# Level lowering for GSp(4) and vanishing cycles on Siegel threefolds

Haining Wang  Department of Mathematics, McGill University, 805 Sherbrooke St
W, Montreal, QC H3A 0B9, Canada<EMAIL_ADDRESS>
###### Abstract.

In this article we prove some level lowering results for Siegel modular forms
of degree $2$ with paramodular level structure by adapting a method of Ribet
in his proof of the epsilon conjecture. The proof relies on the description of
the supersingular locus of a quaternionic Siegel threefold which is obtained
by the author in a previous work. The heart of the proof is the so called
$p,q$ switch trick introduced by Ribet where $p,q$ are two distinct prime
numbers and it relies on the comparison between the vanishing cycle at $q$ on
the paramodular Siegel threefold and the vanishing cycle at $p$ on the
quaternionic Siegel threefold.

###### Key words and phrases:

_Shimura varieties, Siegel modular forms, Vanishing cycles_

###### 2000 Mathematics Subject Classification:

Primary 11G18, Secondary 20G25

###### Contents

  1. 1 Introduction
  2. 2 Review of nearby and vanishing cycles
  3. 3 Automorphic and Galois representations for $\mathrm{GSp}_{4}$
  4. 4 Siegel threefold of paramodular level
  5. 5 Mazur’s principle for paramodular Siegel modular forms
  6. 6 Quaternionic Siegel threefold of paramodular level
  7. 7 Ribet’s principle for paramodular Siegel modular forms

##  1\. Introduction

###  1.1. Motivations

In this article we provide two level lowering principles for Galois
representations attached to a cuspidal automorphic representation of the
symplectic similitude group $\mathrm{GSp}_{4}$ of degree $4$ that has
paramodular fixed vectors. We refer to these as the Mazur’s principle and
Ribet’s principle. These terminologies come from the celebrated work of Ribet
on Serre’s epsilon conjecture [Rib90], [Rib91] and Mazur’s previous work. To
motivate our results, we first recall their results in the $\mathrm{GL}_{2}$
setting. Let

$\rho:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\mathrm{GL}_{2}(k)$

be an irreducible representation of the absolute Galois group over
$\mathbb{Q}$ valued in a finite field $k$ of characteristic $l\geq 3$. Suppose
that $\rho$ is modular of level $N$ which means $\rho$ comes as the reduction
of an $l$-adic representation

$\rho_{\pi,l}:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\mathrm{GL}_{2}(E_{\lambda})$

attached to a weight $2$ newform of level $\Gamma_{0}(N)$ whose associated
automorphic representation which we denote by $\pi$. Here $E_{\lambda}$ is a
suitable finite extension over $\mathbb{Q}_{l}$. Let $p$ be a prime which
exactly divide $N$ and suppose that $\rho$ is unramified at $p$. Then Serre
[Ser87] conjectured that $\rho$ is modular of level $N/p$. This conjecture is
known as the Serre’s epsilon conjecture. The following theorem is the main
result of [Rib90].

######  Theorem (Epsilon conjecture).

Assume $\rho$ is modular of level $N$ and unramified at $p$ with $p\mid\mid
N$. Then $\rho$ is modular of level $N/p$ if $N$ is prime to $l$.

We remark that Ribet is able to prove Fermat’s last theorem using this result
under the assumption that all the elliptic curves over $\mathbb{Q}$ are
modular using Frey’s construction. The proof of the above theorem in fact
breaks into two parts. First one proves the following theorem which is known
as the Mazur’s principle.

######  Theorem (Mazur’s principle).

Assume $\rho$ is modular of level $N$ and unramified at $p$ with $p\mid\mid
N$. Then $\rho$ is modular of level $N/p$ if $N$ is prime to $l$ and
$p\not\equiv 1\mod l$.

Note that by a theorem of Langlands [Lan73], $\rho_{\mid G_{\mathbb{Q}_{p}}}$
is of the form

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

where $\chi$ is the mod $l$ cyclotomic character. The assumption that
$p\not\equiv 1\mod l$ signifies that the two Frobenius eigenvalues of $\rho$
at $p$ are distinct even modulo $l$. One then proves next the following
theorem which we call the Ribet’s principle.

######  Theorem (Ribet’s principle).

Let $p,q$ be two distinct primes which are coprime to $lN$. Assume $\rho$ is
modular of level $pqN$. Assume that $\rho$ is unramified at $p$ but ramified
at $q$. Then $\rho$ is modular of level $qN$.

This result is not quite explicitly proved in [Rib90] but rather it is
formulated and proved in [Rib91]. Combining Mazur’s principle and Ribet’s
principle with level raising results, one can then prove the epsilon
conjecture. The existence of the auxiliary prime $q$ allows Ribet to use the
so-called $p,q$ switch trick. Here the crucial observation is that the
singular locus of a Shimura curve associated to a suitable indefinite
quaternion algebra and the singular locus of a suitable modular curve can be
parametrized by a same discrete Shimura set. We make this more precise now.
Let $B=B_{pq}$ be an indefinite quaternion algebra with discriminant $pq$. Let
$X^{B}_{\Gamma_{0}(N)}$ be the Shimura curve associated to $B$ and level
$\Gamma_{0}(N)$ and let $X_{\Gamma_{0}(pq)}$ be the usual modular curve with
$\Gamma_{0}(pqN)$ level structure. Consider the special fiber
$X^{B}_{\Gamma_{0}(N),\bar{\mathbb{F}}_{p}}$ of $X^{B}_{\Gamma_{0}(N)}$ at
$p$. The singular locus of $X^{B}_{\Gamma_{0}(N),\bar{\mathbb{F}}_{p}}$ can be
identified with the double coset

$B^{\prime\times}(\mathbb{Q})\backslash
B^{\prime\times}(\mathbb{A}_{f})/K^{\prime}(pq)K_{0}(N)$

where $B^{\prime}=B_{q\infty}$ is the definite quaternion algebra with
discriminant $q$, $K_{0}(N)=\hat{\Gamma}_{0}(N)$ is the completion of
$\Gamma_{0}(N)$ and
$K^{\prime}(pq)={\rm{Iw}}(p)\mathcal{O}^{\times}_{B^{\prime}_{q}}$ with
${\rm{Iw}}(p)$ the Iwahori subgroup of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ and
$\mathcal{O}^{\times}_{B^{\prime}_{q}}$ the maximal order of $B^{\prime}_{q}$.
On the other hand, the singular locus of
$X_{\Gamma_{0}(pq),\bar{\mathbb{F}}_{q}}$ at the prime $q$ can be identified
with the same double coset. Therefore the vanishing cycles on
$X^{B}_{\Gamma_{0}(N),\bar{\mathbb{F}}_{p}}$ and on
$X_{\Gamma_{0}(pq),\bar{\mathbb{F}}_{q}}$ can be identified. The starting
point of this article is the observation that the same phenomenon happens for
the Siegel modular threefold with paramodular level structure and its
quaternionic analogue.

###  1.2. Main results

Let $\pi$ be an automorphic reprsentation of $\mathrm{GSp}_{4}$ over
$\mathbb{Q}$ which we assume through out this article that is non-endoscopic
and non-CAP. Suppose $\pi_{\infty}$ is in the holomorphic discrete series and
cohomological of weights $a\geq b\geq 0$. Then one can attach a Galois
representation

$\rho_{\pi,l}:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\mathrm{GSp}_{4}(E_{\lambda})$

for a finite extension $E_{\lambda}$ over $\mathbb{Q}_{l}$. We will denote by

$\bar{\rho}_{\pi,l}:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\mathrm{GSp}_{4}(k)$

its associated residual representation valued in the residue field $k$ of
$E_{\lambda}$. We will assume that the residual characteristic $l>a+b+4$. In
fact this Galois representation is realized in the cohomology of the Siegel
threefold in degree three. That is it is realized in
$\operatorname{Hom}_{\mathrm{GSp}_{4}(\mathbb{A}_{f})}(\pi_{f},\mathrm{H}^{3}_{!}(X_{\bar{\mathbb{Q}}}),\mathbb{V}))$
where

$\mathrm{H}^{3}_{!}(X_{\bar{\mathbb{Q}}}),\mathbb{V})=\varinjlim_{U}\mathrm{H}^{3}_{!}(X_{U,\bar{\mathbb{Q}}},\mathbb{V})$

is the interior cohomology of the Siegel threefolds indexed by the open
compact subgroups $U$ of $\mathrm{GSp}_{4}(\mathbb{A}_{f})$ and $\mathbb{V}$
is a suitable $l$-adic étale automorphic sheaf which we will make it more
precise in the main body of the article. Recall that there are two maximal
parahoric subgroups in $\mathrm{GSp}_{4}(\mathbb{Q}_{p})$ up to isomorphism,
one is known as the hyperspecial subgroup which we will denote by $H$ and the
other is known as the paramodular subgroup which we denote by $K(p)$. Suppose
that the local component of $\pi$ at $p$ is para-spherical which means that
$\pi^{K(p)}_{p}$ is non-zero. In this case, $\pi$ is ramified at $p$. We will
let
$U=K(p)U^{p}\subset\mathrm{GSp}_{4}(\mathbb{Q}_{p})\mathrm{GSp}_{4}(\mathbb{A}^{(p)}_{f})$
be a neat open compact subgroup such that $\pi^{U}\neq 0$. Let
$X_{K(p)}=X_{U}$ be the Siegel threefold with paramodular level at $p$. Let
$\mathbb{T}_{p}$ be a suitable Hecke algebra which we will make it more
explicit in the main body of this article and let $\mathfrak{m}$ be the
maximal ideal in $\mathbb{T}_{p}$ corresponding to $\bar{\rho}_{\pi,l}$. Then
our first result is the analogue of Mazur’s principle for holomorphic Siegel
modular forms of paramodular level.

######  Theorem 1.

Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GSp}_{4}$ that
is cohomological which is non-CAP and non-endoscopic whose component at
infinity $\pi_{\infty}$ is a holomorphic discrete series of Harish Chandra
parameter $(a,b,-a-b+3)$ with $a\geq b\geq 0$ and such that $l>a+b+4$. Let $p$
be a prime distinct from $l$ and such that $p\not\equiv 1\mod l$. Let
$U=K(p)U^{p}\subset\mathrm{GSp}_{4}(\mathbb{Q}_{p})\mathrm{GSp}_{4}(\mathbb{A}^{(p)}_{f})$
be a neat open compact subgroup such that $\pi^{U}\neq 0$. We assume that
$\pi_{p}$ is ramified. Suppose the residual Galois representation
$\bar{\rho}_{\pi,l}$ satisfies the following assumptions

  * •

$\bar{\rho}_{\pi,l}$ is unramified at $p$;

  * •

$\bar{\rho}_{\pi,l}$ is irreducible;

  * •

$\mathrm{H}^{3}_{!}(X_{K(p),\bar{\mathbb{Q}}},\mathbb{V})_{\mathfrak{m}}\otimes
k$ is semisimple as a Galois module.

Then there exists a cuspidal automorphic representation $\pi^{\prime}$ with
the same type as $\pi$ at $\infty$ such that
$\bar{\rho}_{\pi,l}\cong\bar{\rho}_{\pi^{\prime},l}$ and $\pi^{\prime}_{p}$ is
unramified.

We remark that in a previous work [VH19], Van Hoften was able to prove another
version of Mazur’s principle in this setting. Our assumptions are quite
different from his. In particular we are not relying on the vanishing results
of Lan-Suh [LS13] and thus we allow more general weights for the étale
automorphic sheaf $\mathbb{V}$. In addition, we do not require that the
Frobenius eigenvalues of $\bar{\rho}_{\pi,l}$ at $p$ to be distinct instead we
only assume that $p\not\equiv 1\mod l$. However we introduce the additional
assumption that
$\mathrm{H}^{3}_{!}(X_{K(p),\bar{\mathbb{Q}}},\mathbb{V})_{\mathfrak{m}}\otimes
k$ is semisimple. This assumption is the analogue of the main result of
[BLR91]. However the proof of [BLR91] relies on the fact the representation of
interest is $2$ dimensional and therefore their methods do not generalize
directly to our setting. It is nevertheless an interesting question to study
the semi-simplicity of
$\mathrm{H}^{3}_{!}(X_{K(p),\bar{\mathbb{Q}}},\mathbb{V})_{\mathfrak{m}}\otimes
k$ which we hope to treat in another occasion.

Now let $p,q$ be two distinct primes which are different from $l$. We fix
$U^{pq}$ an open compact subgroup of $\mathrm{GSp}_{4}(\mathbb{A}^{(pq)}_{f})$
which is sufficiently small. Suppose now that for $U=K(p)K(q)U^{pq}$,
$\pi^{U}\neq 0$. We consider the Siegel threefold $X_{K(pq)}=X_{U}$ with
paramodular level at $p$ and $q$ . Our second result is the analogue of
Ribet’s principle for holomorphic Siegel modular forms of paramodular level.

######  Theorem 2.

Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GSp}_{4}$ that
is cohomological which is non-CAP and non-endoscopic whose component at
infinity $\pi_{\infty}$ is a holomorphic discrete series of Harish Chandra
parameter $(a,b,-a-b+3)$ with $a\geq b\geq 0$ such that $l\geq a+b+4$. Let
$p,q$ be two distinct primes different from $l$. Let
$U=K(p)K(q)U^{pq}\subset\mathrm{GSp}_{4}(\mathbb{Q}_{p})\mathrm{GSp}_{4}(\mathbb{Q}_{q})\mathrm{GSp}_{4}(\mathbb{A}^{(p)}_{f})$
be a neat open compact subgroup such that $\pi^{U}\neq 0$. We assume that
$\pi_{p}$ and $\pi_{q}$ are both ramified. Suppose the residual Galois
representation $\bar{\rho}_{\pi,l}$ satisfies the following assumptions

  * •

$\bar{\rho}_{\pi,l}$ is unramified at $p$ and is ramified at $q$;

  * •

$\bar{\rho}_{\pi,l}$ is irreducible;

  * •

$H^{3}_{!}(X_{K(pq),\bar{\mathbb{Q}}},\mathbb{V})_{\mathfrak{m}}\otimes k$ is
a semisimple Galois module.

Then there exists a cuspidal automorphic representation $\pi^{\prime}$ with
the same type as $\pi$ at $\infty$ and such that
$\bar{\rho}_{\pi,l}\cong\bar{\rho}_{\pi^{\prime},l}$ and $\pi^{\prime}_{p}$ is
unramified.

The proof of this theorem relies heavily on the interplay between the geometry
of the Siegel threefold $X_{K(pq)}$ and its quaternionic analogue $X^{B}$.
Here the quaternion algebra $B$ is again the indefinite quaternion algebra
over $\mathbb{Q}$ with discriminant $pq$ and $X^{B}$ is the Shimura variety
associated to the quaternionic unitary group $\mathrm{GU}_{2}(B)$. We will
rely on the description of the supersingular locus of $X^{B}$ in [Wang19a]. In
particular the starting point of this work is the observation that the
singular locus of $X_{K(pq),\bar{\mathbb{F}}_{p}}$ and the singular locus
$X^{B}_{\bar{\mathbb{F}}_{q}}$ are parametrized by the same double coset and
therefore one can identify the vanishing cycles on
$X_{K(pq),\bar{\mathbb{F}}_{p}}$ and the vanishing cycles on
$X^{B}_{\bar{\mathbb{F}}_{q}}$. Also the role of $p,q$ is symmetric in the
study of these two Shimura varieties and this give us the flexibility to
permute the two primes. It is natural to ask if one can upgrade this result to
remove the assumption that there exits a prime $q$ such that $\pi_{q}$ is
ramified and $\bar{\rho}_{\pi,l}$ is ramified at $q$. This seems to rely on
suitable level raising results which are unknown to the author.

Our results seem to be the first to generalize Ribet’s techniques to higher
dimensional Shimura varieties. In the case of Hilbert modular forms, there are
works of Javis [Jar99] and Rajaei [Raj01] which work with Shimura curves over
totally real field. Note that there are also other techniques to prove level
lowering results, notably the work of Skinner-Wiles [SW01]. In the setting of
$\mathrm{GSp}_{4}$, Sorensen [Sor09a] is able to prove a potential level
lowering result using the techniques of [SW01].

###  1.3. Acknowledgement.

This work is completed when the author is a postdoctoral fellow at McGill
university and he would like to thank Henri Darmon and Pengfei Guan for their
generous support. We would like to thank Marc-Hubert Nicole for informing him
the work of Van Hoften [VH19] and discussions about paramodular Siegel
threefold.

##  2\. Review of nearby and vanishing cycles

###  2.1. Nearby and vanishing cycles

Let $S=\mathrm{Spec}(R)$ be the spectrum of a henselian DVR. We choose a
uniformizer $\pi$ of $R$. We denote by $s$ the closed point of $S$ and by
$\eta$ the generic point of $S$. We assume the residue field $k(s)$ at $s$ is
of characteristic $p$. Let $\bar{s}$ be a geometric point of $S$ over $s$ and
$\tilde{S}$ be the localization $S_{(\bar{s})}$ of $S$ at $\bar{s}$. We will
denote by $\tilde{s}$ and $\tilde{\eta}$ the closed and generic point of
$\tilde{S}$. Let $\bar{\eta}$ be a seperable closure of $\tilde{\eta}$. We
summarize the above notations in the following diagram.

$\begin{array}[c]{ccccc}\bar{s}&\rightarrow&\bar{S}&\leftarrow&\bar{\eta}\\\
\downarrow&&\downarrow&&\downarrow\\\
\tilde{s}&\rightarrow&\tilde{S}&\leftarrow&\tilde{\eta}\\\
\downarrow&&\downarrow&&\downarrow\\\
s&\rightarrow&S&\leftarrow&\eta\end{array}$

For a morphsim $f:X\rightarrow S$, we obtain by base change the following maps

$\begin{array}[c]{ccccc}X_{\bar{s}}&\xrightarrow{\bar{i}}&X_{\bar{S}}&\xleftarrow{\bar{j}}&X_{\bar{\eta}}\\\
\downarrow&&\downarrow&&\downarrow\\\
X_{\tilde{s}}&\rightarrow&X_{\tilde{S}}&\leftarrow&X_{\tilde{\eta}}\\\
\downarrow&&\downarrow&&\downarrow\\\
X_{s}&\xrightarrow{i}&X&\xleftarrow{j}&X_{\eta}.\end{array}$

Let $K\in D^{+}(X_{\bar{\eta}},\Lambda)$ with coefficient in
$\Lambda=\mathbb{Z}/l^{m}\mathbb{Z}$ or $\mathbb{Z}_{l}$, then we define the
_nearby cycle complex_ $R\Psi(K)\in D^{+}(X_{\bar{s}},\Lambda)$ by

$R\Psi(K)=\bar{i}^{*}\mathrm{R}\bar{j}_{*}(K|X_{\bar{\eta}}).$

For $K\in D^{+}(X,\Lambda)$, the adjunction map defines the following
distinguished triangle

$K|X_{\bar{s}}\rightarrow R\Psi(K|X_{\bar{\eta}})\rightarrow
R\Phi(K)\rightarrow$

The complex $R\Phi(K)$ is known as the _vanishing cycle complex_ and is the
cone of the previous map. If $f$ is proper, then we have

$R\Gamma(X_{\bar{\eta}},K|X_{\bar{\eta}})=R\Gamma(X_{\bar{s}},R\Psi(K|X_{\bar{\eta}}))$

and in general we always have the following long exact sequence

$\cdots\rightarrow\mathrm{H}^{i}(X_{\bar{s}},K|X_{\bar{s}})\xrightarrow{\rm{sp}}\mathrm{H}^{i}(X_{\bar{s}},R\Psi(K|X_{\bar{\eta}}))\rightarrow\mathrm{H}^{i}(X_{\bar{s}},R\Phi(K))\rightarrow\cdots.$

The first map $\rm{sp}$ is called the specialization map and the cohomology of
vanishing cycles $\mathrm{H}^{i}(X_{\bar{s}},R\Phi(K))$ measures the defect of
$\rm{sp}$ from being an isomorphism. If we assume that $f$ is smooth, then
$\mathrm{H}^{i}(X_{\bar{s}},R\Phi(K))$ vanishes and the specialization map is
an isomorphism.

###  2.2. Isolated singularities

Suppose that $f:X\rightarrow S$ be a regular, flat, finite type morphism of
relative dimension $n$ which is smooth outside a finite collection $\Sigma$ of
closed points in $X_{s}$. In this subsection, we only consider the trivial
coefficient $\Lambda$ for simplicity and all the results recalled here can be
extended to more general constructible coefficients. In this case we have

$R\Phi(\Lambda)|X_{s}-\Sigma=0$

and moreover

$R\Phi(\Lambda)=\bigoplus_{x\in\Sigma}R^{n}\Phi(\Lambda)_{x}$

is concentrated at degree $n$. Therefore we have the following exact sequence

(2.1)
$\begin{split}&0\rightarrow\mathrm{H}^{n}(X_{\bar{s}},\Lambda)\xrightarrow{\rm{sp}}\mathrm{H}^{n}(X_{\bar{s}},R\Psi(\Lambda))\xrightarrow{\alpha}\oplus_{x\in\Sigma}R^{n}\Phi(\Lambda)_{x}\rightarrow\\\
&\mathrm{H}^{n+1}(X_{\bar{s}},\Lambda)\rightarrow\mathrm{H}^{n+1}(X_{\bar{s}},R\Psi(\Lambda))\rightarrow
0.\end{split}$

Assume next that for every $x\in\Sigma$ is an _ordinary quadratic singularity_
and furthermore
$\mathrm{H}^{n}(X_{\bar{s}},R\Psi(\Lambda))=\mathrm{H}^{n}(X_{\bar{\eta}},\Lambda)$.
This means $X$ is étale locally near $x$ isomorphic to

  * •

$V(\Sigma_{1\leq i\leq m}x_{i}x_{i+m}+\pi)\subset\mathbb{A}^{2m}_{S}$ if
$n=2m-1$;

  * •

$V(\Sigma_{1\leq i\leq
m}x_{i}x_{i+m}+x^{2}_{2m+1}+\pi)\subset\mathbb{A}^{2m+1}_{S}$ if $n=2m$.

Then in this case

$R^{n}\Phi(\Lambda)_{x}=\Lambda$

noncanonically. Let $I\subset\mathrm{Gal}(\bar{\eta}/\eta)$ be the inertia
group and let $\sigma\in I$. Then we have the variation map
$\mathrm{Var}(\sigma):(R^{n}\Phi\Lambda)_{x}\rightarrow
H^{n}_{x}(X_{\bar{s}},R\Psi(\Lambda))$ and the action of $\sigma-1$ on
$H^{n}(X_{\bar{\eta}},\Lambda)$ can be factored as

$\begin{array}[c]{ccccc}\mathrm{H}^{n}(X_{\bar{\eta}},\Lambda)&\xrightarrow{\alpha}&\bigoplus_{x\in\Sigma}\mathrm{R}^{n}\Phi(\Lambda)_{x}\\\
\Big{\downarrow}\scriptstyle{\sigma-1}&&\Big{\downarrow}\scriptstyle{\mathrm{Var}(\sigma)}\\\
\mathrm{H}^{n}(X_{\bar{\eta}},\Lambda)&\xleftarrow{\beta}&\bigoplus_{x\in\Sigma}\mathrm{H}^{n}_{x}(X_{\bar{s}},\mathrm{R}\Psi(\Lambda)).\end{array}$

Here $\beta$ is the composite

$\bigoplus_{x\in\Sigma}\mathrm{H}^{n}_{x}(X_{\bar{s}},R\Psi(\Lambda))\rightarrow\mathrm{H}^{n}(X_{\bar{s}},\Lambda)\rightarrow\mathrm{H}^{n}(X_{\bar{\eta}},\Lambda)$

where the first map is the Gysin map under the identification

$\bigoplus_{x\in\Sigma}\mathrm{H}^{n}_{x}(X_{\bar{s}},R\Psi(\Lambda))=\bigoplus_{x\in\Sigma}\mathrm{H}^{n}_{x}(X_{\bar{s}},\Lambda)$

and the second map is the specialization map. One has also a Frobenius
equivariant version

$\begin{array}[c]{ccccc}\mathrm{H}^{n}(X_{\bar{\eta}},\Lambda)(1)&\xrightarrow{\alpha}&\bigoplus_{x\in\Sigma}R^{n}\Phi(\Lambda)_{x}(1)\\\
\Big{\downarrow}\scriptstyle{N}&&\Big{\downarrow}\scriptstyle{N_{x}}\\\
\mathrm{H}^{n}(X_{\bar{\eta}},\Lambda)&\xleftarrow{\beta}&\bigoplus_{x\in\Sigma}\mathrm{H}^{n}_{x}(X_{\bar{s}},R\Psi(\Lambda))\end{array}$

where $N$ and $N_{x}$ are the monodromy and local monodromy operator. We will
loosely refer to this diagram as the Picard-Lefschetz formula for
$X_{\bar{\eta}}$.

###  2.3. Nearby cycles of automorphic étale sheaves

Suppose $f:X\rightarrow S$ is proper. Then the proper base change theorem
implies that we have an isomorphism

$\mathrm{H}^{i}(X_{\bar{\eta}},\Lambda)\cong\mathrm{H}^{i}(X_{\bar{s}},R\Psi(\Lambda)).$

However if $f:X\rightarrow S$ is not proper, then it is not always true that

$\mathrm{H}^{i}(X_{\bar{\eta}},\Lambda)\cong\mathrm{H}^{i}(X_{\bar{s}},R\Psi(\Lambda)).$

In the setting of Shimura varieties with good compactification, we do have
such isomorphism. In particular all the Shimura varieties we will consider in
this article have good compactifications. In fact all the Shimura varieties
belong to the case of

  * •

(Nm): A flat integral model defined by taking normalization of a
characteristic $0$ PEL type moduli problem over a producty of good reduction
integral models of smooth PEL type moduli problem

in the classification of [LS18a] and [LS18b]. The following theorem summarizes
the results we need.

######  Theorem 2.2 ([LS18b, Corollary 4.6]).

Suppose $X$ is a Shimura variety that is in the case of (Nm) and let
$\mathbb{V}$ be an automorphic étale sheaf defined as in [LS18b, §3]. Then the
canonical adjunction morphisms

$\mathrm{H}^{i}(X_{\bar{\eta}},\mathbb{V})\rightarrow\mathrm{H}^{i}(X_{\bar{s}},R\Psi(\mathbb{V})).$

and

$\mathrm{H}^{i}_{c}(X_{\bar{s}},R\Psi(\mathbb{V}))\rightarrow\mathrm{H}^{i}_{c}(X_{\bar{\eta}},\mathbb{V})$

are isomorphisms for all $i$.

##  3\. Automorphic and Galois representations for $\mathrm{GSp}_{4}$

###  3.1. The group $\mathrm{GSp}_{4}$

Let $\mathrm{GSp}_{4}$ be the symplectic similitude group defined by the set
of matrices $g$ in $\mathrm{GL}_{4}$ that satisfy $g^{t}Jg=c(g)J$ for

$J=\begin{pmatrix}&&&1\\\ &&1&\\\ &-1&&\\\ -1&&&\\\ \end{pmatrix}$

and some $c(g)\in\mathbb{G}_{m}$. We define the map
$c:\mathrm{GSp}_{4}\rightarrow\mathbb{G}_{m}$ by sending
$g\in\mathrm{GSp}_{4}$ to $c(g)$ and refer to it as the similitude map. The
kernel of this map is by definition the symplectic group $\mathrm{Sp}(4)$.
There are two conjugacy classes of maximal parabolic subgroups given by

  * •

the _Siegel parabolic subgroup_ $P$, whose Levi factor is

$M_{P}=\\{\begin{pmatrix}A&\\\ &uA^{\prime}\\\
\end{pmatrix}:u\in\mathrm{GL}_{1},A\in\mathrm{GL}_{2}\\}\cong\mathrm{GL}_{1}\times\mathrm{GL}_{2}$

where $A^{\prime}=\begin{pmatrix}&1\\\ 1&\\\
\end{pmatrix}(A^{-1})^{t}\begin{pmatrix}&1\\\ 1&\\\ \end{pmatrix}$

  * •

the _Klingen parabolic subgroup_ $Q$, whose Levi factor is

$M_{Q}=\\{\begin{pmatrix}u&\\\ &A&\\\ &&u^{-1}\mathrm{det}(A)\\\
\end{pmatrix}:u\in\mathrm{GL}_{1},A\in\mathrm{GL}_{2}\\}\cong\mathrm{GL}_{1}\times\mathrm{GL}_{2}.$

Let $T$ be the diagonal torus in $\mathrm{GSp}_{4}$ and $X^{*}(T)$ be its
character group which we identify with subset
$\\{(a,b;c)\in\mathbb{Z}^{3}:a+b\equiv c\mod 2\\}$ of $\mathbb{Z}^{3}$ by
associating a triple $(a,b;c)$ the character

(3.1) $\begin{pmatrix}t_{1}&&&\\\ &t_{2}&&&\\\ &&vt^{-1}_{2}&\\\
&&&vt^{-1}_{1}\\\ \end{pmatrix}\rightarrow t^{a}_{1}t^{b}_{2}v^{c}.$

Let $B$ be the Borel subgroup of upper triangular matrices in
$\mathrm{GSp}_{4}$. Then the set of dominant weights $X^{*}(T)^{+}$ with
respect to $B$ is given explicitly by $\\{(a,b;c)\in X^{*}(T):a\geq b\geq
0\\}.$

###  3.2. Parahoric subgroups of $\mathrm{GSp}_{4}(\mathbb{Q}_{p})$

The affine Dynkin diagram of type $\tilde{\mathrm{C}}_{2}$ is given by

$\textstyle{\underset{0}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{1}{\bullet}}$$\textstyle{\underset{2}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

and correspondingly we have the parahoric subgroups given by

  * •

The _Iwahori subgroup_ $I$;

  * •

The _Siegel parahoric subgroup_ $K_{\\{0,2\\}}$, this is obtained from the
Iwahori subgroup by adding the affine root group corresponding to $1$; Its
reduction is the Siegel parabolic $P$.

  * •

The _Kilngen parahoric subgroup_ $K_{\\{0,1\\}}$, this is obtained from the
Iwahori subgroup by adding the affine root group corresponding to $2$; Its
reduction is the Kilingen parabolic $Q$.

  * •

The _paramodular parahoric subgroup_ $K_{\\{1\\}}$, this is obtained from the
Iwahori subgroup by adding the affine root groups corresponding to $0$ and
$2$;

  * •

The _hyperspecial parahoric subgroup_ $K_{\\{0\\}}$ and $K_{\\{2\\}}$, this is
obtained from the Iwahori subgroup by adding the affine root groups
corresponding to $1,2$ or $1,0$. Note that they are conjugate to each other.

In this article we will be particularly interested to representations and
automorphic forms related to the paramodular subgroup and therefore we denote
it by $K(p)$ instead of $K_{\\{1\\}}$. We will also denote by $H$ the
hyperspecial parahoric subgroup. Note that the hyperspecial subgroup and the
paramodular subgroup are the maxiamal parahoric subgroups in
$\mathrm{GSp}_{4}(\mathbb{Q}_{p})$, therefore there is no map between Siegel
threefold of paramodular level to the Siegel threefold of hyperspecial level.
This is one of the difference between our case and the case of modular curves
or Shimura curves.

###  3.3. Hecke algebra

Let $p$ be a prime number. We let $\mathcal{H}_{p}$ be the spherical Hecke
algebra over $\mathbb{Z}$. This is a commutative algebra isomorphic to
$\mathbb{Z}[T_{p,0},T^{-1}_{p,0},T_{p,1},T_{p,2}]$ where

(3.2)
$\begin{split}&T_{p,0}=\mathrm{char}(\mathrm{GSp}_{4}(\mathbb{Z}_{p})\begin{pmatrix}p&&&\\\
&p&&\\\ &&p&\\\ &&&p\\\ \end{pmatrix}\mathrm{GSp}_{4}(\mathbb{Z}_{p})),\\\
&T_{p,1}=\mathrm{char}(\mathrm{GSp}_{4}(\mathbb{Z}_{p})\begin{pmatrix}1&&&\\\
&1&&\\\ &&p&\\\ &&&p\\\ \end{pmatrix}\mathrm{GSp}_{4}(\mathbb{Z}_{p})),\\\
&T_{p,2}=\mathrm{char}(\mathrm{GSp}_{4}(\mathbb{Z}_{p})\begin{pmatrix}1&&&\\\
&p&&\\\ &&p&\\\ &&&p^{2}\\\ \end{pmatrix}\mathrm{GSp}_{4}(\mathbb{Z}_{p})).\\\
\end{split}$

where $\mathrm{char}(\cdot)$ is the characteristic function. The Hecke
polynomial is by definition given by

$Q_{p}(X)=1-T_{p,2}X+p(T_{p,1}+(p^{2}+1)T_{p,0})X^{2}-p^{3}T_{p,2}T_{p,0}X^{3}+p^{6}T^{2}_{p,0}X^{6}.$

To any $\pi_{p}$ irreducible unramified admissible representation of
$\mathrm{GSp}_{4}(\mathbb{Q}_{p})$, we associate a character

$\Theta_{\pi_{p}}:\mathcal{H}_{p}\rightarrow\operatorname{End}(\pi_{p}^{\mathrm{GSp}_{4}(\mathbb{Z}_{p})})=\mathbb{C}$

and a Langlands parameter

$\begin{pmatrix}\alpha&&&\\\ &\beta&&&\\\ &&\gamma&&\\\ &&&\delta\\\
\end{pmatrix}$

considered as an element in
$\mathrm{GSp}_{4}(\mathbb{C})={{}^{L}\mathrm{GSp}_{4}}$. Then we have the
following identity

(3.3) $\Theta_{\pi_{p}}(Q_{p}(X))=(1-p^{2/3}\alpha X)(1-p^{2/3}\beta
X)(1-p^{2/3}\gamma X)(1-p^{2/3}\delta X).$

Given an automorphic representation $\pi$, we let $S$ be the product of primes
at which that $\pi$ is ramified. Then we define the unramified Hecke algebra
to be the restricted tensor product

$\mathbb{T}^{S}=\bigotimes^{\prime}_{p\nmid S}\mathcal{H}_{p}.$

###  3.4. Representations with paramodular fixed vector

Let $\pi_{p}$ be an irreducible smooth representation of
$\mathrm{GSp}_{4}(\mathbb{Q}_{p})$ such that $\pi^{K(p)}_{p}\neq 0$. One can
find the classification of these representations in [Schm05] for example. The
classification shows that there are five types (IIa, IVc, Vb, Vc, VIc) and
only IIa is generic. Let
$|\cdot|:\mathbb{Q}^{\times}_{p}\rightarrow\bar{\mathbb{Q}}_{l}$ be the
normalized absolute value such that $|p|=\frac{1}{p}$ and let $\sigma$ be an
unramified character $\mathbb{Q}^{\times}_{p}$ and $\chi$ be character of
$\mathbb{Q}^{\times}_{p}$ such that $\chi\neq|\cdot|^{\pm},|\cdot|^{\pm 3/2}$.
Then the type IIa irreducible representation
$\chi\mathrm{St}_{\mathrm{GL}_{2}}\rtimes\sigma$ corresponds to the Weil-
Deligne representation given by

(3.4) $\begin{pmatrix}\chi^{2}\sigma&&&\\\ &|\cdot|^{1/2}\chi\sigma&&\\\
&&|\cdot|^{-1/2}\chi\sigma&\\\ &&&\sigma\\\ \end{pmatrix},$

with monodromy operator of rank one given by

(3.5) $N=\begin{pmatrix}0&&&\\\ &0&1&\\\ &&0&\\\ &&&0.\\\ \end{pmatrix}$

under the local Langlands correspondence for $\mathrm{GSp}_{4}$ established in
[GT11]. In other words, the $L$-parameter is given by the
$\chi^{2}\sigma\oplus|\cdot|^{-1/2}\chi\sigma\otimes\mathrm{Sp}_{2}\oplus\sigma$.
We will later refer to $|\cdot|^{-1/2}\chi\sigma\otimes\mathrm{Sp}_{2}$ as the
twisted Steinberg part of the Weil-Deligne representation.

We define the following elements in the paramodular Hecke algebra

$\begin{split}&U_{p,1}=\mathrm{char}(K(p)\begin{pmatrix}1&&&\\\ &1&&\\\
&&p&\\\ &&&p\\\ \end{pmatrix}K(p)),\\\
&U_{p,2}=\mathrm{char}(K(p)\begin{pmatrix}1&&&\\\ &p&&\\\ &&p&\\\ &&&p^{2}\\\
\end{pmatrix}K(p)).\\\ \end{split}$

The element which we call the Atkin-Lehner operator

$u_{p}=\begin{pmatrix}&&&1\\\ &&-1&\\\ &p&&\\\ -p&&&\\\ \end{pmatrix}$

normalizes the paramodular subgroup $K(p)$ and thus will acts on the space of
$K(p)$ fixed vectors.

######  Lemma 3.6.

Let $\pi_{p}=\chi\mathrm{St}_{\mathrm{GL}_{2}}\rtimes\sigma$ be a
representation of type IIa with $\chi$ and $\sigma$ as above. Then
$\pi^{K(p)}_{p}$ is one dimensional on which $u_{p}$ acts by the scalar
$\chi\sigma(p)$.

###### Proof.

This is [VH19, Lemma 2.2.3]. ∎

Let $p$ and $q$ be two distinct odd primes. Then we define the following Hecke
algebras which we will also use in addition to $\mathbb{T}^{S}$

(3.7) $\mathbb{T}_{p}=\mathbb{T}^{S}[U_{p,0},U_{p,1},u_{p}].$

(3.8)
$\mathbb{T}_{pq}=\mathbb{T}^{S}[U_{p,0},U_{p,1},u_{p},U_{q,0},U_{q,1},u_{q}].$

###  3.5. Cohomology of Siegel threefolds

Let
$h:\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m}=\mathbb{C}^{\times}\rightarrow\mathrm{GSp}_{4}(\mathbb{R})$
sending $x+iy$ to

$\begin{pmatrix}xI_{2}&yS\\\ -yS&xI_{2}\end{pmatrix}$

where $I_{2}$ is the identity matrix of size $2$ and $S=\begin{pmatrix}0&1\\\
1&0\\\ \end{pmatrix}$. Let $K^{h}$ be the centralizer of $h$ in
$\mathrm{GSp}_{4}(\mathbb{R})$. Then $K^{h}=K_{\infty}\mathbb{R}^{\times}$
where $K_{\infty}$ is the maximal compact subgroup of
$\mathrm{GSp}_{4}(\mathbb{R})$. Let $U\subset\mathrm{GSp}_{4}(\mathbb{A}^{f})$
be an open compact subgroup and we have the Siegel threefold with level $U$
denoted by $X_{U}$. This is a quasi-projective variety whose
$\mathbb{C}$-points are given by

$X_{U}(\mathbb{C})=\mathrm{GSp}_{4}(\mathbb{Q})\backslash(\mathrm{GSp}_{4}(\mathbb{R})/K^{h}\times\mathrm{GSp}_{4}(\mathbb{A}_{f}))/U.$

Let $\mu=(a,b;c)\in X^{*}(T)$ and $V_{\mu}$ be the corresponding irreducible
representation of highest weight $\mu$. We fix an isomorphism
$\mathbb{C}\cong\bar{\mathbb{Q}}_{l}$ and let $E_{\lambda}/\mathbb{Q}_{l}$ be
a large enough coefficient field with ring of intergral elements
$\mathcal{O}=\mathcal{O}_{E_{\lambda}}$ and uniformizer $\varpi$. We can
consider $V_{\mu}$ as a representation of $\mathrm{GSp}_{4}$ over
$E_{\lambda}$ and we fix a lattice $V^{\mathrm{int}}_{\mu}$ in $V_{\mu}$. We
review the construction of an automorphic étale sheaf $\mathbb{V}_{\mu}$ on
$X_{U}$. For $m$ sufficiently large, let $U(l^{m})$ be the open compact
subgroup in $U_{l}=\mathrm{GSp}_{4}(\mathbb{Z}_{l})$ that acts trivially on
$V^{\mathrm{int}}_{\mu}/\varpi^{m}$. Then $\mathbb{V}^{\mathrm{int}}_{\mu}$ is
defined to be

(3.9) $(\varprojlim X_{U(l^{m})U^{l}}\times
V^{\mathrm{int}}_{\mu}/\varpi^{m})/U.$

We that $l>a+b+4$ and this guarantees that the étale automorphic sheaf is
independent of the choice of lattice. We denote its associated
$\mathbb{Q}_{l}$-local system
$\mathbb{V}^{\mathrm{int}}_{\mu}\otimes\mathbb{Q}_{l}$ by $\mathbb{V}_{\mu}$
and its associated $\mathbb{C}$-local system
$\mathbb{V}_{\mu,\mathbb{C}}=\mathbb{V}^{\mathrm{int}}_{\mu}\otimes\mathbb{C}$
. One is usually interested to the following cohomology theories of $X_{U}$.

  * •

The _intersection cohomology_
$\mathrm{IH}^{*}(X_{U}(\mathbb{C}),\mathbb{V}_{\mu})$;

  * •

The _$L^{2}$ -cohomology_
$\mathrm{H}^{*}_{(2)}(X_{U}(\mathbb{C}),\mathbb{V}_{\mu,\mathbb{C}})$ which
can be computed by the Matsushima formula

(3.10)
$\mathrm{H}^{*}_{(2)}(X_{U}(\mathbb{C}),\mathbb{V}_{\mu,\mathbb{C}})=\bigoplus_{\pi}m(\pi)\pi^{U}_{f}\otimes\mathrm{H}^{*}(\mathfrak{g},K_{\infty},\mathbb{V}_{\mu,\mathbb{C}}\otimes\pi_{\infty})$

where $\pi$ runs through all the automorphic representations
$\pi=\pi_{f}\otimes\pi_{\infty}$ that occurs in the discrete spectrum of
$L^{2}(\mathrm{GSp}_{4}(\mathbb{Q})\backslash\mathrm{GSp}_{4}(\mathbb{A}))$
with multiplicity $m(\pi)$. There is a natural direct summand
$\mathrm{H}^{*}_{\mathrm{cusp}}(X_{U}(\mathbb{C}),\mathbb{V}_{\mu,\mathbb{C}})$
of cuspidal cohomology which is the direct sum over cuspidal automorphic
representations $\pi$.

  * •

The _interior cohomology_

(3.11)
$\mathrm{H}^{*}_{!}(X_{U}(\mathbb{C}),\mathbb{V}_{\mu})=\mathrm{Im}(\mathrm{H}^{*}_{c}(X_{U}(\mathbb{C}),\mathbb{V}_{\mu})\rightarrow\mathrm{H}^{*}(X_{U}(\mathbb{C}),\mathbb{V}_{\mu})).$

Moreover we have an isomorphism

$\mathrm{IH}^{*}(X_{U}(\mathbb{C}),\mathbb{V}_{\mu,\mathbb{C}})=\mathrm{H}^{*}_{(2)}(X_{U}(\mathbb{C}),\mathbb{V}_{\mu,\mathbb{C}}).$

###  3.6. Galois representations

Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GSp}_{4}$ which
is not CAP, whose $\pi_{\infty}$ is a holomorphic discrete series of Harish
Chandra parameter $(a,b,-a-b+3)$ with $a\geq b\geq 0$. Let $S$ be the product
of primes $p$ such that $\pi_{p}$ is ramified. We review the construction of
the Galois representation attached to $\pi$. We define

$\mathrm{H}^{i}_{!}(X_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu})=\varinjlim_{U}\mathrm{H}^{i}_{!}(X_{U,\bar{\mathbb{Q}}},\mathbb{V}_{\mu}).$

In this case, it is known that $\pi$ is concentrated in degree $3$ in
$\mathrm{H}^{i}_{!}(X_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu})$. Then we define
the Galois module $W_{\pi,l}$ by

(3.12)
$W_{\pi,l}=\operatorname{Hom}_{\mathrm{GSp}_{4}(\mathbb{A}_{f})}(\pi_{f},\mathrm{H}^{3}_{!}(X_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu})).$

In fact by [Weis09, Theorem 1.1], we have

(3.13)
$\begin{split}W_{\pi,l}&=\operatorname{Hom}_{\mathrm{GSp}_{4}(\mathbb{A}_{f})}(\pi_{f},\mathrm{H}^{3}_{!}(X_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu}))\\\
&=\operatorname{Hom}_{\mathrm{GSp}_{4}(\mathbb{A}_{f})}(\pi_{f},\mathrm{H}_{c}^{3}(X_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu}))\\\
&=\operatorname{Hom}_{\mathrm{GSp}_{4}(\mathbb{A}_{f})}(\pi_{f},\mathrm{H}^{3}(X_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu})).\\\
\end{split}$

This will allow us pass from the usual cohomology
$\mathrm{H}^{3}(X_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu})$ or the cohomology with
compact support $\mathrm{H}_{c}^{3}(X_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu})$ to
the interior cohomology
$\mathrm{H}_{!}^{3}(X_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu})$ after localizing
at a suitable maximal ideal in the Hecke algebra.

The following theorem summarizes the properties of $W_{\pi,l}$ and the proof
of it can be found in [Tay93], [Laum05], [Weis05], [Sor10] and [Mo14].

######  Theorem 3.14.

Let $\pi$ be as above and $\Theta_{\pi}:\mathbb{T}^{S}\rightarrow\mathbb{C}$
be the character giving the action of $\mathbb{T}^{S}$ on $\pi$. Then
$W_{\pi,l}$ gives rise to a Galois representation

$\rho_{\pi,l}:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\mathrm{GSp}_{4}(\bar{\mathbb{Q}}_{l})$

satisfying

  * •

The representation $\rho_{\pi,l}$ is unramified at a prime $p$ for $p\neq l$
and $\pi_{p}$ is unramified. Moreover we have

$\mathrm{det}(1-\rho_{\pi,l}(\mathrm{Frob}_{p})X)=\Theta_{\pi}(Q_{p}(X)).$

  * •

At all places $p\neq l$ we have the local-global compatibility of Langlands
correspondence, this means

$\mathrm{WD}(\rho_{\pi,l\mid
G_{\mathbb{Q}_{p}}})^{\mathrm{F-ss}}\cong\mathrm{rec}_{p}(\pi_{p}\otimes|\cdot|^{-3/2})$

where $\mathrm{rec}_{p}$ is the local Langlands reciprocity map of [GT11] and
$\mathrm{F-ss}$ denotes the Frobenius semi-simplification of the Weil-Deligne
representation.

###  3.7. Some inner forms of $\mathrm{GSp}_{4}$

Let $B$ be a quaternion algebra over $\mathbb{Q}$. If $B$ split at $\infty$,
then we call $B$ _indefinite_ and otherwise we call $B$ _definite_. For an
element $b\in B$, we denote by $\bar{b}$ the image of $b$ under the main
involution of $B$. We choose an element $\tau\in B$ such that
$\bar{\tau}=-\tau$ if $B$ is indefinite and $\tau=1$ if $B$ is definite. We
define a new involution $*$ on $B$ by putting $b^{*}=\tau\bar{b}\tau^{-1}$.
Then consider the form $(\cdot,\cdot)$ on $V=B\oplus B$ defined by

(3.15) $(x,y)=\mathrm{tr}^{0}(\tau^{-1}(x_{1}y^{*}_{1}+x_{2}y^{*}_{2}))$

where $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$ are elements in $V$. Then we
define the _quaternionic unitary similitude group of degree $2$_ by

(3.16)
$\mathrm{GU}_{2}(B)(\mathbb{Q})=\\{g\in\mathrm{GL}_{2}(B)(\mathbb{Q}):(gx,gy)=c(g)(x,y),c(g)\in\mathbb{Q}^{\times}\\}.$

We write $D$ the quaternion division algebra over $\mathbb{Q}_{p}$. If
$B_{p}=D$ at place $p$, then
$\mathrm{GU}_{2}(B)(\mathbb{Q}_{p})=\mathrm{GU}_{2}(D)$ where
$\mathrm{GU}_{2}(D)$ is defined similarly as in (3.16) for $V=D\oplus D$. If
If $B_{p}=\mathrm{M}_{2}(\mathbb{Q}_{p})$, then
$\mathrm{GU}_{2}(B)(\mathbb{Q}_{p})=\mathrm{GSp}_{4}(\mathbb{Q}_{p}).$ Since
$D$ splits over $\mathbb{Q}_{p^{2}}$, we also have an identification

$\mathrm{GU}_{2}(D)(\mathbb{Q}_{p^{2}})\cong\mathrm{GSp}_{4}(\mathbb{Q}_{p^{2}}).$

There are two kinds of parahoric subgroups of $\mathrm{GU}_{2}(D)$. This can
be explained using the affine Dynkin diagram of $\mathrm{GU}_{2}(D)$. The
affine Dynkin diagram is still of type $\tilde{\mathrm{C}}_{2}$ but with
Frobenius acting non-trivially on the diagram by switching the nodes $0$ and
$2$ while fixing $1$.

$\textstyle{\underset{0}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{1}{\bullet}}$$\textstyle{\underset{2}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Therefore we have the following parahoric subgroups of $\mathrm{GU}_{2}(D)$.

  * •

The _Iwahori subgroup_ $I^{\prime}$ which corresponds to the Iwahori subgroup
$I$ in $\mathrm{GSp}_{4}$ when base change to $\mathbb{Z}_{p^{2}}$;

  * •

The _Siegel parahoric_ $H^{\prime}=K^{\prime}_{\\{02\\}}$ which corresponds to
the Siegel parahoric $K_{\\{02\\}}$ in $\mathrm{GSp}_{4}$ when base changed to
$\mathbb{Z}_{p^{2}}$;

  * •

The _paramodular parahoric_ $K^{\prime}(p)=K^{\prime}_{\\{1\\}}$ which
corresponds to the paramodular parahoric $K(p)=K_{\\{1\\}}$ in
$\mathrm{GSp}_{4}$ when base changed to $\mathbb{Z}_{p^{2}}$.

##  4\. Siegel threefold of paramodular level

###  4.1. Integral models

Let $U^{p}$ be a fixed prime to $p$ open compact subgroup of
$\mathrm{GSp}_{4}(\mathbb{A}_{f})$. Let $H=\mathrm{GSp}_{4}(\mathbb{Z}_{p})$
be the hyperspecial subrgoup of $\mathrm{GSp}_{4}(\mathbb{Q}_{p})$ and recall
that $K(p)$ is the paramodular subgroup. In this section, we will omit $U^{p}$
from all the notations. For example we have $X_{H}=X_{HU^{p}}$ and
$X_{K_{p}}=X_{K(p)U^{p}}$. These Shimura varieties adimits natural integral
models $\mathcal{X}_{H}$ and $\mathcal{X}_{K(p)}$ over $\mathbb{Z}_{p}$ which
are well documented in the literature and we will only recall their moduli
interpretations.

  * •

$\mathcal{X}=\mathcal{X}_{H}$ classifies triples of the form
$(A,\lambda,\eta)$ up to isomorphism where $A$ is an abelian scheme of
relative dimension $2$ over a test scheme $S$ over $\mathbb{Z}_{p}$ equipped
with a principal polarization $\lambda$ and $U^{p}$-level structure $\eta$.

  * •

$\mathcal{X}_{K(p)}$ classifies triples of the form $(A,\lambda,\eta)$ up to
isomorphism where $A$ is an abelian scheme of relative dimension $2$ and
$\lambda$ is a polarization on $A$ such that $\ker(\lambda)[p]$ has rank $2$.
Again $\eta$ is a prime to $p$ level structure.

Next we explain the structure of the supersingular locus and the singular
locus of the special fiber of $\mathcal{X}_{K(p)}$. First we recall a result
of Yu [Yu11a], let $X_{K(p),\bar{\mathbb{F}}_{p}}$ be the special fiber of
$\mathcal{X}_{K(p)}$ base changed to $\bar{\mathbb{F}}_{p}$ where
$\bar{\mathbb{F}}_{p}$ is an algebraically closed field containing
$\mathbb{F}_{p}$.

######  Theorem 4.1.

The scheme $X_{K(p),\bar{\mathbb{F}}_{p}}$ has isolated quadratic
singularities. The singular locus $\Sigma_{p}(K(p))$ consists of those
$(A,\lambda,\mu)$ such that $\ker(\lambda)\subset A[p]$ and $A$ is
superspecial.

###### Proof.

The first assertion is proved by a local model computation in [Yu11a]. For the
second assertion, the singular locus consists of those $(A,\lambda,\mu)$ such
that $\ker(\lambda)\cong\alpha_{p}\times\alpha_{p}$ by [Yu11a, Theorem 4.7].
Thus $\ker(\lambda)\subset A[p]$ and moreover it implies that $A$ is
superspecial. ∎

Next we recall the description of the supersingular locus $S_{K(p)}$ of
$X_{K(p),\bar{\mathbb{F}}_{p}}$ using the Bruhat-Tits stratification in
[Wang19a]. We fix a $p$-divisible group $\mathbb{X}$ of dimension $2$ which is
isoclinic of slope $1/2$ equipped with a polarization of height $2$. We denote
by $N$ its associated isocrystal. The polarization induces an alternating form
$(\cdot,\cdot)$ on $N$. Let $b\in B(\mathrm{GSp}_{4})$ be the element
corresponds to $N$ in the Kottwitz set $B(\mathrm{GSp}_{4})$ of
$\mathrm{GSp}_{4}$. Let $J_{b}$ be the group functor over $\mathbb{Q}_{p}$
defined by assigning each $\mathbb{Q}_{p}$-algebra $R$ the group

(4.2)
$J_{b}(R)=\\{g\in\mathrm{GSp}_{4}(R\otimes_{\mathbb{Q}_{p}}K_{0}):g(b\sigma)=(b\sigma)g\\}.$

It is well known that in this case $J_{b}\cong\mathrm{GU}_{2}(D)$ in our case.
In fact if we denote by $C=N^{\tau=1}$ with $\tau=F^{-1}V$, then $C$ can be
viewed as an $D$-module equipped with an alternating form $(\cdot,\cdot)$
which is the restriction of $(\cdot,\cdot)$ on $N$. Then we have
$J_{b}\cong\mathrm{GU}_{2}(C)$. We consider the Rapoport-Zink space
$\mathcal{N}_{K(p)}$ for $\mathrm{GSp}_{4}(\mathbb{Q}_{p})$ with paramodular
level structure. Let $W_{0}=W(\bar{\mathbb{F}}_{p})$ and denote by
$(\mathrm{Nilp})$ the category of $W_{0}$-schemes $S$ on which $p$ is locally
nilpotent. This is the set valued functor on $(\mathrm{Nilp})$ that classifies
the data $(X,\lambda_{X},\rho_{X})$ where

  * •

$X$ is a two dimensional $p$-divisible group over $S$ of height $4$;

  * •

$\lambda_{X}:X\rightarrow X^{\vee}$ is a polarization of height $2$;

  * •

$\rho_{X}:\mathbb{X}\times_{\bar{\mathbb{F}}_{p}}\bar{S}\rightarrow
X\times_{S}\bar{S}$ is a quasi-isogeny.

Here $\bar{S}$ is the special fiber of $S$ at $p$. This functor is
representable by a formal scheme $\mathcal{N}_{K(p)}$ over
$\mathrm{Spf}(W_{0})$ and $J_{b}(\mathbb{Q}_{p})$ acts on $\mathcal{N}_{K(p)}$
naturally. The formal scheme $\mathcal{N}_{K(p)}$ can be decomposed into
connected components

(4.3) $\mathcal{N}_{K(p)}=\bigsqcup_{i\in\mathbb{Z}}\mathcal{N}^{(i)}_{K(p)}$

where $\mathcal{N}^{(i)}_{K(p)}$ classifies those $(X,\lambda_{X},\rho_{X})$
such that $\rho_{X}$ has height $i$. We denote by $\mathcal{M}_{K(p)}$ the
underlying reduced scheme of $\mathcal{N}^{(0)}_{K(p)}$.

We recall the notion of vertex lattices in $C$. A _vertex lattice_ $L$ is by
definition a $\mathbb{Z}_{p}$-lattice $L\subset C$ such that
$pL^{\perp}\subset L\subset L^{\perp}$ where $L^{\perp}$ is the integral dual
of $L$ in $C$. They are naturally divided into three kinds according to the
indices of the inclusions:

  * •

$L$ is a vertex lattice of type $0$ if
$pL^{\perp}\subset^{4}L\subset^{0}L^{\perp}$;

  * •

$L$ is a vertex lattice of type $1$ if
$pL^{\perp}\subset^{2}L\subset^{2}L^{\perp}$;

  * •

$L$ is a vertex lattice of type $2$ if
$pL^{\perp}\subset^{0}L\subset^{4}L^{\perp}$.

We also define a vertex lattice of type $02$ by:

  * •

A pair $(L_{0},L_{2})$ is a vertex lattice of type $02$ if $L_{0}$ is a vertex
lattice of type $0$ and $L_{2}$ is a vertex lattice of type $2$.

The vertex lattices of type $02$ and type $1$ are called vertex lattices of
$\mathrm{GU}_{2}(C)$.

######  Remark 4.4.

Each vertex lattice $L_{i}$ for $i=0,1,2$ gives rise to a maximal parahoric
subgroup in $\mathrm{GSp}_{4}(\mathbb{Q}_{p})$ by taking the stabilizer. The
vertex lattices of type $0$ and $2$ correspond to the hyperspecial parahorics
$H$ and the vertex lattice of type $1$ corresponds to the paramodular
parahoric $K(p)$.

On the other hand, vertex lattices of type $02$ and type $1$ can be linked to
the maximal parahoric subgroups of $\mathrm{GU}_{2}(D)$. Here vertex lattices
of type $02$ will correspond to the Siegel parahoric $H^{\prime}$ and vertex
lattice of type $1$ will correspond to the paramoular parahoric
$K^{\prime}(p)$.

For each vertex lattice $L_{02}=(L_{0},L_{2})$ of type $02$ and each vertex
lattice $L_{1}$ of type $1$, we have defined in [Wang19a] projective
subschemes $\mathcal{M}_{K(p),L_{02}}$ and $\mathcal{M}_{K(p),L_{1}}$ of
$\mathcal{M}_{K(p)}$. In fact, each $\mathcal{M}_{K(p),L_{02}}$ is isomorphic
to a projective line $\mathbb{P}^{1}$ and $\mathcal{M}_{K(p),L_{1}}$ is
isomorphic to a reduced point which can be defined over $\mathbb{F}_{p^{2}}$.
We will refer to these projective schemes as the lattice strata. We also
define the open lattice strata by

  * •

$\mathcal{M}^{\circ}_{K(p),L_{02}}=\mathcal{M}_{L_{02}}-\cup_{L_{1}}\mathcal{M}_{L_{1}}$.

where the index $L_{1}$ runs through all the vertex lattices of type $1$. This
allows us to define the _Bruhat-Tits strata_ of $\mathcal{M}_{K(p)}$ by

  * •

$\mathcal{M}^{\circ}_{K(p),\\{02\\}}=\bigcup_{L_{02}}\mathcal{M}^{\circ}_{K(p),L_{02}}$;

  * •

$\mathcal{M}_{K(p),\\{1\\}}=\bigcup_{L_{1}}\mathcal{M}_{K(p),L_{1}}$.

Then by [Wang19a, Theorem 7.3] we have the following description of the scheme
$\mathcal{M}_{K(p)}$.

######  Proposition 4.5.

The scheme $\mathcal{M}_{K(p)}$ admits the Bruhat-Tits stratification

$\mathcal{M}_{K(p)}=\mathcal{M}^{\circ}_{K(p),\\{02\\}}\sqcup\mathcal{M}_{K(p),\\{1\\}}.$

The irreducible components of $\mathcal{M}_{K(p)}$ are given by the lattice
strata of the form $\mathcal{M}_{K(p),L_{02}}$ for a vertex lattice $L_{02}$
of type $02$. The singular locus of $\mathcal{M}_{K(p)}$ is given by
$\mathcal{M}_{K(p),\\{1\\}}$.

The Rapoport-Zink uniformization theorem [RZ96, Theorem 6.30] furnishes an
isomorphism

(4.6) $S_{K(p)}\cong
I(\mathbb{Q})\backslash\mathcal{N}_{K(p)}\times\mathrm{GSp}_{4}(\mathbb{A}^{(p)}_{f})/U^{p}.$

Here $I(\mathbb{Q})\cong\mathrm{GU}_{B^{\prime}}(\mathbb{Q})$ and
$B^{\prime}=B_{p\infty}$ is the quaternion algebra over $\mathbb{Q}$ which
ramifies at $p$ and infinity.

######  Proposition 4.7.

  * •

The irreducible components of $S_{K(p)}$ can be parametrized by the double
coset

(4.8)
$\mathrm{GU}_{B^{\prime}}(\mathbb{Q})\backslash\mathrm{GU}_{B^{\prime}}(\mathbb{A}_{f})/H^{\prime}U^{p}.$

  * •

The singular locus of $S_{K(p)}$ can be parametrized by the double coset

(4.9)
$\mathrm{GU}_{B^{\prime}}(\mathbb{Q})\backslash\mathrm{GU}_{B^{\prime}}(\mathbb{A}_{f})/K^{\prime}(p)U^{p}.$

###### Proof.

By Proposition 4.5, the irreducible components of the scheme
$\mathcal{M}_{K(p)}$ corresponds to vertex lattices of type $02$. The derived
group of $\mathrm{GU}_{2}(D)$ acts transitively on these lattices and the
stabilizer corresponds to the Siegel parahoric $H^{\prime}$. Therefore the
irreducible components of $\mathcal{N}_{K(p)}$ is parametrized by
$\mathrm{GU}_{2}(D)/H^{\prime}$. Now we apply (4.6) to conculde the proof of
(4.8).

By Proposition 4.5, the singular locus of the scheme $\mathcal{M}_{K(p)}$
corresponds to vertex lattices of type $1$. Then we proceed similarly as above
to prove (4.9). ∎

##  5\. Mazur’s principle for paramodular Siegel modular forms

###  5.1. Mazur’s principle

We recall that we are concerned with a cuspidal automorphic representation
$\pi$ for $\mathrm{GSp}_{4}$ which is non-CAP and non-endoscopic whose
component at infinity $\pi_{\infty}$ is a holomorphic discrete series of
Harish Chandra parameter $(a,b,-a-b+3)$ with $a\geq b\geq 0$. We also assume
that the representation $\pi$ is para-spherical and its local component
$\pi_{p}$ at $p$ is of type IIa under Schmidt’s classification. In this case,
its associated Galois representation $\rho_{\pi,l}$ is realized in the Galois
module $W_{\pi,l}$ introduced in (3.12). The integral cohomology
$\mathrm{H}^{3}_{!}(X_{\bar{\mathbb{Q}}},\mathbb{V}^{\mathrm{int}}_{\mu})$
gives a Galois stable lattice $\Lambda_{\pi,l}\subset W_{\pi,l}$ such the
natural reduction of $\Lambda_{\pi,l}$ gives rise to a representation of
$\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ valued in $k$ for some finite
field of characteristic $l$

$\bar{\rho}_{\pi,l}:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\mathrm{GSp}_{4}(k)$

which we will refer to as the residual representation of $\rho_{\pi,l}$. We
will make the following assumption throughout the rest of this article that

(Irr) $\bar{\rho}_{\pi,l}$ is irreducible.

The residual Galois representation $\bar{\rho}_{\pi,l}$ corresponds to a
maximal ideal $\mathfrak{m}\subset\mathbb{T}_{p}$ with residue field $k$. We
make the following assumption

(Semi)
$\mathrm{H}^{3}_{!}(X_{K(p),\bar{\mathbb{Q}}},\mathbb{V}^{\mathrm{int}}_{\mu})_{\mathfrak{m}}\otimes
k$ is semisimple as a Galois module.

This implies that

$\mathrm{H}^{3}_{!}(X_{K(p),\bar{\mathbb{Q}}},\mathbb{V}^{\mathrm{int}}_{\mu})_{\mathfrak{m}}\otimes
k=\bar{\rho}^{\oplus s}$

for some positive number $s$.

In this section, we will prove the following theorem which we will refer to as
the Mazur’s principle.

######  Theorem 5.1.

Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GSp}_{4}$ that
is cohomological and is non-CAP and non-endoscopic whose component at infinity
$\pi_{\infty}$ is a holomorphic discrete series of Harish Chandra parameter
$(a,b,-a-b+3)$ with $a\geq b\geq 0$ and $l>a+b+4$. Let $p$ be a prime distinct
from $l$ and such that $p\not\equiv 1\mod l$. Let
$U=K(p)U^{p}\subset\mathrm{GSp}_{4}(\mathbb{Q}_{p})\mathrm{GSp}_{4}(\mathbb{A}^{(p)}_{f})$
be a neat open compact subgroup such that $\pi^{U}\neq 0$. We assume that
$\pi_{p}$ is ramified of type IIa in the classification of Schmidt. Suppose
the residual Galois representation $\bar{\rho}_{\pi,l}$ satisfies the
following assumptions

  * •

$\bar{\rho}_{\pi,l}$ is unramified at $p$;

  * •

$\bar{\rho}_{\pi,l}$ satisfies (Irr);

  * •

$\bar{\rho}_{\pi,l}$ satisfies (Semi);

Then there exists a cuspidal automorphic representation $\pi^{\prime}$ with
the same type as $\pi$ at $\infty$ such that
$\bar{\rho}_{\pi,l}\cong\bar{\rho}_{\pi^{\prime},l}$ and $\pi^{\prime}_{p}$ is
unramified.

###  5.2. Picard-Lefschetz formula

We consider $X=\mathcal{X}_{K(p)}$ in the setting of §2. Here
$\bar{\eta}=\mathrm{Spec}(\bar{\mathbb{Q}}_{p})$ and
$\bar{s}=\mathrm{Spec}(\bar{\mathbb{F}}_{p})$, we will set
$X_{K(p),\bar{\mathbb{Q}}_{p}}=\mathcal{X}_{K(p),\bar{\eta}}$,
$X_{K(p),\bar{\mathbb{F}}_{p}}=\mathcal{X}_{K(p),\bar{s}}$ and
$\Sigma=\Sigma_{p}(K(p))$. Recall also that $S_{K(p)}$ is the supersingular
locus of $X_{K(p),\bar{\mathbb{F}}_{p}}$. Furthermore, we will let
$\mathbb{V}=\mathbb{V}^{\mathrm{int}}_{\mu}$ to further light up the notation.
In this subsection, we are mainly concerned with analyzing the following
diagram which we refer to as the Picard-Lefschetz formula for
$X_{K(p),\bar{\mathbb{Q}}_{p}}$

(5.2)
$\begin{array}[c]{ccccc}\mathrm{H}^{3}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})(1)&\xrightarrow{\alpha}&\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x}(1)\\\
\Big{\downarrow}\scriptstyle{N}&&\Big{\downarrow}\scriptstyle{N_{x}}\\\
\mathrm{H}^{3}_{c}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})&\xleftarrow{\beta}&\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V}).\end{array}$

By general theory, the map $\beta$ and $\alpha$ are dual to each other and
therefore $\beta$ is injective if and only if $\alpha$ is surjective. We prove
first the following important proposition

######  Proposition 5.3.

The map $\beta$ localized at $\mathfrak{m}$ is injective

(5.4)
$\beta:\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\rightarrow\mathrm{H}^{3}_{c}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}$

and therefore

(5.5)
$\alpha:\mathrm{H}^{3}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}(1)\rightarrow\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}(1)$

is surjective.

###### Proof.

Since
$\mathrm{H}^{3}_{c}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})\rightarrow\mathrm{H}^{3}_{c}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})$
is injective, we need to show the Gysin map

$\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})\rightarrow\mathrm{H}^{3}_{c}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})$

is injective after localizing at $\mathfrak{m}$. Note the kernel of this
morphism is the image of the connecting homomorphism

$\mathrm{H}^{2}_{c}(U,\mathbb{V})\rightarrow\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})$

where $U=X_{K(p),\bar{\mathbb{F}}_{p}}-\Sigma$. We claim that
$\mathrm{H}^{2}_{c}(U,\mathbb{V})=0$ after localizing at $\mathfrak{m}$. To
prove this, we use the following excision exact sequence

$\mathrm{H}^{2}_{S_{K(p)}-\Sigma}(X_{K(p),\bar{\mathbb{F}}_{p}}-\Sigma,\mathbb{V})\rightarrow\mathrm{H}^{2}_{c}(U,\mathbb{V})\rightarrow\mathrm{H}^{2}_{c}(X_{K(p),\bar{\mathbb{F}}_{p}}-S_{K(p)},\mathbb{V}).$

Note that $X_{K(p),\bar{\mathbb{F}}_{p}}-\Sigma$ is now smooth and
$S_{K(p)}-\Sigma$ is of codimension $2$ in
$X_{K(p),\bar{\mathbb{F}}_{p}}-\Sigma$. Then by purity,
$\mathrm{H}^{2}_{S_{K(p)}-\Sigma}(X_{K(p),\bar{\mathbb{F}}_{p}}-\Sigma,\mathbb{V})$
vanishes. Also notice that $X_{K(p),\bar{\mathbb{F}}_{p}}-S_{K(p)}$ is the
ordinary locus. Let $X^{*}_{K(p),\bar{\mathbb{F}}_{p}}$ be the minimal
compactification of $X_{K(p),\bar{\mathbb{F}}_{p}}$, then
$X^{*}_{K(p),\bar{\mathbb{F}}_{p}}-S_{K(p)}$ is affine and therefore

$\mathrm{H}^{2}_{c}(X^{*}_{K(p),\bar{\mathbb{F}}_{p}}-S_{K(p)},\mathbb{V})=0$

by Artin vanishing. Since the difference between

$\mathrm{H}^{2}_{c}(X^{*}_{K(p),\bar{\mathbb{F}}_{p}}-S_{K(p)},\mathbb{V})=0$

and

$\mathrm{H}^{2}_{c}(X_{K(p),\bar{\mathbb{F}}_{p}}-S_{K(p)},\mathbb{V})=0$

is supported on the boundary of the minimal compactification, the difference
vanishes after localizing at $\mathfrak{m}$ by our assumption (Irr). Therefore
$\mathrm{H}^{2}_{c}(X_{K(p),\bar{\mathbb{F}}_{p}}-S_{K(p)},\mathbb{V})$
vanishes after localizing at $\mathfrak{m}$. ∎

By Proposition 5.3, we have the following fundamental diagram

(5.6)
${\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes{E}}$${\mathrm{H}^{3}_{c}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
E}$${\mathrm{H}^{3}_{!}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
E}$${\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
E.}$$\scriptstyle{\alpha}$

We have

(5.7)
$\mathrm{H}^{3}_{!}(X_{K(p),\bar{\mathbb{Q}}/\mathbb{Q}},\mathbb{V})_{\mathfrak{m}}\otimes
E=\bigoplus_{\pi^{\prime}}\rho^{\oplus m(\pi^{\prime})}_{\pi^{\prime},l}$

where $\pi^{\prime}$ runs through all the autormorphic representation
$\pi^{\prime}$ whose Galois representation $\rho_{\pi^{\prime},l}$ is
congruent to $\rho_{\pi,l}$ and $m(\pi^{\prime})$ is the multiplicity of
$\pi^{\prime}$ which could be zero. Let $\pi^{\prime}$ be an automorphic
representation such that $\pi^{\prime}_{p}$ is ramified. Then we have the
following description of $\rho_{\pi^{\prime},l}$ in terms of the fundamental
diagram (5.6).

######  Proposition 5.8.

Let $\pi^{\prime}$ be a representation appearing in (5.7) such that
$\pi^{\prime}_{p}$ is ramified. Then

  * •

$\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes{E}$
contributes to $\rho_{\pi^{\prime},l\mid G_{\mathbb{Q}_{p}}}$ as a
$1$-dimensional subspace;

  * •

$\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes E$
contributes to $\rho_{\pi^{\prime},l\mid G_{\mathbb{Q}_{p}}}$ as a
$1$-dimensional quotient;

  * •

$\mathrm{H}^{3}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
E/\mathrm{im}(\beta)$ contributes to $\rho_{\pi^{\prime},l\mid
G_{\mathbb{Q}_{p}}}$ as a $2$-dimensional subspace.

###### Proof.

The local-global compatibility of Langlands correspondence as in Theorem 3.14
implies that the monodromy operator $N$ has one dimensional image. Thus the
Picard-Lefschetz formula in (7.19) implies that
$\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})$
has one dimensional contribution to $\rho_{\pi^{\prime},l}$. Since the local
monodromy map $N_{x}$ are isomorphisms,
$\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes E$
contributes to $\rho_{\pi^{\prime},l\mid G_{\mathbb{Q}_{p}}}$ as a
$1$-dimensional quotient. Then the third claim is also clear from what we have
proved. ∎

###  5.3. Proof of the Mazur’s principle

Now we come to the proof of Theorem 5.1. We will prove the theorem by
contradiction. Assume that we can not find such a $\pi^{\prime}$ in the
statement of the theorem. This means that all the $\pi^{\prime}$ appears in
(5.7) are ramified at $p$. We first prove a congruence relation on
$\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}$ under this
assumption.

######  Lemma 5.9.

Suppose that all the $\pi^{\prime}$ appears in (5.7) are ramified at $p$. The
Frobenius action $\mathrm{Frob}_{p}$ on the Galois module

$\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}$

is given by the Atkin-Lehner operator $u_{p}$.

###### Proof.

Examine the Weil-Deligne representation corresponding to
$\rho_{\pi^{\prime},l}$ in (3.4) and the shape of the monodromy operator
(3.5),
$\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes{E}$
contributes to the Galois representation $\rho_{\pi^{\prime},l\mid
G_{\mathbb{Q}_{p}}}$ as the character $\chi\sigma|\cdot|$. Since the local
mondromy operator $N_{x}$ induces an isomorphism between
$\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}(1)\otimes E$ and
$\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes{E}$.
It follows that Then the result follows from Lemma 3.6. ∎

We continue with the proof of the Mazur’s principle. The reduction modulo $l$
of (5.7) gives

(5.10)
$\mathrm{H}^{3}_{!}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k=\bar{\rho}^{s}_{\pi,l\mid G_{\mathbb{Q}_{p}}}$

which is unramified by our assumption. Therefore the monodromy operator
$N\otimes k$ degenerates to $0$. We calculate this monodromy operator using
again the Picard-Lefschetz formula

(5.11)
$\begin{array}[c]{ccccc}\mathrm{H}^{3}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}(1)\otimes
k&\xrightarrow{\alpha\otimes
k}&\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}(1)\otimes k\\\
\Big{\downarrow}\scriptstyle{N\otimes
k}&&\Big{\downarrow}\scriptstyle{N_{x}}\otimes k\\\
\mathrm{H}^{3}_{c}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k&\xleftarrow{\beta\otimes
k}&\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k.\end{array}$

By Proposition (5.3), $\alpha\otimes k$ is surjective and $N_{x}\otimes k$ is
an isomorphism. Therefore $\beta\otimes k$ has to be $0$. Now we examine the
mod $l$ version of the fundamental diagram (5.6)

(5.12)
${\bigoplus_{x\in\Sigma}\mathrm{H}^{3}_{x}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes{k}}$${\mathrm{H}^{3}_{c}(X_{K(p),\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k}$${\mathrm{H}^{3}_{!}(X_{K(p),\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k}$${\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
k.}$$\scriptstyle{\beta^{\prime}}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha\otimes
k}$

The map $\gamma$ here is still injective. This can be proved by snake lemma
and the fact that
$\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}$ is obviously
torsion free. Therefore
$\bigoplus_{x\in\Sigma}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}$ contributes to
$\bar{\rho}_{\pi,l\mid G_{\mathbb{Q}_{p}}}$ as the two dimensional quotient
which is isomorphic to the reduction modulo $l$ of the twisted Steinberg
$\chi\mathrm{St}_{\mathrm{GL}_{2}}$ part of Weil-Deligne representation in
(3.4). In particular the determinant of the element $\mathrm{Frob}_{p}$ on
this two dimensional quotient is of the form $\chi^{2}\sigma^{2}|\cdot|$
modulo $l$. On the other hand, by the congruence relation in Lemma 5.9,
$\mathrm{Frob}_{p}$ acts by the character $\chi\sigma$ modulo $l$ on this two
dimensional quotient. This implies that the its determinant is of the form
$\chi^{2}\sigma^{2}$. It then follows that $p\equiv 1\mod l$ which is a
contradiction. This finishes the proof of Theorem 5.1.

##  6\. Quaternionic Siegel threefold of paramodular level

###  6.1. Integral model

In this section we review some results obtained in [Wang19a] about the
supersingular locus of the quaternionic Siegel threefold. In light of the
forthcoming application, we let $B=B_{pq}$ be the indefinite quaternion
algebra with discriminant $pq$ over $\mathbb{Q}$. Here $p$ and $q$ are two
distinct primes that are disctinct from $l$. Let $*$ be a _neben-involution_
on $B$ defined as in [KR00, A.4] and $\mathcal{O}_{B}$ be a maximal order
fixed by $*$. We set $V=B\oplus B$ and let

$(\cdot,\cdot):V\times V\rightarrow\mathbb{Q}$

be the alternating form defined in (3.15). This form satisfies the following
equation

$(av_{1},v_{2})=(v_{1},a^{*}v_{2})$

for all $v_{1},v_{2}\in V$ and $a\in B$. Let $G=\mathrm{GU}_{2}(B)$ be the
quaternionic unitrary group of degree $2$ defined as in (3.16). Since $B$
splits over $\mathbb{R}$. We have the identification
$G(\mathbb{R})\cong\mathrm{GSp}_{4}(\mathbb{R})$. Let
$\mathbb{S}=\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m}$ be the Deligen
torus and let $h:\mathbb{S}\rightarrow G_{\mathbb{R}}$ be the Hodge
cocharacter that it induces the miniscule cocharacter

$\mu:\mathbb{G}_{m,\mathbb{C}}\rightarrow{G}_{\mathbb{C}}\cong\mathrm{GSp}(4)_{\mathbb{C}}$

that sends $z\in\mathbb{G}_{m,\mathbb{C}}$ to
$\text{diag}(z,z,1,1)\in\mathrm{GSp}(4)_{\mathbb{C}}$. Let
$U^{pq}\subset\mathrm{GSp}_{4}(\mathbb{A}^{pq}_{f})$ be a sufficiently small
subgroup which we will fix throughout the rest of this article. Let
$\mathcal{L}_{p}=\mathcal{O}_{B_{p}}\oplus\mathcal{O}_{B_{p}}$ and let
$\mathcal{L}_{q}=\mathcal{O}_{B_{q}}\oplus\mathcal{O}_{B_{q}}$. Then the
stabilizer of this lattice in $G(\mathbb{Q}_{p})$ and $G(\mathbb{Q}_{q})$ are
the paramodular subgroups denoted by $K^{\prime}(p)$ and $K^{\prime}(q)$ in
§3.7. Then the tuple

(6.1)
$(B,\mathcal{O}_{B},V,(\cdot,\cdot),\mu,\mathcal{L}_{p},U^{p}=K^{\prime}(q)U^{pq})$

gives a PEL-data over $\mathbb{Z}_{p}$ and the tuple

(6.2)
$(B,\mathcal{O}_{B},V,(\cdot,\cdot),\mu,\mathcal{L}_{q},U^{q}=K^{\prime}(p)U^{pq})$

gives a PEL-data over $\mathbb{Z}_{q}$. They define PEL-type moduli problems
$\mathcal{X}^{B}_{K^{\prime}(p)}$ over $\mathbb{Z}_{p}$ and
$\mathcal{X}^{B}_{K^{\prime}(q)}$ over $\mathbb{Z}_{q}$. As the role of $p$
and $q$ are completely symmetric in this section, we will only state the
moduli problem $\mathcal{X}^{B}_{K^{\prime}(p)}$ over $\mathbb{Z}_{p}$. For
each $\mathbb{Z}_{p}$ scheme $S$, a point of the functor
$\mathcal{X}^{B}_{K^{\prime}(p)}$ with value in $S$ is given by the following
data up to isomorphism.

  * •

A is an abelian scheme over $S$ of dimension $4$;

  * •

$\iota:\mathcal{O}_{B}\rightarrow\operatorname{End}_{S}(A)$ is a ring
homomorphism;

  * •

$\lambda:A\rightarrow A^{\vee}$ is a prime to $p$ polarization;

  * •

$\eta:\mathrm{H}_{1}(A,\mathbb{A}^{p}_{f})\cong V\otimes\mathbb{A}^{p}_{f}\mod
U^{p}$ is a $U^{p}$-level structure that respects the bilinear forms on both
sides up to a constant in $(\mathbb{A}^{p}_{f})^{\times}$.

The quadruple $(A,\iota,\lambda,\eta)$ is required to satisfy the following
additional conditions

  * •

The Kottwitz condition

(6.3)
$\mathrm{det}(T-\iota(a);\mathrm{Lie}(A))=(T^{2}-\mathrm{Trd}(a)T+\mathrm{Nrd}(a))^{2}$

for all $a\in\mathcal{O}_{B}$;

  * •

$\lambda\circ\iota(a)\circ\lambda^{-1}=\iota(a^{*})^{\vee}$.

This functor is representable by a quasi-projective scheme
$\mathcal{X}^{B}_{K^{\prime}(p)}$ over $\mathbb{Z}_{p}$ of relative dimension
$3$. We remark that $\mathrm{GU}_{2}(B)$ has semi-simple $\mathbb{Q}$-rank one
and therefore $\mathcal{X}^{B}_{K^{\prime}(p)}$ is not proper. The boundary of
the minimal compactification of $\mathcal{X}^{B}_{K^{\prime}(p)}$ consists of
cusps. The scheme $\mathcal{X}^{B}_{K^{\prime}(p)}$ is regular but not smooth
over $\mathrm{Spec}(\mathbb{Z}_{p})$. Since the group $G$ can be identified
with $\mathrm{GSp}_{4}$ over $\mathbb{Z}_{p^{2}}$ and the parahoric subgroup
$K^{\prime}(p)$ can be identified with $K(p)$ over the same extension, the
local model for $\mathcal{X}^{B}_{K^{\prime}(p)}$ agrees with that of
$\mathcal{X}_{K(p)}$ after the base change to $\mathbb{Z}_{p^{2}}$. Therefore
$\mathcal{X}^{B}_{K^{\prime}(p)}$ also has isolated singularities which are
ordinary and quadratic. We denote by
$X^{B}_{K^{\prime}(p),\bar{\mathbb{Q}}_{p}}$ the generic fiber of
$\mathcal{X}^{B}_{K^{\prime}(p)}$ base changed to $\bar{\mathbb{Q}}_{p}$. The
set of singular points $\Sigma_{p}(B)$ again agree with the superspecial
points on $X^{B}_{K^{\prime}(p),\bar{\mathbb{F}}_{p}}$.

######  Lemma 6.4.

The singular locus $\Sigma_{p}(B)$ consists of those
$(A,\iota,\lambda,\eta)\in X^{B}_{K^{\prime}(p),\bar{\mathbb{F}}_{p}}$ such
that $A$ is superspecial.

###### Proof.

This follows from [Wang19a, Lemma 2.1] and the discussion on local model
following it. ∎

Next we recall the description of the supersingular locus of
$X^{B}_{K^{\prime}(p),\bar{\mathbb{F}}_{p}}$ using the Bruhat-Tits
stratification in [Wang19a]. Let $S_{K^{\prime}(p)}$ be the supersingular
locus considered as a reduced closed subscheme of
$X^{B}_{K^{\prime}(p),\bar{\mathbb{F}}_{p}}$. We fix a $p$-divisible group
$\mathbb{X}$ over $\bar{\mathbb{F}}_{p}$ with an action
$\iota:\mathcal{O}_{D}\rightarrow\operatorname{End}(\mathbb{X})$ and a
polarization $\lambda:\mathbb{X}\rightarrow\mathbb{X}^{\vee}$. We assume that
$\mathbb{X}$ is isoclinic of slope $\frac{1}{2}$ and we denote by $N$ its
associated isocrystal. Let $b\in B(G)$ be the element associated to $N$ in the
Kottwitz set $B(G)$. Then the group $J_{b}$ defined as in (4.2) can be
identified with the split $\mathrm{GSp}_{4}/\mathbb{Q}_{p}$ in this case. In
fact let $C=N^{\tau=1}$ with $\tau=\Pi^{-1}F$ equipped with the restriction of
the alternating form $(\cdot,\cdot)$ to $C$. Then we have
$J_{b}=\mathrm{GSp}_{4}(C)$. We consider the Rapoport-Zink space
$\mathcal{N}_{K^{\prime}(p)}$ for the group $G$. This is the set valued
functor on $(\mathrm{Nilp})$ which assigns each $S\in(\mathrm{Nilp})$ the set
of isomorphism classes of quadruples $(X,\iota_{X},\lambda_{X},\rho_{X})$
where

  * •

$X$ is a $4$-dimensional $p$-divisible group over $S$;

  * •

$\iota_{X}:\mathcal{O}_{D}\rightarrow\operatorname{End}_{S}(X)$ is an action
of $\mathcal{O}_{D}$ on $X$;

  * •

$\lambda_{X}:X\rightarrow X^{\vee}$ is a principal polarization;

  * •

$\rho_{X}:\mathbb{X}\times_{\bar{\mathbb{F}}_{p}}\bar{S}\rightarrow
X\times_{S}\bar{S}$ is a quasi-isogeny where $\bar{S}$ is the special fiber of
$S$ at $p$.

This functor $\mathcal{N}_{K^{\prime}(p)}$ is representable by a formal scheme
locally formally of finite type over $\mathrm{Spf}(W_{0})$. It carries an
action of $J_{b}\cong\mathrm{GSp}_{4}(\mathbb{Q}_{p})$. Again
$\mathcal{N}_{K^{\prime}(p)}$ can be decomposed into connected components

(6.5)
$\mathcal{N}_{K^{\prime}(p)}=\bigsqcup_{i\in\mathbb{Z}}\mathcal{N}^{(i)}_{K^{\prime}(p)}$

where $\mathcal{N}^{(i)}_{K^{\prime}(p)}$ classifies those
$(X,\iota_{X},\lambda_{X},\rho_{X})$ such that $\rho_{X}$ has height $i$. We
denote by $\mathcal{M}_{K^{\prime}(p)}$ the reduced scheme of the formal
scheme $\mathcal{N}^{(0)}_{K^{\prime}(p)}$. Then we define vertex lattices in
$C$ similarly as before. Recall that they are naturally divided into three
kinds

  * •

$L$ is a vertex lattice of type $0$ if
$pL^{\perp}\subset^{4}L\subset^{0}L^{\perp}$;

  * •

$L$ is a vertex lattice of type $1$ if
$pL^{\perp}\subset^{2}L\subset^{2}L^{\perp}$;

  * •

$L$ is a vertex lattice of type $2$ if
$pL^{\perp}\subset^{0}L\subset^{4}L^{\perp}$.

Each vertex lattice gives rise to a _lattice stratum_ which is a projective
sub scheme of $\mathcal{M}_{K^{\prime}(p)}$, see [Wang19a, Theorem 5.1].

  * •

For a vertex lattice $L_{0}$ of type $0$, the scheme
$\mathcal{M}_{K^{\prime}(p),L_{0}}$ is a projective surface of the form

$x^{p}_{3}x_{0}-x^{p}_{0}x_{3}+x^{p}_{2}x_{1}-x^{p}_{1}x_{2}=0$

for a projective coordinate $[x_{0}:x_{1}:x_{2}:x_{3}]$ of $\mathbb{P}^{3}$;

  * •

For a vertex lattice $L_{2}$ of type $2$, the scheme
$\mathcal{M}_{K^{\prime}(p),L_{2}}$ is a projective surface of the form

$x^{p}_{3}x_{0}-x^{p}_{0}x_{3}+x^{p}_{2}x_{1}-x^{p}_{1}x_{2}=0$

for a projective coordinate $[x_{0}:x_{1}:x_{2}:x_{3}]$ of $\mathbb{P}^{3}$;

  * •

For a vertex lattice $L_{1}$ of type $1$, the scheme
$\mathcal{M}_{K^{\prime}(p),L_{1}}$ consists of a superspecial point.

There are two types of irreducible components of
$\mathcal{M}_{K^{\prime}(p)}$. They are given by lattice strata
$\mathcal{M}_{K^{\prime}(p),L_{0}}$ of type $0$ and lattice strata
$\mathcal{M}_{K^{\prime}(p),L_{2}}$ of type $2$. The Rapoport-Zink
uniformization theorem [RZ96, Theorem 6.30] then yields

(6.6) $S_{K^{\prime}(p)}\cong
I(\mathbb{Q})\backslash\mathcal{N}_{K^{\prime}(p)}\times\mathrm{GU}_{2}(B)(\mathbb{A}^{(p)}_{f})/U^{p}.$

Here $I(\mathbb{Q})\cong\mathrm{GU}_{B^{\prime\prime}}(\mathbb{Q})$ and
$B^{\prime\prime}=B_{q\infty}$ is the quaternion algebra over $\mathbb{Q}$
which ramifies at $q$ and infinity.

######  Proposition 6.7.

  * •

The irreducible components of $S_{K^{\prime}(p)}$ can be parametrized by the
union of two double cosets

(6.8)
$\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{Q})\backslash\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{A}_{f})/K_{\\{0\\}}U^{p}\bigsqcup\mathrm{GU}_{2}(B^{\prime\prime})\backslash\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{Q})(\mathbb{A}_{f})/K_{\\{2\\}}U^{p}.$

  * •

The singular locus $\Sigma_{p}(B)$ can be parametrized by the following double
coset

(6.9)
$\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{Q})\backslash\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{A}_{f})/K(p)U^{p}.$

###### Proof.

Recall there are two kinds of irreducible components of
$\mathcal{M}_{K^{\prime}(p)}$ and they correspond to vertex lattices of type
$0$ and vertex lattice of type $2$. Therefore by (6.6) the irreducible
components of type $0$ can be parametrized by

(6.10) $I(\mathbb{Q})\backslash
J_{b}(\mathbb{Q}_{p})/{K_{\\{0\\}}}\times\mathrm{GU}_{2}(B)(\mathbb{A}^{(p)}_{f})/U^{p}.$

One then sees that this is exactly

$\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{Q})\backslash\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{A}_{f})/K_{\\{0\\}}U^{p}.$

Similarly, the irreducible components of type $2$ can be parametrized by

(6.11) $I(\mathbb{Q})\backslash
J_{b}(\mathbb{Q}_{p})/{K_{\\{2\\}}}\times\mathrm{GU}_{2}(B)(\mathbb{A}^{(p)}_{f})/U^{p}.$

One then sees that this is exactly

$\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{Q})\backslash\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{A}_{f})/K_{\\{2\\}}U^{p}.$

By Lemma 6.4, the singular locus is exactly the superspecial locus which
correspond to vertex lattices of type $1$. Therefore by (6.6), it is
parametrized by the double coset

(6.12) $I(\mathbb{Q})\backslash
J_{b}(\mathbb{Q}_{p})/{K(p)}\times\mathrm{GU}_{2}(B)(\mathbb{A}^{(p)}_{f})/U^{p}.$

One then sees that this is exactly

(6.13)
$\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{Q})\backslash\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{A}_{f})/K(p)U^{p}.$

∎

Recall our notations for the quaternion algebras: $B=B_{pq}$,
$B^{\prime}=B_{p\infty}$ and $B^{\prime\prime}=B_{q\infty}$. The following
observation will be important in the subsequent discussions.

######  Corollary 6.14.

The singular locus $\Sigma_{p}(K(pq))$ of $X_{K(pq),\bar{\mathbb{F}}_{p}}$ is
parametrized by the same set as the singular locus $\Sigma_{q}(B)$ of
$X^{B}_{K^{\prime}(q),\bar{\mathbb{F}}_{q}}$. More precisely, we have

(6.15)
$\Sigma_{p}(K(pq))=\mathrm{GU}_{2}(B^{\prime})(\mathbb{Q})\backslash\mathrm{GU}_{2}(B^{\prime})(\mathbb{A}_{f})/K^{\prime}(p)K(q)U^{pq}=\Sigma_{q}(B).$

Symmetrically, The singular locus $\Sigma_{q}(K(pq))$ of
$X_{K(pq),\bar{\mathbb{F}}_{q}}$ is parametrized by the same set as the
singular locus $\Sigma_{p}(B)$ of
$X^{B}_{K^{\prime}(p),\bar{\mathbb{F}}_{p}}$. More precisely, we have

(6.16)
$\Sigma_{q}(K(pq))=\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{Q})\backslash\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{A}_{f})/K(p)K^{\prime}(q)U^{pq}=\Sigma_{p}(B).$

###### Proof.

This follows easily from Proposition 4.7 and Proposition 6.7. ∎

##  7\. Ribet’s principle for paramodular Siegel modular forms

###  7.1. Ribet’s principle

Now we come to state the second main theorem of this article which we call
Ribet’s principle for the paramodular Siegel modular forms. As in the proof of
the Mazur’s prinicple we will make the following semi-simplicity assumption

(Semi-pq)
$\mathrm{H}^{3}_{!}(X_{K(pq),\bar{\mathbb{Q}}},\mathbb{V}^{\mathrm{int}}_{\mu})_{\mathfrak{m}}\otimes
k$ is semisimple as a Galois module.

######  Theorem 7.1.

Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GSp}_{4}$ that
is cohomological which is non-CAP and non-endoscopic whose component at
infinity $\pi_{\infty}$ is a holomorphic discrete series of Harish Chandra
parameter $(a,b,-a-b+3)$ with $a\geq b\geq 0$ such that $l\geq a+b+4$. Let
$p,q$ be two distinct primes different from $l$. Let
$U=K(p)K(q)U^{pq}\subset\mathrm{GSp}_{4}(\mathbb{Q}_{p})\mathrm{GSp}_{4}(\mathbb{Q}_{q})\mathrm{GSp}_{4}(\mathbb{A}^{(pq)}_{f})$
be a neat open compact subgroup such that $\pi^{U}\neq 0$. We assume that
$\pi_{p}$ and $\pi_{q}$ are both ramified of type IIa. Suppose the residual
Galois representation $\bar{\rho}_{\pi,l}$ satisfies the following assumptions

  * •

$\bar{\rho}_{\pi,l}$ is unramified at $p$ and is ramified at $q$;

  * •

$\bar{\rho}_{\pi,l}$ satisfies (Irr);

  * •

$\bar{\rho}_{\pi,l}$ satisfies (Semi-pq);

Then there exists a cuspidal automorphic representation $\pi^{\prime}$ with
the same type as $\pi$ at $\infty$ such that
$\bar{\rho}_{\pi,l}\cong\bar{\rho}_{\pi^{\prime},l}$ and $\pi^{\prime}_{p}$ is
unramified.

###  7.2. Proof of Ribet’s principle

Recall that we have defined the Hecke algebra $\mathbb{T}_{pq}$ in (3.8). Let
$\mathfrak{m}$ be the maximal ideal corresponding to $\bar{\rho}_{\pi,l}$ in
$\mathbb{T}_{pq}$. Then by our assumption (Semi-pq), we have

(7.2)
$\mathrm{H}^{3}_{!}(X_{K(pq),\bar{\mathbb{Q}}},\mathbb{V})_{\mathfrak{m}}\otimes
k=\bar{\rho}^{\oplus s}_{\pi,l}.$

for some positive integer $s$.

Let $U$ be an open compact subgroup of $\mathrm{GU}_{2}(B)(\mathbb{A}_{f})$.
Consider $X^{B}_{U}$ the quaternionic Siegel threefold whose
$\mathbb{C}$-points is given by the symmetric space

(7.3)
$X^{B}_{U}(\mathbb{C})=\mathrm{GU}_{2}(B)(\mathbb{Q})\backslash(\mathrm{GSp}_{4}(\mathbb{R})/K^{h}\times\mathrm{GU}_{2}(B)(\mathbb{A}_{f}))/U.$

Then $X^{B}_{U}$ has a canonical model over $\mathbb{Q}$. Since
$\mathrm{GU}_{2}(B)(L)=\mathrm{GSp}_{4}(L)$ for a suitable quadratic extension
$L/\mathbb{Q}$, we can define a $\mathbb{Q}_{l}$-local system
$\mathbb{V}_{\mu}$ exactly the same way as in (3.9) for each $\mu=(a,b;c)\in
X^{*}(T)$ on $X^{B}_{U}$. As before we impose the assumption that $l>a+b+4$.
Then under this assumption, we define
$\mathbb{V}=\mathbb{V}^{\mathrm{int}}_{\mu}$ an integral structure of
$\mathbb{V}_{\mu}$ on $X^{B}_{U}$.

Now let $U^{pq}$ be the small open compact subgroup that we have fixed before.
Let $U=K(p)K(q)U^{pq}$ and $U^{\prime}=K^{\prime}(p)K^{\prime}(q)U^{pq}$. We
set $X^{B}=X^{B}_{U^{\prime}}$ and $X_{K(pq)}=X_{U}$. By the Jacquet-Langlands
correspondence for $\mathrm{GSp}_{4}$ [RW19], we have an injection

$\mathrm{H}^{3}_{!}(X^{B}_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu})\lhook\joinrel\longrightarrow\mathrm{H}^{3}_{!}(X_{K(pq),\bar{\mathbb{Q}}},\mathbb{V}_{\mu}).$

The subspace $\mathrm{H}^{3}_{!}(X^{B}_{\bar{\mathbb{Q}}},\mathbb{V}_{\mu})$
is preserved by the Hecke algebra $\mathbb{T}_{pq}$. Therefore

(7.4)
$\mathrm{H}^{3}_{!}(X^{B}_{\bar{\mathbb{Q}}},\mathbb{V})_{\mathfrak{m}}\otimes
k=\bar{\rho}_{\pi,l}^{\oplus t}$

for some non-negative integer $t$ under the assumption (Semi-pq). Note that
$t$ is positive in our case, since in the decomposition

(7.5)
$\mathrm{H}^{3}_{!}(X^{B}_{\bar{\mathbb{Q}}},\mathbb{V})_{\mathfrak{m}}\otimes
E=\bigoplus_{\pi^{\prime}}\rho^{m(\pi^{\prime})}_{\pi^{\prime},l}$

where $\pi^{\prime}$ runs through all the representations that is congruent to
$\pi$, there is at least one $\pi^{\prime}$ such that $\pi^{\prime}_{p}$ and
$\pi^{\prime}_{q}$ are both ramified. Our strategy is to prove the theorem by
contradiction and therefore we assume that all the representation
$\pi^{\prime}$ appear in the following equations

(7.6)
$\begin{split}&\mathrm{H}^{3}_{!}(X^{B}_{\bar{\mathbb{Q}}},\mathbb{V})_{\mathfrak{m}}\otimes
E=\bigoplus_{\pi^{\prime}}\rho^{n(\pi^{\prime})}_{\pi^{\prime},l}\\\
&\mathrm{H}^{3}_{!}(X_{K(pq),\bar{\mathbb{Q}}},\mathbb{V})_{\mathfrak{m}}\otimes
E=\bigoplus_{\pi^{\prime}}\rho^{m(\pi^{\prime})}_{\pi^{\prime},l}\\\
\end{split}$

are ramified at $p$. As we have assumed that $\bar{\rho}_{\pi^{\prime},l}$ is
ramified at $q$, $\pi_{q}$ is necessarily ramified. Therefore it follows that
in particular $t$ in (7.4) is positive. Our analysis relies on the Picard-
Lefschetz formulas for $X_{K(pq),\bar{\mathbb{Q}}_{q}}$ and
$X^{B}_{\bar{\mathbb{Q}}_{p}}$, that is the following diagrams

(7.7)
$\begin{array}[c]{ccccc}\mathrm{H}^{3}(X_{K(pq),\bar{\mathbb{Q}}_{q}},\mathbb{V})(1)&\xrightarrow{\alpha}&\bigoplus_{x\in\Sigma_{q}(K(pq))}R^{3}\Phi(\mathbb{V})_{x}(1)\\\
\Big{\downarrow}\scriptstyle{N}&&\Big{\downarrow}\scriptstyle{N_{x}}\\\
\mathrm{H}^{3}_{c}(X_{K(pq),\bar{\mathbb{Q}}_{q}},\mathbb{V})&\xleftarrow{\beta}&\bigoplus_{x\in\Sigma_{q}(K(pq))}\mathrm{H}^{3}_{x}(X_{K(pq),\bar{\mathbb{F}}_{q}},\mathbb{V}).\end{array}$

(7.8)
$\begin{array}[c]{ccccc}\mathrm{H}^{3}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})(1)&\xrightarrow{\alpha}&\bigoplus_{x\in\Sigma_{p}(B)}R^{3}\Phi(\mathbb{V})_{x}(1)\\\
\Big{\downarrow}\scriptstyle{N}&&\Big{\downarrow}\scriptstyle{N_{x}}\\\
\mathrm{H}^{3}_{c}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})&\xleftarrow{\beta}&\bigoplus_{x\in\Sigma_{p}(B)}\mathrm{H}^{3}_{x}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V}).\end{array}$

######  Proposition 7.9.

Suppose that all the $\pi^{\prime}$ are ramified in (7.6). The map $\beta$
localized at $\mathfrak{m}$ in (7.6) is injective

(7.10)
$\beta:\bigoplus_{x\in\Sigma_{p}(B)}\mathrm{H}^{3}_{x}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\rightarrow\mathrm{H}^{3}_{c}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}$

and therefore

(7.11)
$\alpha:\mathrm{H}^{3}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}(1)\rightarrow\bigoplus_{x\in\Sigma_{p}(B)}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}(1).$

is surjective.

###### Proof.

Since
$\mathrm{H}^{3}_{c}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})\rightarrow\mathrm{H}^{3}_{c}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})$
is injective, we need to show the Gysin map

$\bigoplus_{x\in\Sigma_{p}(B)}\mathrm{H}^{3}_{x}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})\rightarrow\mathrm{H}^{3}_{c}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})$

is injective after localizing at $\mathfrak{m}$. Note the kernel of this
morphism is the image of the connecting homomorphism

$\mathrm{H}^{2}_{c}(U,\mathbb{V})\rightarrow\bigoplus_{x\in\Sigma_{p}(B)}\mathrm{H}^{3}_{x}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})$

where $U=X^{B}_{\bar{\mathbb{F}}_{p}}-\Sigma_{p}(B)$. We claim that
$\mathrm{H}^{2}_{c}(U,\mathbb{V})=0$ after localizing at $\mathfrak{m}$. To
prove this, we use the following excision exact sequence

$\mathrm{H}^{2}_{S_{K^{\prime}(p)}-\Sigma_{p}(B)}(X^{B}_{\bar{\mathbb{F}}_{p}}-\Sigma_{p}(B),\mathbb{V})\rightarrow\mathrm{H}^{2}_{c}(U,\mathbb{V})\rightarrow\mathrm{H}^{2}_{c}(X^{B}_{\bar{\mathbb{F}}_{p}}-S_{K^{\prime}(p)},\mathbb{V}).$

Note that $X^{B}_{\bar{\mathbb{F}}_{p}}-\Sigma_{p}(B)$ is now smooth and
$S_{K^{\prime}(p)}-\Sigma_{p}(B)$ is of codimension $1$ in
$X^{B}_{\bar{\mathbb{F}}_{p}}-\Sigma_{p}(B)$. Then by purity,
$\mathrm{H}^{2}_{S_{K^{\prime}(p)}-\Sigma_{p}(B)}(X^{B}_{\bar{\mathbb{F}}_{p}}-\Sigma_{p}(B),\mathbb{V})=H^{0}(S_{K^{\prime}(p)}-\Sigma_{p}(B),\mathbb{V})$.
Note the set of components of $S_{K^{\prime}(p)}-\Sigma_{p}(B)$ can be
identified with the union of discrete Shimura varieties

$X^{B^{\prime\prime}}_{K_{\\{0\\}}}\sqcup X^{B^{\prime\prime}}_{K_{\\{2\\}}}$

where

$\begin{split}&X^{B^{\prime\prime}}_{K_{\\{0\\}}}=\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{Q})\backslash\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{A}_{f})/K_{\\{0\\}}U^{p}\\\
&X^{B^{\prime\prime}}_{K_{\\{2\\}}}=\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{Q})\backslash\mathrm{GU}_{2}(B^{\prime\prime})(\mathbb{A}_{f})/K_{\\{2\\}}U^{p}.\\\
\end{split}$

Note that we also have a natural injection

$H^{0}(S_{K^{\prime}(p)}-\Sigma_{p}(B),\mathbb{V})\hookrightarrow
H^{0}(X^{B^{\prime\prime}}_{K_{\\{0\\}}},\mathbb{V})\oplus
H^{0}(X^{B^{\prime\prime}}_{K_{\\{2\\}}},\mathbb{V}).$

Since $K_{\\{0\\}}$ and $K_{\\{2\\}}$ are both hyperspecial, the set
$H^{0}(S_{K^{\prime}(p)}-\Sigma_{p}(B),\mathbb{V})$ only supports those
representations that are unramified at $p$. Therefore
$H^{0}(S_{K^{\prime}(p)}-\Sigma_{p}(B),\mathbb{V})$ vanishes when localized at
$\mathfrak{m}$ by our assumption. Also notice that
$X^{B}_{\bar{\mathbb{F}}_{p}}-S_{K^{\prime}(p)}$ is the ordinary locus. Let
$X^{B*}_{\bar{\mathbb{F}}_{p}}$ be the minimal compactification of
$X^{B}_{\bar{\mathbb{F}}_{p}}$, then
$X^{B*}_{\bar{\mathbb{F}}_{p}}-S_{K^{\prime}(p)}$ is affine and therefore

$\mathrm{H}^{2}_{c}(X^{B*}_{\bar{\mathbb{F}}_{p}}-S_{K^{\prime}(p)},\mathbb{V})=0$

by Artin vanishing. Since the difference between

$\mathrm{H}^{2}_{c}(X^{B*}_{\bar{\mathbb{F}}_{p}}-S_{K^{\prime}(p)},\mathbb{V})=0$

and

$\mathrm{H}^{2}_{c}(X^{B}_{\bar{\mathbb{F}}_{p}}-S_{K^{\prime}(p)},\mathbb{V})=0$

is supported on the boundary of the minimal compactification, the difference
vanishes after localizing at $\mathfrak{m}$ by our assumption (Irr). Therefore
$\mathrm{H}^{2}_{c}(X^{B}_{\bar{\mathbb{F}}_{p}}-S_{K^{\prime}(p)},\mathbb{V})$
vanishes after localizing at $\mathfrak{m}$. Then we conclude that
$\mathrm{H}^{2}_{c}(U,\mathbb{V})=0$ vanishes after localizing at
$\mathfrak{m}$. ∎

The above Proposition gives the following fundamental diagram

(7.12)
${\bigoplus_{x\in\Sigma_{p}(B)}\mathrm{H}^{3}_{x}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}}$${\mathrm{H}^{3}_{c}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}}$${\mathrm{H}^{3}_{!}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}}$${\bigoplus_{x\in\Sigma_{p}(B)}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}.}$$\scriptstyle{\beta^{\prime}}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha}$

######  Proposition 7.13.

Suppose that those representation $\pi^{\prime}$ appearing in (7.5) are all
ramified at $p$. Then

(7.14)
$\dim_{k}\bigoplus_{x\in\Sigma_{p}(B)}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
k=2t.$

###### Proof.

Take the fundamental diagram modulo $l$ gives

${\bigoplus_{x\in\Sigma_{p}(B)}\mathrm{H}^{3}_{x}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k}$${\mathrm{H}^{3}_{c}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k}$${\mathrm{H}^{3}_{!}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k}$${\bigoplus_{x\in\Sigma_{p}(B)}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
k.}$$\scriptstyle{\beta^{\prime}\otimes k}$$\scriptstyle{\gamma\otimes
k}$$\scriptstyle{\alpha\otimes k}$

Since
$\mathrm{H}^{3}_{!}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k=\bar{\rho}_{\pi,l\mid G_{\mathbb{Q}_{p}}}^{\oplus t}$ and
$\bar{\rho}_{\pi,l\mid G_{\mathbb{Q}_{p}}}$ is unramified at $p$, the rank one
monodromy $N\otimes k$ degenerates and has rank $0$. By the mod $l$ version of
the Picard-Lefschetz formula

(7.15)
$\begin{array}[c]{ccccc}\mathrm{H}^{3}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}(1)\otimes
k&\xrightarrow{\alpha\otimes
k}&\bigoplus_{x\in\Sigma_{p}(B)}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}(1)\otimes
k\\\ \Big{\downarrow}\scriptstyle{N\otimes
k}&&\Big{\downarrow}\scriptstyle{N_{x}}\otimes k\\\
\mathrm{H}^{3}_{c}(X^{B}_{\bar{\mathbb{Q}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k&\xleftarrow{\beta\otimes
k}&\bigoplus_{x\in\Sigma_{p}(B)}\mathrm{H}^{3}_{x}(X^{B}_{\bar{\mathbb{F}}_{p}},\mathbb{V})_{\mathfrak{m}}\otimes
k\end{array}$

and the fact that
$\bigoplus_{x\in\Sigma_{p}(B)}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}$ is
obviously torsion free, we obtain that $\beta^{\prime}\otimes k$ is zero and
$\gamma\otimes k$ is injective exactly the same way as in the proof of the
Mazur’s principle. Therefore
$\bigoplus_{x\in\Sigma_{p}(B)}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes k$
contributes to $\bar{\rho}_{\pi,l\mid G_{\mathbb{Q}_{p}}}$ as the two
dimensional quotient which is isomorphic to the reduction modulo $l$ of the
twisted Steinberg part of the Weil-Deligne representation in (3.4). Then it
follows that

(7.16)
$\dim_{k}\bigoplus_{x\in\Sigma_{p}(B)}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
k=2t.$

∎

######  Proposition 7.17.

We have

(7.18)
$\dim_{k}\bigoplus_{x\in\Sigma_{q}(K(pq))}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
k=s.$

###### Proof.

Consider the mod $l$ fundamental diagram in this case

${\bigoplus_{x\in\Sigma_{q}(K(pq))}\mathrm{H}^{3}_{x}(X_{K(pq),\bar{\mathbb{F}}_{q}},\mathbb{V})_{\mathfrak{m}}\otimes
k}$${\mathrm{H}^{3}_{c}(X_{K(pq),\bar{\mathbb{F}}_{q}},\mathbb{V})_{\mathfrak{m}}\otimes
k}$${\mathrm{H}^{3}_{!}(X^{B}_{\bar{\mathbb{Q}}_{q}},\mathbb{V})_{\mathfrak{m}}\otimes
k}$${\bigoplus_{x\in\Sigma_{q}(K(pq))}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
k}$$\scriptstyle{\beta^{\prime}\otimes k}$$\scriptstyle{\gamma\otimes
k}$$\scriptstyle{\alpha\otimes k}$

and the mod $l$ version of the Picard-Lefschetz formula

(7.19)
$\begin{array}[c]{ccccc}\mathrm{H}^{3}(X_{K(pq),\bar{\mathbb{Q}}_{q}},\mathbb{V})_{\mathfrak{m}}(1)\otimes
k&\xrightarrow{\alpha\otimes
k}&\bigoplus_{x\in\Sigma_{q}(K(pq))}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}(1)\otimes
k\\\ \Big{\downarrow}\scriptstyle{N\otimes
k}&&\Big{\downarrow}\scriptstyle{N_{x}}\otimes k\\\
\mathrm{H}^{3}_{c}(X_{K(pq),\bar{\mathbb{Q}}_{q}},\mathbb{V})_{\mathfrak{m}}\otimes
k&\xleftarrow{\beta\otimes
k}&\bigoplus_{x\in\Sigma_{q}(K(pq))}\mathrm{H}^{3}_{x}(X_{K(pq),\bar{\mathbb{F}}_{q}},\mathbb{V})_{\mathfrak{m}}\otimes
k.\end{array}$

Since $\bar{\rho}_{\pi,l\mid G_{\mathbb{Q}_{q}}}$ is ramified, $N\otimes k$
does not degenerate and still has rank one. Therefore $\beta\otimes k$ is in
fact injective and contributes to $\bar{\rho}_{\pi,l\mid G_{\mathbb{Q}_{q}}}$
as a one dimensional subspace. It follows then that

$\bigoplus_{x\in\Sigma_{q}(K(pq))}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}(1)\otimes
k$

contributes to $\bar{\rho}_{\pi,l\mid G_{\mathbb{Q}_{q}}}$ as a one
dimensional quotient since $N_{x}\otimes k$ is an isomorphism. Then

$\dim_{k}\bigoplus_{x\in\Sigma_{q}(K(pq))}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
k=s.$

∎

Recall that in Corollary 6.14, we have proved that $\Sigma_{p}(B)$ and
$\Sigma_{q}(K(pq))$ can be identified. It follows immediately that $s=2t$.
Note also that if we switch the role of $pq$ in the previous arguments, we
will obtain that

$\begin{split}&\dim_{k}\bigoplus_{x\in\Sigma_{p}(K(pq))}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
k=2s\\\
&\dim_{k}\bigoplus_{x\in\Sigma_{q}(B)}R^{3}\Phi(\mathbb{V})_{x,\mathfrak{m}}\otimes
k=t.\\\ \end{split}$

By Corollary 6.14 again, we have $t=2s$. Therefore we obtain that $s=t=0$
which is obviously impossible. Hence Ribet’s principle is proved.

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