We study the Picard-Lefschetz formula for the Siegel modular threefold of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology with arbitrary automorphic coefficients. We give some applications to the Langlands programme: Using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet-Langlands correspondence between $\gsp_4$ and a definite inner form, proving a conjecture of Ibukiyama <cit.>. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of $\gsp_4$.

§ INTRODUCTION

In this paper we will study the cohomology of paramodular Siegel threefolds, using geometric results of <cit.> and the Picard-Lefschetz formula of <cit.>. Our first main result (Theorem <ref>) is a geometric Jacquet-Langlands correspondence for such paramodular Siegel threefolds, which proves a conjecture of Ibukiyama <cit.> that we will explain below. Our second main result (Theorem <ref>) is a level lowering result for cohomological cuspidal automorphic representations of $\gsp_4$, in the spirit of Mazur's level lowering results for modular forms. Our main technical result (Theorem <ref>) is the weight-monodromy conjecture for the inner cohomology of Siegel threefolds over $\mathbb{Q}_p$ with paramodular level structure at $p$.

§.§ Ibukiyama's conjecture

Let $D$ be the quaternion algebra over $\qqq$ that is non-split over $\rr$ and $\qqq_p$ for a prime $p$ and split at all other places. Let $S_k[\Gamma_0(p)]^{\text{new}}$ be the space of modular forms of weight $k \ge 2$ and level $\Gamma_0(p)$ that are $p$-new. A classical result of Eichler gives a Hecke equivariant injection (surjective for $k \ge 3$)

\begin{align}
S_k[\Gamma_0(p)]^{\text{new}} \xhookrightarrow{} \mathcal{A}^{D^{\times}}_k[\mathcal{O}_D^{\times}], \label{new}
\end{align}

where $\mathcal{A}^{D^{\times}}_k[\mathcal{O}_D^{\times}]$ denotes the space of algebraic modular forms for $D^{\times}$ of weight $k$ and level $\mathcal{O}_D^{\times}$. These can be described explicitly as `functions' on the (finite) class set of $D$ which are easier to understand computationally than modular forms. In fact this description is used in practice to compute bases of newforms for spaces of modular forms. This correspondence is a special case of the Jacquet-Langlands correspondence between the algebraic group $\gl_2/\qqq$ and its inner form $D^{\times}/\qqq$, proven in <cit.> using the trace formula.

The quaternionic similitude group $G=GU_2(D)$ associated with $D$ is an inner form of $\gsp_4$, such that $G(\rr)$ is compact modulo centre. General conjectures of Langlands predict a transfer from automorphic representations of $G$ to automorphic representations of $\gsp_4$. A particular instance of this transfer was first conjectured by Ihara and Ibukiyama <cit.> and later extended by Ibukiyama. We will state a slight reformulation of Conjecture 5.1 of <cit.> below. Let $S_{k,j}[\para(p)]$ denote the space of Siegel modular forms of weight $k,j$ and level $\para(p)$, where $\para(p)$ is the paramodular group (c.f. Section <ref>). Let $\mathcal{A}^G_{k,j}[K_2(p)]$ denote the space of algebraic modular forms for $G$ of weight $k,j$ and level $K_2(p)$, where $K_2(p)$ is an analogue of the paramodular group (c.f. Section <ref>).

For $k \ge 0$ and for $j \ge 3$ there is an injective map

\begin{align}
S_{k,j}[\para(p)]^{\text{new}} \xhookrightarrow{} \mathcal{A}^G_{k,j}[K_2(p)], \label{newer}
\end{align}

which is Hecke-equivariant for the prime-to-$p$ Hecke operators.

This conjecture can be used to do explicit computations with Siegel modular forms, we give some examples: In <cit.> the author constructs an algorithm computing Hecke eigenvalues of Siegel modular forms, assuming a form of Conjecture <ref>; in <cit.> the authors do extensive computations with algebraic modular forms for $G$; in <cit.> the author uses the conjecture to find computational evidence for Harder's conjecture on congruences between Siegel modular forms and elliptic modular forms.

Our approach to this conjecture is via the characteristic $p$ geometry of paramodular Siegel threefolds $X/\zz_p$. The singularities of $X$ were studied by Yu in <cit.> and his results allow us to use the Picard-Lefschetz formula of <cit.>. Using results of <cit.> and <cit.> we can identify the finite set of singular points of $X \otimes \mathbb{F}_p$ with an adelic double quotient of the form

\begin{align}
G(\qqq) \setminus G(\mathbb{A}^{\infty})/(U^p K_2(p)).
\end{align}

Algebraic modular forms for $G$ are basically functions on this finite set and the Picard-Lefschetz formula gives us a map from the space of these algebraic modular forms to the middle cohomology of $X_{\overline{\qqq}_p}$. This map has an interpretation in terms of the action of the inertia group and our first result concerns this action.

Let $X/\qqq_p$ be the Siegel threefold of neat level $U=U^p K(p)$ and let $\mathbb{V}$ be an automorphic local system of $L$-vector spaces where $L/\qqq_\ell$ is a finite extension ($\ell \not=p)$. Then the weight-monodromy conjecture holds for $H^3_{!}(X_{\overline{\qqq}_p}, \mathbb{V})$.
For automorphic local systems of sufficiently regular weight, our proof is geometric and uses the cohomological vanishing theorems of <cit.>. For automorphic local systems of singular weight we have to combine our geometric argument with an automorphic characterisation of the cohomology of Siegel threefolds (c.f. <cit.>). In particular we use the fact that the inner cohomology is `equal' to the cuspidal cohomology. We note that it is not a new idea to use automorphic characterisations of cohomology to prove the weight-monodromy conjecture for the cohomology of Shimura varieties, c.f. <cit.>.

At this point we have all the ingredients to prove Conjecture <ref>, but we will actually prove a little bit more. We can work with arbitrary level away from $p$ to transfer certain cuspidal automorphic representations $\pi$ of $\gsp_4$ to $G$, which is a geometric incarnation of the Jacquet-Langlands correspondence. We will state a short version of our main theorem below, see Theorem <ref> for a more general statement.

* Let $\pi$ be a non-endoscopic cohomological cuspidal automorphic representation of $\gsp_4$ such that $\pi_{\infty}$ is in the discrete series and such that $\pi_p$ is ramified and $K(p)$-spherical. Then there is a cuspidal automorphic representation $\sigma$ of $G$ such that $\pi_v \cong \sigma_v$ for finite places $v \not=p$, such that $\sigma_p$ is $K_2(p)$-spherical and with $\sigma_{\infty}$ determined by $\pi_{\infty}$. Moreover, $\sigma$ occurs with multiplicity one in the cuspidal spectrum of $G$.

* Let $k \ge 0, j \ge 3$ and let $N$ be a squarefree integer with $p \mid N$, then there is an injective map

\begin{align}
\varphi:S_{k,j}[\para(N)]^{p-\text{new}} \xhookrightarrow{} \mathcal{A}^G_{k,j}[K_2(N)]
\end{align}

equivariant for the prime-to-$p$ Hecke operators, which proves Conjecture <ref>.

Sorensen constructs a Jacquet-Langlands transfer in <cit.> using the stable trace formula. However he has to assume that there is a certain auxiliary prime $q \not=p$ where $\sigma_q$ is ramified enough, which is never satisfied for paramodular forms of squarefree level. Moreover, while I was writing up this paper, Weissauer and Rösner informed me that they can construct a general Jacquet-Langlands correspondence between $\gsp_4$ and its inner forms, also using trace formula methods, see their forthcoming article <cit.>.

Ibukiyama gives a conjectural characterisation of the image of $\varphi$ when $N=p$. We will give a different characterisation of its image in Theorem <ref>, which is probably easier to verify in practice. We will also discuss the image of the map $\pi \mapsto \sigma$ in Theorem <ref>.

§.§ Mazur's principle

In this section we describe a level lowering result for cohomological cuspidal automorphic representations of $\gsp_4$. We start by recalling a classical result of Mazur concerning levels of modular forms, which answers the following question: Given a normalised eigenform $f \in S_2[\Gamma_0(Np)]$ that is $p$-new, is there a congruence $f \cong g \mod \ell$, for some $\ell \not=p$, with $g \in S_2[\Gamma_0(N)]$? We can translate this congruence into an isomorphism of mod $\ell$ Galois representations $\overline{\rho}_{f,l} \cong \overline{\rho}_{g,l}$. Since the Galois representation $\rho_{g,l}$ is unramified at $p$, a necessary condition for such a congruence to exist is that the Galois representation $\overline{\rho}_{f,l}$ is unramified at $p$, and Serre conjectured in <cit.> that the converse should be true. Ribet proved this conjecture in <cit.>, which was famously used to show that modularity of semi-stable elliptic curves implies Fermat's last theorem. Below we state a slightly weaker version of Ribet's theorem, due to Mazur:
[Mazur's principle, Theorem 6.1 in <cit.>]
Assume that $\overline{\rho_{f,l}}$ is irreducible, unramified at $p$ and that it has two distinct $\frob_p$ eigenvalues (equivalently. $p \not=1 \mod \ell$). Then there is a normalised eigenform $g \in S_2[\Gamma_0(N)]$ with $\overline{\rho_{f,l}} \cong \overline{\rho_{g,l}}$.

Our second main result is the following analogue of Mazur's principle for cohomological cuspidal automorphic representations of $\gsp_4$:

Let $\pi$ be a cuspidal automorphic representation of $\gsp_4$ that is cohomological of weight $a > b > 0$ such that $\pi_p$ is ramified and $K(p)$-spherical. Let $U=U^p \cdot K(p) \subset \gsp_4(\mathbb{A}^{\infty}) \times \gsp_4(\qqq_p)$ be a neat compact open subgroup such that $\pi^U \not=0$. Let $\ell \not=p$ be a prime such that $\overline{\rho_{\pi, \ell}}$ is irreducible, such that the group $U_{\ell}$ is hyperspecial at $\ell$ and such that $\ell>a+b+4$.
Then if $\overline{\rho_{\pi, \ell}}$ is unramified at $p$ and has four distinct $\frob_p$ eigenvalues, there exists a cuspidal automorphic representation $\pi'$, of the same weight and level $U^p$ away from $p$, such that $\overline{\rho_{\pi, \ell}}=\overline{\rho_{\pi',\ell}}$ and such that $\pi'_p$ is unramified.

An essential ingredient in the proof of Mazur's principle is the fact that the component group of the Jacobian of the modular curve $X_0(Np)$ is Eisenstein. Following Jarvis and Rajaei <cit.> we define a cohomological analogue of the component group and show that it vanishes under the assumptions on $a,b,\ell$ and $K_{\ell}$. This vanishing can be seen as an integral refinement of the weight-monodromy conjecture because the component group is defined as the cokernel of a certain monodromy operator.

We need the assumption that $a>b>0$, that $\ell>a+b+4$ and that is $K_{\ell}$ hyperspecial to apply Theorem 10.1 of <cit.>, which is a torsion vanishing result that is used to show that the component group vanishes. Since our proof only needs a torsion vanishing result after localising at a `nice' maximal ideal of the Hecke algebra, it should be possible to relax these assumptions. For example one could hope that an analogue of the torsion-vanishing theorem of Caraiani and Scholze <cit.> holds in this context. The assumption that $\overline{\rho_{\pi, \ell}}$ is irreducible and has four distinct Frobenius eigenvalues is crucial for the actual level lowering argument.

Gee and Geraghty <cit.> prove more general level lowering results for cuspidal automorphic representations of $\gsp_4$, under the assumption that $\overline{\rho}_{\pi,l}$ has large image and is `ordinary' (see Theorem 7.5.2 of op. cit. for a precise statement). Moreover Sorensen proves a potential level lowering result for $\gsp_4$ in <cit.>.

The paramodular group $\para(Np)$ is not neat, so a level lowering result from $S_{k,j}[\para(Np)]$ to $S_{k,j}[\para(N)]$ does not follow. However, it should be possible to prove such a result by cleverly choosing some auxiliary level structure (but we would need some extra conditions on $\overline{\rho_{\pi, \ell}}$, c.f. the main theorem of <cit.>).

§.§ Overview of the paper

We start by giving an overview of the theory of Siegel modular forms and automorphic representations of $\gsp_4(\mathbb{A})$ in Section <ref>. In Section <ref> we discuss the Vogan-Zuckerman classification and Arthur's classification and use these to describe the cohomology of Siegel threefolds. We define algebraic modular forms for $G$ in Section <ref>. In Section <ref> we survey some results on integral models of Siegel threefolds and describe the combinatorics of their supersingular loci explicitly in terms of Shimura sets for $G$. In Section <ref> we discuss nearby cycles and the Picard-Lefschetz formula, where we crucially use results of <cit.> to get off the ground. Section <ref> is the technical heart of the paper where we prove the vanishing of the component group. We deduce the main theorems from this in Section <ref>.

§ AUTOMORPHIC FORMS

In this section we will discuss (cuspidal) automorphic representations of $\gsp_4(\mathbb{A})$. We will first describe classical Siegel modular forms of vector-valued weight for $\gsp_4$, which give rise to such automorphic representations. We then carry out some local computations needed in our proof of Mazur's principle. We end by recalling the Galois representations associated with cohomological cuspidal automorphic representations of $\gsp_4(\mathbb{A})$.

§.§ Siegel modular forms of genus 2

We define the group $\gsp_4$ as the group scheme over $\mathbb{Z}$ defined by the functor sending a commutative ring $R$ to

\begin{align}
\left\{(g, \lambda) \in \operatorname{GL}_4(R) \times R^{\times} \; | \; J = \lambda g J g^t \right\},
\end{align}

where $J=\TJM{0}{1_2}{-1_2}{0}$ and $1_2$ is the $2 \times 2$ identity matrix. The projection map $(g, \lambda) \mapsto \lambda$ is called the similitude character and is denoted by

\begin{align}
\operatorname{sim}:\gsp_4 \to \mathbf{G}_m.
\end{align}

The group $\gsp_4(\rr)$ acts transitively on the Siegel upper half space $\mathbb{H}_2^{\pm}$ defined as

\begin{align}
\{z=x+iy \in \operatorname{Mat}_{2 \times 2}(\mathbb{C}) \; | \; z \text{ symmetric and } y \text{ positive- or negative definite} \}
\end{align}

\begin{align}
\JM{a}{b}{c}{d} \cdot z:=(az+b)(cz+d)^{-1}.
\end{align}

We will define Siegel modular forms as holomorphic functions on $\mathbb{H}_2^{\pm}$ satisfying certain transformation rules. The weights will be given by irreducible representations $\rho_{k,j}$ of $\gl_2(\cc)$ which are of the form

\begin{align}
W_{k,j}:=\sym^k V \otimes \operatorname{det}^j V
\end{align}

for $k \ge 0, j \ge 0$, where $V$ is the standard representation of $\gl_2(\cc)$ on a $2$-dimensional vector space. The levels are given by certain discrete subgroups $\Gamma \subset \gsp_4(\qqq)$.

A Siegel modular form of weight $k,j$ and level $\Gamma$ is a holomorphic function

\begin{align}
f: \mathbb{H}_2 \to W_{k,j}
\end{align}

such that for all $z \in \mathbb{H}_2$ and all $\TJM{a}{b}{c}{d} \in \Gamma$ we have

\begin{align}
f \left(\frac{az+b}{cz+d} \right)&=\rho_{k,j}(cz+d) f(z).
\end{align}

A Siegel modular form is called a cusp-form if $\Phi(f)=0$, where $\Phi(f)$ is the Siegel operator (c.f. <cit.>) and we write $S_{k,j}[\Gamma]$ for the $\cc$-vector space of cusp forms of weight $k,j$ and level $\Gamma$.

We will be particularly interested in the case that $\Gamma=\para(N)$ for some squarefree $N$, where

\begin{align}
\para(N)=\left \{ g \in \gsp_4(\qqq) \cap
\begin{pmatrix}
\mathbb{Z} &  N \zz & \zz &  \zz \\
\zz & \zz & \zz &  N^{-1} \zz \\
\zz & N \zz & \zz &  \zz \\
N \zz & N \zz & N \zz &  \mathbb{Z}
\end{pmatrix} \right \}.
\end{align}

We usually like to think of our levels as compact open subgroups in $\gsp_4(\mathbb{A}^{\infty})$ when translating Siegel modular forms into automorphic representations. For a prime $p$ we define

\begin{align}
H=H(p)&:=\gsp_4(\mathbb{Z}_p) \\
K=K(p)&:= \left \{ g \in \gsp_4(\mathbb{Q}_p) \cap
\begin{pmatrix}
\mathbb{Z}_p & p \mathbb{Z}_p & \mathbb{Z}_p &  \mathbb{Z}_p \\
\mathbb{Z}_p & \mathbb{Z}_p & \mathbb{Z}_p &  p^{-1} \mathbb{Z}_p \\
\mathbb{Z}_p & p\mathbb{Z}_p & \mathbb{Z}_p &  \mathbb{Z}_p \\
p \mathbb{Z}_p & p \mathbb{Z}_p & p \mathbb{Z}_p &  \mathbb{Z}_p
\end{pmatrix}
\; : \operatorname{sim}(g) \in \mathbb{Z}_p^{\times} \right \} \\
Q=Q(p)&:= K(p) \cap H(p),
\end{align}

we cal $K$ the paramodular group and $Q$ the Klingen parahoric. The paramodular group $K(N)$ now corresponds to the compact open subgroup

\begin{align}
\prod_{p \mid N} K(p) \prod_{p \nmid N} \gsp_4(\zz_p) \subset \gsp_4(\mathbb{A}^{\infty}).
\end{align}

We will implictly use the following result throughout the paper (c.f. <cit.>):

Given a normalised eigenform $f \in S_{k,j}(\para(N))$ with $N$ squarefree then there is an associated cuspidal automorphic representation $\pi$ of $\gsp_4(\mathbb{A})$ such that $\pi_{\infty}$ is in the holomorphic discrete series and such that $\pi_{f}$ is unramified for $p \nmid N$ and $\pi_{f}^{K(p)} \not=0$ for $p \mid N$.

§.§ The local Langlands correspondence

This section is about irreducible smooth representations $\pi$ of $\gsp_4(\qqq_p)$, that are ramified and $K(p)$-spherical, and their associated Weil-Deligne representations under the local Langlands correspondence of <cit.>. Table A.13 of <cit.> describes all of these representations and their associated Weil-Deligne representations can be found in Chapter 2.4 of op. cit. These representations break up into $5$ types (IIa, IVc, Vb, Vc, VIc) of which only one (type IIa) is generic (c.f. Table A1 of op. cit.).

Let $v$ be the normalised absolute value $v:\qqq_p^{\times} \to \overline{\qqq_{\ell}}$, let $\sigma$ be an unramified character of $\qqq_p^{\times}$ and let $\chi$ be a character of $\qqq_p^{\times}$ such that $\chi^2 \not= v^{\pm 1}$ and $\chi \not= v^{\pm 3/2}$. We consider all of these as character of the Weil group via local class field theory (uniformizers go to geometric Frobenius elements). The representation $\chi \operatorname{St}_{\gl_2} \rtimes \sigma$ of type IIa is generic and its associated Weil-Deligne representation is given by:

\begin{align}
\begin{pmatrix}
\chi^2 \sigma & 0 & 0 & 0 \\
0 & v^{1/2} \chi \sigma &  0 & 0 \\
0 & 0 & v^{-1/2} \chi \sigma & 0 \\
0 & 0 & 0 & \sigma
\end{pmatrix}, \qquad
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}.
\end{align}

We note that the central character of this representation is $\chi^2 \sigma^2$ which is also equal to the similitude character of the Weil-Deligne representation. Furthermore, the Weil-Deligne representation satisfies the weight-monodromy conjecture, i.e., it is pure in the sense of Taylor-Yoshida <cit.>. More concretely, this means that the monodromy operator has rank `as large as possible'.

The other four types (IVc, Vb, Vc, VIc) are not generic, and their associated Weil-Deligne representations do not satisfy the weight-monodromy conjecture. We will use this observation to deduce genericity of local components of automorphic representations $\pi$ associated with certain paramodular forms of squarefree level $\para(N)$, by proving weight-monodromy for the associated Galois representations.

§.§.§ Atkin-Lehner eigenvalues

The element

\begin{align}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
p & 0 & 0 & 0 \\
0 & -p & 0 & 0
\end{pmatrix}
\end{align}

normalises the paramodular group $K$ and so it acts on the space of invariants $\pi^K$. Moreover since $u^2=p I$ is central we know that $\pi(u^2)$ acts via the central character of $\pi$.

Let $\pi=\chi \operatorname{St}_{\gl_2} \rtimes \sigma$ be a representation of type IIa with $\chi, \sigma$ unramified and central character $\chi^2 \sigma^2$. Then $\pi^K$ is one-dimensional and $u$ acts via the scalar $(\chi \sigma)(p)$

When $\chi \sigma$ is trivial, Roberts and Schmidt compute that
$u$ acts by (c.f. Table A.12 in <cit.>)

\begin{align}
(\chi \sigma)(p)=1,
\end{align}

one can reduce to this case by twisting and using the fact that $\chi, \sigma$ are unramified.

§.§ Galois representations

Galois representations associated with cohomological cuspidal automorphic representations of $\gsp_4(\mathbb{A})$ were constructed in <cit.>, <cit.>, <cit.>. They satisfy local-global compatibility with the local Langlands correspondence of <cit.> by work of Mok <cit.>. To summarise we state the following version of the result of Mok:
[Theorem 3.5 of <cit.>]
Let $\pi$ be a cohomological cuspidal automorphic representation of $\gsp_4(\mathbb{A}_{\qqq})$ that is not CAP. Then for every prime number $\ell$ there exists a continuous Galois representation

\begin{align}
\rho_{\pi, \ell}:\galq \to \gsp_4(\overline{\mathbb{Q}_{\ell}})
\end{align}

* The representation is unramified at primes $p \not= \ell$ where $\pi_p$ is unramified, de Rham at $\ell$ and crystalline at $\ell$ if $\pi_{\ell}$ is unramified.

* For all primes $p \not= \ell$ we have local-global compatibility with the local Langlands correspondence of <cit.>. To be precise we have

\begin{align}
\operatorname{WD} \left( \restr{\rho_{\pi, \ell}}{G_{\mathbb{Q}_p}} \right)^{\text{F-ss}} \cong \operatorname{rec}_p(\pi_p \otimes \operatorname{sim}(v^{3/2})),
\end{align}

where $\operatorname{rec}_p$ is the Langlands reciprocity map of <cit.> and $\text{F-ss}$ denotes Frobenius semi-simplification of a Weil-Deligne representation.

* The representation satisfies

\begin{align}
\rho_{\pi, \ell}^{\vee} \cong \rho_{\pi, \ell} \otimes \operatorname{sim}(\rho_{\pi, \ell})
\end{align}

where $\operatorname{sim}(\rho_{\pi, \ell})$ is the similitude character of $\rho_{\pi, \ell}$.

If $\pi$ is CAP then the above theorem also holds except that local global compatibility is only true up to semi-simplification (this is Proposition 3.4 of <cit.>).

§ COHOMOLOGY OF SIEGEL THREEFOLDS

In this section we will describe the inner cohomology $H^3_{!}$ of Siegel threefolds in terms of cuspidal automorphic representations of $\gsp_4$ using the Vogan-Zuckerman classification (c.f.<cit.>). We then make use of Arthur's classification <cit.> of these cuspidal automorphic representations to make our discussion explicit.

§.§ Automorphic local systems

Let $Y_U$ be the Shimura variety for $(\gsp_4, \mathbb{H}_2^{\pm})$ of level $U=U^{\ell} U_{\ell} \subset \gsp_4(\mathbb{A}^{\infty})$. Irreducible representations $V$ of $\gsp_4(\cc)$ give rise to local systems $\mathbb{V}$ on $Y_U$, called automorphic local systems, as follows (c.f. <cit.>, <cit.>): Choose an isomorphism $\cc \cong \overline{\qqq}_{\ell}$ and consider $V$ as a representation of $\gsp_4(L)$ for some finite extension $L/\qqq_{\ell}$ with ring of integers $\mathcal{O}$ and uniformiser $\varpi$. Let $H=\gsp_4(\mathcal{O})$ be the standard hyperspecial subgroup of $\gsp_4(L)$, then there is a lattice $V_{\mathcal{O}} \subset V$ that is stable under the action of $H$. For every $m \gg 0$ there is an open compact subgroup $H_{\ell}(m)$ such that $H_{\ell}(m)$ acts trivially on $V_{\mathcal{O}}/\varpi^m$ and such that $U^p H_{\ell}(m)  \subset U$ is a normal subgroup. We define a local system $\mathbb{V}_m$ on $Y_U$ by descending the trivial local system $\underline{V_{\mathcal{O}}/\varpi^m}$ along the Galois cover

\begin{align}
Y_{U^{\ell} H_{\ell}(m)} \to Y_U.
\end{align}

This means that the sections of $\mathbb{V}_m$ over an étale map $T \to Y$ are given by functions

\begin{align}
f:\pi_0 \left(Y_{U^{\ell} H(n)}\times_{Y_U} T \right) \to V_{\mathcal{O}}/\varpi^m
\end{align}

satisfying $f(g t)=g^{-1} f(t)$ for $g \in U_{\ell}$. Taking the inverse limit over $m$ we obtain local systems $\mathbb{V}_{\mathcal{O}}$ and $\mathbb{V}=\mathbb{V}_{\mathcal{O}} \otimes_{\mathcal{O}} L$.

Recall that irreducible representations $V$ of $\gsp_4(\cc)$ are parametrized by tuples $(a \ge b \ge 0, c)$, where $a \ge b \ge 0$ determines the restriction of $V$ to $\operatorname{Sp}_4$ and $c$ determines the action of the similitude character. If $\ell>a+b+4$ then the lattice $V_{\mathcal{O}}$ is unique up to homothety because $V_{\mathcal{O}}/\varpi$ is an irreducible representation of $H$, which makes the local system $\mathbb{V}_{\mathcal{O}}$ independent on the choice of lattice. Changing $c$ amounts to Tate-twisting the local system, which has no interesting effect on the cohomology. Therefore we will set $c=a+b$, which normalises things such that $\mathbb{V}_{1,0}$ is the local system corresponding to the (dual of the) relative $\ell$-adic Tate module of the universal abelian variety. In general the local system $\mathbb{V}_{a,b} \otimes L$ occurs as a direct summand of

\begin{align}
\mathbb{V}_{1,0}^{\otimes (a+b)},
\end{align}

and these local systems are self-dual up to a Tate twist.

§.§ Cohomology theories

Before studying the Galois representations that occur in the étale cohomology of $Y_U$, we will first study the Hodge structure on the singular cohomology. Write $\mathbb{V}_{\cc}$ for the $\cc$-linear local system on $Y_U(\mathbb{C})$ of weight $a \ge b \ge 0$. One is usually interested in the intersection cohomology

\begin{align}
H^{\bullet}(\overline{Y}_U(\mathbb{C}), j_{!\ast} \mathbb{V}_{\cc}),
\end{align}

where $j:Y_U \to \overline{Y}_U$ is the inclusion of $Y_U$ into its Baily-Borel compactification $\overline{Y}_U$. By the Zucker conjecture (proven in <cit.> and <cit.>) this is equal to the $L^2$ cohomology

\begin{align}
H^{\bullet}_{(2)}(Y_U(\mathbb{C}), \mathbb{V}_{\cc}),
\end{align}

which can be described in terms of of automorphic representations as follows:

\begin{align}
H^{\bullet}_{(2)}(Y_U(\mathbb{C}), \mathbb{V}_{\cc}) = \bigoplus_{\pi} m(\pi) \pi_{\text{fin}}^U \otimes H^{\bullet}(\mathfrak{g}, K_{\infty} ; \mathbb{V}_{\cc} \otimes \pi_{\infty})
\end{align}

where the sum runs over automorphic representations $\pi = \pi_{\infty} \otimes \pi_{\text{fin}}$ occurring in the discrete spectrum of $L^2(\gsp_4(\qqq) \setminus \gsp_4(\mathbb{A}))$ with multiplicity $m(\pi)$. There is a natural direct summand $H^{\bullet}_{\text{cusp}}(Y_U(\mathbb{C}), \mathbb{V}_{\cc})$ of cuspidal cohomology which is the direct sum over cuspidal automorphic representations $\pi$. Moreover it follows from the results announced in <cit.> and proven in <cit.> that $m(\pi) \le 1$ for all $\pi$.
In this article we will work with the inner cohomology which is defined to be the image of the natural map

\begin{align}
H^{\bullet}_{!}(Y_U(\cc), \mathbb{V}_{\cc}):= \operatorname{Im} \bigg(H^{\bullet}_c(Y_U(\cc), \mathbb{V}_{\cc}) \to H^{\bullet}(Y_U(\cc), \mathbb{V}_{\cc}) \bigg)
\end{align}

where subscript $c$ denotes compactly supported cohomology. In middle degree there is an isomorphism (c.f. <cit.>)

\begin{align}
H^3_{!}(Y_U(\cc), \mathbb{V}_{\cc}) \cong H^3_{\text{cusp}}(Y_U(\cc), \mathbb{V}_{\cc}).
\end{align}

§.§ Vogan-Zuckerman classification

The Vogan-Zuckerman classification (c.f. <cit.>) describes the possible $\pi_{\infty}$ for which

\begin{align}
H^{i}(\mathfrak{g}, K_{\infty} ; \mathbb{V}_{\cc} \otimes \pi_{\infty})
\end{align}

is nonzero, where $\mathbb{V}_{\cc}$ is a local system of weight $a \ge b \ge 0$. If $\pi_{\infty}$ is in the discrete series, then it is always cohomological, and we have listed the other cohomological representations $\pi_{\infty}$ below.

* When $a>b>0$ then the only cohomological $\pi_{\infty}$ are in the discrete series. There are precisely two choices of $\pi_{\infty}$ in the discrete series, which we will denote by $\pi^H$ and $\pi^W$, where the $H$ stands for `holomorphic' and the $W$ for `Whittaker'. The $(\mathfrak{g}, K_{\infty})$-cohomology of these representations is two-dimensional and occurs in degree $3$ with Hodge numbers $(3,0),(0,3)$ and $(2,1),(1,2)$ respectively. We note that the representations $\pi$ with $\pi_{\infty}=\pi^H$ correspond to holomorphic cuspidal Siegel modular forms of weight $a-b, b+3$ (c.f. Proposition <ref>).

* When $b=0$ there is a representation $\pi^1$ whose $(\mathfrak{g}, K^{\infty})$-cohomology is two-dimensional in degrees $2$ and $4$, with Hodge numbers $(2,0), (0,2),(3,1),(1,3)$.

* When $a=b$ there are representations $\pi^{\pm}$ whose $(\mathfrak{g}, K^{\infty})$-cohomology is one-dimensional in degrees $2,4$ with Hodge numbers $(1,1),(2,2)$.

* When $a=b=0$ there are two one-dimensional representations which have one-dimensional $(\mathfrak{g}, K^{\infty})$-cohomology in degrees $0,2,4,6$ with Hodge numbers $(0,0),(1,1),(2,2),(3,3)$.

§.§ Arthur's classification

Arthur has classified the cuspidal automorphic representations $\pi$ in <cit.> into six different types. Only three of these contribute to the inner cohomology in middle degree:

* The cuspidal automorphic representations $\pi$ of general type, which occur with multiplicity one. These are stable which means that both $\pi_{\text{fin}} \otimes \pi^H$ and $\pi_{\text{fin}} \otimes \pi^W$ are cuspidal automorphic. This implies that the four-dimensional Galois representation associated with $\pi$ occurs in the the inner cohomology of the Siegel threefold.

* The cuspidal automorphic representations $\pi$ of Yoshida type which occur with multiplicity $m(\pi)=1$. These are unstable which means that precisely one of $\pi_{\text{fin}} \otimes \pi^H$ and $\pi_{\text{fin}} \otimes \pi^W$ is cuspidal automorphic. These are related to pairs $(\pi_1, \pi_2)$ of cuspidal automorphic representations of $\gl_2$, which we will make precise in Section <ref>.

* The cuspidal automorphic representations $\pi$ of Saito-Kurokawa type which are CAP and occur with multiplicity one in the cuspidal spectrum. They only occur in scalar weight (so $a=b$) and have $\pi_{\infty}$ isomorphic to either $\pi^H, \pi^W, \pi^{+}, \pi^{-}$. These lifts come from cuspidal automorphic representations $\sigma$ of $\gl_2$ of weight $a+b+4$ and the associated Galois representation is

\begin{align}
\rho_{\pi, \ell}=\rho_{\sigma,\ell} \oplus \mathbb{Q}_{\ell}(-b-2) \oplus \mathbb{Q}_{\ell}(-b-1).
\end{align}

This Galois representation cannot occur in the middle degree cohomology by purity (outside a finite set of places the inner cohomology will be pure of weight $a+b+4$). In fact we will only see the two-dimensional Galois representation $\rho_{\sigma,\ell}$ in the middle cohomology.

§.§ Yoshida lifts

We will now describe Yoshida lifts in terms of classical modular forms. Let $\pi_1, \pi_2$ be cuspidal automorphic representations of $\gl_2(\mathbb{A})$ with the same central character and suppose that $\pi_{1, \infty}, \pi_{2,\infty}$ are in the holomorphic discrete series with Harish-Chandra parameters $r_1-1, r_2-2$ satisfying $r_1 > r_2 \ge 2$ with $r_1=a+b+4$ and $r_2=a-b+2$.

Let $\pi$ be a cuspidal automorphic representation of $\gsp_4(\mathbb{A})$ that is not CAP and such that $\pi_{\infty}$ is in the discrete series. We say that $\pi$ is a Yoshida lift of $(\pi_1, \pi_2)$ if for almost all places $v$ the Satake parameter $\pi_v$ agrees with the Satake parameter of $(\pi_1, \pi_2)$ via the inclusion of dual groups $\gl_2 \times_{\mathbb{G}_m} \gl_2 \to \gsp_4$.

Alternatively, we can explicitly construct an $L$-packet $L(\pi_1, \pi_2)$ as the restricted tensor product of local $L$-packets $L(\pi_{1,v}, \pi_{2,v})$ for all places $v$. These local $L$-packets are constructed using the $\theta$-correspondence between $\gsp_4$ and $(\gl_2 \times \gl_2/\mathbb{G}_m)$ and have size two if $v$ is infinite or if $v$ is finite and both $\pi_{1,v}$ and $\pi_{2,v}$ are essentially square-integrable. In all other cases the local $L$-packet has size one and moreover always contains a generic member. However not all members of this $L$-packet are cuspidal, which is quantified in the following result of Weissauer.
[c.f. Theorem 5.2 of <cit.>]
A cuspidal automorphic representation $\pi$ is a Yoshida lift of $(\pi_1, \pi_2)$ if and only if it is a cuspidal member of the $L$-packet $L(\pi_1, \pi_2)$. Moreover, elements $\pi$ of this $L$-packet occur in the cuspidal spectrum with multiplicity

\begin{align}
\end{align}

where $c(\pi)$ is the number of places where $\pi_v$ is non-generic. Moreover $\pi_{\infty}$ is cohomological with respect to the local system of weight $a,b$.

The following result describes what Galois representation occur in the inner cohomology:
[c.f. Corollary 4.2 of <cit.>]
Let $\pi$ be a Yoshida lift of $(\pi_1, \pi_2)$, then $\pi$ is in the discrete series which means that the $(\mathfrak{g}, K_{\infty})$-cohomology of $\pi_{\infty}$ is two-dimensional. The Galois action on this two-dimensional piece is given by

\begin{align}
\twopartdef{\rho_{\pi_{1,l}}}{\pi_{\infty}=\pi^H}{\rho_{\pi_{2,l}}(-b-1)}{\pi_{\infty}=\pi^W}.
\end{align}

We call $\pi$ an irrelevant Yoshida lift if $\pi_p$ is ramified but the associated two-dimensional Galois representation occurring in cohomology is unramified.

If $\pi \in L(\pi_1, \pi_2)$ has nonzero invariant vectors for $\para(N)$ for squarefree $N$, then $L(\pi_1, \pi_2)$ has precisely one cuspidal member $\pi$, with $\pi_v$ generic for all places $v$. In particular there are no holomorphic Yoshida lifts of level $\para(N)$ and so the Galois representation that we see in the cohomology is

\begin{align}
\rho_{\pi_{2,l}}(-b-1).
\end{align}

This is just Remark 3.5 in Section 3.3 of <cit.>.

§.§ Vanishing theorems

In this section we discuss vanishing theorems for

\begin{align}
H^{\bullet}_c(Y_U, \mathbb{V}_{a,b}),
\end{align}

where we will now take $\mathbb{V}_{a,b}$ to be a $\mathcal{O}$-linear local system, note that these are only well defined if $\ell>a+b+4$.

[Theorem 10.1, Corollary 10.2 of <cit.>]
Let $Y_U$ be the Siegel threefold of neat level $U$ and let $\mathbb{V}=\mathbb{V}_{a,b}$ be an automorphic local system of $\mathcal{O}$-modules. Assume that $a > b > 0$, that $U_{\ell}$ is hyperspecial and that $\ell>a+b+4$. Then

\begin{align}
H^3_{!}(Y_{U,\overline{\qqq}}, \mathbb{V}_{\mathcal{O}})
\end{align}

is a free $\mathcal{O}$-module of finite rank and the natural map (with $\mathbb{F}=\mathcal{O}/\varpi$)

\begin{align}
H^3_{!}(Y_{U,\overline{\qqq}}, \mathbb{V}_{\mathcal{O}}) \otimes_{\mathcal{O}} \mathbb{F} \to H^3_{!}(Y_{U,\overline{\qqq}}, \mathbb{V}_{\mathcal{O}} \otimes_{\mathcal{O}} \mathbb{F})
\end{align}

is surjective. Moreover

\begin{align}
H^i_{c}(Y_{U,\overline{\qqq}}, \mathbb{V}_{\mathcal{O}}))&=0 \text{ for } i>d \\
H^i(Y_{U,\overline{\qqq}}, \mathbb{V}_{\mathcal{O}}))&=0 \text{ for } i<d.
\end{align}

Choosing $\ell$ sufficiently large and using comparison results between singular cohomology and étale cohomology we get results for local systems with coefficients in $L$, without restrictions on $\ell$.

If we now use $\mathbb{V}_{a,b}$ to denote an $L$-linear local system of weight $a>b>0$ then we have

\begin{align}
H^i_{c}(Y_{U, \overline{\qqq}}, \mathbb{V}_{a,b})&=0 \text{ for } i>d \\
H^i(Y_{U, \overline{\qqq}}, \mathbb{V}_{a,b})&=0 \text{ for } i<d.
\end{align}

Theorem 10.1 of op. cit. has the assumptions that $\mu=(a,b,c) \in X_{\gsp_{4}}^{++,<\ell}$, that $|\mu|_{\text{re},+}<\ell$ and that $|\mu|'_{\text{comp}} \le \ell-2$. Working through the definitions we find the first condition is equivalent to $a>b>0$ and $\ell \ge a+b+5$, the second condition is equivalent to $\ell \ge a+b+2$ and the third condition is equivalent to $\ell \ge a+b+4$. We conclude that the conditions $a>b>0$ and $\ell>a+b+4$ imply all the hypotheses of Theorem 10.1 of <cit.> in the case of Siegel modular threefolds.

§ ALGEBRAIC MODULAR FORMS

In this section we will discuss algebraic modular forms on the group $G=GU_2(D)$. The general reference for this section is <cit.>, we have also drawn from chapter 3 of <cit.>.

Let $G/\qqq$ be a connected reductive group such that $G(\rr)$ is connected and compact modulo centre, in this paper we will use $G=D^{\times}$ and $G=GU_2(D)$ with $D/\qqq$ a definite quaternion algebra. Let $L/\qqq_{\ell}$ be a finite extension as before and let $V$ be an irreducible algebraic $L$-linear representation of $G(\qqq_{\ell})$. Let $U=U^{\ell}U_{\ell}$ be a compact open subgroup of $G(\mathbb{A}^{\infty})$, then we define the $L$-vector space of algebraic modular forms of level $U$ with coefficients in $V$

\begin{align}
A^{G}_{V}[U]:=\left\{ F: G(\qqq) \setminus G(\mathbb{A}^{\infty})/U^{\ell} \to V \; | \; F(g \cdot u_{\ell}) = u_{\ell}^{-1} F(g) \text{ for } u_{\ell} \in U_{\ell} \right \}.
\end{align}

This vector space is finite dimensional because the double coset spaces

\begin{align}
G(\qqq) \setminus G(\mathbb{A}^{\infty})/U
\end{align}

are finite (Proposition 4.3 of <cit.>). The following result is well known and can be proven as in <cit.>:

Choose an embedding $L \xhookrightarrow{} \mathbb{C}$, then there is a Hecke-equivariant isomorphism

\begin{align}
A^{G}_{V}[U] \otimes_{E} \mathbb{C} \cong \bigoplus_{\sigma} m(\sigma) \sigma^U
\end{align}

where $\sigma$ runs over cuspidal automorphic representations of $G$ such that $\sigma_{\infty} \cong V \otimes_{E} \mathbb{C}$ and $m(\sigma)$ is the multiplicity with which $\sigma$ occurs in the discrete spectrum of $G$.

§.§ The inner form

Let $D$ be the  definite quaternion algebra over $\qqq$ ramified at a single prime $p$ with canonical involution $d \mapsto \overline{d}$ and consider the algebraic group $G/\qqq$ whose $R$ points are given by

\begin{align}
G(R):=\left\{ (g, \lambda) \in \gl_2(D \otimes_{\qqq} R) \times R^{\times} \; | \; ^t\overline{g}g = \lambda I \right\},
\end{align}

where $\overline{g}$ is entry-wise application of $d \mapsto \overline{d}$ and $I$ is the identity matrix. One can show that $G(\rr)/Z(G(\rr)) \cong \operatorname{USp}(4)/ \{ \pm I \}$ where $\operatorname{USp}(4)$ is the compact $\rr$-form of $\operatorname{Sp}_4$ and that for primes $q \not=p$ there are isomorphisms $G(\qqq_q) \cong \gsp_4(\qqq_q)$. Both of these results are worked out in detail in Section 3.5 of Daniel Fretwell's thesis <cit.>. The statement of Conjecture <ref> concerns algebraic modular forms of weight $k,j$ and level $K_2(p)$ and we will discuss these now.

§.§ Weights

If we identify $G_\mathbb{C}$ with $\gsp_4$ and $\cc$ with $\overline{\qqq}_{\ell}$ then for every irreducible representation $V$ of $\gsp_4$ we get an irreducible representation $V$ of $G$ defined over a finite extension $L$ of $\qqq_{\ell}$. These irreducible representations are parametrized by integers $(a \ge b \ge 0, c)$, just like the local systems on Siegel threefolds. Siegel modular forms of weight $k,j$ correspond to the local system of weight $k+j-3, j-3, k+2j-6$ and we define

\begin{align}
\mathcal{A}^G_{k,j}[U]:=\mathcal{A}^G_{V}[U],
\end{align}

where $V$ denotes the irreducible representation of highest weight $(k+j-3,j-3, k+2j-6)$. We will later see that cohomology of $\mathbb{V}_{a,b}$ with support in the supersingular locus of the Siegel modular threefold can be identified with a space of algebraic modular forms for $G$ of weight $a-b,b+3$.

§.§ Levels

Our level subgroups $U \subset G(\mathbb{A}^{\infty})$ will be of the form $U^p U_p$. The level $U^p$ will come from $\gsp_4$ under the isomorphism $G(\mathbb{A}^{\infty,p}) \cong \gsp_4(\mathbb{A}^{\infty,p})$, the level $U_p$ will be a maximal compact subgroup of $G(\qqq_p)$ which we will describe now. Let $B$ be the non-split quaternion algebra over $\mathbb{Q}_p$ with standard involution $b \mapsto \overline{b}$ and uniformiser $\varpi_B$, then $G(\qqq_p)$ can also be described as (because all Hermitian forms on $B^{\oplus 2}$ are equivalent)

\begin{align}
G(\qqq_p) = \left \{(g, \lambda) \in \gl_2(B) \times \qqq_p^{\times} : \overline{g} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} g = \lambda \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right \}.
\end{align}

The algebra $B$ has a unique maximal order $\mathcal{O}_B$ which defines a (maximal) compact open subgroup

\begin{align}
K_1 = \left  \{(g, \lambda) \in \gl_2(\mathcal{O}_B) \times \zz_p^{\times} : \overline{g} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} g = \lambda \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right\}.
\end{align}

which is the stabiliser of the self-dual lattice $\mathcal{O}_B^{2} \subset B^{2}$. There is another maximal compact open subgroup $K_2$ which can be described explicitly as

\begin{align}
K_2&=\left \{(g, \lambda) \in  G(\qqq_p) : g \in \begin{pmatrix} \mathcal{O}_B & \varpi_B \mathcal{O}_B \\ \varpi_B^{-1} \mathcal{O}_B & \mathcal{O}_B \end{pmatrix}, \; \lambda \in \zz_p^{\times} \right \}.
\end{align}

or as the stabiliser of the lattice $\mathcal{O}_B \oplus \varpi_B \mathcal{O}_B$. Now let $N$ be a squarefree integer with $p \mid N$, then we define

\begin{align}
K_2(N):=K_2(p) \times \prod_{v \mid \tfrac{N}{p}} K(v) \prod_{v \nmid N} \gsp_4(\zz_v).
\end{align}

§.§ Theta lifts

In section 3 of <cit.> Ibukiyama defines a notion of '$\theta$-lift' and suggests that these are precisely the algebraic modular forms that do not correspond to holomorphic Siegel modular forms. He predicts (Conjecture 5.3 of <cit.>) that the $\theta$-lifting gives injective maps to $\mathcal{A}^G_{k,j}[K_2(p)]$ from

\begin{align} \label{lifts}
\twopartdef{S_{2j-2+k}[\Gamma_0(1)] \times S_{k+2}[\Gamma_0(p)]^{\text{new}}}{k>0 \text{ or } j \text{ odd}}{S_{2j-2}[\Gamma_0(1)] \times M_{2}[\Gamma_0(p)] }{k=0 \text{ and } j \text{ even}}.
\end{align}

We remark that $\operatorname{dim} M_{2}[\Gamma_0(p)]=1+\operatorname{dim} S_{2}[\Gamma_0(p)]$ so that Ibukiyama's formula essentially predicts a compact version of the Saito-Kurokawa lifting

\begin{align}
S_{2j-2}[\Gamma_0(1)] \to \mathcal{A}^G_{0,j}[K_2(N)]
\end{align}

for odd $j$. We will actually prove that there are injective lifts from (<ref>) to cuspidal automorphic representations of $G$, but it is unclear to us that these are $\theta$-lifts in the sense of Ibukiyama (which is why we do not prove Conjecture 5.1 of <cit.> but rather a slightly weaker statement statement). It will follow from our arguments that most of these lifts (all if $k>0,j>3$) will be weakly endoscopic in the following sense:

A cuspidal automorphic representation $\pi$ of $G$ is called weakly endoscopic if there is a cuspidal automorphic representation $\pi_1 \boxtimes \pi_2$ of $\gl_2(\mathbb{A}) \times \gl_2(\mathbb{A})$ such that the Satake parameters of $\pi$ and $\pi_1 \boxtimes \pi_2$ agree for almost all primes via the inclusion of dual groups $\gl_2 \times_{\mathbb{G}_m} \times \gl_2 \to \gsp_4$.

§ INTEGRAL MODELS OF SIEGEL MODULAR THREEFOLDS

In this section we will discuss three different parahoric level structures at a fixed prime $p$, define integral models and describe their singularities, following <cit.> and <cit.>. We will then study the natural maps between these models and compute the fibers in characteristic $p$. The supersingular locus of the Siegel threefold with good reduction is well understood by classical work of Katsura-Oort <cit.> (c.f. <cit.>). Combining this description with our understanding of the fibers of the natural maps allows us to describe the supersingular loci of the other Siegel threefolds (c.f.<cit.>).

§.§ Moduli functors

Let $U^p \subset \gsp_4(\mathbb{A}^{p,\infty})$ be a fixed compact open subgroup away from $p$. Following section 2 of <cit.>, we define moduli problems for three different parahorics levels at p, which are defined over $\mathbb{Z}_p$.

* The moduli functor $F_H$ of hyperspecial level $H$ which parametrizes prime-to-$p$ isogeny classes of triples $(A, \lambda, \eta)$ where $A$ is an abelian scheme of relative dimension two, $\lambda$ is a prime-to-$p$ polarisation and $\eta$ is a $U^p$ level structure (c.f. <cit.>).

* The moduli functor $F_K$ of paramodular level $K$ which parametrizes prime-to-$p$ isogeny classes of triples $(A, \lambda, \eta)$ where $A$ is an abelian scheme of relative dimension two, $\lambda$ is a polarisation such that $\left(\ker \lambda \right)[p]$ has rank $p^2$ and $\eta$ is a $U^p$ level structure.

* The moduli functor $F_Q$ of Klingen level $Q$ which parametrizes prime-to-$p$ isogeny classes of quadruples $(A, \lambda, \eta, H)$ where $A$ is an abelian scheme of relative dimension two, $\lambda$ is a prime-to-$p$ polarisation, $\eta$ is a $U^p$ level structure and $H \subset A[p]$ is a finite locally free group scheme of rank $p$.

There are natural maps

\begin{equation}
\begin{tikzcd}
& F_{Q} \arrow{dr}{b} \arrow{dl}[swap]{a} \\
F_H && F_K
\end{tikzcd}
\end{equation}

where $a$ the forgetful map and $b$ is the map that takes $A$ to $A/H$ with the induced polarisation and level structure. To elaborate, a sub-group scheme $H \subset A[p]$ of rank $p$ is automatically isotropic for the Weil pairing induced by $\lambda$ and hence the polarisation $\lambda$ descends uniquely to a polarisation on $A/H$ (c.f. Proposition 11.25 of <cit.>). The following result is standard (c.f. <cit.>)

If $U$ is neat (c.f. <cit.>), then the functors $F_{\star}$ are representable by quasi-projective $\mathbb{Z}_p$ schemes $A_H, A_Q, A_K$. The first one is smooth while the latter two have smooth generic fibres. Moreover the maps $a$ and $b$ are proper and finite etale on the generic fibre.

It is important to note that our schemes $A_{\star}$ don't necessarily have geometrically connected generic fibers. The connected components are parametrized by adelic double cosets coming from the theory of Shimura varieties. It is true, however, that the maps $a$ and $b$ induce bijections on connected components of geometric generic fibers.

§.§ Singularities

In this section we will describe the bad reduction of $A_{K}$ and $A_Q$ following <cit.> and <cit.>.

[Theorem 3 of <cit.>]
The geometric special fiber of the Siegel modular threefold of Klingen level decomposes as $\aksp=X^{\text{e}} \cup X^{\text{m}}$, where $X^{\text{e}}, X^{\text{m}}$ are smooth and intersect in a smooth surface $S$. Both $X^{\text{e}}$ and $X^{\text{m}}$ have a unique connected component lying over each connected component of $\ahsp$. The locus $X^{\text{e}} \setminus X^{\text{m}}$ parametrizes those quadruples where $H$ is étale, the locus $X^{\text{m}} \setminus X^{\text{e}}$ parametrizes those quadruples where $H$ is multiplicative and $S$ parametrizes the quadruples where $H$ is étale locally isomorphic to $\alpha_p$.

[Theorem 1.3 of <cit.>]
The scheme $A_{K} / \mathbb{Z}_p$ is regular, and it is smooth over $\mathbb{Z}_p$ away from a finite set of closed points $\Sigma$ in the special fiber. At the singular points in the geometric special fibre $\apsp$ the completed local ring is isomorphic to

\begin{align}
\overline{\mathbb{F}}_p \llbracket X,Y,Z,W \rrbracket/(XY-ZW).
\end{align}

The singular points $\Sigma$ correspond precisely to the polarized abelian surfaces $(A, \lambda)$ where $(\ker \lambda)[p]$ is etale locally isomorphic to $\alpha_p \times \alpha_p$. Moreover for every connected component of $\apgen$ there is a unique connected component of $\apsp$.

§.§ Geometry of Hecke correspondences

In this section we will compute the fibers of the maps $a,b$ between $ \aksp, \ahsp$ and $\apsp$, which is essentially an exercise in the theory of finite flat group schemes. We let $I_{1,1}$ denote the $p$-torsion groupscheme of a superspecial elliptic curve over $\overline{\mathbb{F}}_p$ and $I_{2,1}$ the $p$-torsion of a supersingular and not superspecial abelian surface over $\overline{\mathbb{F}}_p$ (c.f. Example 2.3 of <cit.>).

§.§.§ The fibers of a

If $(A, \lambda, \eta)$ is a $k$ point of $\ahsp$ with $k$ algebraically closed then the fiber of $a:\aksp(k) \to \ahsp(k)$ over this point is parametrized by sub-group schemes $H \subset A[p]$ of rank $p$. In characteristic zero the $p$-torsion of $A$ is isomorphic to $(\mathbb{Z}/p \mathbb{Z})^4$ which has $p^3+p^2+p+1=(p^2+1)(p+1)$ sub-group schemes of order $p$, so the degree of $a$ in characteristic zero is $p^3+p^2+p+1$.

Let $(A, \lambda, \eta)$ be an $\overline{\mathbb{F}}_p$ point of $\ahsp$, let $a^{-1}(A, \lambda,\eta)$ denote the scheme theoretic fiber. The underlying reduced scheme of $a^{-1}(A, \lambda, \eta)$ has: cardinality $2(p+1)$ if $A$ is ordinary, cardinality $3$ if $A$ has $p$-rank one, cardinality $1$ when $A$ is supersingular but not superspecial. If $A$ is superspecial then $a^{-1}(A, \lambda,\eta)$ is isomorphic to $\mathbb{P}^1$.

We will compute the fibers of the map $a$ over the following classes of geometric points (or Ekedahl-Oort strata):

* Points $(A, \lambda,\eta)$ with $A$ ordinary or equivalently $A[p] \cong \mu_p^2 \times (\zz / p \zz )^2$. In this case we see that every sub-group scheme of order $p$ must be isomorphic to either $\mu_p$ or $\mathbb{Z}/p \mathbb{Z}$ and there are $p+1$ possibilities for both of them.

* Points $(A, \lambda,\eta)$ where $A$ has $p$-rank one or equivalently $A[p] \cong \mu_p \times \zz / p \zz \times I_{1,1}$. In this case there are three sub-group schemes of order $p$, either $\mu_p, \mathbb{Z}/p \mathbb{Z}$ or $\alpha_p \subset I_{1,1}$.

* Points $(A, \lambda,\eta)$ where $A$ is supersingular but not superspecial or equivalently $A[p] \cong I_{2,1}$. Then $\dim_{\mathbb{F}_p} \hom(\alpha_p, A[p])=1$ and so there is only one sub-group scheme of order $p$.

* Points $(A, \lambda,\eta)$ where $A$ is superspecial or equivalently $A[p] \cong I_{1,1}^2$. Theorem 15.2 of <cit.> and the example following it show that the $\overline{\mathbb{F}}_p$ points of the fiber can be identified with $\mathbb{P} \left(\operatorname{Lie}(A) \right)$. Yu proves in Section 4 of <cit.> that the fiber is actually isomorphic to $\mathbb{P}^1$.

§.§.§ The fibers of b

Let us start with recalling the following fact:
[Proposition 11.25 of <cit.>]

Let $X$ be an abelian variety over a field, let $\alpha:X \to X^t$ be a polarisation and $g:X \to Y$ an isogeny.

* There exists a polarisation $\beta:Y \to Y^t$ such that $\alpha=g^t \circ \beta \circ g$ if and only if $\ker g$ is contained in $\ker \alpha$ and is totally isotropic with respect to the natural pairing $e_{\alpha}: \ker \alpha \times \ker \alpha \to \mathbf{G}_m$.  If such an isogeny $\beta$ exists then it is unique.

Now let $(A, \lambda, \eta)$ be an $k$-point of $A_K$ with $k$ an algebraically closed field. Then the fiber of $b:A_Q(k) \to A_K(k)$ over $(A, \lambda, \eta)$ parametrises quadruples $(B, \mu, H, \eta)$ where $\mu$ is a prime-to-$p$ polarisation of $B$ and $H$ is a sub-group scheme of rank $p$ such that $B/H \cong A$ compatibly with $\lambda$ and $\mu$. To be precise we want the following diagram to commute:

\begin{equation} \label{fiber-a}
\begin{tikzcd}
B \arrow{r}{\mu} \arrow{d}{f} & B^t \\
A \arrow{r}{\lambda} & A^t \arrow{u}{f^t}
\end{tikzcd}
\end{equation}

Proposition <ref> tells us that the fiber of $b$ over $(A, \lambda, \eta)$ is in bijection with sub-group schemes $G \subset \ker \lambda^t[p]$ of rank $p$ that are isotropic with respect to the natural pairing. However any $G \subset (\ker \lambda^t)[p]$ is automatically isotropic (because a line inside a two-dimensional space is always isotropic). In characteristic zero we know that $(\ker \lambda^t)[p]$ is isomorphic to $(\mathbb{Z}/p \mathbb{Z})^2$ which contains $p+1$ sub-group schemes of rank $p$, so the degree of $b$ is $p+1$. In characteristic $p$ we have the following description:

Let $(A, \lambda, \eta)$ be an $\overline{\mathbb{F}}_p$ point of $\apsp$ and let $b^{-1}(A, \lambda,\eta)$ denote the scheme-theoretic fiber. We will describe the underlying reduced scheme of $b^{-1}(A, \lambda,\eta)$ for various $(A, \lambda,\eta)$. It consists of two points when $A$ has $p$-rank two or one, it consists of a single point when $A$ is supersingular but $(A, \lambda,\eta) \not \in \Sigma$ (recall that $\Sigma$ is the singular locus of $\apsp$). For $(A, \lambda,\eta) \in \Sigma$ the fiber $b^{-1}(A, \lambda, \eta)$ is isomorphic to $\mathbb{P}^1$.

Using the same strategy as before, we classify order $p$ sub-group schemes of $\mu_p \times \mathbb{Z}/p \mathbb{Z}, I_{1,1}$ and $\alpha_p \times \alpha_p$.

* When $(\ker \lambda)[p] = \mu_p \times \mathbb{Z}/p \mathbb{Z}$ then there are two sub-group schemes of order $p$, one isomorphic to $\mu_p$ and one isomorphic to $\mathbb{Z}/p \mathbb{Z}$

* When $(\ker \lambda)[p] = I_{1,1}$ there one sub-group scheme of order $p$, since $\dim_{\mathbb{F}_p} \hom(\alpha_p, I_{1,1})=1$.

* When $A[p] \cong I_{2,1}$ then $\dim_{\mathbb{F}_p} \hom(\alpha_p, A[p])=1$ and the inclusion $\alpha_p \subset A[p]$ automatically factors over $(\ker \lambda)[p]$. This allows us to conclude that there is only one sub-group scheme of order $p$.

* When $(\ker \lambda)[p] = \alpha_p \times \alpha_p$ then as before Theorem 15.2 of <cit.> shows that the $\overline{\mathbb{F}}_p$ points of the fiber can be identified with $\mathbb{P} \left(\operatorname{Lie}(A) \right)$. It is a result of Yu (Section 4 of <cit.>) that the fiber is actually isomorphic to $\mathbb{P}^1$.

§.§ Supersingular loci

In this section we will give a description of the supersingular loci of $\aksp, \ahsp$ and $\apsp$. These are uniformised by the special fibers of Rapoport-Zink spaces which will allow us to describe their combinatorial structure in terms of Shimura sets for the inner form $G=GU_2(D)$ of $\gsp_4$. This description is worked out carefully for $\ahsp$ in <cit.> and we can use their results to describe the supersingular locus of $\aksp$ and $\apsp$ using Lemma's <ref> and <ref>.

It is a classical result due to Katsura-Oort <cit.> that all the irreducible components of the supersingular locus of the (coarse) moduli space of principally polarized abelian surfaces are isomorphic to $\mathbb{P}^1$. These irreducible components intersect in a finite number of points, which correspond to superspecial abelian surfaces. Moreover $p+1$ irreducible components pass through every such superspecial point and their intersections are transverse. In Chapter 4 of <cit.> the same result is proven for a Siegel modular threefold of arbitrary neat level $U^p$ away from $p$ (but hyperspecial at $p$) and moreover they give a description of the sets of irreducible components.

The authors of op. cit. start by defining a set $X$ of lattices on which $G(\qqq_p)$ acts and prove $G(\qqq_p)$ acts transitively on the set $X_0$ of superspecial lattices and the set $\tilde{X}$ of distinguished lattices. They use this to prove $G(\qqq_p)$-equivariant bijections

\begin{align}
\tilde{X} &= \zz \times \left \{ \text{vertices of the first kind in }\mathcal{B}(G_{\text{ad}}, \qqq_p) \right \} \\
X_0&= \zz \times \left \{\text{vertices of the second kind in } \mathcal{B}(G_{\text{ad}}, \qqq_p) \right \},
\end{align}

where $\mathcal{B}(G_{\text{ad}}, \qqq_p)$ denotes the Bruhat-Tits building of the adjoint group of $G$. This is an inhomogenous tree where the vertices of the first kind have valency $p^2+1$ and the vertices of the second kind have valency $p+1$. Moreover they define a morphism (where the superscript `ss' denotes the supersingular locus)

\begin{align}
\coprod_{\tilde{L} \in \tilde{X}} \mathbb{P}_{\tilde{L}} \to \ahsp^{\text{ss}}
\end{align}

which induces a map

\begin{align}
G(\qqq) \setminus \left[\left(\coprod_{\tilde{L} \in \tilde{X}} \mathbb{P}_{\tilde{L}} \right) \times G(\mathbb{A}^{\infty,p})/U^p \right] \to \ahsp^{\text{ss}} \label{uniformisation}
\end{align}

that turns out to be the normalisation of $\ahsp^{\text{ss}}$. The lines $\tilde{L}$ have a natural $\mathbb{F}_{p^2}$ structure and the superspecial lattices $L \in X_0$ provide the $\mathbb{F}_{p^2}$ points of $\mathbb{P}_{\tilde{L}}$ for a certain $\tilde{L}$ which map to the superspecial points in $\ahsp^{\text{ss}}$. From (<ref>) we get a bijection between the irreducible components of $\ahsp^{\text{ss}}$ and

\begin{align}
N=G(\qqq) \setminus G(\mathbb{A}^{\infty})/ U^p K_2
\end{align}

which is equivariant for the action of prime-to-$p$ Hecke operators. Here $K_2$ is the compact open subgroup $K_2$ described in section <ref>, which turns out be the stabilizer in $G(\qqq_p)$ of $(0, x)$ where $x$ is a certain vertex of the first kind in $\mathcal{B}(G_{\text{ad}}, \qqq_p)$. Similarly there is an equivariant bijection between the superspecial points of $\ahsp^{\text{ss}}$ and the Shimura set

\begin{align}
M=G(\qqq) \setminus G(\mathbb{A}^{\infty})/ U^p K_1
\end{align}

which completes the description of $\ahsp^{\text{ss}}$. Our computation of the fibers of $\aksp \to \ahsp$ now shows that the supersingular locus of $\aksp$ has irreducible components in bijection with

\begin{align}
N \coprod M.
\end{align}

We will usually write

\begin{align}
\aksp^{\text{ss}}=E \cup F \label{defofE}
\end{align}

where $E$ corresponds to $N$ and $F$ corresponds to $M$.
[Theorem 1.2 of <cit.>]
The irreducible components of $E$ are pairwise disjoint, and so are the irreducible components of $F$. The map $b:\aksp \to \apsp$ contracts the irreducible components of $E$ to points, identifying the singular points of $\apsp$ with $N$.

The proof in <cit.> is for a specific choice of level $U^p$, but this is not used anywhere except to prove formulas for the number of components.

§ THE PICARD-LEFSCHETZ FORMULA

In this section we will briefly recall the general machinery of nearby cycles for finite type schemes $X/S$ where $S=\spec R$ with $R$ an Henselian DVR, in practice we will take $R=\zz_p$. Furthermore, we discuss results of Lan and Stroh <cit.> which allow us to use this machinery for non-compact Shimura varieties without having to deal with compactifications. We then discuss the Picard Lefschetz formula for $A_K$, following <cit.> Exposé XV.

§.§ Generalities on nearby cycles

We start with some notation which will be used throughout the paper. Let $\ell$ be a prime invertible in $R$ and let $L/\mathbb{Q}_{\ell}$ be a finite extension of $\qqq_{\ell}$ with ring of integers $\mathcal{O}$ and residue field $\mathbb{F}$. We will use $\Lambda$ to denote either $L, \mathcal{O}$ or $\mathbb{F}$.

Let $\overline{K}$ be a seperable closure of $K=\operatorname{Frac} R$, let $\breve{K}$ be the completion of the maximal unramified extension of $K$ and let $\breve{S}=\spec \mathcal{O}_{\breve{K}}$. For a scheme $X/S$ of finite type consider the following diagram

\begin{equation}
\begin{tikzcd}
X_{\overline{\eta}} \arrow{r}{\overline{j}} \arrow{d} & X_{\breve{S}}  \arrow{d} & X_{\overline{s}} \arrow{d} \arrow{l}{\overline{i}} \\
X_{\eta} \arrow{r}{j} & X  & X_s \arrow{l}{i}.
\end{tikzcd}
\end{equation}

Here $X_s$ is the special fiber, $X_{\eta}$ is the generic fibre, $X_{\overline{\eta}}=X_{\overline{K}}$ the geometric generic fibre and $X_{\overline{s}}=X_{\overline{k}}$ the geometric special fiber. If $\mathcal{F}$ is a constructible sheaf of $\Lambda$-modules on $X_{\eta}$, considered as a Galois-equivariant constructible sheaf on $X_{\overline{\eta}}$, then we define the complex of nearby cycles

\begin{align}
R \Psi \mathcal{F}:=\overline{i}^{\ast} R \overline{j}_{\ast} \mathcal{F},
\end{align}

which is also $\gal(\overline{K}/K)$-equivariant. If $\mathcal{F}$ comes from the restriction of a constructible sheaf $\mathcal{F}$ on $X$ then there is an adjunction map (of complexes) $\mathcal{F} \to R \overline{j}_{\ast} \overline{j}^{\ast} \mathcal{F}$ which can be extended to a distinguished triangle

\begin{align}
\overline{i}^{\ast} \mathcal{F} \to R \Psi \mathcal{F} \to R \Phi \mathcal{F}    \label{fundamentaltriangle}
\end{align}

on $X_{\overline{s}}$. The cone $R \Phi \mathcal{F}$ is called the complex of vanishing cycles.

[Basic Properties] Let $X/S$ and $\mathcal{F}$ be as above, then

* The complex $R \Psi \mathcal{F}$ has constructible cohomology sheaves.

* If $f$ is proper then $H^{\bullet}(X_{\overline{s}}, R \Psi \mathcal{F})=H^{\bullet}(X_{\overline{\eta}}, \mathcal{F})$, equivariant for the Galois action.

* If $f$ is smooth then $R \Psi \mathcal{F} = \mathcal{F}_{\overline{s}}$.

Part 2 of this result does not necessarily hold for non-proper maps. For example start with a proper scheme $X/S$ and define $Y=X \setminus X_{s}$. Then we still have $Y_{\overline{\eta}}=X_{\overline{\eta}}$ but there is no way to compute the cohomology of the generic fibre of $Y$ from the cohomology of the (empty) special fiber of $Y$. However, when $X/S$ is a 'good' integral model of a Shimura variety (c.f. Section 2 of <cit.>), for example the ones defined in the previous section, then we have the following result:

[Corollary 4.6 of <cit.>]
When $X/S$ is a 'good' integral model of a Shimura variety and $\mathcal{F}$ is an automorphic local system of $\Lambda$-modules then the natural maps induce isomorphisms

\begin{align}
H^{\bullet}(X_{\overline{s}}, R \Psi \mathcal{F})&=H^{\bullet}(X_{\overline{\eta}}, \mathcal{F}) \\
H^{\bullet}_c(X_{\overline{s}}, R \Psi \mathcal{F})&=H^{\bullet}_c(X_{\overline{\eta}}, \mathcal{F}).
\end{align}

§.§ The exact sequence

Let us now specialise to the case that $R=\zz_p, X=A_K$ and $\mathcal{F}=\mathbb{V}=\mathbb{V}_{\mathcal{O}}$ is an automorphic local system of $\mathcal{O}$-modules (so that Proposition <ref> applies). Proposition 3.1.2. of Exposé XV of <cit.> tells us that the sheaves $R^i \Phi \mathbb{V}$ are zero for $i \not=3$ and supported on the singular locus $\Sigma$ for $i=3$. This means in particular that

\begin{align}
R^3 \Phi \mathbb{V} = \bigoplus_{x \in \Sigma} i_{x, \ast}\left(R^3 \Phi \mathbb{V} \right)_x
\end{align}

and so the cohomology of $R^3 \Phi \mathbb{V}$ is concentrated in degree zero where we find

\begin{align}
H^3_c(\apsp, R \Phi \mathbb{V})=H^3(\apsp, R \Phi \mathbb{V})= H^0_c(\apsp, R^3 \Phi \mathbb{V}) = H^0(\apsp, R^3 \Phi \mathbb{V}) = \bigoplus_{x \in \Sigma} \left(R^3 \Phi \mathbb{V}\right)_x.
\end{align}

Proposition <ref> now tells us that (using the triangle (<ref>))

\begin{align}
H^i_c(\apsp, \mathbb{V}) & = H^i_c(\apgen, \mathbb{V}) \\
H^i(\apsp, \mathbb{V}) & = H^i(\apgen, \mathbb{V})
\end{align}

for $i \not=3,4$ and gives us the following diagram of exact sequences:

\begin{equation}
\begin{tikzcd}[column sep=small]
0 \arrow{r} & H^3_c(\apsp, \mathbb{V}) \arrow{r} \arrow{d} &  H^3_c(\apgen, \mathbb{V}) \arrow{r}{\alpha} \arrow{d} & H^3_c(\apsp, R \Phi \mathbb{V}) \arrow{r} \arrow[d, equals]& H^4_c(\apsp, \mathbb{V}) \arrow{r} \arrow{d} &  H^4_c(\apgen, \mathbb{V}) \arrow{r} \arrow{d} & 0 \\
0 \arrow{r} & H^3(\apsp, \mathbb{V}) \arrow{r} &  H^3(\apgen, \mathbb{V}) \arrow{r} & H^3(\apsp, R \Phi \mathbb{V}) \arrow{r} & H^4(\apsp, \mathbb{V}) \arrow{r} &  H^4(\apgen, \mathbb{V}) \arrow{r} & 0
\end{tikzcd} \label{double-exact}
\end{equation}

§.§ The action of inertia

Let $\mathbb{Z}_{\ell}(1)$ with generator $\sigma$ be the maximal pro-$\ell$ quotient of the inertia group $I_p$ of $G_{\qqq_p}$, then the general theory tells us that the action of the inertia group factors through $\mathbb{Z}_{\ell}(1)$ and moreover can be computed as follows:

\begin{equation}
\begin{tikzcd}
H^3(\apgen, \mathbb{V}) \arrow{r}{\alpha} \arrow{d}{\sigma-1} & \displaystyle \bigoplus_{x \in \Sigma} \left(R^3 \Phi \mathbb{V} \right)_x \arrow{d}{\oplus \operatorname{Var}_x(\sigma)} \\
H^3_c(\apgen, \mathbb{V})  & \displaystyle \bigoplus_{x \in \Sigma} H^3_x(\apsp, R \Psi \mathbb{V}) \arrow{l}{\beta}
\end{tikzcd}
\end{equation}

Here $\operatorname{Var}_x(\sigma)$ is the local variation map, which is an isomorphism, and $H_x(\apsp, -)$ denotes cohomology with support in the closed subscheme $\{x\}$. The map $\sigma-1$ is not equivariant for the action of Frobenius, but we can fix this by defining a monodromy operator $N$ as the logarithm of $\sigma$. Since $(\sigma-1)^2=0$ we have $N=\sigma-1$ and so the following diagram is $\frob_p$-equivariant:

\begin{equation}
\begin{tikzcd}
H^3(\apgen, \mathbb{V}) \arrow{r}{\alpha}(1) \arrow{d}{N} & \displaystyle  \bigoplus_{x \in \Sigma} \left(R^3 \Phi \mathbb{V} \right)_x(1) \arrow{d}{\oplus N_x} \\
H^3_c(\apgen, \mathbb{V})  & \displaystyle  \bigoplus_{x \in \Sigma} H^3_x(\apsp, R \Psi \mathbb{V}) \arrow{l}{\beta}.
\end{tikzcd}
\end{equation}

Moreover, the general theory tells us that $\beta$ is the Poincare dual of $\alpha$ up to a Tate twist, since $\mathbb{V}$ is self dual up to a Tate twist. The natural map (induced by (<ref>))

\begin{align}
H^3_x(\apsp, \mathbb{V}) \to H^3_x(\apsp, R \Psi \mathbb{V})
\end{align}

is an isomorphism by <cit.>. Taking the direct sum over $x \in \Sigma$ we get an isomorphism

\begin{align}
H^3_{\Sigma}(\apsp, \mathbb{V}) \cong H^3_x(\apsp, R \Psi \mathbb{V})
\end{align}

and under this identification $\beta$ factors as

\begin{equation}
\begin{tikzcd}
H^3_{\Sigma}(\apsp, \mathbb{V}) \arrow{r} \arrow[drr, bend right=40, "\beta", below] \arrow{dr} & H^3_c(\apsp, \mathbb{V}) \arrow{d} \arrow{r} & H^3_c(\apgen, \mathbb{V}) \arrow{d} \\
& H^3(\apsp, \mathbb{V}) \arrow{r} & H^3(\apgen, \mathbb{V})
\end{tikzcd}
\end{equation}

since cohomology with support in a proper subscheme factors through compactly supported cohomology.

§.§ The weight filtration

Now let $\mathbb{V}$ be an automorphic local system of $L$-vector spaces of pure weight $k$. We will show that the weight filtration on $H^3_!(\apgen, \mathbb{V})$ has nonzero graded pieces only of weights $k+2, k+3, k+4$ by explicitly writing down the weight filtration. Here the weight filtration is defined as usual in terms of the action of $\frob_p$ on $H^3_{!}(\apgen, \mathbb{V})$.

The weight filtration on $H^3_{!}(\apgen, \mathbb{V})$ is given by

\begin{align}
\im \beta \subset H^3_{!}(\apsp, \mathbb{V}) \subset H^3_{!}(\apgen, \mathbb{V}).
\end{align}

The quotient $H^3_{!}(\apgen, \mathbb{V})/H^3_{!}(\apsp, \mathbb{V})$ is a subspace of $H^3(\apsp, R \Phi \mathbb{V})$ by (<ref>). The derived projection formula tells us that (with tensor products derived)

\begin{align}
R \Psi \mathbb{V} = \mathbb{V} \otimes_{L} R \Psi L,
\end{align}

where $L/\qqq_{\ell}$ is our coefficient field. This implies that $R \Phi \mathbb{V} = \mathbb{V} \otimes_L R \Phi L$ and the latter is given by

\begin{align}
R \Phi L \cong \bigoplus_{x \in \Sigma} i_{x, \ast} L(-2)[-3],
\end{align}

by the discussion preceding <cit.> Exposé XV 2.2.5.7 (m=1 in our case). This shows that

\begin{align}
R \Phi \mathbb{V} = \bigoplus_{x \in \Sigma} i_{x, \ast} \mathbb{V}_x(-2)[-3],
\end{align}

and so

\begin{align}
H^3(\apsp, R \Phi \mathbb{V})=\bigoplus_{x \in \Sigma} \mathbb{V}_x(-2)
\end{align}

is pure of weight $k+4$. We deduce from this that $\im \beta$ is pure of weight $k+2$ using the isomorphism

\begin{align}
H^3(\apsp, R \Phi \mathbb{V})(1) \cong H^3_{\Sigma}(\apsp, \mathbb{V}),
\end{align}

coming from the local monodromy maps. We are left to show that $H^3_{!}(\apsp, \mathbb{V})/\im \beta$ is pure of weight $k+3$. We know it has weight at most $k+3$ by Weil II, so it suffices to show the weight is at least $k+3$. Let $U=\apsp \setminus \Sigma$ be the smooth locus of $\apsp$, then there is a long exact sequence

\begin{equation}
\begin{tikzcd}[row sep=small]
&\cdots \arrow{r} & H^3_{\Sigma}(\apsp, \mathbb{V}) \arrow{r}{\beta} & H^3(\apsp, \mathbb{V}) \arrow{r} & H^3(U, \mathbb{V}) \arrow{r} & \cdots
\end{tikzcd}
\end{equation}

This implies that $H^3_{!}(\apsp, \mathbb{V})/\im \beta$ is contained in $H^3(U, \mathbb{V})$, which has weight at least $k+3$ by smoothness of $U$.

§.§ The cokernel of monodromy

Now let $\mathbb{V}=\mathbb{V}_{\mathcal{O}}$ be an automorphic local system of $\mathcal{O}$-modules, then we can compute the monodromy operator using the following diagram :

\begin{equation} \label{componentgroupdef}
\begin{tikzcd}
H^3_!(\apgen, \mathbb{V})(1) \arrow[r,twoheadrightarrow, "\alpha"] \arrow{d}{N} & \im \alpha \arrow[r, hook] \arrow{d}{\gamma} & \displaystyle  \bigoplus_{x \in \Sigma} \left(R^3 \Phi \mathbb{V} \right)(1)_x \arrow{d}{\oplus N_x} \\
H^3_!(\apgen, \mathbb{V})  & \im \beta \arrow[l, hook] &\displaystyle \bigoplus_{x \in \Sigma} H^3_x(\apsp, R \Psi \mathbb{V}) \arrow[l, twoheadrightarrow, "\beta"].
\end{tikzcd}
\end{equation}

Let $\Theta$ be the cokernel of $\gamma$ (which depends on the weight, the level and $\ell$), we call it the component group.

The weight monodromy conjecture predicts that $\gamma$ is an isomorphism after inverting $\ell$, i.e., that $\Theta$ is torsion. We call $\Theta$ the component group because it is a direct generalisation of the component group of the Jacobian of the modular curve, in its cohomological incarnation (c.f. <cit.>). We will show that it is trivial under the assumptions of Theorem <ref>. The same proof will show that $\gamma$ is an isomorphism after inverting $\ell$, under the assumption that the weight is sufficiently regular (now using Corollary <ref>). Showing that $\gamma$ is an isomorphism after inverting $\ell$ in singular weight requires automorphic input.

§ THE COMPONENT GROUP VANISHES

§.§ Statement of the main result

In this section we will prove that the component group vanishes under the assumptions of Theorem <ref>, it will be useful to give these assumptions a name.

Assume that $a > b > 2$, that $U_{\ell}$ is hyperspecial and that $\ell>a+b+4$.

If Assumption <ref> holds then

\begin{align}
\Theta=0
\end{align}

The strategy of the proof is to show that the map

\begin{align}
\alpha:H^3_{c}(\apgen, \mathbb{V}) \to H^3_c(\apsp, R \Phi \mathbb{V})
\end{align}

is surjective (which then shows that $\beta$ is injective by duality). The cokernel is given by the image of the map

\begin{align}
H^3_c(\apsp, R \Phi \mathbb{V}) \to H^4_c(\apsp, \mathbb{V}), \label{cokernel}
\end{align}

which has a geometric interpretation in terms of the cycle classes of irreducible components of the supersingular locus of $\ahsp$. To be precise we will identify the image of (<ref>) with the image of the map

\begin{align}
H^4_{Z}(\ahsp, \mathbb{V}) \to H^4_c(\ahsp, \mathbb{V}),
\end{align}

where $Z$ is the supersingular locus of $\ahsp$. Now since $\ahsp$ is smooth we know that

\begin{align}
H^4_c(\ahsp, \mathbb{V}) &= H^4_c(\ahgen, \mathbb{V})
\end{align}

and we can now apply our vanishing theorems to the latter. The proof proceeds in two steps, which we formulate as two lemmas: We first relate the image of (<ref>) to the cycle class of the supersingular locus on $A_Q$. More precisely we will relate it to the image of the cycle class map $H^0(E, \mathbb{V})(-2) \to H^4(X^e, \mathbb{V})$, where we recall that $\aksp^{\text{ss}}=E \cup F \subset X^{\text{e}}$. Since $E$ and $X^e$ are smooth we can instead rewrite this using cohomology with support in $E$, which will appear in the following lemma.

There is an isomorphism making the following diagram commute

\begin{equation}
\begin{tikzcd}
H^4_E(X^{\text{e}}, \mathbb{V}) \arrow{r} & H^4_c(X^{\text{e}}, \mathbb{V}) \\
H^3(\apsp, R \Phi \mathbb{V}) \arrow[u, densely dotted] \arrow{r} & H^4_c(\apsp, \mathbb{V}) \arrow{u}{j}
\end{tikzcd}
\end{equation}

and moreover the natural map $j$ is an isomorphism.

Next we relate this to the cycle class map for the supersingular locus $Z$ of $\ahsp$. Because $a_1^{-1}(Z)=E \cup F$ (where $a_1:X^{\text{e}} \to \ahsp$ is the map induced by $a$) there is no natural map $H^4_Z(\ahsp, \mathbb{V}) \to H^4_E(X^{\text{e}}, \mathbb{V})$, however we can produce one as follows:

The natural map $H^4_{Z}(\ahsp, \mathbb{V}) \to H^4_{E \cup F}(X^{\text{e}}, \mathbb{V})$ factors through $H^4_{E}(X^{\text{e}}, \mathbb{V})$ via an isomorphism. To be precise, there is an isomorphism making the following diagram commute:

\begin{equation}
\begin{tikzcd}
H^4_{Z}(\ahsp, \mathbb{V}) \arrow{r} \arrow[dr, densely dotted, "\sim"] & H^4_{E \cup F}(X^{\text{e}}, \mathbb{V})  \\
& H^4_{E}(X^{\text{e}}, \mathbb{V}). \arrow{u}
\end{tikzcd}
\end{equation}

Here $Z$ is the supersingular locus of $\ahsp$ and $E \cup F$ is the supersingular locus of $\aksp$.

Lemma <ref> and Lemma <ref> give us a commutative diagram

\begin{equation}
\begin{tikzcd}
H^3(\apsp, R \Phi \mathbb{V}) \arrow{r} \arrow[d, "\sim"] & H^4_c(\apsp, \mathbb{V}) \arrow[d, "\sim"] \\
H^4_{E}(X^{\text{e}}, \mathbb{V}) \arrow{r} & H^4_c(X^{\text{e}}, \mathbb{V})  \\
H^4_{Z}(\ahsp, \mathbb{V}) \arrow[u, "\sim"] \arrow{r} & H^4_c(\ahsp, \mathbb{V}) \arrow[u]
\end{tikzcd}
\end{equation}

To show that the top map is zero it suffices to show that the bottom map is zero. Now we note that

\begin{align}
H^4_c(\ahsp, \mathbb{V})=H^4_c(\ahgen, \mathbb{V}),
\end{align}

by smoothness of $A_H$. Moreover by Assumption <ref> we can apply Theorem <ref> to deduce that $H^4_c(\ahgen, \mathbb{V})=0$. We have shown that the map $\zeta$ in the following exact sequence is zero

\begin{equation}
\begin{tikzcd}
H^3_c(\apgen, \mathbb{V}) \arrow{r}{\alpha} & H^3(\apsp, R \Phi \mathbb{V}) \arrow{r}{\zeta} & H^4_c(\apsp, \mathbb{V})
\end{tikzcd}
\end{equation}

which shows that $\alpha$ is surjective and by Poincaré duality we conclude that $\beta$ is injective. This shows that the component group is zero, because the local monodromy operators $N_x$ in (<ref>) are isomorphisms.

If Assumption <ref> holds then the map

\begin{align}
\beta \otimes \mathbb{F}: H^3_{\Sigma}(\apsp, \mathbb{V}  )  \otimes_{\mathcal{O}} \mathbb{F} \to H^3_!(\apgen, \mathbb{V}  ) \otimes_{\mathcal{O}} \mathbb{F}
\end{align}

is injective.

Theorem <ref> says that

\begin{align}
H^3_{!}(\apgen, \mathbb{V}) \otimes_{\mathcal{O}} \mathbb{F} \to H^3_{!}(\apgen, \mathbb{V} \otimes_{\mathcal{O}} \mathbb{F})
\end{align}

is surjective. Moreover since $R \Phi \mathbb{V}$ only has cohomology in degree $3$ we find that

\begin{align}
H^3(\apsp, R \Phi \mathbb{V}) \otimes_{\mathcal{O}} \mathbb{F} \to H^3(\apsp, R \Phi \mathbb{V} \otimes_{\mathcal{O}} \mathbb{F} )
\end{align}

is an isomorphism. Now these natural maps fit into a commutative cube (which we won't draw) with the maps

\begin{align}
\overline{\alpha}:H^3_!(\apgen, \mathbb{V} \otimes_{\mathcal{O}} \mathbb{F}) &\to H^3(\apsp, R \Phi \mathbb{V} \otimes_{\mathcal{O}} \mathbb{F}) \\
\overline{\beta}:H^3_{\Sigma}(\apsp, \mathbb{V} \otimes_{\mathcal{O}} \mathbb{F}) &\to H^3_!(\apgen, \mathbb{V} \otimes_{\mathcal{O}} \mathbb{F}).
\end{align}

The fact that $\alpha \otimes \mathbb{F}$ is surjective implies that $\overline{\alpha}$ is surjective. By duality we find that $\overline{\beta}$ is injective which shows that $\beta \otimes \mathbb{F}$ is injective by chasing the cube.

Let $\mathbb{V}=\mathbb{V}_{a,b}$ be a local system of $L$-vector spaces and suppose that the weight is sufficiently regular, i.e., that $a>b>0$. Then the map

\begin{align}
\alpha:H^3_{!}(\apgen, \mathbb{V}) \to H^3(\apsp, R \Phi \mathbb{V})
\end{align}

is surjective. It now follows from (<ref>) that the monodromy operator induces an isomorphism between $\im \alpha$ and $\im \beta$, i.e., that the weight monodromy conjecture holds for $H^3_{!}(\apgen, \mathbb{V})$.

The proof is the same as the proof of Proposition <ref>, except that we note that the vanishing of $H^4_c(\ahgen, \mathbb{V})$ holds without restrictions on $\ell$  (this is Corollary <ref>).

§.§ Proof of Lemma <ref>

We apply the functors $R \Gamma_c$ and $R \Gamma_{\Sigma}$ to the triangle $\mathbb{V} \to R \Psi \mathbb{V} \to R \Phi \mathbb{V}$ to get a commutative diagram with exact rows:

\begin{equation}
\begin{tikzcd}
H^3_{\Sigma}(\apsp, R \Psi \mathbb{V}) \arrow{d} \arrow{r}{h} &  H^3_{\Sigma}(\apsp, R \Phi \mathbb{V}) \arrow{d}{f} \arrow{r}{g} & H^4_{\Sigma}(\apsp, \mathbb{V})  \arrow{d} \arrow{r} & H^4_{\Sigma}(\apsp, R \Psi \mathbb{V}) \arrow{d} \\
H^3_c(\apsp, R \Psi \mathbb{V}) \arrow{r} & H^3_c(\apsp, R \Phi \mathbb{V}) \arrow{r} \arrow[d, equal] \ & H^4_c(\apsp, \mathbb{V}) \arrow{r} & H^4_c(\apsp, R \Psi \mathbb{V}) \\
& H^3(\apsp, R \Phi \mathbb{V}).
\end{tikzcd}
\end{equation}

The map $h$ is zero by <cit.> Exposé XV 2.2.5.8 showing that $g$ is injective. The group $H^4_{\Sigma}(\apsp, R \psi \mathbb{V})$ vanishes by the discussion preceding op. cit. 2.2.5.1, showing that $g$ is an isomorphism. Since the sheaf $R \Phi \mathbb{V}$ is supported on $\Sigma$ we conclude that $f$ is an isomorphism using the long exact sequence

\begin{align}
\cdots \to H^2(U, R \Phi \mathbb{V}) \to H^3_{\Sigma}(\apsp, R \Phi \mathbb{V}) \to H^3(\apsp, R \Phi \mathbb{V}) \to H^3(U, R \Phi \mathbb{V}) \to \cdots
\end{align}

Using the fact that $a_1^{-1}(\Sigma)=E$ we get a morphism of cohomology with supports $H^4_{\Sigma}(\apsp, \mathbb{V}) \to H^4_E(X^{\text{e}}, \mathbb{V})$ which fits into the following diagram:

\begin{equation}
\begin{tikzcd}
H^3(\apsp, R \Phi \mathbb{V}) \arrow[r, "\cong"] & H^3_{\Sigma}(\apsp, R \Phi \mathbb{V}) \arrow[r, "\cong"] & H^4_{\Sigma}(\apsp, \mathbb{V}) \arrow{r} \arrow{d} & H^4_c(\apsp, \mathbb{V}) \arrow{d}{j} \\
& & H^4_{E}(X^{\text{e}}, \mathbb{V}) \arrow{r} & H^4_c(X^{\text{e}}, \mathbb{V})
\end{tikzcd}
\end{equation}

The map $\pi=a_1$ induces a map $\mathbb{V} \to R \pi_{\ast} \pi^{\ast} \mathbb{V}$ which we extend to a distinguished triangle

\begin{align}
\mathbb{V} \to R \pi_{\ast} \pi^{\ast}\mathbb{V} \to Q.
\end{align}

Using the proper base change theorem and the fact that the pullback of $\mathbb{V}$ to $E_x \cong \mathbb{P}^1$ is constant and isomorphic to $\mathbb{V}_x$ we find that

\begin{align} \label{fibercohomology}
Q \cong \bigoplus_{x \in \Sigma} i_{x, \ast} \mathbb{V}_x(-1)[-2]
\end{align}

where $i_x: \{x \} \to X$ is the inclusion of the point $x$. This means in particular that $H^{\bullet}_{\Sigma}(\apsp, Q)=H^{\bullet}(\apsp, Q)$ since the restriction of $Q$ to $U$ is zero. The long exact sequence in cohomology with support now tells us that
$H^4_{\Sigma}(\apsp, \mathbb{V}) \to H^4_{E}(X^{\text{e}}, \mathbb{V})$ is an isomorphism since the cohomology of $Q$ is concentrated in degree $2$. By a similar long exact sequence argument it follows that $j$ is an isomorphism.

§.§ Proof of Lemma <ref>

The lemma will follow from the fact that $E \to Z$ induces a bijection on irreducible components and the fact that the sheaf $\mathbb{V}$ is constant on $E$ and $Z$ (because the components are projective lines). However, since there is no natural map $H^4_{Z}(\ahsp, \mathbb{V}) \to H^4_E(X^{\text{e}}, \mathbb{V})$ we have to pass to the smooth locus of $Z$ which makes the proof look more involved than it is.

Let $V=\ahsp \setminus Z$ be the complement of $Z$, let $U$ be the complement of the singular locus of $Z$ and let $V'=X^{\text{e}} \setminus \left(E \cup F \right)$. Then we have a commutative diagram of maps of pairs

\begin{equation}
\begin{tikzcd}
&  (X^{\text{e}}, X^{\text{e}} \setminus E)  \\
(X^{\text{e}} \setminus F, V') \arrow{rr} \arrow{ur}  \arrow{d} &  &  (X^{\text{e}}, V') \arrow{ul} \arrow{d}
\\ (U,V) \arrow{rr} & & (\ahsp,V).
\end{tikzcd}
\end{equation}

There are induced (contravariant) maps in cohomology with support

\begin{equation}
\begin{tikzcd}
& H^4_{E}(X^{\text{e}}, \mathbb{V}) \arrow{dl} \arrow{dr}{\psi} \\
H^4_{E \cup F}(X^{\text{e}}, \mathbb{V}) \arrow{rr} &  & H^4_{E \setminus F}(X^{\text{e}} \setminus F, \mathbb{V}) \\
H^4_{Z}(\ahsp, \mathbb{V}) \arrow{u} \arrow{rr}{\phi} & &  H^4_{Z^{\text{sm}}}(U, \mathbb{V})  \arrow{u}{\tau}
\end{tikzcd}
\end{equation}

and we are going to show that $\phi, \psi$ and $\tau$ are all isomorphisms, providing the required factorisation. The map $\phi$ fits into a long exact sequence for $\ahsp \supset U \supset V$ (c.f. Chapter 23 of <cit.>)

\begin{align}
\cdots \to H^4_{Z^{\text{sing}}}(\ahsp, \mathbb{V}) \to H^4_{Z}(\ahsp, \mathbb{V}) \to H^4_{Z^{\text{sm}}}(U, \mathbb{V}) \to H^5_{Z^{\text{sing}}}(\ahsp, \mathbb{V}) \to \cdots
\end{align}

Since $Z^{\text{sing}}$ is smooth of codimension $3$ in $\ahsp$ we find that

\begin{align}
H^i_{Z^{\text{sing}}}(\ahsp, \mathbb{V})=H^{i-6}(Z^{\text{sing}}, \mathbb{V})
\end{align}

which is zero for $i=4,5$, so $\phi$ is an isomorphism. A similar argument shows that $\psi$ is an isomorphism.
Now we note that both $Z^{\text{sm}}$ and $E \setminus F$ are smooth of codimension $2$ in $U$ and $X^{\text{e}} \setminus F$ respectively so we can use cohomological purity twice to produce a commutative diagram

\begin{equation}
\begin{tikzcd}
H^4_{E \setminus F}(X^{\text{e}} \setminus F, \mathbb{V}) \arrow[r, "\sim"] & H^0(E \setminus F, \mathbb{V})(-2) \\
H^4_{Z^{\text{sm}}}(U, \mathbb{V})  \arrow{u}{\tau} \arrow[r, "\sim"]& H^0(Z^{\text{sm}}, \mathbb{V})(-2) \arrow{u}{\sigma}.
\end{tikzcd}
\end{equation}

The map $\sigma$ is an isomorphism because the map $E \setminus F \to Z^{\text{sm}}$ induces a bijection on irreducible components and the sheaf $\mathbb{V}$ is constant on each of those components (because the components are isomorphic to $\mathbb{P}^1$).

§ PROOFS OF THE MAIN RESULTS

§.§ Mazur's principle

Let us recall the theorem that we are trying to prove:

Assume that there are no congruences to unramified forms $\pi'$, in other words, that the conclusion of the theorem does not hold. We will prove that $\overline{\rho_{\pi, \ell}}$ has at most three Frobenius eigenvalues, showing that the assumptions of the theorem are not satisfied. Let $\mathbb{T}$ be the Hecke algebra containing the unramified Hecke operators and the Hecke operators at $p$ and let $\mathfrak{m}$ be the maximal ideal of the Hecke algebra corresponding to $\overline{\rho_{\pi, \ell}}$, then we have a decomposition

\begin{align}
H^3_{!}(\apgen, \mathbb{V})_{\mathfrak{m}} \otimes_{\mathcal{O}} L= \bigoplus_{} \rho_{\pi',\ell} \label{decompositionfive},
\end{align}

where $\rho_{\pi',\ell}$ is congruent modulo $\ell$ to $\rho_{\pi, \ell}$ and might appear multiple times in the direct sum. Note that Yoshida lifts and Saito-Kurokawa lifts don't contribute to the direct sum because the associated Galois representations are reducible and so cannot be congruent to $\rho_{\pi, \ell}$. This means that $\eqref{decompositionfive}$ is really a direct sum of irreducible four-dimensional Galois representations. By assumption all the $\pi'_p$ over which the sum is indexed are ramified at $p$ and so by weight-monodromy (Corollary <ref>) and local-global compatibility we know that $\pi'_p$ is of type IIa. This means that the monodromy operator $N:\rho_{\pi',\ell}(1) \to \rho_{\pi',\ell}$ has a one-dimensional image, which shows that

\begin{align}
\alpha(\rho_{\pi',\ell})
\end{align}

is a one-dimensional subspace of $H^3(\apsp, R \Phi \mathbb{V})_{\mathfrak{m}} \otimes_{\mathcal{O}} L$. Since $\alpha$ is moreover surjective we find that the dimension of $H^3(\apsp, R \Phi \mathbb{V})_{\mathfrak{m}} \otimes_{\mathcal{O}} L$ is equal to $n$, the number of $\pi'$ appearing in the direct sum (<ref>).

By Lemma <ref> we know that Frobenius acts on $\alpha(\rho_{\pi',\ell})$ by the scalar $p \lambda_{\pi'}$, where $\lambda_{\pi'}$ is the eigenvalue of the $u$-operator on $(\pi'_p)^{K(p)}$, introduced in Section <ref>. Localising at $\mathfrak{m}$ means fixing a mod $\ell$ eigenvalue of $u$ and so Frobenius acts with a single eigenvalue on

\begin{align}
H^3(\apsp, R \Phi \mathbb{V})_{\mathfrak{m}} \otimes_{\mathcal{O}} \mathbb{F}.
\end{align}

We also know that $H^3_{!}(\apgen, \mathbb{V})_{\mathfrak{m}} \otimes_{\mathcal{O}} \mathbb{F}$ is isomorphic (up to semi-simplification) to

\begin{align}
\overline{\rho_{\pi, \ell}}^{\oplus n}
\end{align}

for the same $n$ as before (by vanishing of torsion). Since $\overline{\rho_{\pi, \ell}}$ has four distinct Frobenius eigenvalues, we know that the space $H^3_{!}(\apgen, \mathbb{V})_{\mathfrak{m}} \otimes_{\mathcal{O}} \mathbb{F}$ decomposes into four generalised Frobenius eigenspaces, each of dimension $n$. Now $\alpha \otimes \mathbb{F}$ is surjective hence the kernel of $\alpha \otimes \mathbb{F}$ only contains three Frobenius eigenvalues. Because $\overline{\rho_{\pi, \ell}}$ is irreducible we can find a copy

\begin{align}
\overline{\rho_{\pi, \ell}} \subset H^3_{!}(\apgen, \mathbb{V})_{\mathfrak{m}} \otimes_{\mathcal{O}} \mathbb{F},
\end{align}

which is contained in the kernel of $N \otimes \mathbb{F}$ since $\overline{\rho_{\pi, \ell}}$ is unramified. As usual we compute the monodromy operator using the following diagram:

\begin{equation}
\begin{tikzcd}
H^3_{!}(\apgen, \mathbb{V})(1)_{\mathfrak{m}} \otimes_{\mathcal{O}} \mathbb{F} \arrow[r, "\alpha \otimes \mathbb{F}", twoheadrightarrow] \arrow{d}{N \otimes \mathbb{F}} & H^3(\apsp, R \Phi \mathbb{V})(1)_{\mathfrak{m}} \otimes_{\mathcal{O}} \mathbb{F} \arrow{d} \\
H^3_!(\apgen, \mathbb{V})_{\mathfrak{m}} \otimes_{\mathcal{O}} \mathbb{F}  &  H^3_{\Sigma}(\apsp, \mathbb{V})_{\mathfrak{m}} \otimes_{\mathcal{O}} \mathbb{F} \arrow[l, "\beta \otimes \mathbb{F}", hook]
\end{tikzcd}
\end{equation}

Corollary <ref> tells us that $\beta \otimes \mathbb{F}$ is injective which means that $\ker (N \otimes \mathbb{F}) = \ker (\alpha \otimes \mathbb{F})$. Therefore our copy of $\overline{\rho_{\pi, \ell}}$ must be contained in the kernel of $\alpha \otimes \mathbb{F}$, but then $\overline{\rho_{\pi, \ell}}$ only has three distinct Frobenius eigenvalues, which gives a contradiction.

§.§ The Weight-Monodromy Conjecture

In this section we prove weight-monodromy for the Siegel threefold of paramodular level at $p$. Let us start by restating our theorems:

If $a>b>0$, then Corollary <ref> shows that the weight monodromy conjecture is satisfied for $H^3_{!}(Y_{U, \overline{\qqq}_p}, \mathbb{V})$. We will now give a proof that works for general $a,b$, we start by recalling the following diagrams:

\begin{equation} \label{sidebyside}
\begin{tikzcd}
H^3_!(\apgen, \mathbb{V})(1) \arrow[r,twoheadrightarrow, "\alpha"] \arrow{d}{N} & \im \alpha \arrow[r, hook] \arrow{d}{\gamma} & \displaystyle  \bigoplus_{x \in \Sigma} \left(R^3 \Phi \mathbb{V} \right)(1)_x \arrow{d}{\oplus N_x} \\
H^3_!(\apgen, \mathbb{V})  & \im \beta \arrow[l, hook] &\displaystyle \bigoplus_{x \in \Sigma} H^3_x(\apsp, R \Psi \mathbb{V}) \arrow[l, twoheadrightarrow, "\beta"].
\end{tikzcd}
\begin{tikzcd}
H^3(\apsp, R \Phi \mathbb{V}) \arrow{r}{\delta} \arrow[d, "\sim"] & H^4_c(\apsp, \mathbb{V}) \arrow[d, "\sim"] \\
H^4_{E}(X^{\text{e}}, \mathbb{V}) \arrow{r} & H^4_c(X^{\text{e}}, \mathbb{V})  \\
H^4_{Z}(\ahsp, \mathbb{V}) \arrow[u, "\sim"] \arrow{r} & H^4_c(\ahsp, \mathbb{V}). \arrow[u]
\end{tikzcd}
\end{equation}

We are trying to show that the image of $\alpha$ does not intersect the kernel of $\beta$. Since $\alpha$ and $\beta$ are dual we deduce that the kernel of $\beta$ is dual to the cokernel of $\alpha$, which is equal to the image of $\delta$ (recall that $\oplus_{x \in \Sigma} \left(R^3 \Phi \mathbb{V} \right)_x \cong H^3(\apsp, R \Phi \mathbb{V})$). The diagram on the right shows that the image of $\delta$ is contained in $H^4_c(\ahsp, \mathbb{V})$. Now we will write (where either $\epsilon$ or $\epsilon'$ is zero)

\begin{align}
H^3_{!}(\apgen, \mathbb{V})=\bigoplus_{\pi \text{ CAP}} \rho_{\pi_{1},l} \oplus \bigoplus_{\pi \text{ Yoshida}} \rho_{\pi_{1},l}^{\epsilon} \oplus \rho_{\pi_{2},l}^{\epsilon'}(-b-1) \oplus \bigoplus_{\pi \text{ General}} \rho_{\pi, \ell}, \label{eq:decomposition3}
\end{align}

where $\rho_{\pi_{1},l}, \rho_{\pi_{2},l}$ are two-dimensional Galois representations associated with modular forms and $\rho_{\pi, \ell}$ are four-dimensional Galois representations associated with general type cuspidal automorphic representations of $\gsp_4$. All of these Galois representations might occur multiple times in $H^3_{!}(\apgen, \mathbb{V})$ which we absorb into the direct sum. We note note that weight-monodromy holds for all factors except possibly the factors of general type, because weight-monodromy holds for Galois representations associated with modular forms. Showing that weight-monodromy holds for the factors of general type implies that weight monodromy holds for $H^3_{!}(\apgen, \mathbb{V})$, because there can be no nontrivial monodromy operators between the different factors for weight reasons. So let $\pi$ be a general type automorphic form with associated Galois representation $\rho_{\pi, \ell}$.

If $\pi_p$ is unramified at $p$ then $\rho_{\pi, \ell}$ is unramified at $p$ by local-global compatibility and so satisfies weight monodromy because it occurs in $H^3_{!}(\ahgen, \mathbb{V})$ (which is pure). If $\pi_p$ is ramified then we know that $\rho_{\pi, \ell}$ has conductor one and so the image of the monodromy operator has dimension one. Suppose for the sake of contradiction that weight monodromy fails for $\rho_{\pi, \ell}$, then $\alpha(\rho_{\pi, \ell})$ is two-dimensional and moreover intersects $\ker \beta$ in a one-dimensional subspace.

Furthermore, we know that the Hecke eigenvalues associated with $\pi$ occur in $\ker \beta$ and since Hecke operators are adjoint under Poincaré duality, the Hecke eigenvalues for $\pi$ occur in the cokernel of $\alpha$. But $\operatorname{coker} \alpha=\im \delta$ is contained in $H^4_c(\ahsp, \mathbb{V})$ and so $\pi$ has the same prime-to-$p$ Hecke eigenvalues as an automorphic form $\pi'$ whose Hecke eigenvalues occur in $H^4_c(\ahsp, \mathbb{V})$. By the Chebotarev density theorem this means that the semi-simplification of $\rho_{\pi, \ell}$ is equal to the semi-simplification of $\rho_{\pi'}$. But $\rho_{\pi, \ell}$ is already semi-simple and since $\rho_{\pi'}$ is unramified at $p$ (because $\pi'_{p}$ is unramified), we conclude that $\rho_{\pi, \ell}$ is unramified at $p$, a contradiction.

§.§ A geometric Jacquet-Langlands correspondence

Let us stating a precise version of Theorem <ref>

* Let $\pi$ be a cohomological cuspidal automorphic representation of $\gsp_4$ that is not an irrelevant Yoshida lift such that $\pi_p$ is ramified and $K(p)$-spherical. Then there is a cuspidal automorphic representation $\sigma$ of $G$ such that $\pi_v \cong \sigma_v$ for finite places $v \not=p$, such that $\sigma_p$ is $K_2(p)$-spherical and with $\sigma_{\infty}$ determined by $\pi_{\infty}$. Moreover, $\sigma$ occurs with multiplicity one in the cuspidal spectrum of $G$. Conversely, a given cuspidal automorphic representation $\sigma$ of $G$ comes from such a $\pi$ if $\sigma_{\infty}$ has weight $k>0,j>3$.

* Let $k \ge 0, j \ge 3$ and let $N$ be an integer such that $p$ divides $N$ precisely once, then there is an injective map

\begin{align}
S_{k,j}[\para(N)]^{p-\text{new}} \xhookrightarrow{} \mathcal{A}^G_{k,j}[K_2(N)], \label{eq:beta}
\end{align}

equivariant for the prime-to-$p$ Hecke operators, which proves Conjecture <ref>. For $k>0, j>3$ the image consists precisely of algebraic modular forms that are not weakly-endoscopic.

* Let $k \ge 0, j \ge 3$ and $N=p$ be prime, then there is an injective lift

\begin{align}
S_{2j-2+k}[\Gamma_0(1)] \times S_{k+2}[\Gamma_0(p)]^{\text{new}} \to \mathcal{A}^G_{k,j}[K_2(p)]
\end{align}

and for $k=0$ an injective lift

\begin{align}
S_{2j-2}[\Gamma_0(1)] \to \mathcal{A}^G_{0,j}[K_2(p))].
\end{align}

The image of (<ref>) is a complementary subspace to the space generated by the images of these lifts (and the constant algebraic modular forms if $k=0,j=3$). Moreover for $k \ge 0, j \ge 3$ we have an equality

\begin{align}
\dim \mathcal{A}^G_{k,j}[K_2(p)]&=\dim S_{2j-2+k}[\Gamma_0(1)] \cdot \dim S_{k+2}[\Gamma_0(p)]^{\text{new}} + \dim S_{k,j}[\Gamma_0(p)] - 2 \dim S_{k,j}[\Gamma_0(1)] + \\ & \delta_{k,0}\dim S_{2j-2}[\Gamma_0(1)] +\delta_{k,0} \cdot \delta_{j,0}.
\end{align}

The dimension formula in (iii) is proven by Ibukiyama for $j \not=3,4$ in <cit.> and for $k=0,j=3$ in <cit.>.

The main idea of the proof is the incarnation of the singular locus $\Sigma$ of $\apsp$ as a Shimura set for $G$. We then get a map $\alpha$ from $H^3_{!}(\apgen, \mathbb{V})$, which we can explicitly describe in terms of automorphic forms and Galois representations, to the space of algebraic modular forms. Theorem <ref> combined with local-global compatibility allows us to control the image of $\alpha$ in a Galois-theoretic way. For parts (2) and (3) the main ingredients are the multiplicity one result of <cit.> and the main Theorem of <cit.>, which computes the cohomology of $\ahgen$ if $U=K(p)$. We start by relating spaces of algebraic modular forms to cohomology groups on $\aksp$, using our description of the singular locus:

There is an isomorphism

\begin{align}
H^0(\Sigma, \mathbb{V}_{a,b}) \cong \mathcal{A}^{G}_{a-b,b+3}[U^p K_2(p)],
\end{align}

equivariant for the prime-to-$p$ Hecke operators.

Recall from Section <ref> that we have identified the singular locus $\Sigma$ of $\apsp$ with the Shimura set

\begin{align}
\Sigma=G(\qqq) \setminus G(\mathbb{A}^{\infty}/U^p K_2(p),
\end{align}

equivariant for the prime-to-$p$ Hecke operators. The sections of the local system $\mathbb{V}$ over $\Sigma$ are given by

\begin{align}
\left(\varprojlim_{n} H^0(\Sigma, \mathbb{V}_{\mathcal{O}}/p^n) \right) \otimes_{\mathcal{O}} L \label{eq:sections}
\end{align}

and the sections of $\mathcal{V}_{\mathcal{O}}/p^n$ are given by

\begin{align}
\left \{f:G(\qqq) \setminus G(\mathbb{A}^{\infty})/(U^{p,l} K_2(p)H_{\ell}(n)) \to V_{\mathcal{O}}/\varpi^m   \; | \; f(gt)=g^{-1}f(t) \; \forall g \in U_{\ell} \right\},
\end{align}

where $H_{\ell}(n)$ as in Section <ref>. Since the intersection of $H_{\ell}(n)$ over all $n$ is empty, (<ref>) simplifies to

\begin{align}
\left\{f:G(\qqq) \setminus G(\mathbb{A}^{\infty}/(U^{p,l} K_2(p)) \to V  \; | \; f(gt)=g^{-1}f(t) \; \forall g \in U_{\ell} \right\},
\end{align}

which recovers the definition of $A^{G}_{a-b,b+3}[U^p K_2(p)]$ from Section <ref>.

We can identify the following cohomology groups

\begin{align}
H^3(\apsp, R \Phi \mathbb{V}) = \bigoplus_{x \in \Sigma} (R \Phi \mathbb{V})_x = \bigoplus_{x \in \Sigma} \mathbb{V}_x(-2)=H^0(\Sigma, \mathbb{V})(-2)
\end{align}

and the key player of the proof will be the Hecke equivariant map

\begin{align}
\alpha:H^3_{!}(\apgen, \mathbb{V}) \to H^3(\apsp, R \Phi \mathbb{V})=A^{G}_{a-b,b+3}[U^p K_2(p)].
\end{align}

We can write the domain of $\alpha$ in terms of Hecke modules and Galois representations as follows (using the results of Section <ref> and the multiplicity one result of <cit.>)

\begin{align}
H^3_{!}(\apgen, \mathbb{V}) = \bigoplus_{\pi}  \pi_{\text{fin}}^{U} \otimes \rho_{\pi,\ell}.
\end{align}

We are going to hide some things in the notation for bookkeeping purposes: If $\pi$ is a Yoshida lift or CAP then by $ \rho_{\pi,\ell}$ we mean the two-dimensional Galois representation that occurs in the cohomology. If $\pi$ is of general type with $\pi_{\infty}$ holomorphic then $ \rho_{\pi,\ell}$ is the four-dimensional Galois representation associated with $\pi$ and if $\pi_{\infty}$ is generic then $ \rho_{\pi,\ell}=0$. The reason for doing this is that there is only one four-dimensional Galois representation $\rho_{\pi, \ell}$ for the two automorphic representations $\pi_{\text{fin}} \otimes \pi^W$ and $\pi_{\text{fin}} \otimes \pi^H$ as they both contribute a two-dimensional piece to cohomology.

Because $\alpha$ has an interpretation in terms of monodromy operators, we know that $\rho_{\pi,\ell}/(\ker \alpha \cap \rho_{\pi,\ell})$ is one dimensional for all ramified $ \rho_{\pi,\ell}$ occurring in $H^3_{!}(\apgen, \mathbb{V})$. To be precise this follows from the conjunction of weight-monodromy and local-global compatibility. Therefore we will write

\begin{align}
\im \alpha \cong \bigoplus_{\pi}  \pi_{\text{f}}^{U} \label{eq:decomposition}
\end{align}

where $\pi$ runs over certain cuspidal automorphic representations of $\gsp_4$. Since $\pi_p^{K(p)}$ is one-dimensional we can rewrite this as

\begin{align}
\im \alpha \cong \bigoplus_{\pi}  (\pi_{\text{f}}^{p})^{U^p}. \label{eq:decomposition2}
\end{align}

Similarly we can write

\begin{align}
A^{G}_{a-b,b+3}[U^p K_2(p)]=\bigoplus_{\sigma} m(\sigma) \sigma_{\text{f}}^{U'}
\end{align}

where $\sigma$ runs over certain cuspidal automorphic representations of $G$ and $U'= U^p K_2(p)$. Since $\sigma_p^{K_2(p)}$ is one-dimensional we can also write this as

\begin{align}
A^{G}_{a-b,b+3}[U^p K_2(p)]=\bigoplus_{\sigma} m(\sigma) (\sigma_{\text{f}}^{p})^{U^p}.
\end{align}

Let $\mathbb{T}$ be the Hecke-algebra away from $p$ for both $G$ and $\gsp_4$, which makes sense as soon as we fix an isomorphism $\gsp_4(\mathbb{A}^{\infty,p}) \cong G(\mathbb{A}^{\infty,p})$. Then the spaces $(\pi_{\text{f}}^{p})^{U^p}$ and $(\sigma_{f}^{p})^{U^p}$ are simple $\mathbb{T}$-modules and moreover the map $\alpha$ is equivariant for the action of $\mathbb{T}$.

Now fix a cohomological automorphic representation $\pi$ of $\gsp_4$ that is not an irrelevant Yoshida lift such that $\pi_{\infty}$ is in the discrete series and such that $\pi_p$ is ramified and $K(p)$-spherical. Then we can choose $U^p$ sufficiently small such that $ \rho_{\pi,\ell}$ occurs in $H^3_{!}(\apgen, \mathbb{V}$). Because $\pi$ is not an irrelevant Yoshida lift we know that the summand

\begin{align}
F_{\pi}:= \pi_{\text{f}}^U \subset \im \alpha
\end{align}

is nonzero (because the Galois representation occurring in cohomology is ramified). Now we know that

\begin{align}
F_{\pi} \subset A^{G}_{a-b,b+3}[U^p K_2(p)]=\bigoplus_{\sigma} m(\sigma) \sigma_{\text{f}}^{U'}
\end{align}

and so there is a cuspidal automorphic representation $\sigma$ of $G$
such that $(\sigma^p)^{U^p} \cong (\pi^p)^{U^p}$ which implies that $\sigma_v \cong \pi_v$ for all finite places $v \not=p$. Moreover $\sigma_{\infty}$ is determined by $\pi_{\infty}$ because it is determined by the weights $a-b,b+3$. When $a>b>0$ or equivalently $k>0,j>3$ then $\alpha$ is surjective by Corollary <ref> so every cuspidal automorphic representation of $G$ arises in this way. To prove multiplicity one for $\sigma$ `in the image' of this transfer we first prove a claim:

The summand $P_{\sigma}:=m(\sigma) \sigma_f^{U'} \subset \mathcal{A}^G_{a-b,b+3}$ is in the image of $\alpha$.

Let $\pi$ be a cuspidal automorphic representation of $\gsp_4$ that `maps to $\sigma$'. If $P_{\sigma}$ maps nontrivially to $\coker \alpha$ then the Hecke eigenvalues of $\sigma$ occur in $H^4_c(\ahgen, \mathbb{V})$ and so the Hecke eigenvalues associated with $\pi$ occur there. Now we argue as in the proof of Theorem <ref> to show that $ \rho_{\pi,\ell}$ is unramified at $p$, a contradiction.

From the claim we get cuspidal automorphic representations $\pi_{1}, \cdots, \pi_{m(\sigma)}$ such that $\alpha(\rho_{\pi_i})$ maps to the summand $P_{\sigma}$. Therefore we have $\pi_{i,v} \cong \pi_{j,v}$ for all finite places $v \not=p$ and therefore all the $\pi$ are in the same $L$-packet. When the $L$-packet is not CAP, then the $\pi_{i,p}$ are generic (by weight-monodromy) and this means that $\pi_{i,p} \cong \pi_{j,p}$ because local $L$-packets have unique generic constituents. When the $L$-packet is CAP, then the $\pi_i$ are Saito-Kurokawa lifts and the fact that $\pi_{i,p}$ is $K(p)$-spherical also means that $\pi_{i,p} \cong \pi_{j,p}$ (c.f. Table 2 of <cit.>). We conclude that all the $\pi_i$ are isomorphic and therefore by multiplicity one we deduce that $m(\sigma)=1$.

In the proof of parts (2),(3) of Theorem <ref> we will work with the open compact subgroup $K(N)$, which is not neat. This means that the moduli functors we defined are not representable and so we cannot, strictly speaking, take the étale cohomology of the `Shimura variety of level $K(N)$'. To fix this we choose a neat compact open subgroup $U \subset K(N)$ that is normal in $K(N)$ and then define (with similar definitions for compactly supported and inner cohomology)

\begin{align}
H_{}^{\bullet}(Y_{K(N), \overline{\qqq}}, \mathbb{V}):=H^{\bullet}_{}(Y_{U, \overline{\qqq}}, \mathbb{V})^{H},
\end{align}

where $H$ is the finite group $H=K(N)/U$. Note that because $H$ is finite and $\mathbb{V}$ is a local system of $L$-vector spaces, taking $H$ invariants is exact and so all the exact sequences from the previous sections carry over to this setting. Moreover note that the corresponding space of algebraic modular forms $\mathcal{A}^G[U']$ satisfies

\begin{align}
\mathcal{A}_{k,j}^G[U']^{H}=\mathcal{A}_{k,j}^G[K_2(N)]
\end{align}

for all $k,j$. This means that there is an induced map

\begin{align}
\alpha: H^{3}_!(\apgen, \mathbb{V}) \to \mathcal{A}_{k,j}^G[K_2(N)].
\end{align}

\begin{align}
H^3_{!}(\akgen, \mathbb{V}):=\bigoplus_{\pi} \pi_{\text{fin}}^U \otimes \rho_{\pi, \ell},
\end{align}

with the notation as in the proof of part (1) of Theorem <ref> (so $\rho_{\pi, \ell}$ is either two or four-dimensional). Let $S$ be the subspace of $H^3_{!}(\apgen, \mathbb{V})$ spanned by the summands $\pi_{\text{fin}}^{U} \otimes \rho_{\pi,\ell}$ for $\pi$ such that $\pi_{\infty}$ is holomorphic and such that $\pi_p$ is ramified. Then $\alpha(S)$ has dimension equal to the dimension of $S_{k,j}[\para(N)]^{p-\text{new}}$ which gives us an injective map (after choosing a basis of normalised eigenforms)

\begin{align}
S_{k,j}[\para(N)]^{p-\text{new}} \to \mathcal{A}^G_{k, j}[K_2(N)] \otimes \mathbb{C}. \label{THEMAP}
\end{align}

Since there are no holomorphic Yoshida lifts, it is clear that the algebraic modular forms in the image are not weakly endoscopic. If $a>b>0$ then $\alpha$ is surjective by Corollary <ref> and the image of (<ref>) is complementary to the subspace of weakly endoscopic algebraic modular forms. Indeed, the only cohomological cuspidal automorphic representations of this level that don't come from $S_{k,j}[\para(N)]$ are non-holomorphic Yoshida lifts.

Now suppose that $N=p$, then Theorem 2.1 of <cit.> tells us that $H^4_c(A_{H, \overline{\qqq}}, \mathbb{V}_{a,b})=0 \label{zero}$ unless $a=b$ is even. This means that $\alpha$ will be surjective unless $a=b$ is even and so we can prove dimension formulas if we understand the dimension of the image of $\alpha$.

The dimension of the image of $\alpha$ is equal to:

\begin{align} \label{formula}
\dim \im \alpha &= \dim S_{2j-2+k}[\Gamma_0(1)] \times \dim S_{k+2}[\Gamma_0(p)]^{\text{new}} \\ &+ \dim S_{k,j}[K(p)] - 2 \dim S_{k,j}[K(1)] + \delta_{k,0} \tfrac{1+(-1)^j}{2} \dim S_{2j-2}[\Gamma_0(1)].
\end{align}

The cohomological cuspidal automorphic representations $\pi$ with nonzero invariants under $K(p)$ are either non-holomorphic Yoshida lifts or correspond to holomorphic Siegel cusp forms. We only care about holomorphic Siegel cusp forms that are new at $p$ and similarly about Yoshida lifts with $\pi_p$ ramified that are not irrelevant. Since $\pi^{K(p)}$ is one-dimensional for all these $\pi$ it suffices to simply count them. The results of <cit.> tell us that every eigenform $f \in S_{k,j}[K(1)]$ produces two oldforms in $S_{k,j}[K(p)]$, which are distinct unless $k=0$, $j$ is even and $f$ is in the image of the (injective) Saito-Kurokawa lift $S_{2j-2}[\Gamma_0(1)] \to S_{0,j}[K(1)]$. This means that

\begin{align}
S_{k,j}[K(p)]^{\text{new}}&=S_{k,j}[K(p)]-2 \dim S_{k,j}[K(1)]+\delta_{k,0}\tfrac{1+(-1)^j}{2} \dim S_{2j-2}[\Gamma_0(1)].
\end{align}

Section <ref> tells us that the number of relevant Yoshida lifts is equal to $\dim S_{2j-2+k}[\Gamma_0(1)] \times \dim S_{k+2}[\Gamma_0(p)]^{\text{new}}$ and the formula follows.

If $a>b$ or if $b$ is odd (equivalently $k>0$ or $j$ even) then $\alpha$ is surjective which proves that

\begin{align}
\dim \mathcal{A}^G_{k, j}[K_2(p)]=\dim \im \alpha = (\star),
\end{align}

where $(\star)$ is given by (<ref>).
When $a=b$ is even and $b>0$ (equivalently $k=0$ and $j$ odd) then the dimension formula to be proven follows from Theorem 5.2 of <cit.>. We are left to deal with this case $a=b=0$ (equivalently $k=0, j=3$), in which case the claimed formula is

\begin{align}
\dim \mathcal{A}^G_{k,j}[K_2(p)]=\dim \im \alpha +1 = (\star)+1.
\end{align}

By the claim it clearly suffices to prove that $\coker \alpha$ is one-dimensional. We know that $H^4_c(\ahgen, L)$ is one-dimensional and moreover $\alpha$ cannot be surjective because it cannot map nontrivially to the one-dimensional space of constant algebraic modular forms.

Now let us return to the case that $a=b$ is even. Theorem 2.1 of <cit.> tells us that

\begin{align}
H^4_c(A_{H, \overline{\qqq}}, \mathbb{V}_{a,b}) = L(b-2)^{\oplus s_{a+b+4}}
\end{align}

where $s_{a+b+4}=\dim S_{a+b+4}[\Gamma_0(1)]$. Moreover, the Hecke eigenvalues occurring in $H^4_{c}(A_{H, \overline{\qqq}}, \mathbb{V}_{a,b})=H^4_{c}(\ahgen, \mathbb{V}_{a,b})$ are weakly equivalent to those of automorphic representations parabolically induced from the Siegel parabolic (they are conjecturally non-cuspidal). Theorem 5.2 of <cit.> combined with Claim <ref> tells us the dimension of the cokernel of $\alpha$ is also equal to $\dim S_{a+b+4}[\Gamma_0(1)]$, and so

\begin{align}
H^4_{Z}(\ahgen, \mathbb{V}_{a,b}) \to H^4_{c}(\ahgen, \mathbb{V}_{a,b})
\end{align}

is surjective (recall that $\coker \alpha = \im \gamma$). In any case this means that there is an injective lift $S_{a+b+4}[\Gamma_0(1)] \to \mathcal{A}^G_{0,b+3}[K_2(p)]$ as claimed in the theorem. We conclude that the image of

\begin{align}
S_{k,j}[\para(p)]^{\text{new}} \to \mathcal{A}^G_{k, j}[K_2(p)]
\end{align}

is a complementary subspace to the subspace generated by the lift from

\begin{align}
S_{2j-2+k}[\Gamma_0(1)] \times S_{k+2}[\Gamma_0(p)]^{\text{new}},
\end{align}

the lift from $S_{2j-2}[\Gamma_0(1)]$ and the constant algebraic modular forms (if $k=0,j=3$).

§.§ Acknowledgements

I am very grateful to James Newton for his guidance and for carefully reading preliminary versions of this work. I would like to thank Ana Caraiani for her encouragement and her helpful suggestions. I would also like to thank Andrew Graham and Dougal Davis for many useful discussions. This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London and King's College London.

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