}^{p}(Q_{n}),\mathcal{O}/\varpi^{r})_{\mathfrak{m}_{\bar{r},\chi_{\star}}^{S\cup
Q_{n},r},\mathfrak{m}_{Q_{n}}^{\star}}]^{\vee}\otimes_{R_{\mathcal{S}_{\star,Q_{n}}}}R_{n}^{\star}.$

where $\mathfrak{m}_{\bar{r},\chi_{\star}}^{S\cup Q_{n},r}$ is the image of
$\mathfrak{m}_{\bar{r},\chi_{\star}}^{S\cup Q_{n}}$ inside
$\mathbb{T}_{\chi_{\star}}^{S\cup Q_{n}}(H\cdot
U_{1}^{p}(Q_{n}),\mathcal{O}/\varpi^{r})$ and $\mathfrak{m}_{Q_{n}}^{\star}$
is the maximal ideal of $\bigotimes_{v\in
Q_{n}}\mathcal{O}[T(F_{v})/T(\mathcal{O}_{F_{v}})_{1}]$ in
$\operatorname{End}_{\mathcal{O}}(S_{\chi_{\star}}(H\cdot
U_{1}^{p}(Q_{n}),\mathcal{O}/\varpi^{r}))$ containing $\bigotimes_{v\in
Q_{n}}\mathcal{O}[T(\mathcal{O}_{F_{v}})/T(\mathcal{O}_{F_{v}})_{1}]$ such
that $\beta_{0}(\varpi_{v})\equiv\bar{\chi}_{v}(\varpi_{v})$,
$\beta_{1}(\varpi_{v})\equiv\bar{\alpha}_{v,1}$ and
$\beta_{2}(\varpi_{v})\equiv\bar{\alpha}_{v,1}\bar{\alpha}_{v,2}\mod\mathfrak{m}_{Q_{n}}$
(cf. [BCGP21, §2.4.29]). Note that $S_{\chi_{\star}}(H\cdot
U_{1}^{p}(Q_{n}),\mathcal{O}/\varpi^{r})_{\mathfrak{m}_{\bar{r},\chi_{\star}}^{S\cup
Q_{n},r,\star},\mathfrak{m}_{Q_{n}}^{\star}}$ is a
$R^{S}_{\mathcal{S}_{\star,Q_{n}}}$-module by the natural map

$R_{\mathcal{S}_{\star,Q_{n}}}\rightarrow\mathbb{T}_{\chi_{\star}}^{S\cup
Q_{n}}(H\cdot U_{1}^{p}(Q_{n}),\mathcal{O}/\varpi^{r})$ (4.4.12)

which exists by Proposition 4.2.7, and the action of
$\mathcal{O}[\Delta_{Q_{n}}]$ on $M_{r}^{\star}(H)_{n}$ via
$\mathcal{O}[\Delta_{Q_{n}}]\rightarrow R_{\mathcal{S}_{\star,Q_{n}}}^{S}$
agrees with the canonical action via the surjection
$U^{p}/U_{1}^{p}(Q_{n})\cong\Delta_{Q_{n}}$. Note that (4.4.12) is surjective
by Proposition 4.2.7(ii) (since $\chi_{\star}$ is fixed). Moreover we define

$M_{r}^{\star}(H)_{0}=[S_{\chi_{\star}}(H\cdot
U^{p},\mathcal{O}/\varpi^{r})_{\mathfrak{m}_{\bar{r}}^{S,r,\star}}]^{\vee}.$

It follows from [BCGP21, §2.4.29] that there is an isomorphism

$\alpha_{n}^{\star}:M^{\star}_{r}(H)_{n}/\mathfrak{a}_{\infty}\cong
M^{\star}_{r}(H)_{0}$

compatible among different $r\geq 1$ and $H\leq_{\mathrm{c.o.}}G_{p}$.

######  Lemma 4.4.13.

The data

$(S_{\infty},R^{\star}_{\infty},(R^{\star}_{n},\varphi^{\star}_{n},\\{M^{\star}_{r}(H)_{n}\\}_{r\geq
1,H\leq_{\mathrm{c.o.}}G},\alpha^{\star}_{n})_{n\geq
1},\\{M^{\star}_{r}(H)_{0}\\}_{r\geq 1,H\leq_{\mathrm{c.o.}}G})$

defined in the paragraphs above forms a $G_{p}$-patching datum for both
$\star\in\\{\emptyset,\mathrm{cris}\\}$.

###### Proof.

Axiom (1) follows from the remarks above and axiom (2) holds because $H\cdot
U^{p}$ is sufficiently small. ∎

Hence we may form the patched module $M^{\star}_{\infty}$ corresponding to
this $G$-patching datum in Lemma 4.3.9.

We now take $\mathcal{J}=S_{p}$ and choose an isomorphism
$K^{\mathcal{J}}\simeq K_{p}$. The following is the main result of this
section.

######  Lemma 4.4.14.

The pair $(M_{\infty},M_{\infty}^{\mathrm{cris}})$ is a congruent patched
module pair for
$(\bar{r}|_{G_{F_{v}}},\psi_{v,{\mathrm{p}\text{-}\mathrm{cris}}},\psi_{v,\mathrm{cris}})_{v\in
S_{p}}$ satisfying (PM1)–(PM6), and we have $d=|W|=8$ in (PM3).

###### Proof.

(PM1) and (PM2) follow already from Lemma 4.3.9. Axiom (PM3) follows from
Lemma 2.2.3, 2.2.4 (which can be seen as the inertial local Langlands for
principal series inertial type), Proposition 4.2.7(iii) and Lemma 4.4.11 by
the standard commutative algebra argument; see the proof of Lemma 4.18 of
[CEG+16]. We may take $d=|W|$ because this is the dimension of the space of
pro-$p$ Iwahori invariants in a principals series representation. Axiom (PM4)
follows also from Proposition 4.2.7(iii) similar to the proof of Lemma 4.17 of
[CEG+16]. (PM5) follows from the fact that $M_{r}^{\star}(H)_{n}$ in Lemma
4.4.13 for $\star\in\\{\emptyset,\mathrm{cris}\\}$ are congruent modulo
$\varpi$ together with Lemma 4.3.4. Finally, (PM6) follows from Corollary
4.4.7. ∎

Fix a place $w$ of $F$ dividing $p$. We write $U^{w}=\prod_{v\neq w}U_{v}$.
Recall that

$\displaystyle M_{\infty,w}$
$\displaystyle=\operatorname{Hom}_{\mathcal{O}[\prod_{v\in
S_{p}\backslash\\{w\\}}\mathcal{G}(\mathcal{O}_{F_{v}})]}(M_{\infty},(\sigma(\tau_{0})^{S_{p}\backslash\\{w\\}})^{d})^{d}$
$\displaystyle R_{\infty,w}$
$\displaystyle=R_{\overline{r}|_{G_{F_{w}}}}^{\square,\psi_{w,{\mathrm{p}\text{-}\mathrm{cris}}}}[[x_{1},\ldots,x_{h}]]$

and $\mathfrak{m}_{\infty,w}$ the maximal ideal of $R_{\infty,w}$. By Theorem
3.3.8(1), we know that $M_{\infty,w}$ is non-zero.

######  Lemma 4.4.15.

We have

$(M_{\infty,w}/\mathfrak{m}_{\infty,w})^{\vee}\cong
S_{\chi,0,\sigma(\tau_{0})}(U^{w},\mathbb{F})[\mathfrak{m}_{\bar{r},\chi}^{S}].$

###### Proof.

By Lemma 4.3.9 there is a natural inclusion
$(M_{\infty,w}/\mathfrak{m}_{\infty,w})^{\vee}\hookrightarrow
S_{\chi,0,\sigma(\tau_{0})}(U^{w},\mathbb{F})[\mathfrak{m}_{\bar{r},\chi}^{S}].$
It is surjective because we have arranged the map (4.4.12) to be surjective. ∎

######  Remark 4.4.16.

The $R_{\infty,w}$-module $M_{\infty,w}$ is the patched module constructed by
[CEG+16], adapted to the group $\operatorname{GSp}_{4}$, using the space of
automorphic forms with infinite level at $w$. In contrast, the construction of
the $R_{\infty}$-module $M_{\infty}$ uses the space of mod $p$ automorphic
forms with infinite level at all places dividing $p$. We find it is more
convenient to make a comparison between $M_{\infty}$ and
$M_{\infty}^{\mathrm{cris}}$ rather than between $M_{\infty,w}$ and its
crystalline version to prove Theorem 3.3.8(1). However, it is plausible that
one can obtain the same result using $M_{\infty,w}$ and its crystalline
version.

###  4.5 Mod $p$ local-global compatibility

We can now prove the main result.

######  Theorem 4.5.1.

Let $(F,\overline{r},\chi)$ be suitable of weight $(a_{3},a_{2},a_{1},a_{0})$
with $\chi=\chi_{\mathrm{p}\text{-}\mathrm{cris}}$ as in §4.4, and
$\mathcal{G}$ be as in §4.1. Let $w$ be a place of $F$ dividing $p$. Then the
Fontaine-Laffaille parameters of $\overline{r}|_{G_{F_{w}}}$ can be recovered
from the admissible $\mathbb{F}[\operatorname{GSp}_{4}(F_{w})]$-module
$S_{\chi,0,\sigma(\tau_{0})}(U^{w},\mathbb{F})[\mathfrak{m}_{\bar{r},\chi}^{S}]$
by the recipe described in Remark 3.3.10.

###### Proof.

This follows from Lemmas 4.4.14 and 4.4.15 and Theorem 3.3.8. ∎

##  Appendix A Tables and Figures

$\lambda\in\,$Alcove | $\langle\lambda+\rho^{\prime},\alpha_{0}^{\vee}\rangle$ | $\langle\lambda+\rho^{\prime},\alpha_{1}^{\vee}\rangle$ | $\langle\lambda+\rho^{\prime},(2\alpha_{0}+\alpha_{1})^{\vee}\rangle$ | $\langle\lambda+\rho^{\prime},(\alpha_{0}+\alpha_{1})^{\vee}\rangle$
---|---|---|---|---
$C_{0}$ | $(0,p)$ | $(0,p)$ | $(0,p)$ | $(0,p)$
$C_{1}$ | $(0,p)$ | $(0,p)$ | $(0,p)$ | $(p,2p)$
$C_{2}$ | $(0,p)$ | $(0,p)$ | $(p,2p)$ | $(p,2p)$
$C_{3}$ | $(0,p)$ | $(0,p)$ | $(p,2p)$ | $(2p,3p)$
$D_{0}$ | $(0,p)$ | $(p,2p)$ | $(p,2p)$ | $(2p,3p)$
$D_{1}$ | $(0,p)$ | $(p,2p)$ | $(p,2p)$ | $(3p,4p)$
$E_{0}$ | $(p,2p)$ | $(0,p)$ | $(p,2p)$ | $(p,2p)$
$E_{1}$ | $(p,2p)$ | $(0,p)$ | $(p,2p)$ | $(2p,3p)$
$E_{2}$ | $(p,2p)$ | $(0,p)$ | $(2p,3p)$ | $(2p,3p)$
$E_{3}$ | $(2p,3p)$ | $(0,p)$ | $(2p,3p)$ | $(2p,3p)$
Table A.1: List of some $p$-alcoves for $\operatorname{GSp}_{4}$.
$-\rho^{\prime}$$s_{\alpha_{0}}$$s_{\alpha_{1}}$$s_{2\alpha_{0}+\alpha_{1}}$$s_{\alpha_{0}+\alpha_{1}}$$C_{0}$$C_{1}$$C_{2}$$C_{3}$$D_{0}$$D_{1}$$E_{0}$$E_{1}$$E_{2}$$E_{3}$
Figure A.1: A picture of the dominant Weyl chamber with labelled alcoves and
reflection axes under the dot action.

$\lambda\in\,$Alcove | $\chi(\lambda)=$
---|---
$C_{0}$ | $\chi_{p}(\lambda)$
$C_{1}$ | $\chi_{p}(\lambda)+\chi_{p}(C_{0})$
$C_{2}$ | $\chi_{p}(\lambda)+\chi_{p}(C_{1})$
$C_{3}$ | $\chi_{p}(\lambda)+\chi_{p}(C_{2})$
$D_{0}$ | $\chi_{p}(\lambda)+\chi_{p}(C_{3})$
$D_{1}$ | $\chi_{p}(\lambda)+\chi_{p}(D_{0})+\chi_{p}(E_{1})+\chi_{p}(C_{3})$
$E_{0}$ | $\chi_{p}(\lambda)+\chi_{p}(C_{2})$
$E_{1}$ | $\chi_{p}(\lambda)+\chi_{p}(E_{0})+\chi_{p}(C_{3})+\chi_{p}(C_{2})+\chi_{p}(C_{1})$
$E_{2}$ | $\chi_{p}(\lambda)+\chi_{p}(E_{1})+\chi_{p}(D_{0})+\chi_{p}(C_{3})+\chi_{p}(C_{2})+\chi_{p}(C_{1})+\chi_{p}(C_{0})$
$E_{3}$ | $\chi_{p}(\lambda)+\chi_{p}(E_{2})+\chi_{p}(C_{2})$
Table A.2: This table gives the Jordan-Hölder factors of Weyl modules for
$\operatorname{GSp}_{4}$. In it, we write $\chi_{p}(A)$ instead of
$\chi_{p}(\lambda_{A})$, where $\lambda_{A}$ is the unique weight linked to
$\lambda$ in alcove $A$.

$\lambda\in C_{0}$ | $\operatorname{gr}^{0}$ | $F(x,y)_{0}+F^{\prime}(y+p-1,x)_{2}$
---|---|---
$\operatorname{gr}^{1}$ |  $F(x-2,y)_{0}+F(x-1,y-1)_{0}+F^{\prime}(p-3-y,x)_{1}$ $+F^{\prime}(y+p-1,p-1-x)_{2}+F(y+p-2,p-2-x)_{2}+F(x+p-1,p-1-y)_{3}$
$\operatorname{gr}^{2}$ |  $F^{\prime}(p-2-x,y-1)_{0}+F(x-2,y-2)_{0}+F^{\prime}(p-1-x,y)_{0}$ $+F^{\prime}(p-3-x,y-2)_{0}+F^{\prime}(p-3-x,y)_{0}+F^{\prime}(p-2-y,x-1)_{1}$ $+F(p-3-y,p-1-x)_{1}+F^{\prime}(p-1-y,x)_{1}+F(p-2-y,p-2-x)_{1}$ $+F^{\prime}(y+p-2,x-1)_{2}$
$\operatorname{gr}^{3}$ | $F(p-1-y,p-1-x)_{1}+F(2p-2-x,p-1-y)_{3}$
$\lambda\in C_{1}$ | $\operatorname{gr}^{0}$ | $F(x,y)_{1}+F^{\prime}(y+p-1,x)_{3}$
$\operatorname{gr}^{1}$ |  $F^{\prime}(y-2,p-1-x)_{0}+F^{\prime}(y,p-1-x)_{0}+F(p-3-y,p-3-x)_{0}$ $+F^{\prime}(y-1,p-2-x)_{0}+F^{\prime}(y-2,p-3-x)_{0}+F(x-2,y)_{1}$ $+F(x-1,y-1)_{1}+F^{\prime}(x-1,p-2-y)_{1}+F^{\prime}(x,p-1-y)_{1}$ $+F^{\prime}(2p-3-x,p-2-y)_{2}$
$\operatorname{gr}^{2}$ |  $F(p-3-y,p-1-x)_{0}+F(p-2-y,p-2-x)_{0}+F^{\prime}(x-2,p-1-y)_{1}$ $+F(2p-2-x,y)_{2}+F(2p-3-x,y-1)_{2}+F(2p-2-y,x)_{3}$
$\operatorname{gr}^{3}$ | $F(p-1-y,p-1-x)_{0}+F^{\prime}(2p-2-x,p-1-y)_{2}$
$\lambda\in C_{2}$ | $\operatorname{gr}^{0}$ | $F^{\prime}(y,x-p+1)_{0}+F(x,y)_{2}$
$\operatorname{gr}^{1}$ |  $F^{\prime}(y-1,x-p)_{0}+F^{\prime}(y-2,x-p+1)_{0}+F(2p-4-x,y)_{1}$ $+F^{\prime}(x-1,p-2-y)_{2}+F^{\prime}(x,p-1-y)_{2}+F^{\prime}(y+p-1,2p-2-x)_{3}$
$\operatorname{gr}^{2}$ |  $F^{\prime}(y-2,x-p-1)_{0}+F(p-2-y,x-p)_{0}+F(p-3-y,x-p-1)_{0}$ $+F(p-3-y,x-p+1)_{0}+F(p-1-y,x-p-1)_{0}+F(2p-3-x,y-1)_{1}$ $+F(2p-2-x,y)_{1}+F^{\prime}(2p-3-x,p-2-y)_{1}+F^{\prime}(2p-4-x,p-1-y)_{1}$ $+F(x-1,y-1)_{2}$
$\operatorname{gr}^{3}$ | $F^{\prime}(2p-2-x,p-1-y)_{1}+F(2p-2-y,2p-2-x)_{3}$
$\lambda\in C_{3}$ | $\operatorname{gr}^{0}$ | $F^{\prime}(y,x-p+1)_{1}+F(x,y)_{3}$
$\operatorname{gr}^{1}$ |  $F(x-p-1,p-1-y)_{0}+F(x-p-1,p-3-y)_{0}+F(x-p,p-2-y)_{0}$ $+F^{\prime}(2p-4-x,p-3-y)_{0}+F(x-p+1,p-1-y)_{0}+F^{\prime}(y-1,x-p)_{1}$ $+F(y,2p-2-x)_{1}+F(y-1,2p-3-x)_{1}+F^{\prime}(y-2,x-p+1)_{1}$ $+F(2p-2-y,2p-2-x)_{2}+F(2p-3-y,2p-3-x)_{2}$
$\operatorname{gr}^{2}$ |  $F^{\prime}(2p-3-x,p-2-y)_{0}+F^{\prime}(2p-4-x,p-1-y)_{0}+F(y-2,2p-2-x)_{1}$ $+F^{\prime}(2p-3-y,x-p)_{2}+F^{\prime}(2p-2-y,x-p+1)_{2}+F^{\prime}(3p-3-x,y)_{3}$
$\operatorname{gr}^{3}$ | $F^{\prime}(2p-2-x,p-1-y)_{0}$
|  |
Table A.3: For $\lambda=(x,y;z)$ 7-deep inside a $p$-restricted alcove, this
table gives the characters of
$\operatorname{gr}^{i}:=\overline{M}(\lambda)_{\mathbb{F}}(s_{1}s_{0}s_{1},i)/\overline{M}(\lambda))_{\mathbb{F}}(s_{1}s_{0}s_{1},i+1)$
(we have $\operatorname{gr}^{i}=0$ for $i\geq 4$ by Theorem 3.1.2). We use the
shorthand $F(a,b)_{i}:=F(a,b;z)$ and $F^{\prime}(a,b)_{i}:=F(a,b;z+p-1)$, the
subscript $i$ referring to the alcove $C_{i}$ in which the Serre weight lies.

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