# Control Theorems for Hilbert Modular Varieties

Arshay Sheth<EMAIL_ADDRESS>Mathematics Institute, Zeeman
Building, University of Warwick, Coventry, CV4 7AL, UK

###### Abstract.

We prove an exact control theorem, in the sense of Hida theory, for the
ordinary part of the middle degree étale cohomology of certain Hilbert modular
varieties, after localizing at a suitable maximal ideal of the Hecke algebra.
Our method of proof builds upon the techniques introduced by
Loeffler–Rockwood–Zerbes [LRZ23]; another important ingredient in our proof is
the recent work of Caraiani–Tamiozzo [CT23] on the vanishing of the étale
cohomology of Hilbert modular varieties with torsion coefficients outside the
middle degree. This work will be used in forthcoming work of the author to
show that the Asai–Flach Euler system corresponding to a quadratic Hilbert
modular form varies in Hida families.

The author is supported by funding from the European Research Council under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 101001051 — Shimura varieties and the Birch–Swinnerton-Dyer
conjecture).

##  1\. Introduction

Let $p$ be an odd prime and let $N$ be a positive integer coprime to $p$. A
fundamental theme in Hida theory is to consider the tower of modular curves

$\cdots\rightarrow Y_{1}(Np^{r})\rightarrow\cdots\rightarrow Y_{1}(Np)$

corresponding to the chain of congruence subgroups

$\cdots\subseteq\Gamma_{1}(Np^{r})\subseteq\cdots\subseteq\Gamma_{1}(Np).$

The étale cohomology groups of this tower are packaged into the following
inverse limit

$H^{1}(Np^{\infty}):=\varprojlim_{r}H^{1}_{\textrm{\'{e}t}}(Y_{1}(Np^{r})_{\overline{\mathbb{Q}}},\mathbb{Z}_{p}(1)),$

where the transition maps are taken to be the corresponding pushfoward maps in
étale cohomology. The module $H^{1}(Np^{\infty})$ is equipped with an action
of the adjoint Hecke operators $T^{\prime}_{\ell}$ for $\ell\nmid N$ as well
as the adjoint Atkin operator $U_{p}^{\prime}$ (the usual Hecke operators
$T_{\ell}$ and $U_{p}$ do not commute with the pushforward maps and hence do
not act on the inverse limit). Analogous to the usual Hida projector, we may
define the adjoint or “anti-ordinary" Hida projector by
$e^{\prime}_{\operatorname{ord}}=\lim\limits_{n\to\infty}(U_{p}^{\prime})^{n!}$.
Building on the theory of Hida on $p$-adic families of modular forms, Ohta
[Oht99] proved a control theorem for the anti-ordinary part of
$H^{1}(Np^{\infty})$. To state the theorem, we note that $H^{1}(Np^{\infty})$
is a module over the Iwasawa algebra
$\Lambda:=\mathbb{Z}_{p}[[1+p\mathbb{Z}_{p}]]\cong\mathbb{Z}_{p}[[X]]$ via the
diamond operators. Let $\Sigma$ denote the set of primes dividing $Np$ and let
$G_{\mathbb{Q},\Sigma}$ denote the Galois group of the maximal extension of
$\mathbb{Q}$ unramified outside the primes in $\Sigma$.

######  Theorem 1.1 (Ohta).

The following hold.
(a) We have that $e^{\prime}_{\operatorname{ord}}H^{1}(Np^{\infty})$ is finite
and free as a $\Lambda$-module.
(b) For $r\geq 1$ and $k\geq 0$, let $\mathfrak{p}_{r,k}$ denote the ideal of
$\Lambda$ generated by $(1+X)^{p^{r-1}}-(1+p)^{kp^{r-1}}$. Then there is a
canonical isomorphism

$e^{\prime}_{\operatorname{ord}}H^{1}(Np^{\infty})/\mathfrak{p}_{r,k}\cong
e^{\prime}_{\operatorname{ord}}H^{1}_{\textrm{\'{e}t}}(Y_{1}(Np^{r})_{\overline{\mathbb{Q}}},\operatorname{Sym}^{k}(\mathbb{Z}_{p})(1))$

of $\mathbb{Z}_{p}$-modules that is compatible with the action of
$G_{\mathbb{Q},\Sigma}$ and the Hecke operators.

A remarkable aspect of the above theorem is that the module
$H^{1}(Np^{\infty})$, which is built only from étale cohomology groups with
constant coefficients, also embodies information about étale cohomology with
non-constant coefficients. Indeed, Theorem 1.1 can be thought of as a
cohomological version of the landmark work of Hida [Hid86], where he
constructed a space of Lambda-adic modular forms which $p$-adically
interpolate, as the weights vary, the ordinary parts of spaces of classical
modular forms. Ohta’s control theorem has since been used in a wide variety of
different contexts. For instance, building on the ideas introduced in [Oht99],
Ohta gave a new and streamlined proof of the theorem of Mazur-Wiles [MW84]
(the Iwasawa main conjecture over $\mathbb{Q}$) in a subsequent paper [Oht00].
Ohta’s control theorem has also been used as a crucial input by Lei-Loeffler-
Zerbes [LLZ14] and Kings-Loeffler-Zerbes [KLZ17] to show that the
Beilinson–Flach Euler system associated to the tensor product of two modular
forms varies in Hida families.
The main goal of this paper is to prove an analogous control theorem for the
(anti)-ordinary part of certain Hilbert modular varieties after localizing at
a suitable maximal ideal of the Hecke algebra.

###  1.1. Main results.

Let $F$ be a totally real number field of degree $g=[F:\mathbb{Q}]$ with ring
of integers $\mathcal{O}_{F}$ and discriminant $\Delta_{F}$. We fix an odd
prime $p$ which is unramified in $F$. Let $G=\textrm{Res}_{F/\mathbb{Q}}\hskip
2.84526pt{\operatorname{GL}}_{2}$ and let $E/\mathbb{Q}_{p}$ be a finite
extension with ring of integers $\mathcal{O}$ such that the maximal torus of
diagonal matrices splits over $E$. For each $n\geq 1$, we let

$U_{n,p}:=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in{\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p}):\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\equiv\begin{pmatrix}x&*\\\ 0&x\end{pmatrix}\textrm{ mod
}p^{n}\textrm{ for some }x\in\mathcal{O}_{F}\otimes\mathbb{Z}_{p}\right\\}.$

When $n=0$, we let $U_{n,p}$ denote the Iwahori subgroup i.e. the subgroup of
${\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})$ consisting of
those matrices which are upper triangular mod $p$. Let $\mathfrak{N}$ be an
ideal of $\mathcal{O}_{F}$ which does not divide 2, 3, or $\Delta_{F}$. We fix
a prime-to-$p$ open compact subgroup $U^{(p)}=\\{g\in
G(\mathbb{A}_{f}^{(p)}):g\equiv\begin{pmatrix}*&*\\\ 0&1\end{pmatrix}\textrm{
mod }\mathfrak{N}\\}$ and for all $n\geq 0$, we set $U_{n}=U^{(p)}U_{n,p}$ to
be an open compact subgroup of $G(\mathbb{A}_{f})$. We denote by
$Y_{G}(U_{n})$ the corresponding Hilbert modular variety. The reason for
working with this particular level group is explained in Remark 2.5; briefly,
since our reductive group $G$ does not satisfy the SV5 axiom for Shimura
varieties, we need to work with level groups having fixed and sufficiently
large intersection with the center $Z_{G}$.

As above, we wish to study the étale cohomology of the Hilbert modular
varieties in the tower

$\cdots\rightarrow Y_{G}(U_{r})\rightarrow\cdots\rightarrow Y_{G}(U_{1})$

which we similarly package in the Iwasawa cohomology module

$H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O}):=\varprojlim_{n}H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},\mathcal{O}),$

where the transition maps are taken to be the corresponding pushforward maps
in étale cohomology.
Let $\Sigma$ be the set of places of $F$ containing all primes dividing
$\mathfrak{N}$ and all primes above $p$. Let $\mathbb{T}$ denote the spherical
Hecke algebra generated by the standard Hecke operators $\mathcal{T}_{v}$ and
$\mathcal{S}_{v}^{\pm 1}$ for $v\not\in\Sigma$ (henceforth, we use
calligraphic font for our Hecke operators to avoid confusion with various
level groups appearing in the paper). We let $e^{\prime}_{\textrm{ord}}$
denote Hida’s anti-ordinary projection operator; as above,
$e^{\prime}_{\operatorname{ord}}=\lim\limits_{n\to\infty}(\mathcal{U}_{p}^{\prime})^{n!}$,
where $\mathcal{U}_{p}^{\prime}$ is the Hecke operator corresponding to the
double coset $U_{n,p}\begin{pmatrix}1&0\\\ 0&p\end{pmatrix}U_{n,p}$ (see
Section 3.1) . Both $\mathbb{T}$ and $e^{\prime}_{\textrm{ord}}$ act on the
étale cohomology of $Y_{G}(U_{n})$. Let $\mathfrak{m}$ be a maximal ideal in
the support of
$H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},\mathbb{F}_{p})$
such that the image of the associated Galois representation
$\overline{\rho}_{\mathfrak{m}}$ is not solvable (see Section 4).
Let
$\Lambda=\mathcal{O}\llbracket(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}\rrbracket$
i.e. $\Lambda$ is the Iwasawa algebra over $\mathcal{O}$ corresponding to
$(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}$ and for each $n\geq$ 1, let
$\Lambda_{n}$ denote the Iwasawa algebra over $\mathcal{O}$ corresponding to
elements of $\mathcal{(}\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}$ which
are congruent to 1 mod $p^{n}$. Let $\lambda$ denote a weight of $G$ that is
self-dual in the sense of Section 3.3; in other words $\lambda$ is a character
of the maximal torus of diagonal matrices of $G$ that is trivial on the
subgroup of scalar matrices. Let $V_{\lambda}$ denote the irreducible
representation of $G$ of highest weight $\lambda$. Let $\mathcal{O}[-\lambda]$
denote $\mathcal{O}$ with $\Lambda$ acting via the inverse of the character
$\lambda$.

######  Theorem 1.2.

With the notation as above, the following hold:
(a) For all $n\geq 1$, we have that
$e^{\prime}_{\operatorname{ord}}H^{g}_{\operatorname{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}$
is free as a $\Lambda_{n}$-module.
(b) For all $n\geq 1$, we have an isomorphism of $\mathcal{O}$-modules

$e^{\prime}_{\operatorname{ord}}H^{g}_{\operatorname{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}\otimes_{\Lambda_{n}}\mathcal{O}[-\lambda]\cong
e^{\prime}_{\operatorname{ord}}H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},V_{\lambda})_{\mathfrak{m}}$

that is compatible with the action of $G_{\mathbb{Q},\Sigma}$ and
$\mathbb{T}$.
(c) When $n=0$, we have a similar isomorphism

$e^{\prime}_{\operatorname{ord}}H^{g}_{\operatorname{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}\otimes_{\Lambda}\mathcal{O}[-\lambda]\cong
e^{\prime}_{\operatorname{ord}}H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{0})_{\overline{\mathbb{Q}}},V_{\lambda})_{\mathfrak{m}}.$

This theorem can be regarded as a generalisation to étale cohomology of
Hilbert varieties of Ohta’s work on étale cohomology of modular curves. In
[LRZ23], Loeffler–Rockwood–Zerbes further generalize Ohta’s result to Shimura
varieties associated to arbitrary reductive groups. However, their results do
not apply in this setting since they assume the SV5 axiom for Shimura
varieties, which does not hold for our reductive group
$G=\operatorname{Res}_{F/\mathbb{Q}}{\operatorname{GL}}_{2}$. The techniques
used in this paper build on the methods introduced in [LRZ23], but we face
additional technical difficulties due to the lack of the SV5 axiom in our
setting. Throughout this paper, we have tried to emphasize the places where
the SV5 axiom was needed in op.cit, and the alternative arguments we make in
the absence of this axiom. A key ingredient to obtaining the perfect control
(after localizing at $\mathfrak{m}$) in our main theorem is the recent work of
Caraiani–Tamiozzo [CT23] where they show that the étale cohomology of Hilbert
modular varieties with torsion coefficients is concentrated in the middle
degree after localizing at suitable maximal ideal of the Hecke algebra.

###  1.2. Arithmetic applications.

We expect that this work can find applications in the study of $p$-adic
families of various global cohomology classes in the Hilbert setting such as,
for instance, the Hirzebruch–Zagier cycles considered in [BCF19] and [FJ24].
In particular, the results of the paper will be used in forthcoming work of
the author to show that the Asai–Flach Euler system associated to a quadratic
Hilbert modular form, constructed by Lei–Loeffler–Zerbes in [LLZ18], varies in
Hida families. This in turn will be an important ingredient in forthcoming
work of Grossi–Loeffler–Zerbes on the proof of the Bloch–Kato conjecture in
analytic rank zero for the Asai representation of a quadratic Hilbert modular
form.

###  1.3. Acknowledgements

I would like to thank my advisor David Loeffler for his support and guidance,
as well as very helpful feedback on previous drafts of this paper. I am also
grateful to Rob Rockwood, Matteo Tamiozzo, Chris Williams and Ju-Feng Wu for
helpful discussions in connection with this paper.

##  2\. Background on Hilbert modular varieties

In this section, we establish some basic properties of the Hilbert modular
varieties $Y_{G}(U_{n})$ which we work with. We also follow the method in
[Urb11] to establish a relation between the Betti cohomology of these
varieties and the group cohomology of their corresponding arithmetic
subgroups.

###  2.1. Notation.

We begin by setting some notation that will remain fixed in the paper. Let $F$
be a totally real number field of degree $g$ with ring of integers
$\mathcal{O}_{F}$ and discriminant $\Delta_{F}$. We fix a numbering
$\\{\sigma_{1},\ldots,\sigma_{g}\\}$ of real embeddings of $F$ into
$\mathbb{C}$. We let $F^{\times+}$ (resp. $\mathcal{O}_{F}^{\times+})$ denote
the totally positive elements in $F^{\times}$ (resp.
$\mathcal{O}_{F}^{\times})$. Let $\mathcal{H}$ denote the upper half plane and
let $\mathcal{H}_{F}$ denote the set of elements of $F\otimes\mathbb{C}$ of
totally positive imaginary part; note that $\mathcal{H}_{F}$ can be identified
with the product of $g$ copies of $\mathcal{H}$. We let $p$ be an odd prime
that is unramified in $F$. We let $\mathbb{A}_{f}$ denote the finite adeles of
$\mathbb{Q}$, $\mathbb{A}_{f}^{(p)}$ the finite adeles away from $p$ and
$\mathbb{A}_{F,f}$ the finite adeles of $F$. We let $G$ to be the algebraic
group $\operatorname{Res}_{F/\mathbb{Q}}{\operatorname{GL}}_{2}$ over
$\mathbb{Q}$.

###  2.2. Shimura varieties for $G$

If $K\subseteq G(\mathbb{A}_{f})$ is an open compact subgroup, its
corresponding Shimura variety $Y_{G}(K)$ is a quasi-projective variety with a
canonical model over the reflex field $\mathbb{Q}$ whose complex points are
given by

$Y_{G}(K)(\mathbb{C})=G(\mathbb{Q})^{+}\char
92\relax[G(\mathbb{A}_{f})\times\mathcal{H}_{F}]/U.$

The Shimura varieties $Y_{G}(K)$ are called Hilbert modular varieties.

######  Definition 2.1.

We say than an open compact subgroup $K\subseteq G(\mathbb{A}_{f})$ is
sufficiently small if for every $h\in G(\mathbb{A}_{f})$ the quotient

$\frac{G(\mathbb{Q})^{+}\cap hKh^{-1}}{U\cap\left\\{\begin{pmatrix}u&0\\\
0&u\end{pmatrix}:u\in\mathcal{O}_{F}^{\times}\right\\}}$

acts without fixed points on $\mathcal{H}_{F}$.

######  Remark 2.2.

The above definition is slightly different from [LLZ18, Definition 2.2.1]; we
have used $\mathcal{O}_{F}^{\times}$ in the denominator rather than
$\mathcal{O}_{F}^{\times+}$ used in op.cit.

If $K\subseteq G(\mathbb{A}_{f})$ is sufficiently small, then $Y_{G}(K)$ is
smooth. We also note that if $K_{1}\subseteq K_{2}$ is an inclusion of open
compact subgroups and $K_{2}$ is sufficiently small, then the map
$Y_{G}(K_{1})\rightarrow Y_{G}(K_{2})$ is a finite étale Galois cover.

###  2.3. The Hilbert modular variety $Y_{G}(U_{n})$

######  Definition 2.3.

For each $n\geq 1$, we let

$U_{n,p}:=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in{\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p}):\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\equiv\begin{pmatrix}x&*\\\ 0&x\end{pmatrix}\textrm{ mod
}p^{n}\textrm{ for some }x\in\mathcal{O}_{F}\otimes\mathbb{Z}_{p}\right\\}$

When $n=0$, we let $U_{n,p}$ denote the Iwahori subgroup i.e. the subgroup of
${\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})$ consisting of
those matrices which are upper triangular mod $p$. Let $\mathfrak{N}$ be an
ideal of $\mathcal{O}_{F}$ which does not divide 2, 3, or $\Delta_{F}$. We fix
a prime-to-$p$ open compact subgroup

$U^{(p)}=\left\\{g\in G(\mathbb{A}_{f}^{(p)}):g\equiv\begin{pmatrix}*&*\\\
0&1\end{pmatrix}\textrm{ mod }\mathfrak{N}\right\\}$

and let $U_{n}=U^{(p)}U_{n,p}$. By [Dim09, Lemma 2.1], we have that $U_{n}$ is
sufficiently small for all $n\geq 1$ and that the determinant map
$\det:U_{n}\rightarrow\widehat{\mathbb{Z}}^{\times}$ is surjective.

The inclusion $U_{n+1}\hookrightarrow U_{n}$ induces a map of Shimura
varieties $\phi_{n}:Y_{G}(U_{n+1})\rightarrow Y_{G}(U_{n}).$

######  Proposition 2.4.

The degree of the map $\phi_{n}$ is $[U_{n}:U_{n+1}]$.

###### Proof.

Pick any element $[(x,g)]\in Y_{G}(U_{n})(\mathbb{C})=G(\mathbb{Q})^{+}\char
92\relax[G(\mathbb{A}_{f})\times\mathcal{H}_{F}]/U_{n}$ and note that

$\phi_{n}^{-1}([(x,g)])=\\{[(x,gu_{i})]:i\in I\\},$

where $\\{u_{i}\\}_{i\in I}$ is a set of representatives of $U_{n}/U_{n+1}$.
It suffices to prove that $[(x,gu_{i})]\neq[(x,gu_{j})]$ when $i\neq j$.
Suppose for contradiction that $[(x,gu_{i})]=[(x,gu_{j})]$ when $i\neq j$.
Then there exist $h\in G(\mathbb{Q})^{+}$ and $k\in U_{n+1}$ such that
$(hx,hgu_{i}k)=(x,gu_{j})$. In particular, we conclude that $h\in
G^{+}(\mathbb{Q})\cap gU_{n}g^{-1}$. Since $hx=x$ and since $U_{n}$ is
sufficiently small, we have by definition that

$h\in U_{n}\cap\left\\{\begin{pmatrix}u&0\\\
0&u\end{pmatrix}:u\in\mathcal{O}_{F}^{\times}\right\\}=\left\\{\begin{pmatrix}u&0\\\
0&u\end{pmatrix}:u\in\mathcal{O}_{F}^{\times}\right\\}.$

Thus, $h\in U_{n+1}$ as well. Using the equality, $hgu_{i}k=gu_{j}$, we can
now conclude that $u_{i}hk=u_{j}$. Since $hk$ lies in $U_{n+1}$, this
contradicts the fact that $\\{u_{i}\\}_{i\in I}$ is a set of representatives
for $U_{n}/U_{n+1}$. ∎

######  Remark 2.5.

The SV5 axiom for Shimura varieties (see [Mil17, pp.75]) states that if
$(G,X)$ is a Shimura datum, then the centre $Z$ is isogenous to the product of
a $\mathbb{Q}$-split torus and an $\mathbb{R}$-anisotropic torus. An
equivalent formlaution is that $Z(\mathbb{Q})$ is discrete in
$Z(\mathbb{A}_{f})$. In [Loe22], Loeffler showed that if the SV5 axiom is
satisfied, and if $K_{1}$ and $K_{2}$ are open compact subgroups of
$G(\mathbb{A}_{f})$ with $K_{1}\subseteq K_{2}$, then the degree of the
corresponding map of Shimura varieties $Y_{G}(K_{1})\rightarrow Y_{G}(K_{2})$
equals the index $[K_{2}:K_{1}]$. The group
$G=\textrm{Res}_{F/\mathbb{Q}}{\operatorname{GL}}_{2}$ that we are working
with does not satisfy the SV5 axiom (this is essentially because the unit
group $\mathcal{O}_{F}^{\times}$ is infinite). Nevertheless, with our choice
of level groups $U_{n}$, Proposition 2.4 shows that the desired claim still
holds.

###  2.4. The number of components of $Y_{G}(U_{n})$

Let $I(F)$ denote the group of fractional ideals of $F$, and let
$\textrm{Cl}^{+}(F):=I(F)/\\{(\beta):\beta\textrm{ totally positive}\\}$
denote the narrow class group of $F$. Let $h^{+}$ denote the narrow class
number of $F$.

######  Proposition 2.6.

The Hilbert modular variety $Y_{G}(U_{n})$ has
$h^{+}\cdot|\mathcal{O}_{F}^{\times+}\backslash(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}/\det(U_{n,p})|$
connected components.

###### Proof.

Note that the map
$\displaystyle{\mathbb{A}_{F,f}^{\times}\rightarrow\textrm{Cl}^{+}(F)}$
defined via
$\displaystyle{(\alpha)_{\mathfrak{p}}\mapsto\prod_{\mathfrak{p}}\mathfrak{p}^{v_{\mathfrak{p}}(\alpha_{\mathfrak{p}})}}$
has kernel $\prod_{\mathfrak{p}}\mathcal{O}_{F,\mathfrak{p}}^{\times}\cdot
F^{\times+}$. For each $x\in\textrm{Cl}^{+}(F)$, choose a preimage
$\alpha_{x}\in\mathbb{A}_{F,f}^{\times}$. Hence, we have a decomposition
$\displaystyle{\mathbb{A}_{F,f}^{\times}=\bigsqcup_{x\in\textrm{Cl}^{+}(F)}(\alpha_{x}+\prod_{\mathfrak{p}}\mathcal{O}_{F,\mathfrak{p}}^{\times}\cdot
F^{\times+})}$ as sets. By strong approximation, the connected components of
$Y_{G}(U_{n})$ are indexed by
$F^{\times}_{+}\backslash\mathbb{A}_{F,f}^{\times}/\det(U_{n})$. This set is
in bijection with

$\displaystyle\bigsqcup_{x\in\textrm{Cl}^{+}(F)}F^{\times+}\backslash\prod_{\mathfrak{p}}\mathcal{O}_{F,\mathfrak{p}}^{\times}\cdot
F^{\times+}/\det(U_{n})$
$\displaystyle\cong\bigsqcup_{x\in\textrm{Cl}^{+}(F)}\mathcal{O}_{F}^{\times^{+}}\backslash\prod_{\mathfrak{p}}\mathcal{O}_{F,\mathfrak{p}}^{\times}/\det(U_{n})$
$\displaystyle\cong\bigsqcup_{x\in\textrm{Cl}^{+}(F)}\mathcal{O}_{F}^{\times+}\backslash(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}/\det(U_{n,p}),$

where the last equality follows since the determinant map
$\det:U_{n}\rightarrow\widehat{\mathbb{Z}}^{\times}$ is surjective.

∎

###  2.5. Hecke action

For $K\subseteq G(\mathbb{A}_{f})$ an open compact subgroup, we let
$\mathbb{T}_{K}(G)=\mathbb{Z}[K\backslash G(\mathbb{A}_{f})/K]$ be the Hecke-
algebra of compactly supported bi-invariant functions on $G(\mathbb{A}_{f})$
with multiplication given by convolution. Let $g\in G(\mathbb{A}_{f})$ and let
$K_{g}=K\cap gKg^{-1}$; we have a correspondence $[KgK]$

${Y_{G}(K_{g})}$${Y_{G}(K_{g^{-1}})}$${Y_{G}(K)}$${Y_{G}(K)}$$\scriptstyle{f}$$\scriptstyle{[KgK]}$

where the vertical maps are canonical projections and the upper-half
horizontal map $f$ on complex points is induced by multiplication by $g$. We
obtain an action of $\mathbb{T}_{K}(G)$ on
$H^{i}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{F}_{p})$ and
$H^{i}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})$.
Let $\Sigma$ be the set of places of $F$ containing all primes dividing
$\mathfrak{N}$ and all primes above $p$. Let

$\mathbb{T}:=\bigotimes^{{}^{\prime}}_{v\not\in\Sigma}\mathbb{Z}[{\operatorname{GL}}_{2}(\mathcal{O}_{v})\backslash{\operatorname{GL}}_{2}(F_{v})/{\operatorname{GL}}_{2}(\mathcal{O}_{v})]$

denote the abstract spherical algebra away from $\Sigma$. We note that
$\mathbb{T}$ is a subalgebra of $\mathbb{T}_{K}(G)$; it is also commutative,
generated by the following Hecke operators $\mathcal{T}_{v}$ and
$\mathcal{S}_{v}^{\pm 1}$ for every finite place $v\not\in\Sigma$; for every
such $v$ we choose a uniformizer $\mathcal{\varpi}_{v}$ of $\mathcal{O}_{v}$
and define

  * •

$\mathcal{T}_{v}$ to be the double coset
${\operatorname{GL}}_{2}(\mathcal{O}_{v})\begin{pmatrix}\varpi_{v}&0\\\
0&1\end{pmatrix}{\operatorname{GL}}_{2}(\mathcal{O}_{v}).$

  * •

$\mathcal{S}_{v}$ to the double coset
${\operatorname{GL}}_{2}(\mathcal{O}_{v})\begin{pmatrix}\varpi_{v}&0\\\
0&\varpi_{v}\end{pmatrix}{\operatorname{GL}}_{2}(\mathcal{O}_{v}).$

###  2.6. Betti cohomology

Let $\Gamma$ be a subgroup of $G^{+}(\mathbb{Q})$ and let $\overline{\Gamma}$
be its image in $G(\mathbb{Q})/Z_{G}(\mathbb{Q})$. We assume that
$\overline{\Gamma}$ has no non-trivial elements of finite order; hence, it
acts freely and continuously on $\mathcal{H}_{F}$. In this subsection, we
closely follow [Urb11] to establish a relation between the group cohomology of
$\overline{\Gamma}$ and the Betti cohomology of the corresponding Hilbert
modular variety.

By the work of Borel–Serre, there exists a canonical compactification
$\overline{\Gamma}\backslash\overline{\mathcal{H}_{F}}$, where
$\overline{\mathcal{H}_{F}}$ is a contractible real manifold with corners.
Since $\overline{\Gamma}\backslash\overline{\mathcal{H}_{F}}$ is compact, we
may choose a finite triangulation of
$\overline{\Gamma}\backslash\overline{\mathcal{H}_{F}}$. We may pull it back
to $\overline{\mathcal{H}_{F}}$ via the canonical projection
$\overline{\mathcal{H}_{F}}\rightarrow\overline{\Gamma}\backslash\overline{\mathcal{H}_{F}}$.
Let $C_{q}(\overline{\Gamma})$ be the free $\mathbb{Z}$-module over the set of
$q$-dimensional simplices of the triangulation obtained by pull-back to
$\overline{\mathcal{H}_{F}}$. Since the action of $\overline{\Gamma}$ on
$\overline{\mathcal{H}_{F}}$ is free, and since
$\overline{\Gamma}\backslash\overline{\mathcal{H}_{F}}$ is finite, the
$C_{q}(\overline{\Gamma})$’s are free $\mathbb{Z}[\overline{\Gamma}]$-modules
of finite type. We also note that

$0\rightarrow
C_{d}(\overline{\Gamma})\xrightarrow{\partial_{d-1}}\cdots\xrightarrow{\partial_{1}}C_{0}(\overline{\Gamma})\rightarrow
0$

is a complex computing the homology of $\overline{\mathcal{H}_{F}}$. Since
$\overline{\mathcal{H}_{F}}$ is contractible, this complex is exact except in
degree zero and
$H_{0}(\overline{\mathcal{H}_{F}},\mathbb{Z})=C_{0}(\overline{\Gamma})/\partial_{0}(C_{1}(\overline{\Gamma}))=\mathbb{Z}$.
Thus, in summary, we have that

$0\rightarrow
C_{d}(\overline{\Gamma})\xrightarrow{\partial_{d-1}}\cdots\xrightarrow{\partial_{1}}C_{0}(\overline{\Gamma})\rightarrow\mathbb{Z}\rightarrow
0$

is an exact sequence of finite free $\mathbb{Z}[\overline{\Gamma}]$-modules.
If $M$ is a $\Gamma$-module, we let
$\mathcal{C}^{\bullet}(\overline{\Gamma},M)$ denote the complex

$0\rightarrow\textrm{Hom}(C_{0}(\overline{\Gamma}),M)\rightarrow\cdots\rightarrow\textrm{Hom}(C_{d}(\overline{\Gamma}),M).$

Thus, $H^{i}(\overline{\Gamma},M)$ is the $i$-th cohomology group of the
complex $\mathcal{C}^{\bullet}(\overline{\Gamma},M)$.

Let $K$ be an open compact subgroup of $G(\mathbb{A}_{f})$ which is
sufficently small and let $M$ be a left $K$-module acting via its projection
to $K_{p}$, the image of $K$ in $G(\mathbb{Q}_{p})$. The corresponding Hilbert
modular variety $Y_{G}(K)$ satisfies

$Y_{G}(K)\cong\bigsqcup_{j\in J}\Gamma_{j}\backslash\mathcal{H}_{F},$

where $J$ is a finite set and for each $j\in J$,
$\Gamma_{j}=g_{j}Kg_{j}^{-1}\cap G^{+}(\mathbb{Q})$ for some $g_{j}\in
G(\mathbb{A}_{f})$. As before, we let $\overline{\Gamma_{j}}$ denote the image
of $\Gamma_{j}$ in $G(\mathbb{Q})/Z_{G}(\mathbb{Q})$. We set

$\displaystyle C^{\bullet}(K,M):=\bigoplus_{j\in
J}C^{\bullet}(\overline{\Gamma_{j}},M)$ (1)

Let $\overline{Y_{G}}:=G(\mathbb{Q})^{+}\backslash
G(\mathbb{A}_{f})\times\overline{\mathcal{H}_{F}}$. Then
$\overline{Y_{G}}(K):=\overline{Y_{G}}/K$ is the Borel–Serre compactification
of $Y_{G}(K)$. Let $\pi:\overline{Y_{G}}\rightarrow\overline{Y_{G}}(K)$ denote
the canonical projection. We choose a finite triangulation of
$\overline{Y_{G}}(K)$ and pull it back via $\pi$. Let $C^{q}(K)$ denote the
corresponding chain complex equipped with a right action of $K$. Then

$C^{\bullet}(K,M)=\textrm{Hom}_{K}(C^{q}(K),M).$

Thus, $C^{\bullet}(K,M)$ also computes the cohomology of the local system $M$
on $Y_{G}(K)$ and so we have an isomorphism

$\displaystyle H^{i}(Y_{G}(K),M)\cong\bigoplus_{j\in
J}H^{i}(\overline{\Gamma}_{j},M).$ (2)

##  3\. Construction of a Tor spectral sequence

In this section, we construct a Tor descent spectral sequence which will be an
important tool to relate the Iwasawa cohomology module
$H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})$
to the cohomology of $Y_{G}(U_{n})$ at finite layers.

###  3.1. General notation

We consider the group scheme $\textrm{GL}_{2}$ over $\mathbb{Z}_{p}$ and we
let $B_{2}$, $N_{2}$ and $T_{2}$ to be the subgroups of upper-triangular,
unipotent and diagonal matrices respectively. Following [LRZ23], we set the
following notation:

  * •

$Q=\textrm{Res}^{\mathcal{O}_{F}\otimes\mathbb{Z}_{p}}_{\mathbb{Z}_{p}}\hskip
2.84526ptB_{2}$.

  * •

$N=\textrm{Res}^{\mathcal{O}_{F}\otimes\mathbb{Z}_{p}}_{\mathbb{Z}_{p}}\hskip
2.84526ptN_{2}$.

  * •

$S=\textrm{Res}^{\mathcal{O}_{F}\otimes\mathbb{Z}_{p}}_{\mathbb{Z}_{p}}\hskip
2.84526ptT_{2}$.

  * •

$E/\mathbb{Q}_{p}$ is a finite extension with ring of integers $\mathcal{O}$
such that $S$ splits over $E$.

  * •

$S^{0}=\textrm{Res}^{\mathcal{O}_{F}\otimes\mathbb{Z}_{p}}_{\mathbb{Z}_{p}}\hskip
2.84526pt\mathbb{G}_{m}$, viewed as a subgroup of $S$ via the diagonal
embedding.

  * •

$Q^{0}$ denotes the preimage of $S^{0}$ under the projection $Q\rightarrow S$
i.e. $Q^{0}(\mathbb{Z}_{p})=\left\\{\begin{pmatrix}a&b\\\
0&a\end{pmatrix}\in{\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})|a,b\in\mathcal{O}_{F}\otimes\mathbb{Z}_{p}\right\\}$.

  * •

$\mathfrak{S}=S(\mathbb{Z}_{p})/S^{0}(\mathbb{Z}_{p})$ identified with
$(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}$ via the short exact sequence

$1\rightarrow S^{0}(\mathbb{Z}_{p})\rightarrow
S(\mathbb{Z}_{p})\rightarrow(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}\rightarrow
1.$

where the map

$S(\mathbb{Z}_{p})\rightarrow(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}\hskip
8.53581pt\text{ is given by }\begin{pmatrix}a&0\\\ 0&b\end{pmatrix}\mapsto
ab^{-1}.$

  * •

$\Lambda=\mathcal{O}\llbracket\mathfrak{S}\rrbracket$.

  * •

$\tau=\begin{pmatrix}p&0\\\
0&1\end{pmatrix}\in{\operatorname{GL}}_{2}(\mathbb{Q}_{p}\otimes\mathcal{O}_{F})$.

  * •

$N_{r}=\tau^{r}N(\mathbb{Z}_{p})\tau^{-r}=\left\\{\begin{pmatrix}1&x\\\
0&1\end{pmatrix}\in{\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})|\hskip
2.84526ptx\equiv 0\textrm{ mod }p^{r}\right\\}$.

  * •

$\overline{N_{r}}=\tau^{-r}\overline{N}(\mathbb{Z}_{p})\tau^{r}=\left\\{\begin{pmatrix}1&0\\\
x&1\end{pmatrix}\in{\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})|\hskip
2.84526ptx\equiv 0\textrm{ mod }p^{r}\right\\}$

  * •

$L_{r}=\\{\ell\in S(\mathbb{Z}_{p}):\ell\in S^{0}(\mathbb{Z}_{p})\textrm{ mod
}p^{r}\\}=\left\\{\begin{pmatrix}a&0\\\
0&d\end{pmatrix}\in{\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})|\hskip
2.84526pta\equiv d\textrm{ mod }p^{r}\right\\}$.

  * •

$V_{r}=\overline{N_{r}}L_{r}N_{0}$.

######  Proposition 3.1.

For every $r\geq 1$, we have that $V_{r}=U_{r,p}$.

###### Proof.

This follows from a direct matrix computation. ∎

We also set
$V_{0}={\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})$ and
$V_{0}^{(s)}:=\tau^{-s}V_{0}\tau^{s}\cap V_{0}$ for $s\geq 1$; in other words,
$V_{0}^{(s)}$ is the group of upper-triangular matrices $\textrm{mod }p^{s}:$

$V_{0}^{(s)=}\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in{\operatorname{GL}}_{2}(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})|a,b,c,d\in\mathcal{O}_{F}\otimes\mathbb{Z}_{p}\textrm{
and }c\equiv 0\textrm{ mod }p^{s}\right\\}.$

Similarly, for $s\geq 0$ and $n\geq 0$, we set

$U_{n,p}^{(s)}:=\tau^{-s}U_{n,p}\tau^{s}\cap U_{n,p}.$

For our level groups $U_{n,p}$, we define the Hecke operator
$\mathcal{U}_{p}^{\prime}$ to be the double coset
$U_{n,p}\begin{pmatrix}1&0\\\ 0&p\end{pmatrix}U_{n,p}$ and we define the
analogue of Hida’s anti-ordinary projection operator to be

$e^{\prime}_{\operatorname{ord}}=\lim_{n\to\infty}(\mathcal{U}_{p}^{\prime})^{n!}.$

The operator $e^{\prime}_{\operatorname{ord}}$ acts on the cohomology of
$Y_{G}(U_{n})$. Let $M$ be a $\mathbb{Z}[U_{n}]$-module (acting via projection
to $U_{n,p}$) with a compatible action of $\mathcal{U}_{p}^{\prime}$ (in the
sense that for any $u\in U_{n,p}^{(s)}$, the action of
$\mathcal{U}_{p}^{\prime}$ intertwines the action of $u$ and
$\tau^{-s}u\tau^{s}$). For $K\subseteq G(\mathbb{A}_{f})$ an open compact
subgroup, the operator $e^{\prime}_{\operatorname{ord}}$ also acts on the
complexes $C^{\bullet}(K,M)$ introduced in Section 2.6 by lifting the action
on cohomology (see [LRZ23, pp.6]).

######  Lemma 3.2.

Let $M$ be a $\mathbb{Z}[U_{n}]$-module with a compatible action of
$\mathcal{U}_{p}^{\prime}$. The following diagram commutes on cohomology

${\mathcal{C}^{\bullet}(U^{(p)}U_{n,p}^{(s)},M)}$${\mathcal{C}^{\bullet}(U_{n},M)}$${\mathcal{C}^{\bullet}(U^{(p)}U_{n,p}^{(s)},M)}$${\mathcal{C}^{\bullet}(U_{n},M)}$$\scriptstyle{(\mathcal{U}_{p}^{\prime})^{s}}$$\scriptstyle{\mathrm{cores}}$$\scriptstyle{{[\tau^{s}]}_{*}}$$\scriptstyle{(\mathcal{U}_{p}^{\prime})^{s}}$$\scriptstyle{\mathrm{cores}}$

###### Proof.

This follows from [LRZ23, Lemma 2.7.4] (the SV5 axiom is not needed in the
proof of this lemma). ∎

######  Proposition 3.3.

The corestriction maps induce isomorphisms

$e^{\prime}_{\operatorname{ord}}H^{i}(Y_{G}(U^{(p)}U_{n,p}^{(s)}),M)\cong
e^{\prime}_{\operatorname{ord}}H^{i}(Y_{G}(U_{n}),M).$

###### Proof.

As explained in [LRZ23, Corollary 2.7.5], this follows from the previous
lemma. ∎

###  3.2. Algebraic representations

Let $X^{\bullet}(S)$ denote the character lattice of $S$ and
$X^{\bullet}_{+}(S)$ be the set of dominant weights. For each $\lambda\in
X^{\bullet}_{+}(S)$, there is a unique isomorphism class of irreducible
representations of $(\rho_{\lambda},V_{\lambda})$ of $G$ (over $E$) of highest
weight $\lambda$. A representative of this isomorphism class can be
constructed using the Borel–Weil–Bott theorem, as the space of all polynomials

$\\{f\in E[G]:f(\overline{n}\ell g)=\lambda(\ell)f(g)\hskip
5.69054pt\forall\overline{n}\in\overline{N},\ell\in S,g\in G\\},$

with $G$ acting by right translation. More concretely, each such $\lambda\in
X^{\bullet}_{+}(S)$ can be identified with an integer tuple
$(k_{1},\ldots,k_{g},t_{1},\ldots,t_{g})$ such that the associated
$V_{\lambda}$ is given by the representation
$\textrm{Sym}^{k_{i}}V\otimes\det^{t_{i}}$ at the $i$-th embedding, where $V$
is the standard representation of $G$. All our Hecke operators defined above
also act on cohomology with the $V_{\lambda}$’s as coefficient systems (see
[LRZ23, Definition 2.5.1]).

###  3.3. Modules of measures

In this subsection, for brevity, we let $U$ to be $U_{n,p}$ for some $n\geq
0$. We also fix a character $\lambda=(k_{1},\ldots,k_{g},t_{1},\ldots t_{g})$
such that $k_{1}+2t_{1}=\cdots=k_{g}+2t_{g}=0$.

######  Lemma 3.4.

We have that $\lambda$ is trivial on the subtorus $S^{0}$.

###### Proof.

This follows immediately from our normalization of the weights described in
Section 3.2; namely,

$\lambda\left(\begin{pmatrix}a&0\\\
0&a\end{pmatrix}\right)=\sigma_{1}(a)^{k_{1}+2t_{1}}\cdots\sigma_{g}^{k_{g}+2t_{g}}(a)=1.\qed$

######  Remark 3.5.

The weights we have considered above are exactly those which are self-dual i.e
those for which the dual of $V_{\lambda}$ is isomorphic to itself.

Following [LRZ23], we set the following notation. We define $U_{0,p}$\-
modules of continuous functions

$C_{\lambda,U}:=\\{f:U_{0,p}\rightarrow\mathcal{O}:f\textrm{
continuous},f(\ell ng)=\lambda^{-1}(\ell)f(g)\hskip 5.69054pt\forall
ln\in(S\cap U)N_{0},g\in U_{0,p}\\}$

and

$C_{\textrm{univ}}:=\\{f:U_{0,p}\rightarrow\mathcal{O}:f\textrm{
continuous},f(\ell ng)=f(g)\hskip 5.69054pt\forall ln\in
Q^{0}(\mathbb{Z}_{p}),g\in U_{0,p}\\},$

with $U_{0,p}$ acting by right translation. We endow these spaces with an
action of $\tau$ given by

$\tau\cdot f(n\ell\bar{n})=f(n\ell\tau^{-1}\bar{n}\tau).$

We define modules of bounded distributions

$D_{\lambda,U}:=\textrm{Hom}_{\textrm{cts}}(C_{\lambda,U},\mathcal{O})$

and

$D_{\textrm{univ}}:=\textrm{Hom}_{\textrm{cts}}(C_{\textrm{univ}},\mathcal{O})$

which inherit actions of $V_{0}^{\prime}$ and $\tau^{-1}$ by duality. We let
$\mathfrak{S}_{U}$ denote the image of $U$ in $\mathfrak{S}$; note that
$\mathfrak{S}_{U}=\mathfrak{S}\cong(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}$
when $U=U_{0,p}$ and $\mathfrak{S}_{U}$ is the set of elements of
$\mathcal{(}\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}$ which are
congruent to 1 mod $p^{n}$ when $U=U_{n,p}$ and $n\geq 1$. We let
$\mathcal{O}\llbracket\mathfrak{S}_{U}\rrbracket$ denote the Iwasawa algebra
corresponding to $\mathfrak{S}_{U}$. We have that $D_{\lambda,U}$ and
$D_{\textrm{univ}}$ are modules over
$\Lambda=\mathcal{O}\llbracket\mathfrak{S}\rrbracket$ (with the action given
by inverse translation) and this structure is given explicitly by the
isomorphisms

$D_{\lambda,U}\cong(\mathcal{O}[-\lambda]\otimes_{\mathcal{O}\llbracket\mathfrak{S}_{U}\rrbracket}\Lambda)\hat{\otimes}_{\mathcal{O}}\mathcal{O}\llbracket\bar{N_{0}}\rrbracket$

and

$D_{\textrm{univ}}\cong\Lambda\hat{\otimes}_{\mathcal{O}}\mathcal{O}\llbracket\bar{N_{0}}\rrbracket,$

where $\mathcal{O}[-\lambda]$ denotes $\mathcal{O}$ regarded as an
$\mathfrak{S}$ (and hence $\Lambda$) module via the inverse of $\lambda$. In
particular, we have that

$D_{\textrm{univ}}\hat{\otimes}_{\mathcal{O}\llbracket\mathfrak{S}_{U}\rrbracket}\mathcal{O}[-\lambda]\cong
D_{\lambda,U}$

as $\Lambda$-modules. Since a power series ring in finitely many variables
over a Noetherian ring is flat (see [Bou06, pp.146]) , we have that
$D_{\operatorname{univ}}$ is flat as a $\Lambda$-module.

######  Proposition 3.6.

The anti-ordinary projector $e^{\prime}_{\operatorname{ord}}$ acts on
$\mathcal{C}^{\bullet}(U,D_{\operatorname{univ}})$ such that we have a
decomposition

$C^{\bullet}(U,D_{\operatorname{univ}})=e^{\prime}_{\operatorname{ord}}C^{\bullet}(U,D_{\operatorname{univ}})\oplus(1-e^{\prime}_{\operatorname{ord}})C^{\bullet}(U,D_{\operatorname{univ}}).$

with $\mathcal{U}_{p}^{\prime}$ acting invertibly on the first component and
topologically nilpotently on the second. Moreover, the complex
$e^{\prime}_{\operatorname{ord}}C^{\bullet}(U,D_{\operatorname{univ}})$
consists of flat $\Lambda$-modules.

###### Proof.

See Proposition 2.7.3 in [LRZ23]. ∎

###  3.4. Proof of the Tor spectral sequence

In this subsection, we let $\lambda$ denote a weight that is self-dual in the
sense of Section 3.3. We let $s$ and $n$ be integers with $s\geq n$. Let
$\Gamma_{1}=G(\mathbb{Q})^{+}\cap U^{(p)}V_{0}^{(s)}$ and
$\Gamma_{2}=G(\mathbb{Q})^{+}\cap(U^{(p)}(U_{n,p}\cap V_{0}^{(s)}))$. Let
$\overline{\Gamma_{1}}$ and $\overline{\Gamma_{2}}$ denote the images of
$\Gamma_{1}$ and $\Gamma_{2}$ in $G/Z_{G}(\mathbb{Q})$. Write

$Y_{G}(U^{(p)}V_{0}^{(s)})(\mathbb{C})=\bigsqcup_{j\in
J_{1}}\Gamma_{1}\backslash\mathcal{H}_{F},$

and

$Y_{G}(U^{(p)}(V_{0}^{(s)}\cap U_{n,p}))(\mathbb{C})\cong\bigsqcup_{j\in
J_{2}}\Gamma_{2}\backslash\mathcal{H}_{F}.$

Each $i\in J_{1}$ corresponds to a matrix $g_{i}\in G(\mathbb{A}_{f})$ whose
determinants form a set of representatives for
$F^{\times+}\backslash\mathbb{A}_{F,f}^{\times}/\det(U^{(p)}V_{0}^{(s)})$.
Similarly, each $i\in J_{2}$ corresponds to a matrix $g_{i}\in
G(\mathbb{A}_{f})$ whose determinants form a set of representatives for
$F^{\times+}\backslash\mathbb{A}_{F,f}^{\times}/\det(U^{(p)}(U_{n,p}\cap
V_{0}^{(s)}))$.

Since
$\det(U^{(p)}V_{0}^{(s)}))=(\mathcal{O}_{F}\otimes\hat{\mathbb{Z}})^{\times}$,
we have that $|J_{1}|=h^{+}$, and by Lemmma 2.6, $|J_{2}|=$
$h^{+}\cdot|\mathcal{O}_{F}^{\times^{+}}\backslash(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}/\det(U_{n,p})|$
(as written, Lemma 2.6 can only be applied when $s=n$ but the exact same proof
goes through when $s\geq n$ since $\det(U_{n,p})=\det(U_{n,p}\cap
V_{0}^{(s)})\textrm{ in }(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times})$.

######  Proposition 3.7.

Let $\varphi$ denote the natural map

$\varphi:\Gamma_{1}/\Gamma_{2}\rightarrow V_{0}^{(s)}/(U_{n,p}\cap
V_{0}^{(s)})$

Then $\operatorname{im}(\varphi)$ has index
$|\mathcal{O}_{F}^{\times+}\backslash(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}/\det(U_{n,p})|$
in $V_{0}^{(s)}/(U_{n,p}\cap V_{0}^{(s)})$.

###### Proof.

By strong approximation for the semisimple group
$\textrm{Res}_{F/\mathbb{Q}}\textrm{SL}_{2}$, the diagram

${\Gamma_{1}/\Gamma_{2}}$${V_{0}^{(s)}/(U_{n,p}\cap
V_{0}^{(s)})}$${\mathcal{O}_{F}^{\times+}/(\mathcal{O}_{F}^{\times})^{2}}$${(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}/\det(U_{n,p})}$$\scriptstyle{\det}$$\scriptstyle{\det}$

is cartesian. On the other hand, note that
$\det(U_{n,p})=(\mathcal{O}_{F}^{\times})^{2}$ in
$(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}$ by Hensel’s lemma; thus we
conclude that
$\displaystyle{|\textrm{coker}(\varphi)|=\frac{[(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}:\det(U_{n,p})]}{[\mathcal{O}_{F}^{\times+}:(\mathcal{O}_{F}^{\times})^{2}]}}$
and that the natural map

$(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}/\det(U_{n,p})\rightarrow\mathcal{O}_{F}^{\times+}\backslash(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}/\det(U_{n,p})$

has kernel $\mathcal{O}_{F}^{\times+}/(\mathcal{O}_{F}^{\times})^{2}$. Thus,
$\displaystyle{\frac{[(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}:\det(U_{n,p})]}{[\mathcal{O}_{F}^{\times+}:(\mathcal{O}_{F}^{\times})^{2}]}=|\mathcal{O}_{F}^{\times^{+}}\backslash(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}/\det(U_{n,p})|}$
proving the claim.

∎

######  Proposition 3.8.

We have a Hecke-equivariant isomorphism

$\bigoplus_{j\in
J_{1}}H^{i}(\overline{\Gamma_{1}},\mathcal{O}/(p^{s})[-\lambda][V_{0}^{(s)}/V_{0}^{(s)}\cap
U_{n,p}])\cong\bigoplus_{j\in
J_{2}}H^{i}(\overline{\Gamma_{1}},\mathcal{O}/(p^{s})[-\lambda][\overline{\Gamma_{1}}/\overline{\Gamma_{2}}]).$

###### Proof.

This follows from the previous proposition and the fact that
$|J_{1}|=h^{+}\text{ and }\newline
|J_{2}|=h^{+}\cdot|\mathcal{O}_{F}^{\times+}\backslash(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}/\det(U_{n,p})|.$
∎

######  Theorem 3.9.

We have an isomorphism of $\mathcal{O}$-modules

$e^{\prime}_{\operatorname{ord}}H^{i}(Y_{G}(U_{0}),D_{\lambda,U_{n,p}})\cong
e^{\prime}_{\operatorname{ord}}H^{i}(Y_{G}(U_{n}),V_{\lambda}).$

###### Proof.

As explained in [LRZ23], Lemma 3.2 along with the fact that
$e^{\prime}_{\textrm{ord}}$ contracts the $\overline{N_{0}}$ part of
$D_{\lambda,U}$ implies that we have an isomorphism

$e^{\prime}_{\textrm{ord}}H^{i}(Y_{G}(U_{0}),D_{\lambda,U_{n,p}}/p^{s})\cong
e^{\prime}_{\textrm{ord}}H^{i}(Y_{G}(U^{(p)}V_{0}^{(s)}),\mathcal{O}/(p^{s})[-\lambda]\otimes_{\mathcal{O}[\mathfrak{S}_{U_{n,p}}]}\mathcal{O}/(p^{s})[\mathfrak{S}]).$

Hence,

$\displaystyle
H^{i}(Y_{G}(U^{(p)}V_{0}^{(s)}),\mathcal{O}/(p^{s})[-\lambda]\otimes_{\mathcal{O}[\mathfrak{S}_{U_{n,p}}]}\mathcal{O}(p^{n})[\mathfrak{S}])$
$\displaystyle\cong\bigoplus_{j\in
J_{1}}H^{i}(\overline{\Gamma_{1}},\mathcal{O}/(p^{s})[-\lambda]\otimes_{\mathcal{O}[\mathfrak{S}_{U_{n,p}}]}\mathcal{O}/(p^{s})[\mathfrak{S}])$
$\displaystyle\cong\bigoplus_{j\in
J_{1}}H^{i}(\overline{\Gamma_{1}},\mathcal{O}/(p^{s})[-\lambda][\mathfrak{S}/\mathfrak{S}_{U_{n,p}}])$
$\displaystyle\cong\bigoplus_{j\in
J_{1}}H^{i}(\overline{\Gamma_{1}},\mathcal{O}/(p^{s})[-\lambda][V_{0}^{(s)}/V_{0}^{(s)}\cap
U_{n,p}])$ $\displaystyle\cong\bigoplus_{j\in
J_{2}}H^{i}(\overline{\Gamma_{1}},\mathcal{O}/(p^{s})[-\lambda][\overline{\Gamma_{1}}/\overline{\Gamma_{2}}])$
$\displaystyle\cong\bigoplus_{j\in
J_{2}}H^{i}(\overline{\Gamma_{2}},\mathcal{O}/(p^{s})[-\lambda])$
$\displaystyle\cong H^{i}(Y_{G}(U^{(p)}(V_{0}^{(s)}\cap
U_{n,p})),\mathcal{O}/(p^{s})[-\lambda]).$

Here, the first isomorphism follows from Equation (2), the second is a general
property of tensor products, the third follows the fact that
$\mathfrak{S}/\mathfrak{S}_{U_{n,p}}\cong V_{0}^{(s)}/V_{0}^{(s)}\cap
U_{n,p}$, the fourth follows from Proposition 3.8 , the fifth follows from
Shapiro’s Lemma and the sixth follows from Equation (2) again. As explained in
[LRZ23, Proposition 2.7.6], we can use Proposition 3.3 to conclude that
$e^{\prime}_{\textrm{ord}}H^{i}(Y_{G}(U^{(p)}(V_{0}^{(s)}\cap
U_{n,p})),\mathcal{O}/(p^{s})[-\lambda])$ is in turn isomorphic to
$e^{\prime}_{\textrm{ord}}H^{i}(Y_{G}(U_{n}),V_{\lambda}/p^{s})$ . Thus,
combining our isomorphims, we conclude that

$\displaystyle
e^{\prime}_{\textrm{ord}}H^{i}(Y_{G}(U_{0}),D_{\lambda,U_{n,p}})$
$\displaystyle=\varprojlim_{s}e^{\prime}_{\textrm{ord}}H^{i}(Y_{G}(U_{0}),D_{\lambda,U_{n,p}}/p^{s})$
$\displaystyle\cong\varprojlim_{s}e^{\prime}_{\textrm{ord}}H^{i}(Y_{G}(U_{n}),V_{\lambda}/p^{s})$
$\displaystyle=e^{\prime}_{\textrm{ord}}H^{i}(Y_{G}(U_{n}),V_{\lambda}).$

∎

######  Remark 3.10.

The analogue of this theorem in [LRZ23] ([LRZ23, Proposition 2.7.6]) crucially
used the SV5 axiom for Shimura varieties in the application of Shapiro’s
lemma. Nevertheless, as the proof given above demonstrates, with our choice of
level groups, we do not need to invoke this axiom.

We note that Theorem 3.9 is compatible with the comparision isomorphism
between Betti and étale cohomology; for the rest of the paper, we work with
étale cohomology of Hilbert modular varieties.

######  Corollary 3.11.

We have an isomorphism

$e^{\prime}_{\operatorname{ord}}H^{i}_{\textrm{\'{e}t}}(Y_{G}(U_{0}),D_{\operatorname{univ}})\cong
e^{\prime}_{\operatorname{ord}}\varprojlim_{n}H^{i}_{\textrm{\'{e}t}}(Y_{G}(U_{n}),\mathcal{O})$

###### Proof.

Setting $\lambda$ to be the trivial character we deduce from the previous
theorem that

$\displaystyle
e^{\prime}_{\textrm{ord}}H^{i}_{\textrm{\'{e}t}}(Y_{G}(U_{0}),D_{\textrm{univ}})$
$\displaystyle\cong\varprojlim_{Q^{0}\subseteq
U,n}e^{\prime}_{\textrm{ord}}H^{i}_{\textrm{\'{e}t}}(Y_{G}(U_{0}),D_{\lambda,U}/(p^{n}))$
$\displaystyle\cong\varprojlim_{n}e^{\prime}_{\textrm{ord}}H^{i}_{\textrm{\'{e}t}}(Y_{G}(U_{n}),\mathcal{O}/(p^{n}))$
$\displaystyle\cong
e^{\prime}_{\textrm{ord}}\varprojlim_{n}H^{i}_{\textrm{\'{e}t}}(Y_{G}(U_{n}),\mathcal{O}).$

∎

Define $M^{\bullet}$ to be the image of
$e^{\prime}_{\textrm{ord}}C^{\bullet}(U_{0},D_{\textrm{univ}})$ in the
subcategory $D^{\textrm{flat}}(R)$ of the derived category of $R$-modules
generated by flat objects. We also set
$\Lambda_{n}=\mathcal{O}\llbracket\mathfrak{S}_{U_{n,p}}\rrbracket$ for all
$n\geq 1$.

######  Theorem 3.12.

For all $n\geq 1$, we have a quasi-isomorphism

$M^{\bullet}\otimes^{\mathbb{L}}_{\Lambda_{n}}\mathcal{O}[-\lambda]\cong
e^{\prime}_{\operatorname{ord}}C^{\bullet}(U_{n},V_{\lambda}).$

###### Proof.

Since $M^{\bullet}$ is represented by the flat complex
$e^{\prime}_{\textrm{ord}}\mathcal{C}^{\bullet}(U_{0},D_{\textrm{univ}})$, we
can compute the derived tensor product as

$e^{\prime}_{\textrm{ord}}\mathcal{C}^{\bullet}(U_{0},D_{\textrm{univ}})\otimes_{\Lambda_{n}}\mathcal{O}[-\lambda]=e^{\prime}_{\textrm{ord}}\mathcal{C}^{\bullet}(U_{0},D_{\textrm{univ}}\otimes_{\Lambda_{n}}\mathcal{O}[-\lambda]).$

By Theorem 3.9, this complex is isomorphic to $\
e^{\prime}_{\textrm{ord}}C^{\bullet}(U_{n},V_{\lambda})$. ∎

######  Corollary 3.13.

For all $n\geq 1$, there is a spectral sequence

$E_{2}^{i,j}:{\operatorname{Tor}}_{-i}^{\Lambda_{n}}(e^{\prime}_{\operatorname{ord}}\varprojlim_{n}H^{j}_{\textrm{\'{e}t}}(Y_{G}(U_{n}),\mathcal{O}),\mathcal{O}[-\lambda])\implies
e^{\prime}_{\textrm{ord}}H^{i+j}_{\textrm{\'{e}t}}(Y_{G}(U_{n}),V_{\lambda}).$

###### Proof.

This follows from the previous theorem using the spectral sequence for the Tor
functor. ∎

An analogous argument to the one given in this section allows us to obtain a
similar result at the Iwahori level $U_{0}$.

######  Corollary 3.14.

There is a spectral sequence

$E_{2}^{i,j}:{\operatorname{Tor}}_{-i}^{\Lambda}(e^{\prime}_{\operatorname{ord}}\varprojlim_{n}H^{j}_{\textrm{\'{e}t}}(Y_{G}(U_{0}),\mathcal{O}),\mathcal{O}[-\lambda])\implies
e^{\prime}_{\operatorname{ord}}H_{\textrm{\'{e}t}}^{i+j}(Y_{G}(U_{0}),V_{\lambda}).$

##  4\. Proof of the control theorem

In this section, we use the spectral sequences in Corollaries 3.13 and 3.14 to
give a proof of our control theorem. We begin by recalling the results of
Caraiani–Tamiozzo [CT23], which play a crucial role in proving our control
theorem.

###  4.1. The results of Caraiani–Tamiozzo

Let $K\subseteq G(\mathbb{A}_{f})$ be a neat compact open subgroup and take a
maximal ideal $\mathfrak{m}\subseteq\mathbb{T}$ in the support of
$H^{i}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{F}_{p})$. By the work of
Scholze [Sch15], we have a unique continuous semisimple Galois representation

$\overline{\rho_{\mathfrak{m}}}:{\operatorname{Gal}}(\overline{F}/F)\rightarrow{\operatorname{GL}}_{2}(\overline{\mathbb{F}_{p}})$

such that $\overline{\rho_{\mathfrak{m}}}$ is unramified for all $v$ not in a
finite set of suitable places of $F$ and such that the characteristic
polynomial of $\overline{\rho_{\mathfrak{m}}}(\mathrm{Frob}_{v})$ equals
$X^{2}-\mathcal{T}_{v}X+\mathcal{S}_{v}N(v)\textrm{ mod }\mathfrak{m}$.

######  Theorem 4.1 (Caraiani–Tamiozzo).

For any $i\in\mathbb{N}_{\geq 1}$, let $\mathfrak{m}$ be a maximal ideal in
the support of
$H^{i}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{F}_{p})$.
Assume that the image of $\overline{\rho_{\mathfrak{m}}}$ is not solvable.
Then
$H^{i}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{F}_{p})_{\mathfrak{m}}$
is non-zero only for $i=g$.

###### Proof.

See [CT23, Theorem 7.1.1]. ∎

######  Corollary 4.2.

In the above setting, we have
$H^{i}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}\neq
0$ only for $i=g$. Moreover,
$H^{g}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}$
is free as a $\mathbb{Z}_{p}$-module.

###### Proof.

This is explained in [CT23, Corollary 7.1.2]; we give the proof below for the
convenience of the reader. We consider the short exact sequence

$0\rightarrow\mathbb{Z}_{p}\xhookrightarrow{\cdot
p}\mathbb{Z}_{p}\twoheadrightarrow\mathbb{F}_{p}\rightarrow 0,$

localize the corresponding long exact sequence at $\mathfrak{m}$ and employ
Theorem 4.1 to obtain a surjective map

$H^{i}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}\xrightarrow{\cdot
p}H^{i}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}$

for all $i\neq g$. Since
$H^{i}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}$
is a finitely generated $\mathbb{Z}_{p}$-module, we can apply Nakayama’s lemma
to conclude that
$H^{i}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}=0$
for all $i\neq g$. When $i=g$, we get that

$\frac{H^{g}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{F}_{p})_{\mathfrak{m}}}{\textrm{im}(H^{g}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}\rightarrow
H^{g}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{F}_{p})_{\mathfrak{m}}}=0.$

Since
$H^{g}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{F}_{p})_{\mathfrak{m}}\neq
0$, it follows that
$H^{g}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}\neq
0$ as well. Finally, note that the long-exact sequence also yields an
injection
$H^{g}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}\xhookrightarrow{\cdot
p}H^{g}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}$.
Hence,
$H^{g}_{\textrm{\'{e}t}}(Y_{G}(K)_{\overline{\mathbb{Q}}},\mathbb{Z}_{p})_{\mathfrak{m}}$
has no $p$-torsion, hence is torsion-free and hence free (using the fact that
torsion-free modules over a PID are free). ∎

###  4.2. Proof of the control theorem

We begin by recalling some notation and assumptions. We let
$H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})=\varprojlim_{n}H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},\mathcal{O})$,
$\Lambda=\llbracket(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}\rrbracket$
and $\Lambda_{n}=\mathcal{O}\llbracket\mathfrak{S}_{{U_{n,p}}}\rrbracket$. We
recall that $\Lambda_{n}$ is just the Iwasawa algebra corresponding to
elements of $\mathcal{(}\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}$ which
are congruent to 1 mod $p^{n}$. Let $\mathfrak{m}$ be a maximal ideal in the
support of
$H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},\mathbb{F}_{p})$
such that the image of the associated Galois representation
$\overline{\rho}_{\mathfrak{m}}$ is not solvable. Let $\lambda$ denote a
weight of $G$ that is self-dual in the sense of Section 3.3. Let
$\mathcal{O}[-\lambda]$ denote $\mathcal{O}$ with $\Lambda$ acting via the
inverse of the character $\lambda$.

######  Proposition 4.3.

We have that the $E_{2}^{i,j}$ term in Corollary 3.13 is zero unless
$i\in\\{0,-1,\ldots,-g\\}$.

###### Proof.

This follows from identifying $\Lambda$ with products of copies of
$\mathcal{O}[[T_{1},\ldots,T_{g}]]$ and noting that the resulting Koszul
complex to get a projective resolution of $\mathcal{O}[-\lambda]$ as a
$\Lambda$-module has $g$ terms. ∎

######  Proposition 4.4.

For all $n\geq 1$, we have that

${\operatorname{Tor}}^{\Lambda_{n}}_{1}(e^{\prime}_{\operatorname{ord}}H^{g}_{\operatorname{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}},\mathcal{O})=0.$

###### Proof.

Since localization is exact and commutes with the Tor functor, we deduce from
Corollary 3.13 the following spectral sequence (by setting $\lambda$ to be the
trivial character).

$F_{2}^{i,j}:=(E_{2}^{i,j})_{\mathfrak{m}}={\operatorname{Tor}}^{\Lambda_{n}}_{-i}(e^{\prime}_{\textrm{ord}}H^{j}_{\textrm{\'{e}t}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}},\mathcal{O})\implies
e^{\prime}_{\textrm{ord}}H^{i+j}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}.$

By Proposition 4.3 and Corollary 4.2, we deduce that $F_{2}^{i,j}=0$ unless
$(i,j)\in\\{(0,g),\ldots(-g,g)\\}$. It follows that the spectral sequence
degenerates on the second page. The filtration theorem applied to this
spectral sequence gives us a filtration

$F_{\infty}^{0,g-1}\xhookrightarrow{F_{\infty}^{-1,g}}\hskip
14.22636pt\cdots\xhookrightarrow{F_{\infty}^{-g,2g-1}}e^{\prime}_{\textrm{ord}}H^{g-1}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}.$

Using Corollary 4.2 again, we get
$e^{\prime}_{\textrm{ord}}H^{g-1}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}=0$
which forces that

$F_{\infty}^{-1,g}=F_{2}^{-1,g}={\operatorname{Tor}}^{\Lambda_{n}}_{1}(e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O}_{v})_{\mathfrak{m}},\mathcal{O})=0$

as well. ∎

######  Theorem 4.5.

With the notation as above, the following hold:
(a) For all $n\geq 1$, we have that
$e^{\prime}_{\textrm{ord}}H^{g}_{\operatorname{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}$
is free as a $\Lambda_{n}$-module.
(b) For all $n\geq 1$, we have an isomorphism of $\mathcal{O}$-modules

$e^{\prime}_{\textrm{ord}}H^{g}_{\operatorname{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}\otimes_{\Lambda_{n}}\mathcal{O}[-\lambda]\cong
e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},V_{\lambda})_{\mathfrak{m}}.$

that is compatible with the action of $G_{\mathbb{Q},\Sigma}$ and
$\mathbb{T}$.
(c) When $n=0$, we have a similar isomorphism

$e^{\prime}_{\operatorname{ord}}H^{g}_{\operatorname{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}\otimes_{\Lambda}\mathcal{O}[-\lambda]\cong
e^{\prime}_{\operatorname{ord}}H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{0})_{\overline{\mathbb{Q}}},V_{\lambda})_{\mathfrak{m}}.$

###### Proof.

(a) Let $\pi$ and $k$ denote the uniformizer and residue field of
$\mathcal{O}$ respectively. Consider the short-exact sequence

$0\rightarrow\mathcal{O}\xhookrightarrow{\cdot\pi}\mathcal{O}\twoheadrightarrow
k\rightarrow 0.$

This gives rise to a long-exact sequence of Tor groups:

$\displaystyle\cdots\rightarrow{\operatorname{Tor}}^{\Lambda_{n}}_{1}(e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}},\mathcal{O})\xrightarrow{\cdot\pi}{\operatorname{Tor}}^{\Lambda_{n}}_{1}(e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}},\mathcal{O})\rightarrow$
$\displaystyle{\operatorname{Tor}}^{\Lambda}_{1}(e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}},k)\rightarrow
e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}\otimes_{\Lambda_{n}}\mathcal{O}\xrightarrow{\cdot\pi}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}\otimes_{\Lambda_{n}}\mathcal{O}\rightarrow\cdots$

By Proposition 4.4,

${\operatorname{Tor}}^{\Lambda_{n}}_{1}(e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O}_{v})_{\mathfrak{m}},\mathcal{O})=0.$

Combining this with the fact that
$e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}\otimes_{\Lambda_{n}}\mathcal{O}$
has no $\pi$-torsion, the long exact sequence gives us that
${\operatorname{Tor}}^{\Lambda_{n}}_{1}(e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{\'{e}t}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}},k)=0$.
We can now use the local criterion of flatness to conclude that
$e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}$
is flat as a $\Lambda_{n}$-module. Since a finitely generated flat module over
a noetherian local ring is free, we have that
$e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}}$
is free as a $\Lambda_{n}$-module.
(b) We again use the fact that localization is exact and commutes with the Tor
functor to obtain from Corollary 3.13 the following spectral sequence.

$F_{2}^{i,j}:=(E_{2}^{i,j})_{\mathfrak{m}}={\operatorname{Tor}}^{\Lambda_{n}}_{-i}(e^{\prime}_{\textrm{ord}}H^{j}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O})_{\mathfrak{m}},\mathcal{O}[-\lambda])\implies
e^{\prime}_{\textrm{ord}}H^{i+j}_{\textrm{\'{e}t}}(Y_{G}(U_{n})_{\overline{\mathbb{Q}}},V_{\lambda})_{\mathfrak{m}}.$

By Proposition 4.3 and Corollary 4.2, we deduce that $F_{2}^{i,j}=0$ unless
$(i,j)\in\\{(0,g),\ldots(-g,g)\\}$. It follows that the spectral sequence
degenerates on the second page with

$F_{\infty}^{0,g}=F_{2}^{0,g}=e^{\prime}_{\textrm{ord}}H^{g}_{\textrm{Iw}}(Y_{G}(U_{\infty})_{\overline{\mathbb{Q}}},\mathcal{O}_{v})_{\mathfrak{m}}\otimes_{\Lambda_{n}}\mathcal{O}[-\lambda]$

and the desired isomorphism now follows directly by applying the filtration
theorem.
(c) Using Corollary 3.14, this follows from a similar argument as in part (b).

∎

## References

  * [Bou06] N.Bourbaki, _Algèbre commutative, Chapitres 1 à 4_ , Springer, 2006.
  * [BCF19] I. Blanco-Chacón and M. Fornea, _Twisted triple product p-adic $L$-functions and Hirzebruch-Zagier cycles_, J. Inst. Math. Jussieu (2019).
  * [CT23] A. Caraiani and M. Tamiozzo, _On the étale cohomology of Hilbert modular varieties with torsion coefficients_ , Compos. Math. 159 (2023), 2279-2325.
  * [Dim09] M. Dimitrov, _On Ihara’s lemma for Hilbert modular varieties_ , Compos. Math. 145 (2009), no. 5, 1114–1146.
  * [FJ24] M. Fornea and J. Jin, _Hirzebruch-Zagier classes and rational elliptic curves over quintic fields_ , preprint, 2024, arXiv:2005.02520.
  * [Hid86] H. Hida, _Galois representations into ${\operatorname{GL}}_{2}(\mathbb{Z}_{p}[[X]])$ attached to ordinary cusp forms_, Invent. Math. 85 (1986), 545–613.
  * [KLZ17] G. Kings, D. Loeffler, and S. Zerbes, _Rankin–Eisenstein classes and explicit reciprocity laws_ , Cambridge J. Math. 5 (2017), no.1, 1–122.
  * [LLZ14] A. Lei, D. Loeffler, and S. Zerbes, _Euler systems for Rankin–Selberg convolutions of modular forms_ , Ann. of Math.(2) 180 (2012) no. 2, 653–771.
  * [LLZ18] A. Lei, D. Loeffler, and S. Zerbes,_Euler systems for Hilbert modular surfaces_ , Forum Math. Sigma 6 (2018), no. e.23.
  * [Loe22] D. Loeffler, _Spherical varieties and norm relations in Iwasawa theory_ , J. Th. Nombres Bordeaux 33 (2021), no. 3.2 (Iwasawa 2019 special issue), 1021–1043.
  * [LRZ23] D. Loeffler, R. Rockwood and S. Zerbes, _Spherical varieties and $p$-adic families of cohomology classes_, preprint, 2023, arXiv:2106.16082.
  * [MW84] B. Mazur, A.Wiles, _Class fields of abelian extensions of $\mathbb{Q}$_, Invent. Math. 76 (1984), 179-330.
  * [Mil17] J. Milne, _Introduction to Shimura varieties_ , lecture notes, 2017.
  * [Oht99] M. Ohta, _Ordinary $p$-adic étale cohomology groups attached to towers of elliptic modular curves_, Compos. Math. 115 (1999), no. 3, 241–301.
  * [Oht00] M. Ohta, _Ordinary $p$-adic étale cohomology groups attached to towers of elliptic modular curves. II_, Math. Ann. 318 (2000), no. 3, 557–583.
  * [Sch15] P. Scholze, _On torsion in the cohomology of locally symmetric varieties_ , Ann. of Math. (2) 182 (2015), no. 3, 945-1066.
  * [Urb11] E. Urban, _Eigenvarieties for reductive groups_ , Ann. of Math. (2). 174 (2011), no. 3, 1685-1784.


1Association for the Advancement of Artificial Intelligence
1900 Embarcadero Road, Suite 101
Palo Alto, California 94303-3310 USA
<EMAIL_ADDRESS>
# Bootstrapping Multi-view Representations for Fake News Detection

Written by AAAI Press Staff1
AAAI Style Contributions by Pater Patel Schneider, Sunil Issar,
J. Scott Penberthy, George Ferguson, Hans Guesgen, Francisco Cruz, Marc Pujol-
Gonzalez With help from the AAAI Publications Committee.

###### Abstract

Previous researches on multimedia fake news detection include a series of
complex feature extraction and fusion networks to gather useful information
from the news. However, how cross-modal consistency relates to the fidelity of
news and how features from different modalities affect the decision-making are
still open questions. This paper presents a novel scheme of Bootstrapping
Multi-view Representations (BMR) for fake news detection. Given a multi-modal
news, we extract representations respectively from the views of the text, the
image pattern and the image semantics. Improved Multi-gate Mixture-of-Expert
networks (iMMoE) are proposed for feature refinement and fusion.
Representations from each view are separately used to coarsely predict the
fidelity of the whole news, and the multimodal representations are able to
predict the cross-modal consistency. With the prediction scores, we reweigh
each view of the representations and bootstrap them for fake news detection.
Extensive experiments conducted on typical fake news detection datasets prove
that the proposed BMR outperforms state-of-the-art schemes.

## Introduction

Online Social Networks (OSNs) such as Weibo and Twitter have become important
means of socializing and knowledge sharing. The flourishing of OSN also leads
to the fast spread of fake news and distorted opinions. Fake news are
specified as the news that are intentionally fabricated and can be verified as
false ruchansky2017csi; shu2017fake. In many cases, it is difficult for
ordinary people to identify fake news. For network administrators, manually
removing false news one by one is laborious and expensive. Therefore,
automatic Fake News Detection (FND) has become a hot research topic
allein2021like; shu2020fakenewsnet; xue2021detecting. The FND techniques can
effectively analyze the probability of misconducting information, which is
more convenient for the administrator to block fake news from spreading on the
OSN.

The general paradigm of machine-learning-based fake news detection is to
transform the news into a multidimensional latent representation, and identify
the fidelity of the news using binary classification. Existing methods can be
classified into three categories, namely, unimodal fake news detection TM;
shu2020leveraging, multimodal fake news detection MVAE; xue2021detecting and
dynamic fake news detection, i.e, utilizing propagation graph qian2018neural
or knowledge graph abdelnabi2022open; sun2022ddgcn. Some unimodal fake news
detection methods can achieve good performances. However, most posts in OSN
are in the multimodal style. Hence, the detection based on unimodal features
is far from enough, and accordingly, we focus on multimodal fake news
detection in this paper.

Many existing multimodal FND schemes use the textual and visual features as
integrated representations MVAE; SpotFake; chen2019attention; allein2021like.
Nevertheless, the disentanglement of features from different views has not
been thoroughly investigated. In many cases, the models are at a black-box
level, in which the network designs cannot explicitly highlight the most
contributive components EANN; SpotFake. Besides, many recent works solely rely
on cross-modal correlation to generate fused features wei2022cross; WWW, but
we argue that cross-modal correlation not necessarily play a critical role.
Fig. 1 provides four examples respectively from the well-known English
GossipCop SpotFake and the Chinese Weibo weibo dataset, where both fake and
real news more or less contain cross-modal correlations. Therefore, clarifying
the roles of both unimodal and cross-modal features is vital for improving
FND.

Figure 1: Examples of fake news detection result of BMR on Weibo (left two)
and GossipCop (right two). The Chinese sentences from Weibo are translated
here for references. Row “IP” (Image Pattern), “IS” (Image Semantics), “T”
(Text) respectively show the single-view prediction results provided by BMR,
which suggests the dubious parts of the news. Row “CC” (Cross-modal
Consistency) shows the predictions on cross-modal consistency and Row “Pred.”
provides the ultimate FND results.

Aiming at addressing these issues, we propose an effective scheme that
Bootstraps Multi-view Representations (BMR) for fake news detection. Given a
multi-modal news, we extract representations respectively from the views of
the text, the image pattern and the image semantics. Improved Multi-gate
Mixture-of-Expert networks (MMoE) MMoE, denoted as iMMoE, are proposed for
feature refinement and fusion. Representations from each view are separately
used to coarsely predict the fidelity of the whole news, where the prediction
scores are used for adaptive feature reweighing. Cross-modal consistency
learning further implicitly guides multimodal representation refinement, and
we explicitly disentangle correlation from other cross-modal information by
introducing an independent representation. Finally, we bootstrap multi-view
features for refined fake news detection. In the examples shown in Fig. 1, BMR
not only predicts the fidelity of the news, but also gives confidence scores
based on each view, which provides a new way of understanding how different
roles act in multimodal FND.

The contributions of this paper are three-folded:

  * •

We propose a novel fake news detection scheme of generating multi-view
representations, understanding their individual importance, and optimizing the
fused features.

  * •

We propose to disentangle information within unimodal and multimodal features
by single-view prediction and cross-modal consistency learning, which are then
adaptively reweighed and bootstrapped for better detection.

  * •

The proposed BMR detection not only outperforms state-of-the-art multimodal
FND schemes on popular datasets, but also provides a mechanism for
interpreting the contributions of different representations.

## Related Works

Unimodal Fake News Detection. Both the language-based and the vision-based
fake news detections have received extensive attention. TM TM utilizes lexical
and semantic properties of the text to detect fake news. MWSS
shu2020leveraging exploits multiple weak signals from different sources from
user and content engagements. Jin et al. jin2016novel find that there are
noticeable differences in the distribution of images between real news and
fake news. Cao et al. Leveraging_5 suggest that typical methods for image
manipulation detection MVSS are useful in unveiling traces for news tampering.
Besides, Qi et al. qi2019exploiting jointly use the spatial domain and
frequency domain features of the news for forensics. However, for the
multimodal news, these approaches are unable to detect cross-modal
correlations.

Figure 2: The network architecture of BMR. Representations are respectively
extracted from the views of text, image pattern and image semantics. Improved
MMoE, together with single-view predictions and cross-modal consistency
learning, jointly guides unimodal feature refinement and cross-modal feature
generation. The final decision is made upon bootstrapping these multi-view
representations.

Multimodal Fake News Detection. The critical issue of multimodal fake news
detection is to align linguistic and vision representations. SpotFake SpotFake
integrates pretrained XLNet and ResNet for feature extraction. SAFE
zhou2020mathsf feeds the relevance between news textual and visual information
into a classifier to detect fake news. EANN EANN introduces an additional
discriminator to classify news events so as to suppress the impact of specific
events on classification. MCAN wu2021multimodal stacks multiple co-attention
layers for multimodal feature fusion. MCNN xue2021detecting also incorporates
textual semantic features, visual tampering features, and similarity of
textual and visual information for fake news detection, but the aggregation
process only concatenates all features. Hence, how each of these multi-view
features affects the predictions cannot be measured. In comparison, we
explicitly reweigh multi-view representations based on single-view
classifications and adaptively bootstrap them for fake news detection.

Besides, some methods propose to use cross-modal correlation learning for fake
news detection. CAFE WWW uses a VAE to compress images and texts
representations and measures cross-modal consistency based on their Kullback-
Leibler (KL) divergence. The consistency score then linearly adjusts the
weight of unimodal and multimodal features before final classification.
Likewise, CMC wei2022cross includes a two-staged network that train two uni-
modal networks to learn cross-modal correlation by contrastive learning, and
then finetune the network for fake news detection. However, CAFE severely
penalizes unimodal features when the consistency is high, which might be
doubtful in some news. CMC does not adaptively suppress or augment multi-view
representations, and the fine-tuning stage may erase the cross-modal knowledge
learned in the first stage. Therefore, we propose an improved mechanism to
better utilize cross-modal consistency learning for FND.

## Proposed Approach

Fig. 2 depicts the pipeline of the proposed Bootstrapping Multi-view
Representations approach, which contains four stages, i.e., Multi-view Feature
Extraction, Refine & Fusion, Disentangling & Reweighing, and Bootstrapping. In
the first three stages, there are four branches corresponding to the
representations from four views, including the Image Pattern Branch, Image
Semantics Branch, Text Branch, and Fusion Branch.

### Multi-view Feature Extraction

Let the input multimodal news be
$\mathcal{N}=[\mathbf{I},\mathbf{T}]\in\mathcal{D}$, where
$\mathbf{I},\mathbf{T},\mathcal{D}$ are the image, the text and the dataset,
respectively. We begin with extracting coarse multi-view representations,
including the image pattern, the image semantics, and the text, corresponding
to the Image Pattern Branch, the Image Semantic Branch, and the Text Branch,
respectively. We denote these representations as ${r}_{\emph{ip}}$,
${r}_{\emph{is}}$ and ${r}_{\emph{t}}$.

We believe that the general distribution of the image and the tiny traces left
by tampering or compression can be helpful to expose fake news. Therefore, we
explicitly separate the feature learning paradigm from image pattern and
semantics. In the Image Pattern Branch, we use InceptionNet-V3 inceptionv3 as
the image pattern analyzer. We additionally include a BayarConv
bayar2018constrained to process $\mathbf{I}$ ahead of sending it into the
InceptionNet-V3 network, since BayarConv augments details and suppresses the
main component of $\mathbf{I}$ that contributes largely to semantics. While
CNNs are suitable for vision learning, the transformer models he2022masked
based on masked language/image modeling are reported to be great in
establishing long-range attention ViT. Hence, in the Image Semantic Branch, we
use Masked Autoencoder (MAE) as the image semantics analyzer to extract
${r}_{\emph{is}}$. In the Text Branch, we use BERT BERT to extract
${r}_{\emph{t}}$.

Figure 3: Network architecture of improved MMoE.

### Feature Refinement and Fusion with iMMoE

In the Refine & Fusion Stage, we refine the representations extracted from the
first stage. In the Image Pattern Branch, the representation ${r}_{\emph{ip}}$
is projected to a new representation ${e}_{\emph{ip}}$, where the
classification head of the InceptionNet-V3 is replaced with an MLP-based
projection head. The output size of ${e}_{\emph{ip}}$ equals to that of a
single token representation from MAE. In the Image Semantics Branch and the
Text Branch, we propose the improved MMoE (iMMoE) network to refine
${r}_{\emph{is}}$ and ${r}_{\emph{t}}$. We also generate a new representation
${e}_{\emph{m}}$ in the Fusion Branch.

As sketched in Fig. 3, the proposed iMMoE contains several experts, gates and
token attentions that produces features in separate branches. The MMoE network
MMoE is originally designed to model multi-task relationships from data by
sharing the expert across all tasks. The input $x$ is equally sent into $n$
expert networks, and $k$ gates adaptively weigh the outputs of the experts as
the final output using softmax function. $n$ is a hyper-parameter and $k$ is
the amount of down-stream tasks. The algorithm of MMoE is summarized in Eq.
(1)

$x^{k}=\sum_{i=1}^{n}\emph{softmax}(G^{k}_{i}(x))\cdot~{}E_{i}(x),$ (1)

where $E_{i}$ and $G^{k}_{i}$ are the $i_{\emph{th}}$ expert and the
$i_{\emph{th}}$ output of the gate for task $k$. In BMR, we treat the mining
of multi-view unimodal representations and cross-modal feature fusion as
different subtasks. They should share common features and reserve their own
distinctive features. We improve MMoE network in two aspects. First, we use
token attention to compute the importance scores of each token representation
using an MLP, and perform dimension reduction by aggregating all token
representations into one according to the scores. The aggregated
representation is then sent into the gate to compute the weights for each
expert. Second, we find that the softmax functions in the gates are used to
guarantee that the output weights are all positive and the sum equals one.
This is not necessary according to our experiments. Accordingly, we lift the
softmax constraints and allow the weights to be outside the $[0,1]$ range. In
summary, we revise Eq. (1) as Eq. (2) for iMMoE,

$x^{k}=\sum_{i=1}^{n}(G_{i}^{k}({\sum_{j=1}^{t}\emph{MLP}_{k}(x)})\cdot~{}E_{i}(x)),$
(2)

where $t$ is the amount of tokens and $\emph{MLP}_{k}$ denotes the token
attention for task $k$. $k\in[1,2]$.

With the iMMoE network, we refine ${r}_{\emph{is}}$ and ${r}_{\emph{t}}$ into
the features $[{e}_{\emph{is}}^{0},{e}_{\emph{is}}^{1}]$ and
$[{e}_{\emph{t}}^{0},{e}_{\emph{t}}^{1}]$, respectively. ${e}_{\emph{is}}^{0}$
and ${e}_{\emph{t}}^{0}$ are preserved for single-view prediction. Meanwhile,
the ${e}_{\emph{is}}^{1}$ and ${e}_{\emph{t}}^{1}$ are jointly fed into the
iMMoE net in the Fusion Branch to generate a multimodal feature
${e}_{\emph{m}}$, which is preserved for cross-modal consistency learning and
bootstrapping.

### Disentangling & Reweighing

After the refinement and fusion, we obtain the new representations from four
branches. Next, we handle representations of image pattern, image semantics
and text via single-view prediction, while processing the fused representation
in the Fusion Branch for consistency learning.

Single-view Prediction and Reweighing. Single-view prediction uses the first
token of ${e}_{\emph{is}}^{0}$, the first token of ${e}_{\emph{t}}^{0}$ and
${e}_{\emph{ip}}$, to predict the fidelity of $\mathcal{N}$. We design this
module based on two considerations. On one hand, many fake news contains
obvious abnormality in either image noise, distribution or text independently.
Therefore, making predictions on $\mathcal{N}$ with single-view features is
empirically feasible that allows the network to deeply mine the
characteristics from each view. On the other hand, the prediction scores can
be projected into weights that adaptively reweigh the representations.
Therefore, we use MLP to project each single-view representation into scores,
and use another MLP, i.e., $F(\cdot)$ in Fig. 2, to project the scores into
weights. Take the generation of $S_{\emph{ip}}$ and $w_{\emph{ip}}$ as an
example,

$\begin{gathered}S_{\emph{ip}}=\emph{MLP}_{\emph{ip}}(e_{\emph{ip}}),\\\
w_{\emph{ip}}=\emph{Sigmoid}\left(\emph{F}_{ip}\left(S_{\emph{ip}}\right)\right)\cdot{e}_{\emph{ip}},\end{gathered}$
(3)

where $\emph{MLP}_{\emph{ip}}(\cdot)$ denotes the MLP-based single-view
predictor. In the same way, we generate paired prediction scores and reweighed
representations $\\{S_{\emph{is}},w_{\emph{is}}\\}$,
$\\{S_{m},w_{\emph{m}}\\}$ and $\\{S_{\emph{t}},w_{\emph{t}}\\}$. The
reweighed representations are then ready for bootstrapping.

Figure 4: Training data for cross-modal consistency learning. Real news
borrowed from typical FND datasets are marked positive and synthesized news by
combining image and text from different real news are marked negative.

Cross-modal Consistency Learning. Inspired by several recent works WWW;
wei2022cross, we guide the multimodal feature fusion in the Fusion Branch with
cross-modal consistency learning. Specifically, we train BMR to predict
whether a given text-image pair matches. We begin with crafting a new dataset
$\mathcal{D}^{\prime}=[\mathcal{D}_{\emph{real}},\mathcal{D}_{\emph{syn}}]$ on
the basis of $\mathcal{D}$, where news with correlated texts and images are
labeled $y^{\prime}=1$, otherwise $y^{\prime}=0$. Fig. 4 shows three training
data for cross-modal consistency learning, where the positive examples are
directly the real news borrowed from $\mathcal{D}$ and the negative examples
are synthesized “news” by arbitrarily combining images and texts from
different real news. Though there can be cases that some news in
$\mathcal{D}_{\emph{real}}$ do not contain cross-modal consistency, the
possibility of text-image pairs from $\mathcal{D}_{\emph{syn}}$ having
consistency is much lower in nature. Therefore, the model can still learn the
correlation based on such a hybrid dataset. We feed BMR with
$\mathcal{N}^{\prime}=[\mathbf{I}^{\prime},\mathbf{T}^{\prime}]\in\mathcal{D}^{\prime}$.
After the corresponding $e_{\emph{m}}$ is calculated, we use an MLP-based
predictor to output the consistency score ${S}_{\emph{m}}$, and expect the
score to be close to the label $y^{\prime}$. We use a simple regression loss
as supervision instead of contrastive loss applied in CMC wei2022cross in that
the likelihood of any text from the negative pool matching the query image,
i.e., the false negative rate, is proportional to the size of that pool. The
training process of cross-modal consistency learning only activates a part of
the modules in BMR. The task can be in parallel learned with the main task.

Reweighing Multimodal Representation. The predicted consistency scores on
$\mathcal{N}$ also adjust the multimodal representations. We consider that
multimodal features should not merely represent cross-modal correlation. There
are also many other factors, e.g., joint distribution, emotional difference,
etc., that can help decide whether the news is fake. Therefore we refrain from
weighing $e_{\emph{m}}$ using $S_{m}$. Instead, we explicitly disentangle
correlation from other cross-modal information by letting $S_{m}$ reweigh an
independent trainable token $e_{x}$ that represents cross-modal irrelevance.
$w_{x}={e}_{\emph{x}}\cdot~{}F_{\emph{m}}(S_{m})$ and $w_{m}=e_{m}$, which are
both considered multimodal representations for bootstrapping. Compared to
previous works, we do not separate the cross-modal learning with the detecting
stage wei2022cross or reweighing both unimodal and multimodal representations
by correlation score WWW.

Algorithm 1 Model training of BMR

Input: Dataset: $\mathcal{D}$, Training epochs: $N$, Amount of news for
constructing $\mathcal{D}^{\prime}$: $k$
Output: Model parameters: $\Theta$.

1: Sample $k/2$ real news from $\mathcal{D}$ and store them into
$\mathcal{D}^{\prime}$ as positive examples.

2: for i in range($k$/4): do

3: Sample $\mathcal{N}_{1}=[\mathbf{I}_{1},\mathbf{T}_{1}]$,
$\mathcal{N}_{2}=[\mathbf{I}_{2},\mathbf{T}_{2}]$ from $\mathcal{D}$.

4: Synthesize pseudo-news by shuffling information in $\mathcal{N}_{1}$ and
$\mathcal{N}_{2}$. $\mathcal{N}_{3}=[\mathbf{I}_{1},\mathbf{T}_{2}]$,
$\mathcal{N}_{4}=[\mathbf{I}_{2},\mathbf{T}_{1}]$.

5: Store $\mathcal{N}_{3},\mathcal{N}_{4}$ into $\mathcal{D}^{\prime}$ as
negative examples.

6: end for

7: for i in range(N): do

8: Sample $(\mathbf{I},\mathbf{T},y)$,
$(\mathbf{I}^{\prime},\mathbf{T}^{\prime},y^{\prime})$ from $\mathcal{D}$,
$\mathcal{D}^{\prime}$

9:
$S_{\emph{m}}=\emph{BMR}(\mathbf{I}^{\prime},\mathbf{T}^{\prime},$_train_consist_
$=\emph{True})$

10: Compute loss using $\mathcal{L}_{\emph{BCE}}(y^{\prime},S_{\emph{m}})$.

11: $\mathbf{T}=$“No text provided.” if $\emph{len}(\mathbf{T})<5$.

12: $\mathbf{I}=\emph{zeros\\_like}(\mathbf{I})$ if
$\mathbf{I}.\emph{rows}<64$ or $\mathbf{I}.\emph{cols}<64$

13:
[$\tilde{y},S_{\emph{t}},S_{\emph{is}},S_{\emph{ip}}]=\emph{BMR}(\mathbf{I},\mathbf{T},$_train_consist_
$=\emph{True})$.

14: Compute loss using $\mathcal{L}_{\emph{BCE}}(y,\tilde{y})$,
$\mathcal{L}_{\emph{BCE}}(y,S_{\emph{t}})$,
$\mathcal{L}_{\emph{BCE}}(y,S_{\emph{is}})$,
$\mathcal{L}_{\emph{BCE}}(y,S_{\emph{ip}})$.

15: Update parameters in $\Theta$ using Adam optimizer.

16: end for

### Bootstrapping Stage and Loss Function

The multi-view representations
$[{w}_{\emph{is}},{w}_{\emph{ip}},{w}_{\emph{m}},{w}_{\emph{x}},{w}_{\emph{t}}]$
are bootstrapped using another iMMoE, which further refines the information
critical for the decision-making. The final MLP-based classifier gets the
first token of the output $e_{f}$ to predict $\tilde{y}$, which is expected to
be close to the label $y$.

Fake news detection is a binary classification problem. The Binary Cross-
Entropy loss (BCE) is defined in Eq. (4), where $x$ and $\tilde{x}$ stand for
the ground truth and the prediction, respectively.

$\mathcal{L}_{\emph{BCE}}(x,\tilde{x})=x\log\left(\tilde{x}\right)+\left(1-x\right)\log\left(1-\tilde{x}\right).$
(4)

We apply the BCE loss between the ground-truth label $y$ and the predicted
scores $\tilde{y}$, as well as the coarse classification results
$S_{\emph{ip}}$, $S_{\emph{is}}$, $S_{t}$. There is an extra loss for cross-
modal consistency training, where we also apply the BCE loss between the
ground-truth crafted label $y^{\prime}$ and $S_{\emph{m}}$. Therefore, the
losses are defined as
$\mathcal{L}_{\emph{CC}}=\mathcal{L}_{\emph{BCE}}(y^{\prime},S_{\emph{m}})$,
$\mathcal{L}_{\emph{final}}=\mathcal{L}_{\emph{BCE}}(y,\tilde{y})$,
$\mathcal{L}_{T}=\mathcal{L}_{\emph{BCE}}(y,S_{\emph{t}})$,
$\mathcal{L}_{\emph{ip}}=\mathcal{L}_{\emph{BCE}}(y,S_{\emph{ip}})$ and
$\mathcal{L}_{\emph{is}}=\mathcal{L}_{\emph{BCE}}(y,S_{\emph{is}})$. The
single view FND classification loss is the aggregate of
$\mathcal{L}_{\emph{is}}$, $\mathcal{L}_{\emph{ip}}$ and
$\mathcal{L}_{\emph{t}}$, namely,

$\mathcal{L}_{\emph{coarse}}=(\mathcal{L}_{\emph{is}}+\mathcal{L}_{\emph{ip}}+\mathcal{L}_{\emph{t}})/3,$
(5)

The total loss for BMR is defined as follows.

$\mathcal{L}=\mathcal{L}_{\emph{final}}+\alpha\cdot\mathcal{L}_{\emph{coarse}}+\beta\cdot\mathcal{L}_{\emph{CC}},$
(6)

where the hyper-parameters $\alpha$ and $\beta$ are empirically set as
$\alpha=1$, and $\beta=4$. We provide the pseudo-code of detailed training
processes of BMR in Algo. 1.

| Method | Accuracy | Fake News | Real News
---|---|---|---|---
Precision | Recall | F1-score | Precision | Recall | F1-score
Weibo | EANN* EANN | 0.827 | 0.847 | 0.812 | 0.829 | 0.807 | 0.843 | 0.825
MCNN xue2021detecting | 0.846 | 0.809 | 0.857 | 0.832 | 0.879 | 0.837 | 0.858
MCAN wu2021multimodal |  0.899 |  0.913 |  0.889 |  0.901 |  0.884 |  0.909 | 0.897
CAFE* WWW | 0.840 | 0.855 | 0.830 | 0.842 | 0.825 | 0.851 | 0.837
CMC wei2022cross | 0.893 |  0.940 | 0.869 | 0.899 | 0.876 |  0.945 |  0.907
BMR (Proposed) |  0.918 | 0.882 |  0.948 |  0.914 |  0.942 | 0.870 |  0.904
GossipCop | EANN* EANN | 0.864 | 0.702 | 0.518 | 0.594 | 0.887 | 0.956 | 0.920
Spotfake* SpotFake | 0.858 | 0.732 | 0.372 | 0.494 | 0.866 | 0.962 | 0.914
DistilBert allein2021like | 0.857 |  0.805 | 0.527 | 0.637 | 0.866 | 0.960 | 0.911
CAFE* WWW | 0.867 | 0.732 | 0.490 | 0.587 | 0.887 | 0.957 | 0.921
CMC wei2022cross |  0.893 |  0.826 |  0.657 |  0.692 |  0.920 |  0.963 |  0.935
BMR (Proposed) |  0.895 | 0.752 |  0.639 |  0.691 |  0.920 |  0.965 |  0.936
Weibo-21 | EANN* zhou2020mathsf | 0.870 | 0.902 | 0.825 | 0.862 | 0.841 |  0.912 | 0.875
SpotFake* WWW | 0.851 |  0.953 | 0.733 | 0.828 | 0.786 | 0.964 | 0.866
CAFE* WWW |  0.882 | 0.857 |  0.915 |  0.885 |  0.907 | 0.844 |  0.876
BMR (Proposed) |  0.929 |  0.908 |  0.947 |  0.927 |  0.946 |  0.906 |  0.925

Table 1: Comparison between BMR and state-of-the-art multimodal fake news
detection schemes on Weibo, GossipCop and Weibo-21. *: open-sources. The best
performance is highlighted in bold red and the follow-up is highlighted in
bold blue.

## Experiments

We have conducted many experiments using popular datasets to verify the
proposed BMR approach for fake news detection. The source codes along with
more results are attached as the supplementary material.

### Experimental Setups

Implementation Details. We use the “mae-pretrain-vit-base” model for image
processing, the “bert-base-chinese” model for Chinese dataset, and the “bert-
base-uncased” model for English dataset. The hidden sizes of both MAE and BERT
are $768$. BERT and MAE are kept frozen. All MLPs in BMR contain one hidden
layer, a BatchNorm1D and an ELU activation. We use the single-layer
transformer blocks defined in ViT ViT to implement the experts in iMMoE. Each
iMMoE network holds three experts. We use Adam optimizer with the default
parameters. The batch size is 24 and the learning rate is $1\times 10^{-4}$
with cosine annealing decay. Images are resized into $224\times 224$, and the
max length of text is set as $197$. BMR is also designed to be compatible with
unimodal news by feeding a zero matrix as the image or “No text provided” as
the text. Similar to EANN EANN, we increase the quality of the datasets by
replacing images smaller than $64\times 64$ and texts less than five words.

Data Preparation. We use Weibo weibo, GossipCop shu2020fakenewsnet and
Weibo-21 MDFEND for training and testing. Weibo contains 3749 real news and
3783 fake news for training, 1000 fake news and 996 real news for testing.
GossipCop contains 7974 real news and 2036 fake news for training, 2285 real
news and 545 fake news for testing. Weibo-21 is a newly-released dataset that
contains 4640 real news and 4487 fake news in total, and we split it into
training and testing data at a ratio of $9:1$. Though MediaEval Twitter and
Politifact SpotFake are also popular datasets, they contain only 460 images
and 381 posts in the training set. Hence, and neural nets training with
Politifact can easily overfit with such small amount of data. We train BMR on
each dataset five times with different initial weights and report the averaged
best performance.

According to the theory in imbalance, we determine a fixed threshold for each
dataset based on its distribution. Since the ratio of real news versus fake
news in GossipCop training set is close to 4:1, we set the threshold for
GossipCop as 0.80. Similarly, the thresholds for Weibo and Weibo-21 are set as
0.50. For real-world applications, we can set a default threshold as 0.50.

Figure 5: Heatmap Visualization. Each cell in the heat maps represents the
paired cosine similarity between the 64-dim representations respectively from
the final classifier and the single-view predictors. Tests are done on Weibo.

### Performance Analysis

Heatmap Visualization. In Fig. 5, we measure the discriminative capability of
the multi-view representations using heatmap visualization. We arbitrarily
select ten real news and ten fake news, and then compute paired similarity
between the 64-dim representations from the final classifier and the single-
view predictors. From the figures, the bootstrapped representation shows
strong discriminative capability where intra-class similarity and inter-class
difference are both noticeable. Some other views of representations also have
decent borders, but the heatmap for cross-modal consistency tends to be much
messier, showing that it is not practical to directly use the consistency for
FND.

Comparisons. Table 1 shows the average precision, recall, and accuracy of BMR
on Weibo and GossipCop. We use Accuracy, Precision, Recall, and F1 score for
comprehensive performance measurements. The results are promising, with 91.8%
average accuracy on Weibo, 92.9% on Weibo-21 and 89.5% on GossipCop. We
further compare BMR with state-of-the-art methods. BMR outperforms the other
methods on the datasets. Besides, we rank either $1\emph{st}$ or $2\emph{nd}$
on Recall and F1 score of fake news on all of the datasets. CMC has a very
close performance with BMR on GossipCop. However, on Weibo, BMR outperforms
CMC by a more noticeable 2.5%. We have also trained three open-sourced methods
on Weibo-21 and compared them with BMR. The result shows that BMR leads by
4.2% in overall accuracy and 3.2% in Recall of fake news, which proves the
effectiveness of BMR.

Moreover, we present the t-SNE visualizations of features in Fig. 6, which are
learned by BMR, SpotFake and CAFE on the test set of GossipCop. In BMR, the
dots representing fake news are comparatively farther from differently-labeled
dots, and there are fewer outliers. This indicates that the extracted features
in BMR are more discriminative.

Computational Complexity. It costs around 12 hours to train BMR for 50 epochs
on a single NVIDIA RTX 3090 GPU. The computational complexity study in Table 2
shows that BMR requires less trainable parameters compared to EANN and
SpotFake. CAFE requires amazingly low computational complexity by using simple
auto-encoders and MLPs, but the performance is not good enough.

Figure 6: TSNE visualization of mined features on the test set. Dots with the
same color are within the same label.

| BMR | EANN | SpotFake | CAFE
---|---|---|---|---
Params | 94.39M | 143.70M | 124.37M | 0.68M
FLOPS | 18.42G | 19.63G | 30.42G | 0.01G

Table 2: Comparison of trainable parameters and computational speed. FLOPs:
amount of floating point arithmetics.

### Ablation Study

Contribution of Each View. Fig. 7 and Table. 3 show the curves for testing
accuracy and ultimate training loss. We study the contribution of each view on
the datasets. First, we find that $\mathcal{L}_{\emph{ip}}$,
$\mathcal{L}_{\emph{is}}$ and $\mathcal{L}_{\emph{CC}}$ do not fully converge
on GossipCop. Therefore we closer scrutinized GossipCop dataset and
surprisingly find that close to 50% images are celebrity faces that are even
hard to distinguish for many human viewers. However, BMR still defeats all
compared state-of-the-art FND schemes in the overall accuracy, which indicates
that FND on GossipCop relies severely on text and images might contribute
marginally. Second, we study whether there is under-fitting, i.e., single-view
predictions struggling with the ground truth $y$, or over-fitting, i.e.,
networks remembering the training set. In Fig. 7, the testing accuracies
remain high and the curves go into plateau rather than drop even though we
train 50 epochs. Therefore, there is no overfitting in our training.

Figure 7: Curves of testing accuracy (y-axis) versus training epochs (x-axis).
“CC” curve represents using cross-modal consistency as the judgement of fake
news detection.

Dataset | $\mathcal{L}_{\emph{final}}$ | $\mathcal{L}_{\emph{CC}}$ | $\mathcal{L}_{\emph{t}}$ | $\mathcal{L}_{\emph{is}}$ | $\mathcal{L}_{\emph{ip}}$
---|---|---|---|---|---
Weibo | 0.005 | 0.084 | 0.007 | 0.026 | 0.133
GossipCop | 0.005 | 0.540 | 0.006 | 0.511 | 0.493
Weibo21 | 0.002 | 0.124 | 0.003 | 0.067 | 0.012

Table 3: Averaged ultimate training loss of each term.

Removal of Some Views. In the upper part of Table 4, we feed BMR with
different combinations of views of representations. For example, in row three,
we only feed BMR with the image, preserve the Image Pattern Branch and the
Image Semantics Branch, while disabling the rest of the modules. We find that
if we bootstrap three multi-view unimodal representations without introducing
the fusion branch, the averaged accuracy is 88.6%, which is 3.2% worse than
BMR. Though we find that the contribution of text is large for FND, merely
using texts only guarantee 87.0% accuracy.

Test | Accuracy | F1 Score
---|---|---
Fake News | Real News
IP | 0.789 | 0.774 | 0.806
IS | 0.838 | 0.860 | 0.809
IS+IP | 0.857 | 0.874 | 0.833
T | 0.870 | 0.872 | 0.868
IP+T | 0.878 | 0.876 | 0.876
IS+T | 0.881 | 0.888 | 0.873
IS+IP+T | 0.886 | 0.888 | 0.884
${S}_{\emph{m}}$ reweigh. Multi-view. | 0.864 | 0.854 | 0.809
w/o Feature Reweigh. | 0.887 | 0.880 | 0.892
w/o Coarse Class. | 0.895 | 0.903 | 0.887
using ViT Blocks for Refine. | 0.905 | 0.910 | 0.900
w/o Cross. Correlat. | 0.905 | 0.906 | 0.902
using ViT instead of MAE | 0.910 | 0.911 | 0.900
w/o improving MMoE | 0.907 | 0.910 | 0.899
BMR | 0.918 | 0.914 | 0.904

Table 4: Ablation study on bootstrapping multi-view representations and
network design. The tests are done on Weibo. Results of GossipCop and Weibo-21
are in the supplement.

Cross-modal Consistency Learning.  In Table 4, we mimic CAFE that reweighs
$e_{\emph{ip}},e_{\emph{is}},e_{\emph{t}}$ using ${S}_{\emph{m}}$. We
surprisingly find that the result is even worse than not using multimodal
features. It suggests that granting priority to cross-modal consistency rather
than unimodal representations is not beneficial. Second, we remove cross-modal
consistency learning and therefore add no constraint on multimodal feature
generation. The performance drops 1.3% accordingly. From Table 3 and Fig. 7,
we also find that though the training losses for consistency learning on Weibo
and Weibo-21 are both low, we fail to conduct FND simply based on cross-modal
consistency, where the accuracy is less than 0.6. Since we have ruled out
over-fitting, we conclude that the proposed training methodology for cross-
modal consistency is feasible but the consistency can so far only play an
assistant rather than a critical role in multimodal FND.

Single-view Prediction.  In Table 4, we further test the case of adding no
constraint to the generation of multi-view representation, which results in a
performance that drops 2.3%. Therefore, single-view prediction is a useful
manual bias compared to blindly employing multi-branches for feature
extraction. Besides, if we do not explicitly reweigh the representations
according to the sub-tasks, the accuracy will decrease by 2.9%, indicating
that there would be misleading information if we do not reweigh the
representations.

Fig. 8 shows the learned reweighing functions $F(\cdot)$ on GossipCop and
Weibo. We expected “V”-shaped functions to prefer single-view prediction score
with more confidence, i.e., close to either zero or one. However, many of
these learned functions exhibit a inverted “V“ shape. Besides, $e_{x}$ is only
activated when the cross-modal consistency score is close to 0.5 on Weibo.

Figure 8: Curves of the learned reweighing functions $F(\cdot)$. x-axis:
score, e.g., $S_{\emph{ip}}$, y-axis: corresponding weight.

Network Design. Since we are the first to use MAE in FND, we verify the case
of replacing MAE with ViT. The overall accuracy of BMR drops 0.8% results in
Table. 4, which indicates that MAE is better for the task. Besides, When using
iMMoE instead of separate ViT blocks, the result will increase by 1.3%,
suggesting that sharing some representations is beneficial. We also find that
the internal improvement made in MMoE in BMR helps gaining 1.1%.

## Conclusions

We propose BMR that generates multi-view representations, understands the
individual importance, and optimizes the fused features. Single-view
prediction and cross-modal consistency learning are proposed to disentangle
information within unimodal and multimodal features, which are then adaptively
reweighed and bootstrapped for better detection results. Experiments show that
BMR outperforms state-of-the-art multimodal FND schemes on popular datasets.