$\mathcal{S}:\mathbb{T}_{p}\rightarrow\mathbb{Z}[p^{-1}][\mathrm{X}^{\ast}(\hat{\mathrm{T}})]^{\mathrm{W}_{\mathbf{G}}}$

where $\mathrm{W}_{\mathbf{G}}$ is the absolute Weyl group of $\mathbf{G}$. We
fix a datum $\mathbf{T}\subset\mathbf{B}\subset\mathbf{G}$ of the diagonal
torus contained in the upper triangular Borel subgroup as in the global case.
This determines a positive Weyl chamber $\mathrm{P}^{+}$ in
$\mathrm{X}_{\ast}(\mathrm{T})\cong\mathrm{X}^{\ast}(\hat{\mathrm{T}})$. An
element $\nu\in\mathrm{X}^{\ast}(\hat{\mathrm{T}})$ determines an irreducible
representation $V_{\nu}$ of the dual group
$\hat{\mathrm{G}}=\mathrm{GSpin}(5)$ whose character is denoted by
$\chi_{\nu}\in\mathrm{R}(\hat{\mathrm{G}})=\mathbb{Z}[\mathrm{X}^{\ast}(\hat{\mathrm{T}})]^{\mathrm{W}_{\mathbf{G}}}$.

It is well-known that $\mathbb{T}_{p}$ is isomorphic to
$\mathbb{Z}[\mathrm{T}^{\pm}_{p,0},\mathrm{T}_{p,1},\mathrm{T}_{p,2}]$ where

$\begin{split}&\mathrm{T}_{p,0}=c_{\nu_{0}}=\mathrm{char}(\mathbf{G}(\mathbb{Z}_{p})\begin{pmatrix}p&&&\\\
&p&&\\\ &&p&\\\ &&&p\\\ \end{pmatrix}\mathbf{G}(\mathbb{Z}_{p}))\text{ for
}\nu_{0}=(1,1,1,1)\\\
&\mathrm{T}_{p,2}=c_{\nu_{2}}=\mathrm{char}(\mathbf{G}(\mathbb{Z}_{p})\begin{pmatrix}p&&&\\\
&p&&\\\ &&1&\\\ &&&1\\\ \end{pmatrix}\mathbf{G}(\mathbb{Z}_{p}))\text{ for
}\nu_{2}=(1,1,0,0)\\\
&\mathrm{T}_{p,1}=c_{\nu_{1}}=\mathrm{char}(\mathbf{G}(\mathbb{Z}_{p})\begin{pmatrix}p^{2}&&&\\\
&p&&\\\ &&p&\\\ &&&1\\\ \end{pmatrix}\mathbf{G}(\mathbb{Z}_{p}))\text{ for
}\nu_{1}=(2,1,1,0)\\\ \end{split}$

and where $\mathrm{char}(\cdot)$ is the characteristic function.

The following spherical Hecke operator will appear naturally in later
computations

$\mathrm{T}_{p^{2},2}=c_{2\nu_{2}}=\mathrm{char}(\mathbf{G}(\mathbb{Z}_{p})\begin{pmatrix}p^{2}&&&\\\
&p^{2}&&\\\ &&1&\\\ &&&1\\\ \end{pmatrix}\mathbf{G}(\mathbb{Z}_{p}))\text{ for
}2\nu_{2}=(2,2,0,0).$

We will also need the following two Hecke operators in
$\mathbb{Z}[\mathrm{K}_{\\{2\\}}\backslash\mathbf{G}(\mathbb{Q}_{p})/\mathrm{K}_{\\{0\\}}]$

$\displaystyle\mathrm{T}_{02}=\mathrm{char}(\mathrm{K}_{\\{2\\}}\begin{pmatrix}p&&&\\\
&p&&\\\ &&1&\\\ &&&1\\\ \end{pmatrix}\mathrm{K}_{\\{0\\}})$
$\displaystyle\mathrm{T}_{20}=\mathrm{char}(\mathrm{K}_{\\{0\\}}\begin{pmatrix}p^{-1}&&&\\\
&p^{-1}&&\\\ &&1&\\\ &&&1\\\ \end{pmatrix}\mathrm{K}_{\\{2\\}}).$

In the computations below, we will follow mostly the notations and conventions
in [Gross] and use results from there freely. We have the following useful
formula [Gross, 3.12]

$p^{\langle\nu,\rho\rangle}\chi_{\nu}=\mathcal{S}(c_{\nu})+\sum_{\xi<\nu}d_{\nu}(\xi)\mathcal{S}(c_{\xi}).$

The integers $d_{\nu}(\xi)$ are given by $\mathrm{P}_{\xi,\nu}(p)$ for the
Kazhdan-Lusztig polynomial $\mathrm{P}_{\xi,\nu}$ evaluated at $p$ and $2\rho$
is the sum of the positive roots. Since the weight $\nu_{2}$ is minuscule, we
immediately have

$p^{\frac{3}{2}}\chi_{\nu_{2}}=\mathcal{S}(c_{\nu_{2}})=\mathcal{S}(\mathrm{T}_{p,2}).$

On the other hand for the non-minuscule weight $\nu_{1}$, we have

$p^{2}\chi_{\nu_{1}}=\mathcal{S}(c_{\nu_{1}})+\mathcal{S}(c_{0})=\mathcal{S}(\mathrm{T}_{p,1})+1.$

The following identity is the main result of this subsection. It can perhaps
be proved by a massive double coset computation whereas our proof uses
crucially the Satake isomorphism to simplify the computation. This identity
seems not to exist already in the literature and therefore we give full
details which we think are of independent interest.

######  Proposition 6.9.

The following identity holds in $\mathbb{T}_{p}$

$\mathrm{T}_{p^{2},2}=\mathrm{T}^{2}_{p,2}-(p+1)\mathrm{T}_{p,1}-(p+1)(p^{2}+1).$

###### Proof.

The representation $V_{\nu_{1}}$ is the $5$-dimensional orthogonal
representation and $V_{\nu_{2}}$ is the standard representation. Therefore
$\wedge^{2}V_{\nu_{2}}=V_{\nu_{1}}\oplus\mathbb{C}$. We also have
$\wedge^{2}V_{\nu_{1}}=V_{2\nu_{2}}$ is the adjoint representation of
dimension $10$. Thus we have

$\displaystyle V^{\otimes 2}_{\nu_{2}}$
$\displaystyle=V_{2\nu_{2}}+\wedge^{2}V_{\nu_{2}}$
$\displaystyle=V_{2\nu_{2}}+V_{\nu_{1}}+\mathbb{C}$

Taking characters, we have
$\chi^{2}_{\nu_{2}}=\chi_{2\nu_{2}}+\chi_{\nu_{1}}+1$ and hence
$p^{3}\chi^{2}_{\nu_{2}}=p^{3}\chi_{2\nu_{2}}+p^{3}\chi_{\nu_{1}}+p^{3}$.
Therefore

$\displaystyle p^{3}\chi_{2\nu_{2}}$
$\displaystyle=p^{3}\chi^{2}_{\nu_{2}}-p^{3}\chi_{\nu_{1}}-p^{3}$
$\displaystyle=\mathcal{S}(\mathrm{T}^{2}_{p,2})-p\mathcal{S}(\mathrm{T}_{p,1})-(p+p^{3}).$

On the other hand, we also have

$\displaystyle p^{3}\chi_{2\nu_{2}}$
$\displaystyle=\mathcal{S}(c_{2\nu_{2}})+d_{2\nu_{2}}(\nu_{1})S(c_{\nu_{1}})+d_{2\nu_{2}}(0)$
$\displaystyle=\mathcal{S}(\mathrm{T}_{p^{2},2})+d_{2\nu_{2}}(\nu_{1})S(\mathrm{T}_{p,1})+d_{2\nu_{2}}(0).$

  * •

$d_{2\nu_{2}}(\nu_{1})=1$; This follows from the fact that
$\mathrm{P}_{\nu_{1},2\nu_{2}}$ has degree strictly less than $\langle
2\nu_{2}-\nu_{1},\rho\rangle=1$ and the constant coefficient is $1$.

  * •

$d_{2\nu_{2}}(0)=1+p^{2}$; This follows from the fact that $2\nu_{2}$ is the
highest weight of the adjoint representation, which implies that
$\mathrm{P}_{0,2\nu_{2}}(p)=\sum_{i=1}p^{m_{i}-1}$ where $m_{i}$ are the
exponents of the Weyl group of type $\mathrm{C}_{2}$ which are $1$ and $3$,
see [Gross, 4.6].

Equating the two expressions of $p^{3}\chi_{2\nu_{2}}$ gives the desired
result. ∎

###  6.2. Computing the level raising matrix

We now calculate the entries of the level raising matrix
$\mathcal{T}_{\mathrm{lr}}$ term by term. The main result is the following
proposition.

######  Proposition 6.10.

The level raising matrix $\mathcal{T}_{\mathrm{lr}}$ is given by

$\mathcal{T}_{\mathrm{lr}}=-2\begin{pmatrix}&\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1)&(p+1)\mathrm{T}_{02}\\\
&(p+1)\mathrm{T}_{20}&\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1)\\\
\end{pmatrix}.$

###### Proof.

To begin the proof, note that we can identify both
$\bigoplus\limits_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))$
and
$\bigoplus\limits_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{R}^{3}\Phi_{\sigma}(\mathcal{O}_{\lambda}(2))$
with $\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{Pa}}(\overline{\mathrm{B}})]$
ignoring the Galois actions. The level raising diagram (7.2) reduces to

(6.11)
${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]}$${\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{Pa}}(\overline{\mathrm{B}})]}$${\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]}$$\scriptstyle{\mathrm{inc}^{\\{0\\}}_{!}}$$\scriptstyle{\mathrm{inc}^{\\{2\\}}_{!}}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{0\\}}}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{2\\}}}$

By Proposition 5.8, the map $\alpha$ can be realized as a Gysin map and
$\beta$ can be realized as a restriction map. The situation is similar to that
of [LTXZZ, Proposition 5.8.8]. We proceed to analyze each entry.

####  6.2.1. $\mathcal{T}_{\mathrm{lr},\\{00\\}}$

Since $\mathrm{Z}_{\\{i\\}}(\overline{\mathrm{B}})$ corresponds to the set of
vertex lattices of type $i$ for $i\in\\{0,1,2\\}$. Therefore to understand the
map
$\mathcal{T}_{\mathrm{lr},\\{00\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta\circ\mathrm{inc}^{\\{0\\}}_{!}$,
we need to understand the possible relative position of vertex lattices
$\Lambda_{0}$, $\Lambda_{\mathrm{Pa}}$ and $\Lambda^{\prime}_{0}$ of type $0$,
$1$ and $0$. It not hard to see that the only possible situation is that
$\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda_{0}$ and
$\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda^{\prime}_{0}$. There are two possible
cases.

Case $(1)$: $\Lambda_{0}=\Lambda^{\prime}_{0}$. Then we have
$p\Lambda^{\vee}_{0}\subset^{1}p\Lambda^{\vee}_{\mathrm{Pa}}\subset^{2}\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda_{0}$.
Giving such a $\Lambda_{\mathrm{Pa}}$ is equivalent to giving a complete flag
in $\Lambda_{0}/p\Lambda_{0}$. The number of such flags is given by
$[\mathbf{G}(\mathbb{Z}_{p}):\mathrm{Kl}]=(p+1)(p^{2}+1)$ where $\mathrm{Kl}$
is the Klingen parahoric.

Case $(2)$: $\Lambda_{0}\neq\Lambda^{\prime}_{0}$. Then we have
$\Lambda_{0}\cap\Lambda^{\prime}_{0}=\Lambda_{\mathrm{Pa}}$ and the relative
position between $\Lambda_{0}$ and $\Lambda^{\prime}_{0}$ is given by

$\mathrm{K}_{\\{0\\}}\begin{pmatrix}p&&&\\\ &1&&\\\ &&1&\\\ &&&p^{-1}\\\
\end{pmatrix}\mathrm{K}_{\\{0\\}}.$

This operator is the same as $\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}$ by
identifying $\mathrm{K}_{\\{0\\}}$ with the hyperspecial group
$\mathbf{G}(\mathbb{Z}_{p})$. Therefore the entry
$\mathcal{T}_{\mathrm{lr},\\{00\\}}$ is given by

$\mathcal{T}_{\mathrm{lr},\\{00\\}}=-2(\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1)).$

Here the coefficient $-2$ comes from the self-intersection number of the class
$\eta_{\sigma}$ as in Lemma 5.7.

####  6.2.2. $\mathcal{T}_{\mathrm{lr},\\{22\\}}$

To understand the map
$\mathcal{T}_{\mathrm{lr},\\{22\\}}=\mathrm{inc}^{\ast}_{\\{2\\}}\circ\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta\circ\mathrm{inc}^{\\{2\\}}_{!}$,
we need to understand the possible relative position of vertex lattices
$\Lambda_{2}$, $\Lambda_{\mathrm{Pa}}$ and $\Lambda^{\prime}_{2}$ of type $2$,
$1$ and $2$. It not hard to see that the only possible way is that
$\Lambda_{2}\subset^{1}\Lambda_{\mathrm{Pa}}$ and
$\Lambda^{\prime}_{2}\subset^{1}\Lambda_{\mathrm{Pa}}$. The rest of the
analysis is completely the same as in the case of
$\mathcal{T}_{\mathrm{lr},\\{00\\}}$. Once we identify $\mathrm{K}_{\\{2\\}}$
with the hyperspecial group $\mathbf{G}(\mathbb{Z}_{p})$, the entry
$\mathcal{T}_{\mathrm{lr},\\{22\\}}$ is given by

$\mathcal{T}_{\mathrm{lr},\\{22\\}}=-2(\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1)).$

Here the coefficient $-2$ comes from the self-intersection number of the class
$\eta_{\sigma}$ as in Lemma 5.7.

####  6.2.3. $\mathcal{T}_{\mathrm{lr},\\{02\\}}$

To understand the map
$\mathcal{T}_{\mathrm{lr},\\{02\\}}=\mathrm{inc}^{\ast}_{\\{2\\}}\circ\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta\circ\mathrm{inc}^{\\{0\\}}_{!}$,
we need to understand the possible relative position of vertex lattices
$\Lambda_{0}$, $\Lambda_{\mathrm{Pa}}$ and $\Lambda^{\prime}_{2}$ of type $0$,
$1$ and $2$. We find that the only possible situation is that

$\Lambda_{2}\subset^{1}\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{1}\Lambda^{\vee}_{\mathrm{Pa}}\subset^{1}\Lambda^{\vee}_{2}.$

It follows each such $\Lambda_{\mathrm{Pa}}$ corresponds to a line in the two
dimensional space $\Lambda_{0}/\Lambda_{2}$.

Therefore the entry $\mathcal{T}_{\mathrm{lr},\\{02\\}}$ is given by

$\mathcal{T}_{\mathrm{lr},\\{02\\}}=-2(p+1)\mathrm{char}(\mathrm{K}_{\\{0\\}}\begin{pmatrix}p&&&\\\
&p&&\\\ &&1&\\\
&&&1\end{pmatrix}\mathrm{K}_{\\{2\\}})=-2(p+1)\mathrm{T}_{02}.\ $

####  6.2.4. $\mathcal{T}_{\mathrm{lr},\\{20\\}}$

To understand the map
$\mathcal{T}_{\mathrm{lr},\\{20\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta\circ\mathrm{inc}^{\\{2\\}}_{!}$,
we need to understand the possible relative positions of the vertex lattices
$\Lambda_{0}$, $\Lambda_{\mathrm{Pa}}$ and $\Lambda^{\prime}_{2}$ of type $0$,
$1$ and $2$. We find that the only possible way is that

$\Lambda_{2}\subset^{1}\Lambda_{\mathrm{Pa}}\subset^{1}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{1}\Lambda^{\vee}_{\mathrm{Pa}}\subset^{1}\Lambda^{\vee}_{2}.$

Each $\Lambda_{\mathrm{Pa}}$ corresponds to a line in the two dimensional
space $\Lambda_{0}/\Lambda_{2}$.

Therefore the entry $\mathcal{T}_{\mathrm{lr},\\{20\\}}$ is given by

$\mathcal{T}_{\mathrm{lr},\\{20\\}}=-2(p+1)\mathrm{char}(\mathrm{K}_{\\{0\\}}\begin{pmatrix}p^{-1}&&&\\\
&p^{-1}&&\\\ &&1&\\\
&&&1\end{pmatrix}\mathrm{K}_{\\{2\\}})=-2(p+1)\mathrm{T}_{02}.\ $

This finishes the proof of Proposition 6.10. ∎

We will compute the determinant of the level raising matrix and the
supersingular matrix that we will define in the next section. For this reason,
we will next compute the composite $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ and
$\mathrm{T}_{02}\circ\mathrm{T}_{20}$ in terms of elements in the spherical
Hecke algebra.

######  Lemma 6.12.

The composite $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ and
$\mathrm{T}_{02}\circ\mathrm{T}_{20}$ are both given by

$\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p^{2}+1)(p+1).$

As a result, if $\mathrm{T}_{p,0}$ acts trivially, then
$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}_{02}\circ\mathrm{T}_{20}=\mathrm{T}^{2}_{p,2}$

###### Proof.

The proof of this lemma is similar to that of the above proposition.

####  6.2.5. $\mathrm{T}_{20}\circ\mathrm{T}_{02}$

Let $\Lambda_{0}$ be a vertex lattice of type $0$. Then
$\mathrm{T}_{20}\circ\mathrm{T}_{02}(\Lambda_{0})$ classifies a pair of vertex
lattices $(\Lambda_{2},\Lambda^{\prime}_{0})$ of type $2$ and $0$. We need to
understand all the possible relative position of the vertex lattices
$\Lambda_{0}$, $\Lambda_{2}$ and $\Lambda^{\prime}_{0}$. The only possible
situation is given below

$\displaystyle
p\Lambda^{\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\vee}_{2}$
$\displaystyle
p\Lambda^{\prime\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda^{\prime}_{0}=\Lambda^{\prime\vee}_{0}\subset^{2}\Lambda^{\vee}_{2}.$

Note that $\Lambda_{2}$ determines an isotropic subspace of the symplectic
space
$\Lambda_{0}\cap\Lambda^{\prime}_{0}/p\Lambda^{\vee}_{0}+p\Lambda^{\prime\vee}_{0}$
of possible dimensions in the set $\\{0,2,4\\}$.

When
$\dim_{\mathbb{F}}\Lambda_{0}\cap\Lambda^{\prime}_{0}/p\Lambda^{\vee}_{0}+p\Lambda^{\prime\vee}_{0}=0$,
then

$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{2}\Lambda_{0}$
$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{2}\Lambda^{\prime}_{0}.$

Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the double coset operator

$\mathrm{K}_{\\{0\\}}\begin{pmatrix}p&&&\\\ &p&&\\\ &&p^{-1}&\\\
&&&p^{-1}\end{pmatrix}\mathrm{K}_{\\{0\\}}.$

Once we identify $\mathrm{K}_{\\{0\\}}$ with $\mathbf{G}(\mathbb{Z}_{p})$, we
see this is the same as $\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p^{2},2}$.

When
$\dim_{\mathbb{F}}\Lambda_{0}\cap\Lambda^{\prime}_{0}/p\Lambda^{\vee}_{0}+p\Lambda^{\prime\vee}_{0}=2$,
there are $p+1$ choices of $\Lambda_{2}$. And we have

$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{1}\Lambda_{0}$
$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{1}\subset^{1}\Lambda^{\prime}_{0}.$

Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the double coset operator

$(p+1)\mathrm{K}_{\\{0\\}}\begin{pmatrix}p&&&\\\ &1&&\\\ &&1&\\\
&&&p^{-1}\end{pmatrix}\mathrm{K}_{\\{0\\}}.$

Once we identify $\mathrm{K}_{\\{0\\}}$ with $\mathbf{G}(\mathbb{Z}_{p})$, we
see this is the same as $(p+1)\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p,1}$.

When
$\dim_{\mathbb{F}}\Lambda_{0}\cap\Lambda^{\prime}_{0}/p\Lambda^{\vee}_{0}+p\Lambda^{\prime\vee}_{0}=4$,
there are $[\mathbf{G}(\mathbb{Z}_{p}):\mathrm{Sie})]=(p+1)(p^{2}+1)$ choices
of $\Lambda_{2}$. And we have

$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{0}\Lambda_{0}$
$\displaystyle\Lambda_{0}\cap\Lambda^{\prime}_{0}\subset^{0}\Lambda^{\prime}_{0}.$

Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the constant $(p^{2}+1)(p+1)$. All in all, we obtain the following formula

(6.13)
$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p^{2}+1)(p+1).$

####  6.2.6. $\mathrm{T}_{02}\circ\mathrm{T}_{20}$

The computation is almost the same to the above. Let $\Lambda_{2}$ be a vertex
lattice of type $2$. Then $\mathrm{T}_{02}\circ\mathrm{T}_{20}(\Lambda_{2})$
classifies pair of vertex lattices $(\Lambda_{0},\Lambda^{\prime}_{2})$ of
type $0$ and $2$. We need to understand the relative position of vertex
lattices $\Lambda_{0}$, $\Lambda_{2}$ and $\Lambda^{\prime}_{2}$. The only
possible situation is given below

$\displaystyle
p\Lambda^{\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\vee}_{2};$
$\displaystyle
p\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\prime}_{2}\subset^{2}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\prime\vee}_{2}.$

Note that $\Lambda_{0}$ is determined by an isotropic subspace of the
symplectic space
$\Lambda_{2}\cap\Lambda^{\prime}_{2}/\Lambda^{\vee}_{2}+\Lambda^{\prime\vee}_{2}$
of possible dimensions $\\{0,2,4\\}$.

When
$\dim_{\mathbb{F}}\Lambda_{2}\cap\Lambda^{\prime}_{2}/\Lambda^{\vee}_{2}+\Lambda^{\prime\vee}_{2}=0$,
then in this case we have

$\displaystyle\Lambda_{2}\subset^{2}\Lambda_{2}+\Lambda^{\prime}_{2}$
$\displaystyle\Lambda^{\prime}_{2}\subset^{2}\Lambda_{2}+\Lambda^{\prime}_{2}.$

Hence this case contributes to the $\mathrm{T}_{02}\circ\mathrm{T}_{20}$ by
the double coset operator

$\mathrm{K}_{\\{2\\}}\begin{pmatrix}p^{-1}&&&\\\ &p^{-1}&&\\\ &&p&\\\
&&&p\end{pmatrix}\mathrm{K}_{\\{2\\}}.$

Once we identify $\mathrm{K}_{\\{2\\}}$ with $\mathbf{G}(\mathbb{Z}_{p})$, we
see this is the same as $\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p^{2},2}$.

When
$\dim_{\mathbb{F}}\Lambda^{\vee}_{2}\cap\Lambda^{\prime\vee}_{2}/\Lambda_{2}+\Lambda^{\prime}_{2}=2$,
there are $p+1$ choices of $\Lambda_{0}$. And in this case we have

$\displaystyle\Lambda_{2}\subset^{1}\Lambda_{2}+\Lambda^{\prime}_{2}$
$\displaystyle\Lambda^{\prime}_{2}\subset^{1}\Lambda_{2}+\Lambda^{\prime}_{2}.$

Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the Hecke operator

$(p+1)\mathrm{K}_{\\{2\\}}\begin{pmatrix}p^{-1}&&&\\\ &1&&\\\ &&1&\\\
&&&p\end{pmatrix}\mathrm{K}_{\\{2\\}}.$

Once we identify $\mathrm{K}_{\\{2\\}}$ with the hyperspecial subgroup
$\mathbf{G}(\mathbb{Z}_{p})$, we see this is the same as
$(p+1)\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p,1}$.

When
$\dim_{\mathbb{F}}\Lambda^{\vee}_{2}\cap\Lambda^{\prime\vee}_{2}/\Lambda_{2}+\Lambda^{\prime}_{2}=4$,
there are $[\mathbf{G}(\mathbb{Z}_{p}):\mathrm{Sie})]=(p+1)(p^{2}+1)$ choices
of $\Lambda_{0}$. And in this case we have

$\displaystyle\Lambda_{2}\subset^{0}\Lambda_{2}+\Lambda^{\prime}_{2}$
$\displaystyle\Lambda^{\prime}_{2}\subset^{0}\Lambda_{2}+\Lambda^{\prime}_{2}.$

Hence this case contributes to the $\mathrm{T}_{20}\circ\mathrm{T}_{02}$ by
the constant $(p^{2}+1)(p+1)$. All in all, we obtain the following formula

(6.14)
$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}^{-1}_{0,p}\mathrm{T}_{p,1}+(p^{2}+1)(p+1).$

The last claim in the lemma follows immediately from Proposition 6.9. This
finishes the proof of the lemma. ∎

The final result of this section is the computation of the determinant of the
level raising matrix modulo $\mathfrak{m}$.

######  Proposition 6.15.

Let $(\pi,\Sigma,\mathfrak{m})$ be the datum considered as in the beginning of
the section. Suppose the maximal ideal $\mathfrak{m}$ appears in the support
of $\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$.
Let $[\alpha_{p},\beta_{p},p^{3}\beta^{-1},p^{3}\alpha^{-1}_{p}]$ be the Hecke
parameter of $\pi$ at $p$. Then the determinant of the level raising matrix
modulo $\mathfrak{m}$ is given by

$\mathrm{det}\mathcal{T}_{\mathrm{lr}/\mathfrak{m}}=4p^{-2}\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+p^{3}\alpha^{-1}_{p}-\mathrm{u}p(p+1))(\beta_{p}+p^{3}\beta^{-1}_{p}-\mathrm{u}p(p+1)).$

###### Proof.

Since $\pi$ has trivial central character, $\mathrm{T}_{p,0}$ acts trivially
on
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]/\mathfrak{m}$.
We therefore have
$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}_{02}\circ\mathrm{T}_{20}=\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}_{p,1}+(p^{2}+1)(p+1)=\mathrm{T}^{2}_{p,2}$
by Lemma 6.13. Then

$\displaystyle\mathrm{det}\mathcal{T}_{\mathrm{lr}/\mathfrak{m}}$
$\displaystyle=4((\mathrm{T}_{p,1}+p(p+1))^{2}-(p+1)^{2}\mathrm{T}_{20}\mathrm{T}_{02})$
$\displaystyle=4(\mathrm{T}_{p,1}+p(p+1)-(p+1)\mathrm{T}_{p,2})(\mathrm{T}_{p,1}+p(p+1)+(p+1)\mathrm{T}_{p,2})$
$\displaystyle=4p^{-2}$
$\displaystyle[(\alpha_{p}+p^{3}\alpha^{-1}_{p})(\beta_{p}+p^{3}\beta^{-1}_{p})-p(p+1)(\alpha_{p}+\beta_{p}+p^{3}\alpha^{-1}_{p}+p^{3}\beta^{-1}_{p}+p^{2}(p+1)^{2})]$
$\displaystyle[(\alpha_{p}+p^{3}\alpha^{-1}_{p})(\beta_{p}+p^{3}\beta^{-1}_{p})+p(p+1)(\alpha_{p}+\beta_{p}+p^{3}\alpha^{-1}_{p}+p^{3}\beta^{-1}_{p}+p^{2}(p+1)^{2})]$
$\displaystyle=4p^{-2}\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+p^{3}\alpha^{-1}_{p}-\mathrm{u}p(p+1))(\beta_{p}+p^{3}\beta^{-1}_{p}-\mathrm{u}p(p+1)).$

This finishes the proof. ∎

##  7\. Tate cycles on the quaternionic unitary Shimura variety

In this section, our goal is to establish the following priniciple: the
localized cohomology group
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is generated by cycles coming from the supersingular locus. On the dual side,
similar result also holds for
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$,
however we need to show this cohomology group contains no
$\mathcal{O}_{\lambda}$-torsion first. We will establish the full result
simultaneously with our main theorem on arithmetic level raising in Corollary
9.8. Let $(\pi,\Sigma,\mathfrak{m})$ be the datum considered as in the
beginning of the last section.

######  Construction 7.1.

Consider the supersingular locus
$\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B})$, we have the
following restriction morphism

$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\rightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$

and we denote the composite of this restriction morphism with the natural map

$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\rightarrow\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$

by

$\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}:\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\rightarrow\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1)).$

We also have the natural Gysin map

$\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}:\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\rightarrow\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2)).$

We can fit the two maps
$(\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}},\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!})$
in the above construction in the diagram below.

(7.2)
${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]}$${\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))}$${\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))}$${\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]}$${\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]}$$\scriptstyle{\mathrm{inc}^{\\{0\\}}_{!}}$$\scriptstyle{\mathrm{inc}^{\\{2\\}}_{!}}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}}$$\scriptstyle{\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{0\\}}}$$\scriptstyle{\mathrm{inc}^{\ast}_{\\{2\\}}}$

The above diagram sets up the intersection matrix for the supersingular locus
which we will elaborate below.

######  Construction 7.3.

We obtain naturally the following four maps from the above diagram

$\displaystyle\mathcal{T}_{\mathrm{ss},\\{00\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\mathrm{inc}^{\\{0\\}}_{!}$
$\displaystyle\mathcal{T}_{\mathrm{ss},\\{02\\}}=\mathrm{inc}^{\ast}_{\\{2\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\circ\mathrm{inc}^{\\{0\\}}_{!}$
$\displaystyle\mathcal{T}_{\mathrm{ss},\\{20\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\mathrm{inc}^{\\{2\\}}_{!}$
$\displaystyle\mathcal{T}_{\mathrm{ss},\\{22\\}}=\mathrm{inc}^{\ast}_{\\{2\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\circ\mathrm{inc}^{\\{2\\}}_{!}.$

The resulting matrix

$\mathcal{T}_{\mathrm{ss}}=\begin{pmatrix}&\mathcal{T}_{\mathrm{ss},\\{00\\}}&\mathcal{T}_{\mathrm{ss},\\{02\\}}\\\
&\mathcal{T}_{\mathrm{ss},\\{20\\}}&\mathcal{T}_{\mathrm{ss},\\{22\\}}\\\
\end{pmatrix}$

will be referred to as the supersingular matrix. We can localize the above
diagram at the maximal ideal $\mathfrak{m}$ and write the resulting matrix as
$\mathcal{T}_{\mathrm{ss},\mathfrak{m}}$, however all the matrix entries will
still be denoted by symbols without referring to $\mathfrak{m}$. If we need to
consider this matrix modulo $\mathfrak{m}$, then we will denote it by
$\mathcal{T}_{\mathrm{ss}/\mathfrak{m}}$. The entries of this matrix will be
denoted without referring to $\mathfrak{m}$.

###  7.1. Computing the supersingular matrix

We compute the entries of the supersingular matrix using the same procedure as
we did for computing the level raising matrix.

######  Proposition 7.4.

The supersingular matrix $\mathcal{T}_{\mathrm{ss}}$ is given by

$\mathcal{T}_{\mathrm{ss}}=\begin{pmatrix}&4p^{2}(p+1)^{2}+\mathrm{T}_{20}\circ\mathrm{T}_{02}&-4p(p+1)\mathrm{T}_{02}\\\
&-4p(p+1)\mathrm{T}_{20}&4p^{2}(p+1)^{2}+\mathrm{T}_{02}\circ\mathrm{T}_{20}\\\
\end{pmatrix}.$

###### Proof.

We will prove the proposition in an entry by entry manner.

####  7.1.1. $\mathcal{T}_{\mathrm{ss},\\{00\\}}$

Recall
$\mathcal{T}_{\mathrm{ss},\\{00\\}}=\mathrm{inc}^{\ast}_{\\{0\\}}\circ\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}\circ\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}\circ\mathrm{inc}^{\\{0\\}}_{!}$.
If the maps

$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}}\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))$

factor through
$\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa},\\{0\\}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$.
Then the contribution is $4p^{2}(p+1)^{2}$. This follows from the fact that
for each $\Lambda_{0}\in\mathcal{L}_{\\{0\\}}$ the degree of the variety
$\mathrm{DL}(\Lambda_{0})$ has degree $(p+1)$ and the normal bundle has degree
$-2p$ by Lemma 2.10. If the maps

$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}}\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))$

factor through
$\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa},\\{2\\}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$,
then as we have seen before we need to understand the relative position of the
vertex lattices $\Lambda_{0}$, $\Lambda_{2}$ and $\Lambda^{\prime}_{0}$ of
type $0$, $2$ and $0$. The only possible situation is given by

$\displaystyle
p\Lambda^{\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda_{0}=\Lambda^{\vee}_{0}\subset^{2}\Lambda^{\vee}_{2}$
$\displaystyle
p\Lambda^{\prime\vee}_{0}\subset^{2}\Lambda_{2}\subset^{2}\Lambda^{\prime}_{0}=\Lambda^{\prime\vee}_{0}\subset^{2}\Lambda^{\vee}_{2}.$

As we have seen, this is parametrized by
$\mathrm{T}_{20}\circ\mathrm{T}_{02}$. Therefore this entry is given by

$\mathcal{T}_{\mathrm{ss},\\{00\\}}=(p+1)^{2}+\mathrm{T}_{20}\circ\mathrm{T}_{02}.$

####  7.1.2. $\mathcal{T}_{\mathrm{ss},\\{02\\}}$

If the maps

$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\ast}_{\\{\mathrm{ss}\\}}}\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))\xrightarrow{\mathrm{inc}^{\\{\mathrm{ss}\\}}_{!}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))$

factor through
$\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa},\\{0\\}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$.
This case contributes by $-2p(p+1)\mathrm{T}_{02}$. If the maps factor through
$\mathrm{H}^{2}(\overline{\mathrm{X}}^{\mathrm{ss}}_{\mathrm{Pa},\\{2\\}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))$.
This case contributes by $-2p(p+1)\mathrm{T}_{02}$. Therefore the total
contribution gives the entry

$\mathcal{T}_{\mathrm{ss},\\{02\\}}=-4p(p+1)\mathrm{T}_{02}.$

####  7.1.3. $\mathcal{T}_{\mathrm{ss},\\{20\\}}$

The computation is the same as in the case for
$\mathcal{T}_{\mathrm{ss},\\{02\\}}$. Thus this entry is given by

$\mathcal{T}_{\mathrm{ss},\\{20\\}}=-4p(p+1)\mathrm{T}_{20}.$

####  7.1.4. $\mathcal{T}_{\mathrm{ss},\\{22\\}}$

The computation is exactly the same as in the case for
$\mathcal{T}_{\mathrm{ss},\\{00\\}}$. Thus this entry is given by

$\mathcal{T}_{\mathrm{ss},\\{22\\}}=4p^{2}(p+1)^{2}+\mathrm{T}_{02}\circ\mathrm{T}_{20}.$

This finishes the proof of Proposition 7.4. ∎

######  Proposition 7.5.

Let $(\pi,\Sigma,\mathfrak{m})$ be the datum considered as in the beginning of
the section. Suppose the maximal ideal $\mathfrak{m}$ appears in the support
of $\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$.
Suppose that $[\alpha_{p},\beta_{p},p^{3}\beta^{-1},p^{3}\alpha^{-1}_{p}]$ is
the Hecke parameter of $\pi$ at $p$. Then determinant of the supersingular
matrix modulo $\mathfrak{m}$ is given by

$\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{ss}/\mathfrak{m}}=\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+\beta_{p}+p^{3}\alpha^{-1}_{p}+p^{3}\beta^{-1}_{p}-2\mathrm{u}p(p+1))^{2}.$

###### Proof.

Since $\pi$ has trivial central character, then $\mathrm{T}_{p,0}$ acts
trivially on
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]/\mathfrak{m}$.
We therefore have

$\mathrm{T}_{20}\circ\mathrm{T}_{02}=\mathrm{T}_{02}\circ\mathrm{T}_{20}=\mathrm{T}_{p^{2},2}+(p+1)\mathrm{T}_{p,1}+(p^{2}+1)(p+1)=\mathrm{T}^{2}_{p,2}$

by Lemma 6.13. Then we have

$\displaystyle\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{ss}/\mathfrak{m}}$
$\displaystyle=(\mathrm{T}^{2}_{p,2}+4p^{2}(p+1))^{2}-16p^{2}(p+1)^{2}\mathrm{T}^{2}_{p,2}$
$\displaystyle=(\mathrm{T}^{2}_{p,2}+4p^{2}(p+1)-4p(p+1)\mathrm{T}_{p,2})$
$\displaystyle\phantom{abc}(\mathrm{T}^{2}_{p,2}+4p^{2}(p+1)+4p(p+1)\mathrm{T}_{p,2})$
$\displaystyle=(\alpha_{p}+\beta_{p}+p^{3}\beta^{-1}_{p}+p^{3}\alpha^{-1}_{p}-2p(p+1))^{2}$
$\displaystyle\phantom{abc}(\alpha_{p}+\beta_{p}+p^{3}\beta^{-1}_{p}+p^{3}\alpha^{-1}_{p}+2p(p+1))^{2}$
$\displaystyle=\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+\beta_{p}+p^{3}\beta^{-1}_{p}+p^{3}\alpha^{-1}_{p}-2\mathrm{u}p(p+1))^{2}.$

∎

###  7.2. Tate classes in the quaternionic unitary Shimura variety

We fix the datum $(\pi,\Sigma,\mathfrak{m})$ as in the beginning of this
section. Let $p$ be a level raising special prime for $\pi$ of length $m$. We
make the following assumption regarding to the vanishing of the nearby cycle
cohomology of the quaternionic unitary Shimura variety.

######  Assumption 7.6.

Let
$\mathrm{H}^{d}_{(\mathrm{c})}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$
be the usual or compact support nearby cycle cohomology of the special fiber
$\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B})$. We assume the maximal ideal
$\mathfrak{m}$ satisfies the following properties

  1. (1)

The natural morphism
$\mathrm{H}^{d}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}\xrightarrow{\sim}\mathrm{H}^{d}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$
is an isomorphism for all $d$.

  2. (2)

The cohomology
$\mathrm{H}^{d}_{(\mathrm{c})}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}=0$
for $d\neq 3$, and that the middle degree cohomology
$\mathrm{H}^{3}_{(\mathrm{c})}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$
is a finite free $\mathcal{O}_{\lambda}$ module.

######  Remark 7.7.

The first assumption is usually known in literature as that of $\mathfrak{m}$
being a non-Eisenstein ideal. It is satisfied if $\mathfrak{m}$ is associated
to a cuspidal automorphic representation of general type. The second
assumption is more serious. This is reasonable assumption in light of the
recent work of Caraiani-Scholze [CS17, CS19] and recent advances by Koshikawa
[Kos19, Kos21]. Although they deal with certain unitary Shimura varieties, the
methods are general enough to treat at least our cases: if one assumes that
$\pi$ has an unramified type $\mathrm{I}$ component which is generic modulo
$l$, then combining the proof of Koshikawa [Kos21] with a semi-perversity
result of the pushforward along the Hodge-Tate period map [San22], one can
prove that $\mathfrak{m}$ satisfies $(2)$.

If we make the assumption that $\mathrm{X}_{\mathrm{Pa}}(\mathrm{B})$ has good
reduction at $l$ and that $\pi$ is ordinary at $l$, then the method of [MT02]
could be used to show that the Assumption 7.6 is fulfilled.

We also conjecture that the product of Shimura curves for $\mathrm{B}$
embedded in $\mathrm{Sh}(\mathrm{B},\mathrm{K}_{\mathrm{Pa}})$ has an affine
complement. Then the above assumption can be proved if one assumes that the
residual Galois representation has large image. This is the analogue of the
fact that the Igusa divisor on the classical Siegel threefold has an affine
complement.

######  Definition 7.8.

If $\mathfrak{m}$ satisfies either of the two conditions below, we call
$\mathfrak{m}$ a generic maximal ideal.

  1. (1)

We say $\mathfrak{m}$ is generic level raising at $p$ if it is level raising
special at $p$ and Assumption 7.6 is satisfied.

  2. (2)

We say $\mathfrak{m}$ is generic non-level raising at $p$ if the Hecke
parameters for $\pi$ satisfies

$\displaystyle\alpha_{p}+p^{3}\alpha^{-1}_{p}\not\equiv\pm(p+p^{2})\mod\lambda$
$\displaystyle\beta_{p}+p^{3}\beta^{-1}_{p}\not\equiv\pm(p+p^{2})\mod\lambda$

and Assumption 7.6 is satisfied.

######  Theorem 7.9.

Let $\mathfrak{m}$ be a generic maximal ideal as in the above definition, we
have the following statements.

  1. (1)

The map from Construction 6.4

$(\mathrm{inc}^{\\{0\\}}_{!,\mathfrak{m}}+\mathrm{inc}^{\\{2\\}}_{!,\mathfrak{m}}):\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$

is an isomorphism.

  2. (2)

The map from Construction 6.4

$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}}):\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$

is surjective and whose kernel is the torsion part of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.

###### Proof.

Suppose for the moment that $\mathfrak{m}$ is generic level raising at $p$.
Then the determinant of the supersingular matrix
$\mathrm{det}\mathcal{T}_{\mathrm{ss}/\mathfrak{m}}$ is given by

$\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+\beta_{p}+p^{3}\beta^{-1}_{p}+p^{3}\alpha^{-1}_{p}-2\mathrm{u}p(p+1))^{2}.$

This is non-zero by our definition of the level raising condition. This
immediately implies that we have an injection

$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\hookrightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$

by the Nakayama’s lemma. Note that
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is a free $\mathcal{O}_{\lambda}$-module as it injects into

$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}}$

which is torsion free by the genericness of $\mathfrak{m}$. Therefore to
finish the proof in this case, we need to show that

$2\mathrm{rank}_{\mathcal{O}_{\lambda}}\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\geq\mathrm{rank}_{\mathcal{O}_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}.$

For this purpose, let $\vec{\pi}$ be an automorphic representation of
$\mathbf{G}(\mathbb{A}_{\mathbb{Q}})$ of general type. It suffices to show
that

$2\dim_{\overline{\mathbb{Q}}_{l}}\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})][\iota_{l}\vec{\pi}^{\infty
pq}]\geq\dim_{\overline{\mathbb{Q}}_{l}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
pq}].$

Consider the co-specialization exact sequence

$\displaystyle 0$
$\displaystyle\rightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
pq}]\rightarrow\bigoplus_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]$
$\displaystyle\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
p}]\rightarrow 0$

We can canonically identify the space

$\bigoplus\limits_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})[\iota_{l}\vec{\pi}^{\infty
p}]$

with the space
$\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{Pa}}(\overline{\mathrm{B}})][\iota_{l}\vec{\pi}^{\infty
p}]$. Note that to complete $\vec{\pi}^{p\infty}$ to a representation
$\vec{\pi}$ of $\mathbf{G}(\mathbb{A}_{\mathbb{Q}})$ occuring in
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))$,
$\vec{\pi}_{p}$ must be of type $\mathrm{II}\mathrm{a}$ as a representation of
$\mathbf{G}(\mathbb{Q}_{p})$ by the proof of Theorem 4.5. On the other hand,
those $\vec{\pi}_{p}$ completing $\vec{\pi}^{\infty p}$ in
$\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{Pa}}(\overline{\mathrm{B}})][\iota_{l}\pi^{\infty
p}]$ can be either the unramified principal series or of type
$\mathrm{II}\mathrm{a}$. In the proof of Theorem 4.5, we have seen the the
Jacquet–Langlands correspondence preserves the automorphic multiplicity. Thus
by the uniqueness of local newforms [BS, Theorem 5.6.1] or [Sch05, Theorem
2.3.1], those $\vec{\pi}_{p}$ completing $\vec{\pi}^{\infty p}$ in
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
p}]$ can only be unramified unless the map

$\bigoplus_{\sigma\in\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]\ $

is zero. Howerver this is impossible as otherwise, it would imply the
isomorphism

$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l})(1))[\iota_{l}\vec{\pi}^{\infty
p}]\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
p}]$

which is absurd by considering the inertia action. By the oldforms principle
[BS, Theorem 5.6.1] or [Sch05, Theorem 2.3.1], we have the desired

$2\dim_{\overline{\mathbb{Q}}_{l}}\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})][\iota_{l}\vec{\pi}^{\infty
p}]\geq\dim_{\overline{\mathbb{Q}}_{l}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\overline{\mathbb{Q}}_{l}(1))[\iota_{l}\vec{\pi}^{\infty
p}].$

Next we show the same result under the assumption that $\mathfrak{m}$ is
generic non-level raising. Suppose that $\mathfrak{m}$ is not level raising
special. Then the determinant of the level raising matrix
$\mathrm{det}\mathcal{T}_{\mathrm{lr}/\mathfrak{m}}$ is given by

$\mathrm{det}\mathcal{T}_{\mathrm{lr}/\mathfrak{m}}=4p^{-2}\prod_{\mathrm{u}=\pm
1}(\alpha_{p}+p^{3}\alpha^{-1}_{p}-\mathrm{u}p(p+1))(\beta_{p}+p^{3}\beta^{-1}_{p}-\mathrm{u}p(p+1)).$

This is non-zero by our assumption. This immediately implies that we have an
injection

$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\hookrightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$

by the Nakayama’s lemma. The rest of the argument is the same as in the
previous case. Finally, the second part of the proposition is dual to the
first part. In particular, the map

$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}}):\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$

is surjective. Using the same argument as in the first part, we see that
$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}})$
is an isomorphism up to the torsion of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.
The second part now follows ∎

We finish this section with several remarks about this theorem. First of all,
this result can be understood as the statement that the cohomology of the
special fiber
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is generated by Tate cycles coming from the supersingular locus. This result
along with [LTXZZ, Proposition 6.3.1] can be seen as an analogue of the main
result of [XZ] in the bad reduction case for low degree cohomology groups
localized at generic maximal ideals. It seems that these cohomology groups
always correspond to oldforms. It would be of great interest to formulate some
predictions in a similar manner of [XZ]. Secondly, we have established the
following analogue of the Ihara’s lemma in the setting of definite
quaternionic unitary groups.

######  Corollary 7.10.

There is an injection

$\mathbf{Ih}_{\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\xhookrightarrow{\phantom{aaaa}}\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}.$

###### Proof.

We have shown in the above proof that the composite of the two maps

$\displaystyle\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\xrightarrow{(\mathrm{inc}^{\\{0\\}}_{!,\mathfrak{m}}+\mathrm{inc}^{\\{2\\}}_{!,\mathfrak{m}})}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
$\displaystyle\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}\xhookrightarrow{\phantom{aa}\alpha\phantom{aa}}\bigoplus_{\sigma\in
Z_{\\{1\\}}(\overline{\mathrm{B}})}\mathrm{H}^{3}_{\\{\sigma\\}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))=\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$

gives rise to an injection which we call

$\mathbf{Ih}_{\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\xhookrightarrow{\phantom{aaaa}}\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}.$

∎

Note that there is no natural degeneracy map from neither
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
nor
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
to
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{1\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$.
Instead, these maps should be the analogues of the level raising operators in
the sense of [BS]. Finally suppose that $\mathfrak{m}$ is generic level
raising at $p$, we will show that
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$
is torsion free following the method of [LTXZZ] using Galois deformation
argument.

##  8\. Galois deformation rings and cohomology of Shimura varieties

In this section, we recall the main results in [Wang22a] where we use the
method of [LTXZZa] to show that the cohomology of the quaternionic unitary
Shimura varieties are free over certain universal deformation rings. These
results are necessary as we lack of torsion vanishing results on the
cohomology of the special fibers of these quaternionic unitary Shimura
varieties.

###  8.1. Galois representation for $\mathrm{GSp}_{4}$

First we recall some results on associating Galois representations to cuspidal
automorphic representations of $\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ and
the local-global compatibility of Langlands correspondence. Let $\pi$ be a
cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ with trivial central character. We
assume that $\pi$ has weight $(k_{1},k_{2})$ and trivial central character. In
this article, we will only consider discrete automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ that is of general type. This
means there is a cuspidal automorphic representation $\Pi$ of
$\mathrm{GL}_{4}(\mathbb{A}_{\mathbb{Q}})$ of symplectic type such that for
each place $v$ of $\mathbb{Q}$, the $L$-parameter obtained from
$\mathrm{rec}_{\mathrm{GT}}(\pi_{v})$ by composing with the embedding
$\mathrm{GSp}_{4}\hookrightarrow\mathrm{GL}_{4}$ is precisely
$\mathrm{rec}_{\mathrm{GL}_{4}}(\Pi_{v})$ which is the Langlands parameter
attached to $\Pi_{v}$. Here $\Pi$ is of symplectic type if the partial
$L$-function $L^{\mathrm{S}}(s,\Pi,\wedge^{2})$ has a pole at $s=1$ for any
finite set $S$ of places of $\mathbb{Q}$. In this case, we say $\Pi$ is the
transfer of $\pi$.

We now recall some general results of Mok and Sorensen on the existence of
Galois representations attached to $\pi$. This theorem is very general and we
temporarily drop the assumption that $\pi$ is of general type.

######  Theorem 8.1.

Suppose $\pi$ is a cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ of weight $(k_{1},k_{2})$ where
$k_{1}\geq k_{2}\geq 3$ and $k_{1}\equiv k_{2}\mod 2$. Suppose that $\pi$ has
trivial central character. Let $l\geq 5$ be a fixed prime.Then there is a
continuous semisimple representation

$\rho_{\pi,\iota_{l}}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(\overline{\mathbb{Q}}_{l})$

satisfying the following properties.

  1. (1)

$\mathrm{c}\circ\rho_{\pi,\iota_{l}}=\epsilon^{-3}_{l}$;

  2. (2)

For each finite place $v$, we have

$\mathrm{WD}(\rho_{\pi,\iota_{l}}|_{\mathrm{G}_{\mathbb{Q}_{v}}})^{\mathrm{F}-\mathrm{ss}}\cong\mathrm{rec}_{\mathrm{GT}}(\pi_{v}\otimes|\mathrm{c}|^{-3/2})^{\mathrm{ss}};$

  3. (3)

The local representation $\rho_{\pi,\iota_{l}}|_{\mathrm{G}_{\mathbb{Q}_{l}}}$
is de Rham with Hodge–Tate weights

$(2-\frac{k_{1}+k_{2}}{2},-\frac{k_{1}-k_{2}}{2},1+\frac{k_{1}-k_{2}}{2},-1+\frac{k_{1}+k_{2}}{2}).$

  4. (4)

If $\pi$ is unramified at $l$, then
$\rho_{\pi,\iota_{l}}|_{\mathrm{G}_{\mathbb{Q}_{v}}}$ is moreover crystalline
at $l$.

  5. (5)

If $\rho_{\pi,\iota_{l}}$ is irreducible, then for all finite place $v$,
$\rho_{\pi,\iota_{l}}|_{\mathrm{G}_{\mathbb{Q}_{v}}}$ is pure.

######  Remark 8.2.

Several remarks about this theorem are in order. The construction of the
Galois representation in this case is given in [Lau97, Lau05, Wei05, Tay93]
using the Langlands–Kottwitz method. The statements about local-global
compatibilities are mostly proved in [Sor10] and completed in [Mok05]. In fact
the authors use the strong transfer of cuspidal automorphic representation
from $\mathrm{GSp}_{4}$ to $\mathrm{GL}_{4}$ to construct the desired Galois
representation in the totally real field case. The Harish-Chandra parameter
$(\mu_{1},\mu_{2})$ of $\pi$ used more often has the following relation with
the weight of $\pi$ used in this theorem: $k_{1}=\mu_{1}+1$ and
$k_{2}=\mu_{2}+2$. In this article, we will be interested in the case $\pi$ is
of general type and has trivial central character and weights $(3,3)$. In our
normalization, this is the case when these representation appear in the
cohomology of quaternionic unitary Shimura varieties with trivial
coeffficient.

######  Definition 8.3.

Let $\pi$ be a cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ as in the previous remark. We say
a number field $E\subset\mathbb{C}$ is a strong coefficient field of $\pi$ if
for every prime $\lambda$ of $E$ there exits a continuous homomorphism

$\rho_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(E_{\lambda})$

up to conjugation, such that for every isomorphism
$\iota_{l}:\mathbb{C}\xrightarrow{\sim}\overline{\mathbb{Q}}_{l}$ inducing the
prime $\lambda$, the representations
$\rho_{\pi,\lambda}\otimes_{E_{\lambda}}\overline{\mathbb{Q}}_{l}$ and
$\rho_{\pi,\iota_{l}}$ are conjugate.

Let $p$ be a place at which $\pi$ is an unramified type $\mathrm{I}$
representation. Then the Frobenius eigenvalues of
$\rho_{\pi,\lambda}(\mathrm{Frob}_{p})$ agree with the Hecke parameter
$[\alpha_{p},\beta_{p},\gamma_{p},\delta_{p}]$ of $\pi_{p}$ by Theorem 8.1
$(2)$. We will always make the following assumption on $\rho_{\pi,\lambda}$.

######  Assumption 8.4.

The Galois representation
$\rho_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(E_{\lambda})$
is residually absolutely irreducible.

The above assumption allows us to define the residual Galois representation

(8.5)
$\overline{\rho}_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(k)$

which is unique up to conjugation. Note it also follows that $\pi$ is
necessarily of general type if assumption 8.4 is effective.

###  8.2. Galois deformation rings

We summarize the main results in [Wang22a]. Suppose we are given a Galois
representation
$\overline{\rho}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(k)$. We
will write the restriction
$\overline{\rho}|_{\mathrm{G}_{\mathbb{Q}_{v}}}:\mathrm{G}_{\mathbb{Q}_{v}}\rightarrow\mathrm{GSp}(k)$
by $\overline{\rho}_{v}$.

Let $\Sigma_{\mathrm{min}}$ and $\Sigma_{\mathrm{lr}}$ be two sets of non-
archimedean places of $\mathbb{Q}$ away from $l$ and such that

  * •

$q$ is contained in $\Sigma_{\mathrm{lr}}$;

  * •

$\Sigma_{\mathrm{min}}$, $\Sigma_{\mathrm{lr}}$ and $\\{p\\}$ are mutually
disjoint;

  * •

for every $v\in\Sigma_{\mathrm{lr}}\cup\\{p\\}$, the prime $v$ satisfies
$l\nmid(v^{2}-1)$.

######  Definition 8.6.

For a Galois representation
$\overline{\rho}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}(k)$ and
$\psi=\epsilon^{-3}_{l}$, we say the pair $\overline{\rho}$ is rigid for
$(\Sigma_{\mathrm{min}},\Sigma_{\mathrm{lr}})$ if the followings are
satisfied.

  * •

For every $v\in\Sigma_{\mathrm{min}}$, every lifting of $\overline{\rho}_{v}$
is minimally ramified in the sense of [Wang22a, Definition 3.4];

  * •

For every $v\in\Sigma_{\mathrm{lr}}$, the generalized eigenvalues of
$\overline{\rho}_{v}(\phi_{v})$ contain the pair $\\{v^{-1},v^{-2}\\}$ exactly
once;

  * •

At $v=l$, $\overline{\rho}_{v}$ is regular Fontaine-Laffaille crystalline as
in [Wang22a, Definition 3.10];

  * •

For $v\not\in\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{l\\}$, the
representation $\overline{\rho}_{v}$ is unramified.

Suppose that $\overline{\rho}$ is rigid for
$(\Sigma_{\mathrm{min}},\Sigma_{\mathrm{lr}})$. We consider a global
deformation problem in the sense of [Wang22a, Definition 2.2] of the form

$\
\mathcal{S}^{\ast}=(\overline{\rho},\psi,\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p\\}\cup\\{l\\},\\{\mathcal{D}_{v}\\}_{v\in\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p\\}\cup\\{l\\}})$

where $\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$.

  * •

For $v\in\Sigma_{\mathrm{min}}$, $\mathcal{D}_{v}$ is the local deformation
problem $\mathcal{D}^{\mathrm{min}}_{v}$ classifying all the minimal ramified
liftings defined in [Wang22a, Defintion 3.4];

  * •

For $v\in\Sigma_{\mathrm{lr}}\cup\\{q\\}$, $\mathcal{D}_{v}$ is the local
deformation problem $\mathcal{D}^{\mathrm{ram}}_{v}$ defined as in [Wang22a,
Definition 3.6 (2)] classifying certain ramified liftings;

  * •

For $v=l$, $\mathcal{D}_{v}$ is the local deformation problem
$\mathcal{D}^{\mathrm{FL}}_{v}$ defined as in [Wang22a, Definition 3.1.1]
classifying regular Fontaine–Laffaille crystalline liftings;

  * •

For $v=p$, depending on $\ast\in\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$,
we have

    * –

$\mathcal{D}_{v}$ is the local deformation problem
$\mathcal{D}^{\mathrm{unr}}_{v}$ which is given by [Wang22a, Definition 3.6
(2)] when $\ast=\mathrm{unr}$;

    * –

$\mathcal{D}_{v}$ is the local deformation problem
$\mathcal{D}^{\mathrm{ram}}_{v}$ which is given by [Wang22a, Definition 3.6
(3)] when $\ast=\mathrm{ram}$;

    * –

$\mathcal{D}_{v}$ is the local deformation problem
$\mathcal{D}^{\mathrm{mix}}_{v}$ defined in [Wang22a, Definition 3.6 (1)] when
$\ast=\mathrm{mix}$;

  * •

For each global deformation problem $\mathcal{S}^{\ast}$ with
$\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$, we have its corresponding
universal deformation ring denoted by
$\mathrm{R}^{\ast}=\mathrm{R}^{\mathrm{univ}}_{\mathcal{S}^{\ast}}$ for
$\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$.

######  Construction 8.7.

When $\mathrm{B}$ is indefinite, we choose the level
$\mathrm{K}=\mathrm{K}_{\mathrm{Pa}}$ for the Shimura variety
$\mathrm{Sh}(\mathrm{B},\mathrm{K}_{\mathrm{Pa}})$ in the following way.

  * •

For $v\not\in\Sigma_{\mathrm{lr}}\cup\Sigma_{\mathrm{min}}$ or $v=l$, then
$\mathrm{K}_{v}$ is hyperspecial;

  * •

For $v\in\Sigma_{\mathrm{lr}}-\\{p,q\\}$, then $\mathrm{K}_{v}$ is the
paramodular subgroup of $\mathrm{GSp}(\mathbb{Z}_{v})$;

  * •

For $v\in\Sigma_{\mathrm{min}}$, then $\mathrm{K}_{v}$ is contained in the
pro-$v$ Iwahori subgroup $\mathrm{Iw}_{1}(v)$;

  * •

For $v=p,q$, then $\mathrm{K}_{v}$ is the paramodular subgroup
$\mathbf{G}(\mathrm{B})(\mathbb{Z}_{v})$ of $\mathrm{GU}_{2}(\mathrm{D})$ and
$\mathrm{GU}_{2}(\mathrm{D}^{\prime})$ respectively;

Let
$\Sigma_{\mathrm{Pa}}=\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p,q\\}$.
We denote by $\mathbf{T}^{\mathrm{ram}}$ the image of
$\mathbb{T}^{\Sigma_{\mathrm{Pa}}}$ in

$\operatorname{End}_{\mathcal{O}_{\lambda}}(\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))).$

######  Construction 8.8.

We choose the level $\overline{\mathrm{K}}=\mathrm{K}_{\mathrm{H}}$ for the
Shimura variety $\mathrm{Sh}(\overline{\mathrm{B}},\mathrm{K}_{\mathrm{H}})$
when $\overline{\mathrm{B}}$ is definite in such the following way.

  * •

For $v\not\in\Sigma_{\mathrm{lr}}\cup\Sigma_{\mathrm{min}}$ or $v=l$ or $v=p$,
then $\overline{\mathrm{K}}_{v}$ is hyperspecial;

  * •

For $v\in\Sigma_{\mathrm{lr}}$, then $\overline{\mathrm{K}}_{v}$ is the
paramodular subgroup of $\mathrm{GSp}(\mathbb{Z}_{v})$;

  * •

For $v\in\Sigma_{\mathrm{min}}$, then $\overline{\mathrm{K}}_{v}$ is contained
in the pro-$v$ Iwahori subgroup $\mathrm{Iw}_{1}(v)$;

  * •

For $v=q$, then $\overline{\mathrm{K}}_{v}$ is the paramodular subgroup
$\mathbf{G}(\overline{\mathrm{B}})(\mathbb{Z}_{v})$ of
$\mathrm{GU}_{2}(\mathrm{D}^{\prime})$.

Let
$\Sigma_{\mathrm{H}}=\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{q\\}$.
We denote by $\mathbf{T}^{\mathrm{unr}}$ the image of
$\mathbb{T}^{\Sigma_{\mathrm{H}}}$ in
$\operatorname{End}_{\mathcal{O}_{\lambda}}(\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})])$.

Let $\overline{\rho}$ be a Galois representation

$\overline{\rho}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(k)$

and let $\mathfrak{m}$ be the maximal ideal corresponding to $\overline{\rho}$
in $\mathbb{T}^{\Sigma}$ for
$\Sigma\in\\{\Sigma_{\mathrm{Pa}},\Sigma_{\mathrm{H}}\\}$, in the sense that
the characteristic polynomial
$\mathrm{det}(\mathrm{X}-\overline{\rho}(\mathrm{Frob}_{v}))$ is congruent to
the Hecke polynomial $\mathrm{Q}_{v}(\mathrm{X})$ given by

$\mathrm{X}^{4}-\mathrm{T}_{v,2}\mathrm{X}^{3}+(v\mathrm{T}_{v,1}+(v^{3}+v)\mathrm{T}_{v,0})\mathrm{X}^{2}-v^{3}\mathrm{T}_{v,0}\mathrm{T}_{v,2}\mathrm{X}+v^{6}\mathrm{T}^{2}_{v,0}$

modulo $\mathfrak{m}$ for each $v$ not in $\Sigma$. We will also assume that
the image of $\overline{\rho}$ contains $\mathrm{GSp}_{4}(\mathbb{F}_{l})$ and
that

$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}=\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}.$

The following theorem is the main result of [Wang22a] proved using the Taylor-
Wiles method.

######  Theorem 8.9.

Let $\overline{\rho}$ and $\mathfrak{m}$ be given as above. Suppose the
following assumptions hold.

  * (D1)

the image of $\overline{\rho}(\mathrm{G}_{\mathbb{Q}})$ contains
$\mathrm{GSp}_{4}(k)$;

  * (D2)

$\overline{\rho}$ is rigid for $(\Sigma_{\mathrm{min}},\Sigma_{\mathrm{lr}})$;

  * (D3)

For every finite set $\Sigma^{\prime}$ of nonarchimedean places containing
$\Sigma$ and every open compact subgroup $\mathrm{K}^{\prime}$ of $\mathrm{K}$
satisfying $\mathrm{K}^{\prime}_{v}=\mathrm{K}_{v}$ for all
$v\not\in\Sigma^{\prime}$, we have

$\mathrm{H}^{d}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(k))_{\mathfrak{m}^{\prime}}=0$

for $d\neq 3$ where
$\mathfrak{m}^{\prime}=\mathfrak{m}\cap\mathbb{T}^{\Sigma^{\prime}}$.

Then the following holds true.

  1. (1)

If $\mathbf{T}^{\mathrm{unr}}_{\mathfrak{m}}$ is non-zero, then we have an
isomorphim of complete intersection rings:

$\mathrm{R}^{\mathrm{unr}}=\mathbf{T}^{\mathrm{unr}}_{\mathfrak{m}}$

and
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is a finite free $\mathbf{T}^{\mathrm{unr}}_{\mathfrak{m}}$-module.

  2. (2)

If $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$ is non-zero, then we have an
isomorphim of complete intersection rings:

$\mathrm{R}^{\mathrm{ram}}=\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$

and
$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$
is a finite free $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$-module.

######  Remark 8.10.

Let $\pi$ be a cuspidal automorphic representation of general type. Assume
that $\overline{\rho}_{\pi,\lambda}(\mathrm{G}_{\mathbb{Q}})$ contains
$\mathrm{GSp}_{4}(\mathbb{F}_{l})$ for sufficiently large $l$, then we have
shown that $\overline{\rho}_{\pi,\lambda}$ is indeed rigid, see [Wang22a,
Theorem 5.5].

##  9\. Arithmetic level raising and applications

Now we are ready to prove our main result on arithmetic level raising on the
quaternionic Shimura varieties studied in the previous sections. First we
recall the setting we are in. Let $\pi$ be a cuspidal automorphic
representation of $\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ with weight
$(3,3)$ and trivial central character. Suppose that $\pi$ is of general type
and $E$ be the coefficient field of $\pi$. Let $\Sigma_{\pi}$ be the minimal
set of finite places outside of which $\pi$ is unramified. We fix an
isomorphism $\iota_{l}:\mathbb{C}\xrightarrow{\sim}\overline{\mathbb{Q}}_{l}$
which induces a place $\lambda$ in $E$ over $l$. We fix the following datum.

  * •

A finite set $\Sigma_{\mathrm{min}}$ of places of $\mathbb{Q}$ contained in
$\Sigma_{\pi}$;

  * •

A finite set $\Sigma_{\mathrm{lr}}$ of places of $\mathbb{Q}$ which is
disjoint from $\Sigma_{\mathrm{min}}$ and contained in $\Sigma_{\mathrm{lr}}$;

  * •

A finite set of places $\Sigma$ containing
$\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}$ which is away from $l$.

  * •

We have, by Construction 3.3 $(2)$, a homomorphism

$\phi_{\pi}:\mathbb{T}^{\Sigma}\rightarrow\mathcal{O}_{E}.$

and its $\lambda$-adic avatar
$\phi_{\pi,\lambda}:\mathbb{T}^{\Sigma}\rightarrow\mathcal{O}_{\lambda}$ for
the valuation ring $\mathcal{O}_{\lambda}\subset E_{\lambda}$.

  * •

We can associate to it a Galois representation

$\rho_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(E_{\lambda})$

which we assume is residually absolutely irreducible as in Assumption 8.4.

  * •

Let $p$ be a prime which is level raising special of length $m$ for $\pi$ in
the sense of Definition 3.2.

  * •

Let $q$ be a prime such that $l\nmid q^{2}-1$. Suppose that $\pi$ is of type
$\mathrm{II}\mathrm{a}$ and $q$ is contained in $\Sigma_{\mathrm{lr}}$. In
this case $\pi_{q}$ admits Jacquet-Langlands transfer to a representation of
the group $\mathrm{GU}_{2}(\mathrm{D}^{\prime})$ where $\mathrm{D}^{\prime}$
which is the quaternion division algebra over $\mathbb{Q}_{q}$.

  * •

We choose the level $\mathrm{K}_{\mathrm{Pa}}$ and $\Sigma_{\mathrm{Pa}}$ as
in Construction 8.7 for the Shimura varieity
$\mathrm{Sh}(\mathrm{B},\mathrm{K}_{\mathrm{Pa}})$ whose integral model is
$\mathrm{X}_{\mathrm{Pa}}(\mathrm{B})$ and whose special fiber is
$\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B})$; and choose the level
$\mathrm{K}_{\mathrm{H}}$ and $\Sigma_{\mathrm{H}}$ as in Construction 8.8 for
the Shimura set $\mathrm{Sh}(\overline{\mathrm{B}},\mathrm{K}_{\mathrm{H}})$
which is also denoted by $\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})$.

  * •

We introduced two ideals

(9.1)
$\displaystyle\mathfrak{m}=\mathbb{T}^{\Sigma\cup\\{p\\}}\cap\ker(\mathbb{T}^{\Sigma}\xrightarrow{\phi_{\pi,\lambda}}\mathcal{O}_{\lambda}\rightarrow\mathcal{O}_{\lambda}/\lambda)$
$\displaystyle\mathfrak{n}=\mathbb{T}^{\Sigma\cup\\{p\\}}\cap\ker(\mathbb{T}^{\Sigma}\xrightarrow{\phi_{\pi,\lambda}}\mathcal{O}_{\lambda}\rightarrow\mathcal{O}_{\lambda}/\lambda^{m})$

as in Construction 3.3 $(3)$ and we assume that $\mathfrak{m}$ is generic
level raising at $p$ in the sense of 7.8.

######  Construction 9.2.

We define the following map

$\nabla:\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$

by sending

$(x,y)\rightarrow(\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1))x-(p+1)\mathrm{T}_{20}y$

for
$(x,y)\in\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$.

We also write

$\nabla_{\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$

for the localization of $\nabla$ at $\mathfrak{m}$

Note that the composition of the level raising matrix

$\mathcal{T}_{\mathrm{lr},\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$

with the map $\nabla_{\mathfrak{m}}$ is given by sending
$(x,y)\rightarrow\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}\cdot
x$ for
$(x,y)\in\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$
where we have

$\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}=(\mathrm{T}^{-1}_{p,0}\mathrm{T}_{p,1}+(p+1)(p^{2}+1))^{2}-(p+1)^{2}\mathrm{T}_{20}\circ\mathrm{T}_{02}$

by Proposition 6.10.

######  Proposition 9.3.

Let $p$ be a level raising special prime for $\pi$ and $\mathfrak{m}$ be the
maximal ideal as above. We have a surjective morphism

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}})\twoheadrightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\mathrm{B})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$

whose kernel can be identified with the torsion part of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.

###### Proof.

Note that, by Proposition 5.4, we have an isomorphism

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}})\xrightarrow{\sim}\mathrm{Coker}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}.$

On the other hand, by Proposition 7.9 we have an isomorphism

$(\mathrm{inc}^{\\{0\\}}_{!,\mathfrak{m}}+\mathrm{inc}^{\\{2\\}}_{!,\mathfrak{m}}):\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$

and a surjection

$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}}):\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$

whose kernel is given by the torsion part of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.

It follows that we have a surjection

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}})\rightarrow\frac{\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}}{\mathcal{T}_{\mathrm{lr},\mathfrak{m}}(\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}})}.$

The kernel of this map is given by

$\frac{\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))^{\mathrm{tor}}_{\mathfrak{m}}}{\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}\cap\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))^{\mathrm{tor}}_{\mathfrak{m}}}.$

Composing this with the map
$\nabla_{\mathfrak{m}}:\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$,
we therefore obtain a map

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}})\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}.$

which is surjective by Nakayama’s lemma.

It remains to show that
$\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}\cap\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))^{\mathrm{tor}}_{\mathfrak{m}}$
is zero. By [Nek07, Proposition 4.2.2(1)], the group
$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(E_{\lambda})(1))_{\mathfrak{m}})$
vanishes. It follows that the $\mathcal{O}_{\lambda}$-rank of
$\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}$ is
the same as the $\mathcal{O}_{\lambda}$-rank of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.

Next, using the excision exact sequences, we have isomorphisms

(9.4)
$\displaystyle\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))\xrightarrow{\sim}\mathrm{H}^{4}(\overline{\mathrm{X}}^{\mathrm{sm}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))$
$\displaystyle\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}^{\mathrm{sm}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))\xrightarrow{\sim}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))$

where
$\overline{\mathrm{X}}^{\mathrm{sm}}_{\mathrm{Pa}}(\mathrm{B})=\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B})-\overline{\mathrm{X}}^{\mathrm{sing}}_{\mathrm{Pa}}(\mathrm{B})$,
using the fact that the singular locus
$\overline{\mathrm{X}}^{\mathrm{sing}}_{\mathrm{Pa}}(\mathrm{B})$ is zero
dimensional. Therefore Poincare duality for
$\overline{\mathrm{X}}^{\mathrm{sm}}_{\mathrm{Pa}}(\mathrm{B})$ implies that
we have

$\dim_{E_{\lambda}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))_{\mathfrak{m}}=\dim_{E_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))_{\mathfrak{m}}.$

Since the source
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
of $\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}$
is free over $\mathcal{O}_{\lambda}$, the intersection

$\mathrm{Im}(\alpha\circ\mathrm{N}^{-1}_{\Sigma}\circ\beta)_{\mathfrak{m}}\cap\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))^{\mathrm{tor}}_{\mathfrak{m}}$

is trivial. The proposition is proved. ∎

Now we are ready to state and prove our main result on arithmetic level
raising for the quaternionic Shimura variety studied in this article. We first
recall a list of running assumptions below.

  * •

We assume that
$\mathrm{H}^{i}_{(\mathrm{c})}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}=0$
for $i\neq 3$, and that

$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}\xrightarrow{\sim}\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}$

is a finite free $\mathcal{O}_{\lambda}$ module. This is Assumption 7.6.

  * •

We assume that for $\pi$, the associated Galois representation

$\rho_{\pi,\lambda}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(E_{\lambda})$

is residually absolutely irreducible. This is Assumption 8.4.

######  Theorem 9.5 (Arithmetic level raising for $\mathrm{GSp}_{4}$).

Suppose that $\pi$ is a cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ of general type. We assume that
the assumptions in 7.6, 8.4 recalled above are effective. Let $p$ be a level
raising special prime for $\pi$ of depth $m$. We assume further that

  1. (1)

$l\geq 3$ and $l\nmid p^{2}-1$;

  2. (2)

$\overline{\rho}_{\pi,\lambda}$ is rigid for
$(\Sigma_{\mathrm{min}},\Sigma_{\mathrm{lr}})$ as in Definition 8.6;

  3. (3)

The image of $\overline{\rho}_{\pi,\lambda}(\mathrm{G}_{\mathbb{Q}})$ contains
$\mathrm{GSp}_{4}(\mathbb{F}_{l})$;

  4. (4)

$\mathfrak{m}$ is in the support of
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$ and is
generic level raising.

Then following holds.

  1. (1)

There is an isomorphism

$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr}}.$

  2. (2)

The above isomorphism induces an isomorphism

$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}/\lambda^{m}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathcal{O}_{\lambda}/\lambda^{m}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}.$

###  9.1. Proof of Theorem 9.5

Consider the universal deformation problem

$\mathcal{S}^{\ast}=(\overline{\rho}_{\pi,\lambda},\psi,\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p\\}\cup\\{l\\},\\{\mathcal{D}_{v}\\}_{v\in\Sigma_{\mathrm{min}}\cup\Sigma_{\mathrm{lr}}\cup\\{p\\}\cup\\{l\\}})$

where $\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$. Recall that
$\mathrm{Spf}\phantom{.}\mathrm{R}^{\mathrm{ram}}$ and
$\mathrm{Spf}\phantom{.}\mathrm{R}^{\mathrm{unr}}$ are subspaces of
$\mathrm{Spf}\phantom{.}\mathrm{R}^{\mathrm{mix}}$. Thus we have surjections

$\displaystyle\mathrm{R}^{\mathrm{mix}}\twoheadrightarrow\mathrm{R}^{\mathrm{ram}}$
$\displaystyle\mathrm{R}^{\mathrm{mix}}\twoheadrightarrow\mathrm{R}^{\mathrm{unr}}.$

we define
$\mathrm{R}^{\mathrm{cong}}=\mathrm{R}^{\mathrm{unr}}\otimes_{\mathrm{R}^{\mathrm{mix}}}\mathrm{R}^{\mathrm{ram}}$.
We have a universal lifting

$\rho^{\mathrm{mix}}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(\mathrm{R}^{\mathrm{mix}}).$

It also induces two representations

$\displaystyle\rho^{\mathrm{ram}}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(\mathrm{R}^{\mathrm{ram}})$
$\displaystyle\rho^{\mathrm{unr}}:\mathrm{G}_{\mathbb{Q}}\rightarrow\mathrm{GSp}_{4}(\mathrm{R}^{\mathrm{unr}})$

by the above surjections.

Denote by $\mathrm{P}_{\mathbb{Q}_{p}}$ be the maximal closed subgroup of the
inertia subgroup $\mathrm{I}_{\mathbb{Q}_{p}}\subset G_{\mathbb{Q}_{p}}$ of
pro-order coprime to $l$. Then
$G_{\mathbb{Q}_{p}}/\mathrm{P}_{\mathbb{Q}_{p}}\cong
t^{\mathbb{Z}_{l}}\rtimes\phi^{\widehat{\mathbb{Z}}}_{p}$ is a $p$-tame group.
By definition, $\rho^{\mathrm{mix}}$ is trivial on
$\mathrm{P}_{\mathbb{Q}_{p}}$. Let $\overline{\mathrm{v}}_{0}$ and
$\overline{\mathrm{v}}_{1}$ be eigenvectors in
$k^{4}=(\mathcal{O}_{\lambda}/\lambda)^{4}$ for
$\rho^{\mathrm{mix}}(\phi^{2}_{p})$ with eigenvalues $p^{-2}$ and $p^{-4}$,
respectively. By Hensel’s lemma, they lift to $\mathrm{v}_{0}$ and
$\mathrm{v}_{1}$ in $(\mathrm{R}^{\mathrm{mix}})^{\oplus 4}$ for
$\rho^{\mathrm{mix}}(\phi^{2}_{p})$ with eigenvalues $\mathrm{s}_{0}$ and
$\mathrm{s}_{1}$ lifting $p^{-2}$ and $p^{-4}$. Let
$\mathrm{x}\in\mathrm{R}^{\mathrm{mix}}$ be the unique element such that
$\rho^{\mathrm{mix}}(t)\mathrm{v}_{1}=\mathrm{x}\mathrm{v}_{0}+\mathrm{v}_{1}$.
It follows from the relation

$\rho^{\mathrm{mix}}(\phi^{2}_{p})\rho^{\mathrm{mix}}(t)\rho^{\mathrm{mix}}(\phi^{-2}_{p})=\rho^{\mathrm{mix}}(t)^{p^{2}}$

that $\mathrm{x}(\mathrm{s}-p^{-2})=0$. By the definitions of
$\mathrm{R}^{\ast}$ for $\ast=\\{\mathrm{ram},\mathrm{unr},\mathrm{mix}\\}$,
we have

$\displaystyle\mathrm{R}^{\mathrm{unr}}=\mathrm{R}^{\mathrm{mix}}/(\mathrm{x})$
$\displaystyle\mathrm{R}^{\mathrm{ram}}=\mathrm{R}^{\mathrm{mix}}/(\mathrm{s}_{0}-p^{-2})$
$\displaystyle\mathrm{R}^{\mathrm{cong}}=\mathrm{R}^{\mathrm{mix}}/(\mathrm{x},\mathrm{s}_{0}-p^{-2}).$

By Theorem 8.9,
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is a free over $\mathbf{T}^{\mathrm{unr}}_{\mathfrak{m}}$ and we have an
isomorphism $\mathrm{R}^{\mathrm{unr}}\cong\mathbf{T}_{\mathfrak{m}}$. Thus
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is a free $\mathrm{R}^{\mathrm{unr}}$-module of rank $d_{\mathrm{unr}}$. Note
that the characteristic polynomial of $\rho^{\mathrm{mix}}(\phi^{2}_{p})$ can
written as
$(\mathrm{X}-\mathrm{s}_{0})(\mathrm{X}-p^{-6}\mathrm{s}^{-1}_{0})\mathrm{Q}(\mathrm{X})$
where $\mathrm{Q}(\mathrm{X})$ is a polynomial in
$\mathrm{R}^{\mathrm{unr}}[\mathrm{X}]$ whose reduction in
$\mathcal{O}_{\lambda}/\lambda[\mathrm{X}]=k[\mathrm{X}]$ does not contain
$p^{-2}$ and $p^{-4}$ as its roots. By Proposition 6.15 and the equations

$\displaystyle(\alpha_{p}+p^{3}\alpha^{-1}_{p}-p(p+1))(\alpha_{p}+p^{3}\alpha^{-1}_{p}+p(p+1))=\alpha^{2}_{p}+p^{6}\alpha^{-2}_{p}-p^{2}(p^{2}+1),$
$\displaystyle(\beta_{p}+p^{3}\beta^{-1}_{p}-p(p+1))(\beta_{p}+p^{3}\beta^{-1}_{p}+p(p+1))=\beta^{2}_{p}+p^{6}\beta^{-2}_{p}-p^{2}(p^{2}+1),$

we have

$(\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}})\cdot\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}=(\mathrm{s}_{0}-p^{-2})\cdot\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}.$

Thus we have

$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}\cong\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\otimes_{\mathrm{R}^{\mathrm{unr}}}\mathrm{R}^{\mathrm{cong}}$

and it follows that
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$
is a free $\mathrm{R}^{\mathrm{cong}}$-module of rank $d_{\mathrm{unr}}$.

Since we have a surjection

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\twoheadrightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\mathrm{B})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$

by Proposition 9.3, we have $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$ is non-
zero. Moreover
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
is a free $\mathrm{R}^{\mathrm{ram}}$-module by Theorem 8.9 $(2)$, say of rank
$d_{\mathrm{ram}}$. Consider the $\mathrm{R}^{\mathrm{ram}}$-module

$\mathrm{H}:=\operatorname{Hom}_{\mathrm{G}_{\mathbb{Q}}}((\mathrm{R}^{\mathrm{ram}})^{\oplus
4},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})$

where $\mathrm{G}_{\mathbb{Q}}$ acts on $(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}$ via $\rho^{\mathrm{ram}}$ which is free of rank $d_{\mathrm{ram}}$. It
follows this that we have an isomorphism

$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}))_{\mathfrak{m}}\xrightarrow{\sim}\mathrm{H}\otimes_{\mathrm{R}^{\mathrm{ram}}}(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}.$

It follows from that

$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathrm{H}\otimes_{\mathrm{R}^{\mathrm{ram}}}\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}(1))$

We still denote by $\mathrm{v}_{0}$ and $\mathrm{v}_{1}$ the projections of
$\mathrm{v}_{0}$ and $\mathrm{v}_{1}$ in $(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}$ of $\mathrm{G}_{\mathbb{Q}}$-modules. Then a straightforward computation
shows that

$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},(\mathrm{R}^{\mathrm{ram}})^{\oplus
4}(1))\xrightarrow{\sim}\mathrm{R}^{\mathrm{ram}}\mathrm{v}_{0}/\mathrm{x}\mathrm{v}_{0}\cong\mathrm{R}^{\mathrm{cong}}.$

Hence we have

$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})$

is a free $\mathrm{R}^{\mathrm{cong}}$-module of rank $d_{\mathrm{ram}}$.

######  Lemma 9.6.

We have an equality

$d_{\mathrm{unr}}=d_{\mathrm{ram}}.$

###### Proof.

Let
$\eta^{\mathrm{unr}}\in\mathrm{Spec}\phantom{.}\mathrm{R}^{\mathrm{unr}}[1/l](\overline{\mathbb{Q}}_{l})$
be in the support of
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
which gives rise to an automorphic representation $\pi^{\mathrm{unr}}$ of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ such that
$\overline{\rho}_{\pi,\lambda}\otimes_{k}\overline{\mathbb{F}}_{l}$ and
$\rho_{\pi^{\mathrm{unr}},\iota_{l}}$ are residually isomorphic. Then we have

$d_{\mathrm{unr}}=\dim_{\overline{\mathbb{Q}}_{l}}\overline{\mathbb{Q}}_{l}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})][\iota_{l}\phi_{\pi^{\mathrm{unr}}}].$

Similarly, let
$\eta^{\mathrm{ram}}\in\mathrm{Spec}\phantom{.}\mathrm{R}^{\mathrm{ram}}[1/l](\overline{\mathbb{Q}}_{l})$
be in the support of
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
which gives rise to an automorphic representation $\pi^{\mathrm{ram}}$ of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ such that
$\rho_{\pi^{\mathrm{ram}},\iota_{l}}$ is residually isomorphic to
$\overline{\rho}_{\pi,\lambda}\otimes_{k}\overline{\mathbb{F}}_{l}$. Then we
have

$4d_{\mathrm{ram}}=\dim_{\overline{\mathbb{Q}}_{l}}\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\overline{\mathbb{Q}}_{l}))[\iota_{l}\phi_{\pi^{\mathrm{ram}}}].$

Note that $\pi^{\mathrm{unr}}$ and $\pi^{\mathrm{ram}}$ are necessarily of
general type. By considering the transfer of $\pi^{\mathrm{unr}}$ and
$\pi^{\mathrm{ram}}$ to $\mathrm{GL}_{4}(\mathbb{A}_{\mathbb{Q}})$, we have
$d_{\mathrm{ram}}=d_{\mathrm{unr}}$ by [LTXZZ, Lemma 6.4.2]. ∎

_Proof of Theorem 9.5._ Now we can prove our main theorem on arithmetic level
raising. We have proved that we have a surjection

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\twoheadrightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\mathrm{B})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$

where the target is free of rank $d_{\mathrm{unr}}$ over
$\mathrm{R}^{\mathrm{cong}}$. On the other hand, we have an injection

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\hookrightarrow\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1))_{\mathfrak{m}})$

and the target is free of rank $d_{\mathrm{ram}}$ over
$\mathrm{R}^{\mathrm{cong}}$. Since $d_{\mathrm{unr}}=d_{\mathrm{ram}}$, we
have the desired isomorphism

(9.7)
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr}}.$

This finishes the proof of the first part of the theorem.

For the second part, by the freeness of
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
as a $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$-module. It follows that the
natural map

$\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})/\mathfrak{n}\xrightarrow{\sim}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathfrak{n})$

is an isomorphism. We also have the short exact sequence as in Proposition 6.1

$\displaystyle 0$
$\displaystyle\rightarrow\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\rightarrow\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda})(1))_{\mathfrak{m}})$
$\displaystyle\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathrm{F}_{-1}\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}\rightarrow
0$

which is split by the $\mathrm{G}_{\mathbb{F}_{p^{2}}}$-action. And it follows
that

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})/\mathfrak{n}\xrightarrow{\sim}\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathfrak{n})$

is an isomorphism. Since we have shown above that there is an isomorphism

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}),$

it follows that

$\mathrm{H}^{1}_{\mathrm{sing}}({\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})/\mathfrak{n}\xrightarrow{\sim}\mathrm{H}^{1}_{\mathrm{sing}}({\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathfrak{n}).$

On the other hand, we have
$\mathbf{T}^{\mathrm{ram}}/\mathfrak{n}\cong\mathcal{O}_{\lambda}/\lambda^{m}$
and thus

$\mathrm{H}^{1}_{\mathrm{sing}}({\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}/\mathfrak{n})\cong\mathrm{H}^{1}_{\mathrm{sing}}({\mathbb{Q}_{p^{2}}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}/\lambda^{m}(1)))_{\mathfrak{m}}$

by the freeness of
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
as $\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}$-module. Finally the second claim
follows from the observation that

$\displaystyle\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr}}$
$\displaystyle\cong\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\otimes_{\mathrm{R}^{\mathrm{unr}}}\mathrm{R}^{\mathrm{cong}}$
$\displaystyle\cong\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\otimes_{\mathrm{R}^{\mathrm{unr}}}\mathbf{T}^{\mathrm{ram}}_{\mathfrak{m}}/\mathfrak{n}$
$\displaystyle\cong\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\otimes\mathcal{O}_{\lambda}/\lambda^{m}$

and the isomorphism 9.7. $\square$

######  Corollary 9.8.

We maintain the assumptions in Theorem 9.5.

  1. (1)

$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda})_{\mathfrak{m}}$
is a torsion free $\mathcal{O}_{\lambda}$-module. In particular, the map from
Construction 6.4

$(\mathrm{inc}^{\ast}_{\\{0\\},\mathfrak{m}},\mathrm{inc}^{\ast}_{\\{2\\},\mathfrak{m}}):\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\rightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{0\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}\oplus\mathcal{O}_{\lambda}[\mathrm{Z}_{\\{2\\}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$

is an isomorphism.

  2. (2)

$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda})_{\mathfrak{m}}$
and
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda})_{\mathfrak{m}}$
are both torsion free $\mathcal{O}_{\lambda}$-modules.

###### Proof.

Recall we have proved that the kernel of the natural map

$\mathrm{F}_{-1}\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p^{2}}},\mathrm{H}^{n}_{c}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}})\twoheadrightarrow\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\mathrm{B})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr},\mathfrak{m}}$

can be identified with the torsion part of
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$.
We have shown in the above proof that this surjection is an isomorphism and
the result follows.

For the second part, we have an injection

$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}\hookrightarrow\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(2)))_{\mathfrak{m}}$

and
$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(2)))_{\mathfrak{m}}$
is a torsion free $\mathcal{O}_{\lambda}$-module by our assumption. Thus
$\mathrm{H}^{3}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$
is torsion free.

Next, we show the cohomology group
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is torsion free. We consider the long exact sequence

$\cdots\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}\rightarrow\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}\rightarrow\cdots$

induced by the short exact sequence

$0\rightarrow\mathcal{O}_{\lambda}\rightarrow\mathcal{O}_{\lambda}\rightarrow\mathcal{O}_{\lambda}/\lambda\rightarrow
0.$

We need to show the injection

$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}/\lambda\hookrightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}$

is in fact surjective. For this, note that we have established that
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}$
is torsion free and thus
$\dim_{k}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}/\lambda=\dim_{E_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))_{\mathfrak{m}}$.
But we have

$\displaystyle\dim_{k}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}=\dim_{k}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(2))_{\mathfrak{m}};$
$\displaystyle\dim_{E_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))_{\mathfrak{m}}=\dim_{E_{\lambda}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))_{\mathfrak{m}}.$

by the Poincare duality and (9.4). By the considering the same long exact
sequence and the reasoning as above but applied to
$\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}}$,
we have

$\dim_{k}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(2))_{\mathfrak{m}}=\dim_{E_{\lambda}}\mathrm{H}^{4}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(2))_{\mathfrak{m}}$

and hence

$\dim_{k}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}=\dim_{E_{\lambda}}\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),E_{\lambda}(1))_{\mathfrak{m}}.$

It follows that
$\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}(1))_{\mathfrak{m}}/\lambda\hookrightarrow\mathrm{H}^{2}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathcal{O}_{\lambda}/\lambda(1))_{\mathfrak{m}}$
is in fact surjective and we are done. ∎

###  9.2. Level raising for $\mathrm{GSp}_{4}$

Let $\pi$ be a cuspidal automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ of general type with weight
$(3,3)$ and trivial central character. Let $p$ be a level raising special
prime for $\pi$ of length $m$. From the arithmetic level raising theorem
proved in the last subsection, we can deduce a level raising theorem for $\pi$
which can be applied to deduce level raising theorems in [Wang22b]. This is
the content of this subsection.

Suppose the component $\pi_{q}$ of $\pi$ at the prime $q$ is of type
$\mathrm{I}\mathrm{I}\mathrm{a}$. Then $\pi$ admits a global Jacquet–Langlands
transfer to an automorphic representation of
$\mathbf{G}(\overline{\mathrm{B}})(\mathbb{A}_{\mathbb{Q}})$. In particular
the maximal ideal $\mathfrak{m}$ associated to $\pi$ in the Construction 3.3
$(3)$ lies in the support of
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]$ as a
$\mathbb{T}^{\Sigma\cup\\{p\\}}$-module.

######  Theorem 9.9.

Let $\pi$ be an automorphic representation of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ as above. Suppose that $\pi$
satisfy all the assumptions in Theorem 9.5. Let $p$ be a level raising special
prime for $\pi$ of length $m$.

Then there exists an automorphic representation $\pi^{\prime}$ of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ of general type with weight
$(3,3)$ and trivial central character such that

  1. (1)

the component $\pi^{\prime}_{p}$ at $p$ is of type
$\mathrm{I}\mathrm{I}\mathrm{a}$;

  2. (2)

we have an isomorphism of the residual Galois representation

$\overline{\rho}_{\pi,\lambda}\cong\overline{\rho}_{\pi^{\prime},\lambda}.$

In this case we will say $\pi^{\prime}$ is a level raising of $\pi$.

###### Proof.

As remark in the above, the localization
$\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is non-zero. By Theorem 9.5, we have an isomorphism

$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbb{Q}_{p^{2}},\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}})\xrightarrow{\sim}\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{H}}(\overline{\mathrm{B}})]_{\mathfrak{m}}/\mathrm{det}\phantom{.}\mathcal{T}_{\mathrm{lr}}.$

In particular
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
is non-zero. Since
$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}$
is a free $\mathrm{R}^{\mathrm{ram}}$-module, we can find a point
$\eta^{\mathrm{ram}}\in\mathrm{Spec}\phantom{.}\mathrm{R}^{\mathrm{ram}}[1/l]$
in the support of

$\mathrm{H}^{3}_{\mathrm{c}}(\overline{\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B}),\mathrm{R}\Psi(\mathcal{O}_{\lambda}(1)))_{\mathfrak{m}}.$

Then we can find an automorphic representation $\pi^{\prime}$ of
$\mathrm{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ which satisfy the isomorphism

$\overline{\rho}_{\pi,\lambda}\cong\overline{\rho}_{\pi^{\prime},\lambda}.$

In particular $\pi^{\prime}$ is of general type as
$\overline{\rho}_{\pi^{\prime},\lambda}$ is absolutely irreducible. By the
Picard-Lefschetz formula applied to the Shimura variety
${\mathrm{X}}_{\mathrm{Pa}}(\mathrm{B})$ , we see that the monodromy of the
associated Weil-Deligne representation of $\rho_{\pi^{\prime},\lambda}$ is
necessarily is of the form

$\begin{pmatrix}0&&&\\\ &0&1&\\\ &&0&\\\ &&&0\\\ \end{pmatrix}$

up to conjugation. Since $\pi^{\prime}$ is of general type and hence tempered
at $p$, it follows that $\pi^{\prime}$ is of type
$\mathrm{I}\mathrm{I}\mathrm{a}$ by the local Langlands correspondence for
non-supercuspidal representations, see for example [Sch05, Table 2]. The
theorem is proved. ∎

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