#  Bost-Connes-Marcolli system for the Siegel modular variety

Ismail Abouamal<EMAIL_ADDRESS>
###### Abstract.

We construct a quantum satisitical mechanical system which generalizes the
Connes-Marcolli $GL_{2,\mathbb{Q}}$-system. In particular we introduce the
Connes-Marcolli system associated to the Siegel modular variety of degree $2$.
We classify its $\textmd{KMS}_{\beta}$-states for inverse temperatures
$\beta>0$ and show that a spontaneous phase transition occurs at $\beta=3$.
More precisely, we prove that the system does not admit a
$\textmd{KMS}_{\beta}$ state for $\beta<3$ with $\beta\neq 1$, construct the
explicit extremal Gibbs states for $\beta>4$ and show that a unique
$\textmd{KMS}_{\beta}$ state exists for every $\beta>0$ with $3<\beta\leq 4$.

##  1\. Introduction

By the end of last century, Bost and Connes [1], motivated by the ideas
introduced by B. Julia [14], construed a quantum statistical mechanical system
$(\mathcal{A},(\sigma_{t})_{t\in\mathbb{R}})$ with unexpected connection
between class field theory of $\mathbb{Q}$ and the theory of $C^{*}$-dynamical
systems. Bost and Connes defined a dense rational subalgebra
$\mathcal{A}_{\mathbb{Q}}\subset\mathcal{A}$ such that the evaluation of
equilibrium states at low temperatures on $\mathcal{A}_{\mathbb{Q}}$ generate
the maximal abelian extension $\mathbb{Q}^{\textmd{ab}}$ of $\mathbb{Q}$. As a
quantum statistical $C^{*}$-dynamical system, the Bost-Connes system has also
the following interesting thermodynamical property: it exhibits a phase
transition at inverse temperature $\beta=1$. Moreover, this transition happens
to be spontaneous in the sense that the symmetry of the system
$(\mathcal{A},(\sigma_{t})_{t\in\mathbb{R}})$ changes radically with small
changes of temperature. For $\beta>1$, the system admits $\zeta(\beta)$ as its
partition function and the extremal $\textmd{KMS}_{\beta}$ states have type I
while for $0<\beta\leq 1$ it admits a unique $\textmd{KMS}_{\beta}$ states of
type $\textmd{III}_{1}$.

Over the past several years, several generalizations of the BC system have
been studied. The construction of Bost and Connes was first generalized by
Connes and Marcolli [8] to quadratic number fields. They introduced the so
called the $GL_{2,\mathbb{Q}}$-system and showed that maximal abelian
extensions of quadratic number fields are generated by evaluating the ground
states on a dense rational arithmetic subalgebra. In a subsequent work by
Connes, Marcolli and Ramachandran [9], the connection between the
$GL_{2}$-system and CM-fields was formally studied. The construction of Connes
and Marcolli was further generalized to arbitrary number fields by Ha and
Paugam [12]. The former authors reformulated the $GL_{2,\mathbb{Q}}$ system in
the adelic language, making explicit its relation to the Shimura datum
$(GL_{2},\mathbb{H}^{\pm})$ over any given number field. They generalized the
construction of Bost and Marcolli to an arbitrary Shimura datum $(G,X)$ and
introduced a formal definition of the abstract Bost-Connes-Marcolli system
associated to the pair $(G,X)$. It was shown that these systems admit the
Dedekind zeta function as the partition function and the group of connected
components of the idèle class group acts as the symmetry group.

In [18], Laca, Larsen and Neshveyev gave yet another reformulation of the
$GL_{2,\mathbb{Q}}$-system in terms of groupoid $C^{*}$-algebras. They
recovered the classification results of $\textmd{KMS}_{\beta}$ states obtained
by Connes and Marcolli and proved uniqueness in the critical range
$1<\beta\leq 2$. A by-product of the author’s work is the development of a
general framework for analyzing dynamical systems of the type introduced by
Connes and Marcolli. These tools involve the use of Hecke operators, ergodic
theory and equidistribution of Hecke points [6].

In this paper we combine the approach of Laca, Larsen and Neshveyev [18]
together with the work of Ha and Paugan [12] and generalize the work of Connes
and Marcolli to the explicit case of the Shimura variety
$(GSp_{4},\mathbb{H}_{2}^{\pm})$. There are several complications compared to
the $GL_{\mathbb{Q},2}$-system. First, the action of the group
$Sp_{4}(\mathbb{Z})$ on the set $\mathbb{H}_{2}^{+}\times
GSp_{4}(\mathbb{A}_{\mathbb{Q},f})$ is not free and one can not easily resolve
this issue by excluding the subset $\mathbb{H}_{2}^{+}\times\\{0_{4}\\}$. Our
approach consists of first replacing the homogeneous space
$\mathbb{H}_{2}^{+}$ by the quotient $K\backslash PGSp_{4}^{+}(\mathbb{R})$
(where $K$ is a compact subgroup of $PGSp_{4}(\mathbb{R})^{+}$) and then prove
a one-to-one correspondence of the $\textmd{KMS}_{\beta}$ states between the
two systems. We first establish the correspondence between the set of
$\textmd{KMS}_{\beta}$ states and Borel measures, which allows us to study the
properties of those measures instead of working directly with the
$\textmd{KMS}_{\beta}$ states. The second difficulty arises from the structure
of the Hecke pair $(GSp_{2n}(\mathbb{Q}),Sp_{2n}(\mathbb{Z}))$. As we show in
this paper, the case $n=2$ is already computationally demanding and even in
this case it is not always possible to directly apply some techniques used in
[18] (especially in the critical interval $3<\beta\leq 4$). As a first result
we show in Theorem LABEL:label_68 that the $GSp_{4}$-system does admit any
$\textmd{KMS}_{\beta}$ state for $0<\beta<3$ and $\beta\notin\\{1,2\\}$. We
next show that the extremal states in the region $\beta>4$ correspond to Gibbs
states and give an explicit construction of these states in Theorem 3.9. The
final main result (Theorem 3.17) is uniqueness theorem: we show that in the
region $3<\beta\leq 4$, the $GSp_{4}$-system admits a unique
$\textmd{KMS}_{\beta}$ state. To show this, we split the proof into two parts.
The main ingredient of the first is the convergence of Dirichlet $L$-functions
for non-trivial characters. The second part relies on a variant of the
technique used in [18]. As stated above, the structure of the Hecke pair
$(GSp_{2n}(\mathbb{Q}),Sp_{2n}(\mathbb{Z}))$ becomes less explicit for $n\geq
2$ and in order to compute the number of right representatives in a double
coset one has to work with upper bounds instead of explicit formulas. We
achieve this by using the root datum of the group $GSp_{2n}$ and use the
equidistribution of Hecke points for the group $GSp_{2n}$ to establish the
second ergodicity result.

### Acknowledgment

The author would like to thank his advisor Matilde Marcolli for her guidance
throughout this project. The author is also grateful to George Elliott,
Serguey Neshveyev and Jean Renault for their valuable comments on this work.

This work has been partially supported by the NSERC Postgraduate Scholarship
PGSD 2-535022-2019.

### Notations and conventions

We use the common notations
$\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$ together with
$\mathbb{R}^{*}_{+}=(0,+\infty)$ and $\mathbb{Z}^{+}=R^{*}_{+}\cap\mathbb{Z}$;

If $R$ is a ring, we denote its group of multiplicative units by $R^{\times}$;
We use the notation $\textmd{Mat}_{n}(R)$ for the ring of square matrices with
entries in $R$. We denote by $E_{ij}$ the usual elementary matrix with $1$ in
the $(i,j)$ position and $0$ elsewhere;
The group of units in the ring $\textmd{Mat}_{n}(R)$ is denoted by
$GL_{n}(R)$. If $A$ is a square matrix, then $A^{t}$ stands for its transpose.
If $A_{1},\dots,A_{n}$ are square matrices we denote by
$\textmd{diag}(A_{1},\dots,A_{n})$ the square matrix with $A_{1},\dots,A_{n}$
as diagonal blocks and $0$’s otherwise. We use $1_{n}$ and $0_{n}$ to denote
the $n\times n$ identity matrix and the a rectangular zero matrix;
$\absolutevalue{F}$ denotes the cardinality of a finite set $F$;
the set of prime numbers is denoted by $\mathcal{P}$;
given a nonempty finite set of prime numbers $F\subset\mathcal{P}$, we denote
by $\mathbb{N}(F)$ the unital multiplicative subsemigroup of $\mathbb{N}$
generated by $p\in F$;
we write $f(x)=O(g(x))$ if $\absolutevalue{f(x)}/g(x)$ is bounded at
$+\infty$;
for two sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$, we write $a_{n}\sim b_{n}$
if $\lim_{n}(a_{n}/b_{n})=1$ and

$\sum_{n}a_{n}\sim\sum_{n}b_{n}$

if the two series are simultaneously divergent or convergent;
if $Y$ is subset of $X$, we denote $Y^{c}=X\backslash Y=\\{a\in X:a\notin
Y\\}$;
for a number field $K$, we denote by
$\mathbb{A}_{K}=\mathbb{A}_{K,f}\times\mathbb{A}_{K,\infty}$ the adèle ring of
$K$, where $\mathbb{A}_{K,f}$ is the ring of finite adèles and
$\mathbb{A}_{K,\infty}$ the infinite adèles of $K$. The ring of integers of
$K$ is denoted by $\mathcal{O}_{K}$.

## Background

###  1.1.  Operator algebraic formulation of quantum statistical mechanics

We briefly review the operator algebraic formulation of quantum statistical
mechanics. For a more comprehensive treatment of this material, we refer the
reader to [2], [3].

Given an absract $C^{*}$-algebra $\mathcal{A}$, we know from Gelfand–Naimark
theorem that $\mathcal{A}$ is ∗-isomorphic to a ∗-subalgebra of the algebra of
bounded operators on a Hilbert space. This result, together with the axioms of
quantum mechanics, motivates the following operator algebraic formulation of
quantum statistical mechanics.

######  Definition 1.

A quantum statistical mechanical system
$(\mathcal{A},(\sigma_{t})_{t\in\mathbb{R}})$ is a $C^{*}$-algebra
$\mathcal{A}$ together with a strongly continuous one-parameter group of
automorphisms $(\sigma_{t})_{t\in\mathbb{R}}$; that is, the map

$t\rightarrow\sigma_{t}(a)$

is norm continuous for every $a\in\mathcal{A}$.

We also say that the pair $(\mathcal{A},(\sigma_{t})_{t\in\mathbb{R}})$ is a
$C^{*}$-dynamical system. One should view $\mathcal{A}$ as the algebra of
observables of a quantum system with time evolution implemented by the one-
parameter group $(\sigma_{t})_{t\in\mathbb{R}}$. If the algebra $\mathcal{A}$
is unital (with unit element $e$), a state on $\mathcal{A}$ is a linear
functional $\phi:\mathcal{A}\rightarrow\mathbb{C}$ satisfying the following
normalization and positivity conditions:

$\displaystyle\phi(e)=1,\quad\phi(a^{*}a)\geq 0$

If $a\in\mathcal{A}$ is self adjoint, we have a decomposition of the form
$a=a^{+}-a^{-}$ where $a^{+},a^{-}\in\mathcal{A}^{+}$ and one should think of
$\phi(a)$ as the expectation value of the observable $a$ in the physical state
$\phi$.

When the algebra $\mathcal{A}$ is non-unital we shall always work with weights
(which we will define shortly) first and then replace the normalization
condition $\phi(e)=1$ by

$\norm{\phi}:=\sup_{x\in\mathcal{A},\norm{x}\leq 1}|\phi(x)|=1$ (1)

to get a state. A weight on $\mathcal{A}$ is a function
$\phi:\mathcal{A}^{+}\rightarrow[0,\infty]$ (here $\mathcal{A}^{+}$ is the
convex cone of positive elements in $\mathcal{A}$) such that $\phi(\lambda
a+b)=\lambda\phi(a)$ and $\phi(\lambda a)=\lambda\phi(a)$ for
$\lambda\in\mathbb{R}^{+}$ and all $a,b\in\mathcal{A}^{+}$. If the algebra
$\mathcal{A}$ is unital, any weight can be written as $\phi=\lambda\omega$,
where $\lambda>0$ and $\omega$ is a state. Hence the two notions of states and
weights essentially coincide in the unital case. In general, this is not true
in the nonunital case. As noted in [5], one should think of states as
probability Borel measures on noncommutative spaces while weights correspond
to regular Borel measures in the commutative case.

In statistical mechanics we are interested in the so-called thermal
equilibrium states at different temperatures. The Kubo-Martin-Shwinger (KMS)
[13, 29] condition at inverse temperature $\beta$ was proposed in 1961 by
Haag, Winnik and Hugenholtz as an equilibrium condition in the
$C^{*}$-algebraic setting of statistical mechanics. We recall the notion of a
$\textmd{KMS}_{\beta}$-weight from [7]. See and [2] [3] for a more detailed
discussion on $\textmd{KMS}_{\beta}$-weights.

######  Definition 2.

Let $A$ be a $C^{*}$-algebra, $\phi$ a weight on $A$ and $\sigma_{t}$ a
strongly continuous one-parameter group of automorphisms of $A$. We set
$\mathcal{N}_{\phi}=\\{a\in A\mid\phi(a^{*}a)<\infty\\}$ and let $\beta>0$. We
say that $\phi$ is a $\textmd{KMS}_{\beta}$-weight if:

  1. (1)

$\phi\circ\sigma_{t}=\phi$ for every $t\in\mathbb{R}$

  2. (2)

For every $a,b\in\mathcal{N}_{\phi}\cap\mathcal{N}_{\phi}^{*}$, there exists a
bounded continuous function $F$ on the closed strip
$\Omega=\\{x\in\mathbb{C}\mid 0\leq\textmd{im}\leq\beta\\}$ and holomorphic on
$\Omega^{0}$ such that

$F(t)=\phi(a\sigma_{t}(b)),\quad F(t+i\beta)=\phi(\sigma_{t}(b)a)$

Although this was the original definition of the $\textmd{KMS}_{\beta}$
condition, in practice the following equivalent characterization (See [17,
Theorem 6.36]) is often used.

######  Proposition 1.1.

Let $\phi$ be a weight on a $C^{*}$-dynamical system
$(\mathcal{A},(\sigma_{t})_{t\in\mathbb{R}})$. Then $\phi$ is a
$\textmd{KMS}_{\beta}$ weight if and only if

  1. (1)

$\phi\circ\sigma_{t}=\phi$ for every $t\in\mathbb{R}$

  2. (2)

For every $\sigma$-analytic element $a$ in $\mathcal{A}$, we have

$\phi(aa^{*})=\phi(\sigma_{\frac{i\beta}{2}}(a)\sigma_{\frac{i\beta}{2}}(a)^{*}).$

###  1.2. Groupoid algebras and Hecke pairs

An important class of $C^{*}$-dynamical systems arises as the algebra of
compactly supported functions on (locally compact) topological groupoids. We
review, without proofs, the general results related to these systems and
derived in the first two sections of [18]. For general information about
grouopoids and groupoid $C^{*}$-algebras, we refer the reader to [28].

Consider a countable group $G$ acting on a locally compact second countable
topological space $X$. The transformation groupoid is the space $G\times X$
with unit space $X$ with the source and target maps given by $s(g,x)=x$ and
$t(g,x)=gx$

and the composition is defined by

$(g,x)(h,y)=(gh,y)\quad\text{if}\,\,\,x=hy.$

If $\Gamma$ is a subgroup of $G$ and the action of $\Gamma$ is free and
proper, we introduce a new groupoid $\Gamma\backslash G\times_{\Gamma}X$ by
taking the quotient of $G\times X$ by the following action of
$\Gamma\times\Gamma$:

$(\gamma_{1},\gamma_{2})(g,x):=(\gamma_{1}g\gamma_{2}^{-1},\gamma_{2}x).$ (2)

In all our settings, the main motivation for taking the quotient by this
action is physical. In fact, to obtain a well behaved partition function of
the $C^{*}$-dynamical system we will introduce shortly, one should necessarily
take the quotient by the group $\Gamma\times\Gamma$ (See also [8] for another
motivation based on the $\mathbb{Q}$-lattice picture).
The groupoid $\mathcal{G}=\Gamma\backslash G\times_{\Gamma}X$ is an étale
groupoid, i.e. the source map $s$ (and hence the target map) is a local
homeomorphism. In particular, it has discrete fibers . In this case, we
introduce the algebra $C_{c}(\mathcal{G})$ of continuous compactly supported
functions on the quotient space $\Gamma\backslash G\times_{\Gamma}X$ and we
define the convolution of two such functions by

$(f_{1}*f_{2})(\omega)=\sum_{\omega_{1}\omega_{2}=\omega}f_{1}(\omega_{1})f_{2}(\omega_{2})=\sum_{\omega_{1}\in\mathcal{G}^{t(\omega)}}f_{1}(\omega_{1})f_{2}(\omega_{1}^{-1}\omega).$
(3)

Notice that this is a finite sum since $f_{1}$ has compact support and the
fibers are discrete. If the action of $\Gamma$ is proper but not free, the
quotient space $\Gamma\backslash G\times_{\Gamma}X$ is no longer a groupoid
(cf. Proposition 2.11). Under the assumption that $X$ is a homogeneous space
of the form $\tilde{X}/H$, where now the action of $G$ on $\tilde{X}$ is free
and proper, we can still define a natural convolution algebra from the
groupoid algebra $C_{c}(\Gamma\backslash G\times_{\Gamma}\tilde{X})$ as was
done in the $GL_{2}$-case in [8]. More specifically, viewing the elements of
$C_{c}(\Gamma\backslash G\times_{\Gamma}\tilde{X})$ as
$\Gamma\times\Gamma$-invariant functions, we rewrite the convolution product
(3) as

$(f_{1}*f_{2})(g,x)=\sum_{s\in\Gamma\backslash G}f_{1}(gs^{-1},sx)f_{2}(s,x).$
(4)

We then define a convolution algebra on the quotient $\Gamma\backslash
G\times_{\Gamma}X$ by restricting the convolution product (4) to weight zero
functions on $C_{c}(\Gamma\backslash G\times_{\Gamma}\tilde{X})$, namely
functions satisfying

$f(g,x\alpha)=f(g,x),\quad\forall\alpha\in H.$

Define an involution on $C_{c}(\Gamma\backslash G\times_{\Gamma}X)$ by

$f^{*}(\omega)=\overline{f(\omega^{-1})}.$

For each $x\in X$, we have a $*$-representation $\pi_{x}$ of
$C_{c}(\Gamma\backslash G\times X)$ on the Hilbert space
$l^{2}(\Gamma\backslash G)$ defined by

$(\pi_{x})(f)\delta_{\Gamma h}=\sum_{g\in\Gamma\backslash
G}f(gh^{-1},hx)\delta_{\Gamma g},\quad f\in C_{c}(\Gamma\backslash
G\times_{\Gamma}X).$

One can show that the operators $\pi_{x}(f)$ are uniformly bounded [12, 18]
and we denote by $\mathcal{B}=C_{r}^{*}(\Gamma\backslash G\times X)$ the
completion of $C_{c}(\Gamma\backslash G\times X)$ in the reduced norm

$\norm{f}=\sup_{x\in X}\norm{\pi_{x}(f)}.$ (5)

In fact, it is easy to verify that that (5) defines a $C^{*}$-seminorm. The
fact that we get a norm follows from the identity

$\langle\pi_{x}f\delta_{\Gamma g_{1}},\delta_{\Gamma
g_{2}}\rangle=f(g_{2}g_{1}^{-1},g_{1}x).$

Let $Y$ be any clopen $\Gamma$-invariant subset of $X$ and denote by
$\Gamma\backslash G\boxtimes_{\Gamma}Y$ the quotient of the space

$\\{(g,y)\mid g\in G,\,\,\,y\in Y,\,\,\,gy\in Y\\},$

by the action of $\Gamma\times\Gamma$ defined in (2). We denote by
$C_{c}(\Gamma\backslash G\boxtimes_{\Gamma}Y)$ the algebra of compactly
supported functions on $\Gamma\backslash G\boxtimes_{\Gamma}Y$ with the
convolution product given by

$(f_{1}*f_{2})(g,y)=\sum_{\begin{subarray}{c}s\in\Gamma\backslash G\\\ sy\in
Y\end{subarray}}f_{1}(gs^{-1},sy)f_{2}(s,y),$

and involution

$f^{*}(g,y)=\overline{f(g^{-1},gy)}.$

We let $\mathcal{A}=C_{r}^{*}(\Gamma\backslash G\boxtimes_{\Gamma}Y)$ be the
corner algebra $e\mathcal{B}e$, where $e$ is the
$\Gamma\times\Gamma$-invariant function on $G\times X$ defined by

$e(g,x)=\begin{cases}1\quad\text{if}\,\,\,(g,x)\in\Gamma\times Y\\\
0\quad\text{otherwise}.\end{cases}$

Given $x\in X$, we put

$G_{x}=\\{g\in G\mid gx\in Y\\}.$ (6)

Then we have a representation of $C_{c}(\Gamma\backslash
G\boxtimes_{\Gamma}Y)$ on the Hilbert space
$\mathcal{H}_{x}=l^{2}(\Gamma\backslash G_{x})$ given by

$\pi_{x}(f)\delta_{\Gamma h}=\sum_{g\in\Gamma\backslash
G_{x}}f(gh^{-1},hx)\delta_{\Gamma g},\quad f\in C_{c}(\Gamma\backslash
G\boxtimes_{\Gamma}Y),$

and the algebra $\mathcal{A}$ coincides ([18]) with the completion of
$C_{c}(\Gamma\backslash G\boxtimes_{\Gamma}Y)$ in the norm defined by

$\norm{f}=\sup_{y\in Y}\norm{\pi_{y}(f)}.$

Assume that we are given a homomorphism

$N:G\longrightarrow\mathbb{R}_{+}^{*},$

such that $\Gamma\subseteq\ker(N)$. We then define a one-parameter group of
automorphisms of $\mathcal{B}$ by

$\sigma_{t}(f)(g,x)=N(g)^{it}f(g,x),\quad\text{for}\,\,\,f\in
C_{c}(\Gamma\backslash G\times X).$

The operator on $l^{2}(\Gamma\backslash G)$ given by

$H_{x}\delta_{\Gamma g}=\log N(g)\cdot\delta_{\Gamma g}$

is the Hamiltonian and the dynamics $\sigma_{t}$ is then spatially implemented
as

$\pi_{x}(\sigma_{t}(a))=e^{itH_{x}}\pi_{x}(a)e^{-itH_{x}},\quad\forall x\in
X,\forall a\in\mathcal{B}.$

The following result will be the starting point of our
$\textmd{KMS}_{\beta}$-analysis of the dynamical system
$(\mathcal{A},\sigma_{t})$.

######  Proposition 1.2.

Let $G$, $X$ and $Y$ as described earlier and suppose $\Gamma$ acts freely on
$X$. Then for $\beta>0$ there exists a one-to-one correspondence between
$\textmd{KMS}_{\beta}$ weights $\phi$ on $\mathcal{A}$ with domain of
definition containing $C_{c}(\Gamma\backslash Y)$ and Radon measures $\mu$ on
$Y$ such that

$\mu(gB)=N(g)^{-\beta}\mu(B)$

for every $g\in G$ and every Borel compact subset $B\subseteq Y$ such that
$gB\subseteq Y$. If $\nu$ denotes the induced measure on $\Gamma\backslash Y$,
then the corresponding weight $\phi$ is given by

$\phi(f)=\int_{\Gamma\backslash Y}f(e,y)d\nu(y),\quad f\in\mathcal{A}.$ (7)

###### Proof.

See [18, Proposition 2.1] ∎

Recall that if $G$ be a group and $\Gamma$ a subgroup, the pair $(G,\Gamma)$
is called is called a Hecke pair if for any $a\in G$

$[\Gamma:\Gamma\cap a^{-1}\Gamma a]<\infty.$

If $(G,\Gamma)$ is a Hecke pair then every double coset of $\Gamma$ contains
finitely many right and left cosets of $\Gamma$:

$\Gamma a\Gamma=\bigsqcup_{\gamma\in\Gamma\backslash(\Gamma\cap a\Gamma
a^{-1})}\gamma a\Gamma=\bigsqcup_{\gamma\in\Gamma\backslash(\Gamma\cap
a^{-1}\Gamma a)}\Gamma g\gamma,$

so $\absolutevalue{\Gamma\backslash\Gamma a\Gamma}=[\Gamma:\Gamma\cap
a^{-1}\Gamma a]$. We denote the cardinality of this set by $\deg_{\Gamma}(a)$.

Let $\beta\in\mathbb{R}$ and $S$ is a semisubgroup of $G$ containing $\Gamma$.
Then we define

$\zeta_{S,\Gamma}(\beta):=\sum_{s\in\Gamma\backslash
S}N(s)^{-\beta}=\sum_{s\in\Gamma\backslash
S/\Gamma}N(s)^{-\beta}\deg_{\Gamma}(s).$ (8)

Let $G$ be a group acting on a set $X$ and suppose $(G,\Gamma)$ is a Hecke
pair. The Hecke operator associated to $g\in G$ is the operator $T_{g}$ on
$\Gamma$-invariant functions on $X$ defined by

$(T_{g}f)(x)=\frac{1}{\deg_{\Gamma}(g)}\sum_{h\in\Gamma\backslash\Gamma
g\Gamma}f(hx).$ (9)

###  1.3. Abstract Bost-Connes-Marcolli systems

In this section, we briefly recall, without proofs, the general properties of
abstract Bost-Connes-Marcolli systems introduced in [12].
A BCM datum is a tuple $\mathcal{D}=(G,X,V,M)$ with $(G,X)$ a Shimura datum,
$(V,\psi)$ a faithful representations of $G$ and $M$ an envelopping semigroup
(see definition 3) for $G$ contained in $\textrm{End}(V)$. A level structure
on $\mathcal{D}$ is a triple $\mathcal{L}=(L,K,K_{M})$ with $L\subseteq V$ a
lattice, $K\subseteq G(\mathbb{A}_{f})$ a compact subgroup and $K_{M}\subseteq
M(\mathbb{A}_{f})$ a compact open subsemigroup such that

  * •

$K_{M}$ stabilizes $L\otimes_{\mathbb{Z}}\hat{\mathbb{Z}}$

  * •

$\psi(K)$ is contained in $K_{M}$.

The pair $(\mathcal{D},\mathcal{L})$ is called a BCM pair. We let

$Y_{\mathcal{D},\mathcal{L}}=K_{M}\times\textrm{Sh}(G,X),$

and we denote the points of $Y_{\mathcal{D},\mathcal{L}}$ by $y=(\rho,[z,l])$.
We let $Y^{\times}_{\mathcal{D},\mathcal{L}}$ be the set of invertible
elements $y=(\rho,[z,l])$ in $Y_{\mathcal{D},\mathcal{L}}$. We have a
partially defined action of $G(\mathbb{A}_{f})$ on
$Y_{\mathcal{D},\mathcal{L}}$:

$g\cdot y=(gy,[z,lg^{-1}])\quad\text{for}\,\,\,y=(\rho,[z,l]).$

We consider the subspace

$\mathcal{U}_{\mathcal{D},\mathcal{L}}\subseteq G(\mathbb{A}_{f})\times
Y_{\mathcal{D},\mathcal{L}}$

of pairs $(g,y)$ such that $g\cdot y\in Y_{\mathcal{D},\mathcal{L}}$. This
space is a groupoid with source and target maps
$s:\mathcal{U}_{\mathcal{D},\mathcal{L}}\rightarrow
Y_{\mathcal{D},\mathcal{L}}$ and
$t:\mathcal{U}_{\mathcal{D},\mathcal{L}}\rightarrow
Y_{\mathcal{D},\mathcal{L}}$ given by $s(g,y)=y$ and $t(g,y)=gy$. The unit
space is $Y_{\mathcal{D},\mathcal{L}}$ and composition is defined as

$(g_{1},y_{1})\circ(g_{2},y_{2})=(g_{1}g_{2},y_{2})\quad\text{if}\,\,\,y_{1}=g_{2}y_{2}.$

There is an action of $K^{2}$ on the groupoid
$\mathcal{U}_{\mathcal{D},\mathcal{L}}$ given by

$(\gamma_{1},\gamma_{2})\cdot(g,y):=(\gamma_{1}g\gamma_{2}^{-1},\gamma_{2}y)$

and the quotient stack
$\mathfrak{Z}_{\mathcal{D},\mathcal{L}}=[K^{2}\backslash\mathcal{U}_{\mathcal{D},\mathcal{L}}]$
has the structure of a stack-groupoid (see [12, Appendix A] ).
From now on we suppose that that $(\mathcal{D},\mathcal{L})$ is a BCM pair
such that the Shimura datum $(G,X)$ is classical, i.e

$\textrm{Sh}(G,X)=G(\mathbb{Q})\backslash G(\mathbb{\mathbb{A}}_{f})\times X.$

We let $\Gamma=G(\mathbb{Q})\cap K$ and

$\mathcal{U}^{\textmd{princ}}:=\\{(g,\rho,z)\in G(\mathbb{Q})\times
K_{M}\times X\mid g\rho\in K_{M})\\}.$

We let $X^{+}$ be a connected component of $X$,
$G(\mathbb{Q})^{+}=G(\mathbb{Q})\cap G(\mathbb{R})^{+}$ (where
$G(\mathbb{R})^{+}$ is the identity component of $G(\mathbb{R})$) and
$\Gamma_{+}=G(\mathbb{Q})^{+}\cap K$. Consider the groupoid

$\mathcal{U}^{+}=\\{(g,\rho,z)\in G(\mathbb{Q})^{+}\times K_{M}\times
X^{+}\mid g\rho\in K_{M})\\},$

with the composition given by

$(g_{1},\rho_{1},z_{1})\circ(g_{2},\rho_{2},z_{2})=(g_{1}g_{2},\rho_{2},z_{2})\quad\textmd{if}\,(\rho_{1},z_{1})=(g_{2}\rho_{2},\rho_{2}z_{2})$
(10)

There is a natural action of $\Gamma^{2}$ (resp. $\Gamma_{+}^{2}$) on
$\mathcal{U}^{\textmd{princ}}$ (resp. $\mathcal{U}^{+}$) given by

$(\gamma_{1},\gamma_{2})\cdot(g,\rho,z):=(\gamma_{1}g\gamma_{2}^{-1},\gamma_{2}\rho,\gamma_{2}z)$
(11)

and the quotient stack
$\mathfrak{Z}_{\mathcal{D},\mathcal{L}}^{\textmd{princ}}$ (resp.
$\mathfrak{Z}_{\mathcal{D},\mathcal{L}}^{+}$) of
$\mathcal{U}^{\textmd{princ}}$ (resp. $\mathcal{U}^{+}$) by $\Gamma^{2}$
(resp. $\Gamma_{+}^{2}$) has again the structure of a stack-groupoid.

For an arbitrary BCM pair $(\mathcal{D},\mathcal{L})$, the relation between
the three groupoids $\mathfrak{Z}_{\mathcal{D},\mathcal{L}}$ and
$\mathfrak{Z}_{\mathcal{D},\mathcal{L}}^{princ}$,
$\mathfrak{Z}_{\mathcal{D},\mathcal{L}}^{+}$ is unclear a priori. The
following important result obtained in [11, Propositions 5.2, Proposition 5.3]
provides sufficient conditions for these three stack-groupoids to coincide.

######  Proposition 1.3.

We denote by $h(G,K)$ the cardinality of the finite set
$G(\mathbb{Q})\backslash G(\mathbb{A}_{f})/K$. Assume that $h(G,K)=1$ and the
natural map $\Gamma\rightarrow G(\mathbb{Q})/G(\mathbb{Q})^{+}$ is surjective.
Then the natural maps

$\mathfrak{Z}_{\mathcal{D},\mathcal{L}}^{+}\longrightarrow\mathfrak{Z}_{\mathcal{D},\mathcal{L}}^{princ},\quad\mathfrak{Z}_{\mathcal{D},\mathcal{L}}^{princ}\longrightarrow\mathfrak{Z}_{\mathcal{D},\mathcal{L}}$

are isomorphisms.

Hence when the condition of Proposition 1.3 are satisfied, it is enough to
work with the BCM system associated to one of the three stack-groupoids. Let

$\mathcal{H}(\mathcal{D},\mathcal{L})=C_{c}(Z_{\mathcal{D},\mathcal{L}})$

be the algebra of continuous compactly supported functions on the coarse
quotient $Z_{\mathcal{D},\mathcal{L}}$ of $\mathcal{U}$ by the action of
$K^{2}$. We view its elements as functions on
$\mathcal{U}_{\mathcal{D},\mathcal{L}}$ satisfying the following properties:

$f(\gamma g,y)=f(g,y),\quad f(g\gamma,y)=f(g,\gamma y),\quad\forall\gamma\in
K,\,\,\,g\in G(\mathbb{A}_{f}),\,\,\,y\in Y_{\mathcal{D},\mathcal{L}}$

The convolution product on $\mathcal{H}(\mathcal{D},\mathcal{L})$ is defined
by the expression

$(f_{1}*f_{2})(g,y):=\sum_{\begin{subarray}{c}h\in K\backslash
G(\mathbb{A}_{f})\\\ hy\in
Y_{\mathcal{D,L}}\end{subarray}}f_{1}(gh^{-1},hy)f_{2}(h,y),$ (12)

and the involution is given by

$f^{*}(g,y):=\overline{f(g^{-1},gy)}.$

Let $y=(\rho,[z,l])\in Y_{\mathcal{D},\mathcal{L}}$ and we put $G_{y}=\\{g\in
G(\mathbb{A}_{f})\mid g\rho\in K_{M}\\}$. We define a ∗-representation
$\pi_{y}:\mathcal{H}(\mathcal{D},\mathcal{L})\rightarrow\mathcal{B}(l^{2}(K\backslash
G_{y}))$ by

$(\pi_{y}(f)\xi)(g):=\sum_{h\in K\backslash G_{y}}f(gh^{-1},hy)\xi(h),\quad
f\in\mathcal{H}(\mathcal{D},\mathcal{L})$

where $\xi$ is the standard basis of $l^{2}(K\backslash G_{y})$. The operators
$\pi_{y}(f)$, for $y\in Y_{\mathcal{D},\mathcal{L}}$ and
$f\in\mathcal{H}(\mathcal{D},\mathcal{L})$ are uniformly bounded [12, Lemma
4.16] and we obtain a $C^{*}$-algebra $\mathcal{A}$ after completing
$\mathcal{H}(\mathcal{D},\mathcal{L})$ in the norm

$\norm{f}=\sup_{y\in Y_{\mathcal{D},\mathcal{L}}}\norm{\pi_{y}(f)}.$

Given a homomorphism

$N:\textmd{GL}(V)\rightarrow\mathbb{R}_{+}^{*},$

we define a time evolution on $\mathcal{H}(\mathcal{D},\mathcal{L})$ by

$\sigma_{t}(f)(g,y)=N(\psi(g))^{it}f(g,y),$

so that the operator on $l^{2}(K\backslash G_{y})$ given by

$(H_{y}\zeta)(g)=\log N(\psi(g))\zeta(g)$

is the Hamiltonian. The resulting $C^{*}$-dynamical system
$(\mathcal{A},\sigma_{t})$ is the Bost-Connes-Marcolli system associated to
the BCM pair $(\mathcal{D},\mathcal{L})$.

The zeta function associated to the BCM pair $(\mathcal{D},\mathcal{L})$ plays
an important role in the $\textmd{KMS}_{\beta}$ analysis of the system
$(\mathcal{A},\sigma_{t})$. It is defined as the complex-valued series

$\zeta_{\mathcal{D},\mathcal{L}}(\beta):=\sum_{g\in\textmd{Sym}_{f}^{\times}\backslash\textmd{Sym}_{f}}N(\psi(g))^{-\beta},$
(13)

where Sym${}_{f}(\mathcal{D},\mathcal{L}):=\psi^{-1}(K_{M})$ and
$\textmd{Sym}_{f}^{\times}(\mathcal{D},\mathcal{L})$ denotes the group of
invertible elements in $\textmd{Sym}_{f}(\mathcal{D},\mathcal{L})$. The pair
$(\mathcal{D},\mathcal{L})$ is called summable of there exists
$\beta_{0}\in\mathbb{R}$ such that $\zeta_{\mathcal{D},\mathcal{L}}(\beta)$
converges in the right plane
$\\{\beta\in\mathbb{C}\mid\textmd{Re}(\beta)>\beta_{0}\\}$ and extends to a
meromorphic function on the full complex plane. By [12, Proposition 4.19] we
know that if $y\in Y^{\times}_{\mathcal{D},\mathcal{L}}$ , then
$G_{y}=\textmd{Sym}_{f}(\mathcal{D},\mathcal{L})$ and the zeta function
$\zeta_{\mathcal{D},\mathcal{L}}(\beta)$ coincides with the partition function

$\zeta_{y}(\beta)=\frac{1}{|K\backslash K_{0}|}Tr(e^{-\beta H_{y}}),\quad
K_{0}=\psi^{-1}(K_{M}^{\times}),$

of the system $(\mathcal{A},\sigma_{t})$.

##  2\. Bost-Connes-Marcolli system for the Siegel Modular Variety

###  2.1. The Symplectic Group

Let $n\in\mathbb{N}$ and $R$ be a commutative unital ring. The symplectic
group of similtitudes of degree $n$ is defined by

$GSp_{2n}(R)=\\{g\in GL_{2n}(R):\exists\,\lambda(g)\in
R^{\times}\,|\,g^{t}\,\Omega\,g\,=\,\lambda(g)\,\Omega\\},$

where

$\Omega=\matrixquantity(0&1_{n}\\\ -1_{n}&0),\quad\text{ $1_{n}$ is the
$n\times n$ identity matrix}.$

The function $\lambda:GSp_{2n}(R)\rightarrow R^{\times}$ is called the
multiplier homomorphism. Its kernel is the symplectic group $Sp_{2n}(R)$ and
there is an exact sequence

$1\longrightarrow Sp_{2n}(R)\longrightarrow GSp_{2n}(R)\longrightarrow
R^{\times}\longrightarrow 1.$ (14)

If $g=\matrixquantity(A&B\\\ C&D)\in GSp_{2n}(R)$ then the following
assertions are equivalent:

  1. (i)

$\lambda(g)=\lambda(g^{t})$

  2. (ii)

The inverse of the matrix $g$ is given by:

$g^{-1}=\lambda(g)^{-1}\matrixquantity(D^{t}&-B^{t}\\\ -C^{t}&A^{t})$ (15)

  3. (iii)

The blocks $A,B,C,D$ satisfy the conditions

$A^{t}C=C^{t}A,\,B^{t}D=D^{t}B,\,A^{t}D-C^{t}B=\lambda(g)1_{n}$ (16)

  4. (iv)

The blocks $A,B,C,D$ satisfy the conditions

$A^{t}B=B^{t}A,\,C^{t}D=D^{t}C,\,A^{t}D-B^{t}C=\lambda(g)1_{n}$ (17)

For $r\in R^{\times}$, we put

$S_{n}(r):=\\{g\in GSp_{2n}(R)\mid\lambda(g)=r\\}.$

We then obtain an embedding of symplectic groups of different degrees as
follows. Given $r\in\mathbb{N}$ and $0<j<n$, define the map

$\displaystyle S_{n}(q)\times S_{n-j}(q)\rightarrow S_{n}(q)$
$\displaystyle(M_{1},M_{2})\mapsto M_{1}\odot M_{2},$

where

$M_{1}\odot
M_{2}:=\matrixquantity(A_{1}&0_{j\times(n-j)}&B_{1}&0_{j\times(n-j)}\\\
0_{(n-j)\times j}&A_{2}&0_{(n-j)\times j}&B_{2}\\\
C_{1}&0_{j\times(n-j)}&D_{1}&0_{j\times(n-j)}\\\ 0_{(n-j)\times
j}&C_{2}&0_{(n-j)\times j}&D_{2}),\quad M_{1}=\matrixquantity(A_{1}&B_{1}\\\
C_{1}&D_{1}),\quad M_{2}=\matrixquantity(A_{2}&B_{2}\\\ C_{2}&D_{2}).$

Note that

$(M_{1}\odot M_{2})\cdot(N_{1}\odot N_{2})=(M_{1}N_{1})\odot(M_{2}N_{2})$ (18)

Consider the following elements of $Sp_{2n}(R)$:

(19)

where $\alpha_{1},\dots,\alpha_{5}\in R$. If $F$ is a field, then the group
$Sp_{2n}(F)$ is generated [24] by the matrices given in (19) with
$\alpha_{1},\dots,\alpha_{5}\in F$.

As a connected reductive algebraic group, the center $Z$ of $G=GSp_{2n}$
consists of scalar matrices and the standard maximal torus is

$T=\\{\text{diag}(u_{1},\dots,u_{n},v_{1},\dots,v_{n})\,:\,u_{1}v_{1}=\dots=u_{n}v_{n}\neq
0\\}.$

If $t\in T$, we often write

$t=\textrm{diag}(u_{1},\dots,u_{n},u_{1}^{-1}\lambda(t),\dots,u_{n}^{-1}\lambda(t)),$
(20)

We fix the following characters $e_{i}\in\textrm{Hom}(T,G_{m})$:

$e_{i}(t)=u_{i},\quad i=0,1,\dots,n\,\,\,\textmd{where}\quad
u_{0}:=\lambda(t).$

and cocharacters $f_{i}\in\textrm{Hom}(G_{m},T)$:

$\displaystyle f_{0}(u)$
$\displaystyle=\textrm{diag}(\underbrace{1,\dots,1}_{n},\underbrace{u,\dots,u}_{n}),$
$\displaystyle f_{1}(u)$
$\displaystyle=\textrm{(}\underbrace{u,1,\dots,1}_{n},\underbrace{u^{-1},1,\dots,1}_{n}),$
$\displaystyle\vdots$ $\displaystyle f_{n}(u)$
$\displaystyle=\textrm{(}\underbrace{1,\dots,1,u}_{n},\underbrace{1,\dots,1,u^{-1}}_{n}).$

######  Proposition 2.1.

The root datum of $GSp_{2n}$ is described as follows. We set

$\displaystyle X$
$\displaystyle=\mathbb{Z}e_{0}\oplus\mathbb{Z}e_{1}\oplus\dots\mathbb{Z}e_{n}.$
$\displaystyle X^{\vee}$
$\displaystyle=\mathbb{Z}f_{0}\oplus\mathbb{Z}f_{1}\oplus\dots\mathbb{Z}f_{n}.$

and let $\langle\cdot,\cdot\rangle$ the natural pairing on $X\times X^{\vee}$:

$\langle e_{i},f_{j}\rangle=\delta_{ij}.$

Then we have the following set of simple roots:

$\alpha_{1}(t)=u_{n-1}^{-1}u_{n},\quad\dots\quad\alpha_{n-1}(t)=u_{1}^{-1}u_{2},\quad\alpha_{n}(t)=u_{1}^{2}u_{0}^{-1},$

where $t$ has the form in (20). In terms of the basis $e_{i},$
$i=0,1,\dots,n$, we have

$\alpha_{1}=e_{n}-e_{n-1},\quad\dots\quad\alpha_{n-1}=e_{2}-e_{1},\quad\alpha_{n}=2e_{1}-e_{0}.$

The corresponding coroots are

$\alpha_{1}^{\vee}=f_{n}-f_{n-1},\quad\dots\quad\alpha_{n-1}^{\vee}=f_{2}-f_{1},\quad\alpha_{n}^{\vee}=f_{1}.$

Let $R=\\{\alpha_{1},\dots,\alpha_{n}\\}$ and
$R^{\vee}=\\{\alpha_{1}^{\vee},\dots,\alpha_{n}^{\vee}\\}$. Then

$(X,R,X^{\vee},R^{\vee})$

is the root datum of $GSp_{2n}$. The Cartan matrix is given by

$\langle\alpha_{i},\alpha_{j}^{\vee}\rangle=\matrixquantity(2&-1&&&&&\\\
-1&2&-1&&&&\\\ &-1&2&-1&&&\\\ &&\ddots&\ddots&\ddots\\\ &&&-1&2&-1\\\
&&&&-1&2&-1\\\ &&&&&-2&2).$

###### Proof.

See [31, page 134-136] ∎

###  2.2. The Symplectic envelopping semigroup

As noted in section 1.3, the abstract definition of the Bost-Connes-Marcolli
system associated to a general Shimura datum $(G,X)$ requires the notion of an
enveloping semigroup which plays the role of $\textmd{Mat}_{2,\mathbb{Q}}$ in
the the case of the $GL_{2,\mathbb{Q}}$-system.

######  Definition 3.

Let $G$ be a reductive group over a field $F$. An enveloping semigroup for $G$
is a multiplicative semigroup $M$ which is irreducible and normal and such
that $M^{\times}=G$.

It is always possible to construct enveloping semigroup (see [12, Appendix
B.2]). For the case $G=GSp_{2n}$ we are considering in the paper, we have the
following explicit description of $M$. Given a commutative
$\mathbb{Q}$-algebra $R$ we have

$M(R):=MSp_{2n}(R)=\\{m\in\textmd{Mat}_{2n}(R)\mid\exists\,\lambda(m)\in
R,m^{t}\Omega m=\lambda(m)\Omega\\},$ (21)

since $m\in MSp_{2n}(R)^{\times}$ if and only if $\lambda(m)\in R^{\times}$.

###  2.3. The Siegel modular group

The group $\Gamma_{n}=Sp_{2n}(\mathbb{Z})$ is called the Siegel modular group
of degree $n$. For $m\in\mathbb{N}$, we denote by $GL_{m}(\mathbb{Z})$ the
unimodular group of degree $m$ and note that $\Gamma_{n}\subseteq U_{2n}$ with
equality if $n=1$. We let $MSp_{2n}^{+}(\mathbb{Z})=\\{M\in
MSp_{2n}(\mathbb{Z})\mid\lambda(M)>0\\}$. Then given any $M\in
MSp_{2n}^{+}(\mathbb{Z})$, we denote by $\Gamma_{n}M\Gamma_{n}$ the double
coset generated by $M$ and put

$\mathcal{D}(\Gamma_{n}M\Gamma_{n}):=\\{D\in\textrm{Mat}_{n}(\mathbb{Z})\,\textmd{such
that there exists}\,\matrixquantity(A&B\\\ 0&D)\in\Gamma_{n}M\Gamma_{n}\\}.$

For each $D\in\mathcal{D}(\Gamma M\Gamma)$, we set

$\mathcal{B}(D,\Gamma_{n}M\Gamma_{n}):=\\{B\in\textrm{Mat}_{n}(\mathbb{Z})\,\textmd{such
that there exists}\matrixquantity(A&B\\\ 0&D)\in\Gamma_{n}M\Gamma_{n}\\}.$

We define the following equivalence relation on $\mathcal{B}(D)$:

$B\sim
B^{\prime}\Leftrightarrow(B-B^{\prime})D^{-1}\in\textrm{Sym}_{n}(\mathbb{Z}),$
(22)

and we write $B\equiv B^{\prime}\mod D$ if $B\sim B^{\prime}$. For
$r\in\mathbb{N}$, we put

$S_{n}(r):=\\{g\in GSp_{2n}(\mathbb{Z})\mid\lambda(g)=r\\}.$

Let $M\in\textrm{Mat}_{m}(\mathbb{Z})$ with $\det(M)>0$ and $N\in S_{n}(r)$.
Then by the Elementary Divisor Theorems (See [16, Theorem 2.2, Chapter V]) the
double cosets $U_{m}MU_{m}$ and $\Gamma_{n}N\Gamma_{n}$ contain unique
representatives of the form

$\displaystyle\textmd{Elm}(M)=\textrm{diag}(a_{1},a_{2},\dots,a_{m}),\quad
a_{1},a_{2},\dots,a_{n}\in\mathbb{N},$ (23)

with $a_{1}\mid a_{2}\mid\dots\mid a_{m}$ and

$\textmd{Elm}(N)=\textrm{diag}(a_{1},\dots,a_{n},d_{1},\dots,d_{n}),\quad
a_{1},\dots,a_{n},d_{1},\dots,d_{n}\in\mathbb{N},$ (25)

such that $a_{i}d_{i}=r$, $i=1,\dots,n$ and $a_{1}\mid a_{2}\mid\dots
a_{n}\mid d_{n}\mid\dots\mid d_{n-1}\mid d_{1}$.

######  Theorem 2.2.

Let $M\in MSp_{2n}^{+}(\mathbb{Z})$. Then a set of representatives of the
right cosets relative to $\Gamma_{n}$ in $\Gamma_{n}M\Gamma_{n}$ is given by
the matrices

$\matrixquantity(A&B\\\ 0&D),\quad A=\lambda(M)(D^{t})^{-1}$

where

  1. (1)

$D$ runs through a set of representatives of
$GL_{n}(\mathbb{Z})\backslash\mathcal{D}(\Gamma_{n}M\Gamma_{n});$

  2. (2)

$B$ runs through a set of representatives of $\mod D$ incogruent matrices in
$\mathcal{B}(D,\Gamma_{n}M\Gamma_{n}).$

###### Proof.

See [16, Theorem 3.4, Chapter VI] ∎

######  Proposition 2.3.

Let $p$ be a prime number and $l\in\mathbb{N}$. Then the set $S_{n}(p^{l})$
decomposes into finitely many right cosets relative to $\Gamma_{n}$. A set of
representatives is given by

$\matrixquantity(p^{l}(D^{t})^{-1}&B\\\ 0&D)$ (26)

where $D$ runs through a set of representatives of

$GL_{n}(\mathbb{Z})\backslash\Big{\\{}D\in\textrm{Mat}_{n}(\mathbb{Z})\mid\textrm{Elm}(D)=\textrm{diag}(d_{1},d_{2},\dots,d_{n})\,\,\,\textmd{and}\,\,\,d_{i}\mid
p^{l}\,\,\,\textmd{for all}\,\,\,i=1,2,\dots,n\Big{\\}}$

and $B$ runs through a set of representatives of $\mod D$ incongruent matrices
in

$B(D):=\\{B\in\textrm{Mat}_{n}(\mathbb{Z})\mid B^{t}D=D^{t}B\\}$ (27)

###### Proof.

Every right coset $\Gamma_{n}M$ contains a representatives of the form in
(26). Suppose we have two representatives $N$ and $M$ of this form:

$N=\matrixquantity(A&B\\\ 0&D),\quad
M=\matrixquantity(A^{\prime}&B^{\prime}\\\ 0&D^{\prime}),$

with $\Gamma_{n}N=\Gamma_{n}N$. Then since

$\Big{\\{}\matrixquantity(A&B\\\
0&D)\in\Gamma_{n}\Big{\\}}=\Big{\\{}\matrixquantity(U^{t}&0\\\
0&U^{-1})\matrixquantity(1_{n}&S\\\ 0&1_{n})\mid U\in
GL_{n}(\mathbb{Z})\,,S\in\textrm{Sym}_{n}(\mathbb{Z})\Big{\\}},$ (28)

we obtain from the conditions in (16)-(17) that there exists $U\in
GL_{n}(\mathbb{Z})$ such that $D(A^{\prime})^{t}=\lambda(M)U^{-1}$ and hence

$D(D^{\prime})^{-1}=U^{-1}.$

This shows that $D=D^{\prime}$ and consequently $A=A^{\prime}$. Moreover from
the equality $\eqref{label 10}$ we know that there exists
$S\in\textrm{Sym}_{n}(\mathbb{Z})$ such that

$\displaystyle-A(B^{\prime})^{t}+B(A)^{t}$
$\displaystyle=-A(B^{\prime})^{t}+AB^{t},$ $\displaystyle=\lambda(M)S.$

Since $A^{t}D=\lambda(M)=\lambda(N)$ we obtain that $B-B^{\prime}=SD$, i.e
$B\equiv B^{\prime}\mod D$. ∎

######  Lemma 2.4.

Let $p$ be a prime number, $l\in\mathbb{N}$,
$D\in\textrm{Mat}_{n}(\mathbb{Z})$ such that
$\textrm{Elm}(D)=\textrm{diag}(d_{1},d_{2},\dots,d_{n})$ with $d_{i}|p^{l}$
for $i=1,2,\dots,n$ and $B(D)$ is as in equation (27). Given $U,V\in
GL_{n}(\mathbb{Z})$, then

  1. (1)

$|B(D)\mod D|=|B(UDV)\mod UDV|$,

  2. (2)

$|B(D)\mod D|=d_{1}^{n}d_{2}^{n-1}\dots d_{n}$.

###### Proof.

Since $U,V\in\mathcal{U}_{n}(\mathbb{Z})$, we have the following bijection

$\displaystyle B(D)$ $\displaystyle\rightarrow B(UDV)$ $\displaystyle M$
$\displaystyle\mapsto(U^{t})^{-1}MV$

This proves the first claim. Hence we can suppose that $D=\textrm{Elm}(D)$ to
prove the second assertion. Since $d_{1}|\dots|d_{n}$, we can write

$B(D)=\\{M={(b_{jk})\mid
b_{jk}\in\mathbb{Z}},b_{jk}=b_{kj}\frac{d_{k}}{d_{l}}\,\,\,\textmd{for}\,\,\,j\leq
k\\}.$

By definition of the relation in $\eqref{label 19}$ the entries $b_{jk}$ may
be reduced $\mod d_{k}$. This shows that $B(D)$ consists exactly of
$d_{1}^{n}d_{2}^{n-1}\dots d_{n}$ equivalence classes $\mod D$. ∎

Let $r\in\mathbb{N}$ and define

$R_{\Gamma_{n}}(r):=\sum_{g\in\Gamma_{n}\backslash
S_{n}(r)/\Gamma_{n}}\deg_{\Gamma_{n}}(g).$

######  Proposition 2.5.

The function $R_{\Gamma_{n}}:\mathbb{N}\rightarrow\mathbb{N}$ is a
multiplicative function, i.e given relatively prime numbers
$q,r\in\mathbb{N}$, we have

$R_{\Gamma_{n}}(qr)=R_{\Gamma_{n}}(q)R_{\Gamma_{n}}(r).$

###### Proof.

Observe first that if $g\in S_{n}(qr)$, then
$\textrm{Elm}(g)=\textrm{Elm}(g_{1})\textrm{Elm}(g_{2})$ for some $g_{1}\in
S_{n}(q)$ and $g_{2}\in S_{n}(r)$. Hence

$R_{\Gamma_{n}}(qr)=\sum_{g\in\Gamma_{n}\backslash
S_{n}(qr)/\Gamma_{n}}\deg_{\Gamma_{n}}(g)=\sum_{g_{2}\in\Gamma_{n}\backslash
S_{n}(q)/\Gamma_{n}}\sum_{g_{1}\in\Gamma_{n}\backslash
S_{n}(r)/\Gamma_{n}}\deg_{\Gamma_{n}}(g_{1}g_{2}),$

so it is enough to show that $\deg(g_{1}g_{2})=\deg(g_{1})\deg(g_{2})$. We
decompose the double cosets $\Gamma_{n}\backslash g_{1}/\Gamma_{n}$ and
$\Gamma_{n}\backslash g_{2}/\Gamma_{n}$ into finitely many right cosets
$\Gamma_{n}Q_{i},i=1,\dots\deg(g_{1})$ and
$\Gamma_{n}R_{i},i=1,\dots\deg(g_{2})$ and consider the right cosets given by
$\Gamma Q_{i}R_{j}$. Suppose hat $\Gamma Q_{i}R_{j}=\Gamma Q_{k}R_{l}$. Then
there exists some $\gamma\in\Gamma_{n}$ and a matrix $M\in
GSp_{2n}(\mathbb{Q})$ such that

$M=Q_{k}^{-1}\gamma
Q_{i}=R_{l}R_{j}^{-1},\quad\lambda(Q_{k})=\lambda(Q_{i})=q,\quad\lambda(R_{k})=\lambda(R_{i})=r.$

Recall from (15) that $Q_{k}=\lambda(Q_{k})^{-1}\Omega^{-1}Q_{k}^{t}\Omega$
and so after writing $M=\\{\frac{n_{ij}}{m_{ij}}\\}_{ij}$ where
$(n_{ij},m_{ij})=1$, we see that the integers $m_{ij}$ divide
$\lambda(Q_{k})=q$ and similarly $m_{ij}$ divide $\lambda(R_{j})=r$. By
assumption $(q,r)=1$ so $M\in\Gamma_{n}$ since $\lambda(A)=1$. This shows that
$\Gamma_{n}Q_{k}=\Gamma_{n}Q_{i}$ and $\Gamma_{n}R_{l}=\Gamma_{n}R_{k}$, in
other words $i=k$ and $l=j$. To conclude we simply observe that the cosets
$\Gamma Q_{i}R_{j}$ form a partition of $\Gamma_{n}\backslash
g_{1}g_{2}/\Gamma_{n}$. ∎

Let $p$ be a prime and $l\in\mathbb{N}$. For arbitrary $n$, it is in general
not possible to obtain a closed formula for $\deg(a)$ if $a\in S_{p^{l}}$ is
given in its elementary form. On the other hand, an upper bound of $\deg(a)$
will be enough in most of our calculations. We first suppose that $a$ is given
by

$a=\textmd{diag}(p^{k_{1}},p^{k_{2}},\dots,p^{k_{n}},p^{l-k_{1}},\dots,p^{l-k_{n}}),\quad[l/2]\leq
k_{1}\leq k_{2}\leq\dots\leq k_{n}$

Note that this is not the elementary sympletic form of $a$ since
$k_{1}\geq[l/2]$. We shall use the root datum of $GSp_{2n}$ given in
Proposition 2.1. The set $\Phi^{+}$ of positive roots is given by (see [31,
page 167])

$\displaystyle e_{j}-e_{i},\quad 1\leq i<j\leq n$ $\displaystyle
e_{j}+e_{i}-e_{0},\quad 1\leq i<j\leq n$ $\displaystyle 2e_{i}-e_{0},\quad
1\leq i\leq n.$

where we have used our choice of the basis $e_{i},i=1,\dots,n$ (the choice of
the basis used in [31] is different but the computations are essentially the
same). Hence

$2\rho=\sum_{\alpha\in\Phi^{+}}\alpha=2\sum_{i=0}^{i=n-1}(n-i)e_{n-i}-\frac{1}{2}n(n+1)e_{0}.$

We set

$\lambda=\sum_{i=1}^{i=n}k_{i}f_{i}+lf_{0}.$

Observe that $\langle\lambda,\alpha\rangle\geq 0$ for all $\alpha\in\Phi^{+}$.
Then using the degree formula in Proposition $7.4$ in [10] (see also the proof
of Corollary $1.9$ in [6] for a similar result in the case of $GL_{n}$) we
obtain

$\deg(a)=p^{(\sum_{i=0}^{i=n-1}2(n-i)k_{n-i})-\frac{1}{2}n(n+1)l}(1+O(p^{-1}))$
(29)

where the big $O$ depends only on $n$. If $a\in S_{p^{l}}$ is given in its
elementary symplectic form (23) , we apply left and right (symplectic)
permutations matrices to $a$ (which leaves invariant the degree) and use the
formula (29).

###  2.4.  Structure theorems of the symplectic group

As seen in the previous section, the class number $h(G,K)$ plays a detrimental
role in the definition of the abstract BCM system. The aim of this section is
to show that for $G=GSp_{2n}$ we have that $h(G,K)=1$ where $K$ is any open
compact subgroup of $GSp_{4}(\hat{\mathbb{Z}})$. The proof relies on the
notion of strong approximation in algebraic groups, which we briefly review.
Given a linear reductive group $G$ over a global field $K$ and a non-empty
finite set $S$ of places of $K$, we denote by $\mathbb{A}_{S}$ the ring of
$S$-adèles and let $G(\mathbb{A}_{S})$ be the ring of $S$-adèles of $G$

$G(\mathbb{A}_{S}):=\\{g=(g_{\upsilon})\in\prod_{\upsilon\notin
S}G(K_{\upsilon})\mid g_{\upsilon}\in
G(\mathcal{O}_{\upsilon})\,\,\,\textmd{for almost all}\,\,\,\upsilon\notin
S\\}.$

For any given $S$, there is a canonical embedding $G(K)\hookrightarrow
G(\mathbb{A}_{S})$. An algebraic group $G$ over a global field $K$ has the
strong approximation with respect to $S$ if $G(K)$ is dense in
$G(\mathbb{A}_{S})$. It is well known that strong approximation does not hold
in general (for example one can take the group $G=GL_{n}$ with
$S=\\{\infty\\}$). The following theorem provides a necessary and sufficient
condition for the strong approximation theorem to hold for algebraic groups.

######  Theorem 2.6.

(See [15], [26] in characteristic zero and [27],[20] [19] in positive
characteristic), Let $G$ be an absolutely almost simple simply connected
algebraic group over a field $K$ and $S$ a finite nonempty set of places of
$K$. Then $G$ has strong approximation with respect to $S$ if and only if the
group $G_{S}=\prod_{\upsilon\in S}G(K_{\upsilon})$ is noncompact.

We can conclude from Theorem 2.6 that the symplectic group $G=Sp_{2n}$ (over
$\mathbb{Q}$) has the strong approximation with respect to $S=\\{\infty\\}$.
In the rest of this section, we provide an elementary proof of this result
using matrix factorization. For this, we first put

$\Gamma_{n}(N)=\\{\gamma\in\Gamma_{n}\mid\gamma^{t}\,\Omega\,\gamma\equiv\Omega\,\,\mod
N\\},$ (30)

for every positive integer $N$

######  Lemma 2.7.

Let $\pi_{N}$ be the projection $\pi_{N}:Sp_{2n}(\mathbb{Z})\rightarrow
Sp_{2n}(\mathbb{Z}/{N\mathbb{Z}})$ defined by $\pi_{N}(\gamma)=\gamma\mod N$.
Then the sequence

$1\longrightarrow\Gamma_{n}(N)\longrightarrow\Gamma_{n}\overset{\pi_{N}}{\longrightarrow}Sp_{2n}(\mathbb{Z}/{N\mathbb{Z}})\longrightarrow
1$

is exact.

###### Proof.

The only nontrivial part is the surjectivity of the map $\pi_{N}$. This
follows directly from [23, Theorem 1] ∎

We denote by $\hat{\mathbb{Z}}=\varprojlim_{N>1}\mathbb{Z}/N\mathbb{Z}$ the
ring of profinite integers and we write $M_{2n}(\hat{\mathbb{Z}})$ for the
ring of $2n\times 2n$-matrices with coefficients in $\hat{\mathbb{Z}}$. The
profinite compact group
$GL_{2n}(\hat{\mathbb{Z}})=\varprojlim_{N>1}GL_{2n}(\mathbb{Z}/N\mathbb{Z})$
is a subset of $M_{2n}(\hat{\mathbb{Z}})$ and consists of invertible matrices.
The subgroup $Sp_{2n}(\hat{\mathbb{Z}})\triangleleft
GL_{2n}(\hat{\mathbb{Z}})$ is defined by exactness of the sequence (14).

######  Proposition 2.8.

$Sp_{2n}(\mathbb{Z})$ is dense in $Sp_{2n}(\hat{\mathbb{Z}})$.

###### Proof.

Since
$M_{2n}(\hat{\mathbb{Z}})=\varprojlim_{N>1}M_{2n}(\mathbb{Z}/N\mathbb{Z})$ is
a profinite ring, a system of neighborhoods of the zero matrix is given by
$\\{NM_{2n}(\hat{\mathbb{Z}})\mid N\in\mathbb{N}\\}$ and thus a system of
neighborhood of $\mathbb{1}_{2n}$ in $Sp_{2n}(\hat{\mathbb{Z}})$) is given by
$\\{U_{N}\mid N\in\mathbb{N}\\}$ where
$U_{N}=\\{1+NM_{2n}(\hat{\mathbb{Z}})\\}\cap Sp_{2n}(\hat{\mathbb{Z}})\\}$.
Given $N\in\mathbb{N}$, consider the projection map
$\pi_{N}:Sp_{2n}(\hat{\mathbb{Z}})\rightarrow Sp_{2n}(\mathbb{Z}/N\mathbb{Z})$
and note that $\ker{\pi_{N}}=U_{N}$. By Lemma 2.7 the projection $\pi_{N}$
restricted to ${Sp_{2n}(\mathbb{Z})}$ is surjective. Hence for any given $x\in
Sp_{2n}(\hat{\mathbb{Z}})$ and $N\in\mathbb{N}$, we choose $\gamma_{N}\in
Sp_{2n}(\mathbb{Z})$ such that $\pi_{N}(\gamma_{N})=\pi_{N}(x)$.Then
$x^{-1}\gamma_{N}\in U_{N}$, that is $\gamma_{N}\in xU_{N}$. Taking $N$ large
enough shows that $Sp_{2n}(\mathbb{Z})$ is dense in
$Sp_{2n}(\mathbb{\hat{\mathbb{Z}}})$. ∎

For a prime $p$, we denote by $\mathbb{Q}_{p}$ the field of $p$-adic numbers
and $\mathbb{Z}_{p}$ its compact subgring of $p$-adic integers. We consider
$\mathbb{A}_{\mathbb{Q},f}$ the ring of finite adèles of $\mathbb{Q}$, that
is, the restricted product of the fields $\mathbb{Q}_{p}$ with respect to
$\mathbb{Z}_{p}$ and we denote by
$I_{\mathbb{Q}}=\mathbb{A}_{\mathbb{Q},f}^{\times}$ the idèle group.

######  Theorem 2.9.

The algebraic group $G=Sp_{2n}$ (over $\mathbb{Q}$) has the strong
approximation with respect to the infinite place $S=\\{\infty\\}$.

###### Proof.

We denote by $H$ the closure of $Sp_{2n}(\mathbb{Q})$ in
$Sp_{2n}(\mathbb{A}_{\mathbb{Q},f})$. We have a dense diagonal embedding of
$\mathbb{Q}$ inside $\mathbb{A}_{\mathbb{Q},f}$, whence the subgroup $H$
contains the group generated by the matrices of the form (19) with
$\alpha_{1},\dots,\alpha_{5}\in\mathbb{A}_{\mathbb{Q},f}$. In particular,
given any prime number $p$, the subgroup $H$ contains the set of matrices of
the form (19) with $(\alpha_{i})_{q}=1$ for $q\neq p$ and $i=1,\dots,5$. Since
$Sp_{4}(\mathbb{Q}_{p})$ is generated by these type of matrices, we see that
$H$ contains the elements $M=(M_{p})_{p}\in
Sp_{2n}(\mathbb{A}_{\mathbb{Q},f})$, with $M_{p}\in Sp_{2n}(\mathbb{Q}_{p})$
and $M_{q}=1$ for $q\neq p$. Hence for any finite set of primes $F$, we have
the inclusion

$\\{(x)_{q}\in Sp_{2n}(\mathbb{A}_{\mathbb{Q},f})\mid\forall q\in F,x_{q}\in
Sp_{2n}(\mathbb{Q}_{p})\,\,\,\textmd{and}\,\,\,x_{q}=1\,\,\,\textmd{if}\,\,\,q\notin
F\\}\subseteq H.$

The result follows since the union of these subsets over all finite set of
primes $F$ is dense in $Sp_{4}(\mathbb{A}_{\mathbb{Q},f})$. ∎

######  Corollary 2.10.

Let $K$ be an open compact subgroup of $GSp_{2n}(\mathbb{A}_{f})$. Then
$\lambda(K)\subseteq\hat{\mathbb{Z}}^{\times}$ and if
$\lambda(K)=\hat{\mathbb{Z}}^{\times}$, we have

$GSp_{2n}(\mathbb{A}_{f})=K\cdot
GSp_{2n}^{+}(\mathbb{Q})=GSp_{2n}^{+}(\mathbb{Q})\cdot K\,\,\,.$ (31)

In particular we get that $h(G,K)=1$ for the maximal open compact subgroup
$K=GSp_{2n}(\hat{\mathbb{Z}})$.

###### Proof.

The first assertion follows from the fact that the map
$\lambda:GSp_{2n}(\mathbb{A}_{f})\rightarrow\mathbb{A}_{f}^{\times}$ is
continuous and $\mathbb{Z}_{p}^{\times}$ is the unique maximal subgroup of
$\mathbb{Q}_{p}^{\times}$. To show (31), let $g\in GSp_{2n}(\mathbb{A}_{f})$
so that $\lambda(g)\in\mathbb{A}_{f}^{\times}$. Since
$\mathbb{A}_{\mathbb{Q},f}^{\times}=\mathbb{Q}^{\times}\hat{\mathbb{Z}}^{\times}$
, we can write

$\lambda(g)=\alpha\cdot x,$

for some $\alpha\in\mathbb{Q}^{\times}$ (we choose $\alpha>0$ if necessary)
and $x\in\hat{\mathbb{Z}}^{\times}=\lambda(K)$. Thus we can choose $k\in K$
such that $\lambda(k)=x$. Consider the matrix

$g^{\prime}=\textrm{diag}(\underbrace{\alpha^{-1},1,\dots,\alpha^{-1},1}_{n},\underbrace{1,\alpha^{-1},\dots,1,\alpha^{-1}}_{n})\,gk^{-1}.$

Observe that $g^{\prime}\in Sp_{2n}(\mathbb{A}_{f})$ and by Theorem 2.9 the
open set $g^{\prime}\cdot Sp_{2n}(\hat{\mathbb{Z}})\subseteq
Sp_{2n}(\mathbb{A}_{f})$ contains $\eta\in Sp_{2n}(\mathbb{Q})$ such that
$\eta=g^{\prime}\cdot h$ for some $h\in Sp_{2n}(\hat{\mathbb{Z}})$. Moreover
by Proposition 2.8 the group $Sp_{2n}(\mathbb{Z})$ is dense in
$Sp_{2n}(\hat{\mathbb{Z}})$, hence we can find $\gamma\in Sp_{2n}(\mathbb{Z})$
such that $\gamma\in g^{\prime-1}\eta U$, where $U=K\cap
Sp_{2n}(\hat{\mathbb{Z}})$. This shows that $g\in K\cdot
GSp_{2n}^{+}(\mathbb{Q})$ as desired. Considering the automorphism $x\mapsto
x^{-1}$ for $x\in GSp_{2n}(\mathbb{A}_{f})$, we see that
$GSp_{2n}(\mathbb{A}_{f})=GSp_{2n}^{+}(\mathbb{Q})\cdot K$. ∎

###  2.5. Siegel upper half plane

######  Definition 4.

The Siegel upper half plane of degree $n$ consists of all symmetric complex
$n\times n$-matrices whose imaginary part is positive definite:

$\mathbb{H}_{n}^{+}=\\{\tau=\tau_{1}+i\tau_{2}\in\textmd{Mat}_{n}(\mathbb{C})\mid\tau^{t}=\tau,\quad\tau_{2}>0\\}$
(32)

Let $M=\matrixquantity(A&B\\\ C&D)\in GSp_{2n}^{+}(\mathbb{R})$ and
$\tau\in\mathbb{H}_{n}^{+}$. Then the matrix $C\tau+D$ is invertible and if we
define

$g\cdot\tau:=(A\tau+B)(C\tau+D)^{-1},$

then the map

$\tau\mapsto g\cdot\tau$

is an action of $GSp_{2n}^{+}(\mathbb{R})$ on $\mathbb{H}_{n}^{+}$ [25] . If
we write $\tau=\tau_{1}+i\tau_{2}\in\mathbb{H}_{n}^{+}$ and
$d\tau=d\tau_{1}d\tau_{2}$ is the Euclidean measure, then the element of
volume on $\mathbb{H}_{n}^{+}$ given by

$d^{*}\tau:=\det(\tau_{2})^{-(n+1)}d\tau,$ (33)

is invariant under all transformations of the group
$GSp_{2n}^{+}(\mathbb{R})$, i.e

$d^{*}(g\cdot\tau)=d^{*}\tau,\quad\textmd{for all}\,g\in
GSp_{2n}^{+}(\mathbb{R}).$

Given an element $\tau=\tau_{2}+i\tau_{2}\in\mathbb{H}_{n}^{+}$, the relation

$\matrixquantity(1_{n}&\tau_{1}\\\
0_{n}&1_{n})\matrixquantity(\tau_{2}^{1/2}&0_{n}\\\
0_{n}&\tau_{2}^{-1/2})\cdot i1_{n}=\tau$

shows that the action of $GSp_{2n}^{+}(\mathbb{R})$ is transitive. The
stabilizer of $i1_{n}$ is the subgroup

$S=\Big{\\{}\matrixquantity(A&B\\\ -B&A)\in GL_{2n}(\mathbb{R})\Big{\\}}\cap
GSp_{2n}^{+}(\mathbb{R}),$

Hence the group $Z(\mathbb{R})\backslash GSp_{2n}^{+}(\mathbb{R})$ (where
$Z(\mathbb{R})$ denotes the center of $GSp_{2n}^{+}(\mathbb{R})$ ) acts
transitively on $\mathbb{H}_{n}^{+}$ and we have the following identification

$\mathbb{H}_{n}^{+}=PGSp_{2n}^{+}(\mathbb{R})/K,$ (34)

where $K$ is the compact group $K=Z(\mathbb{R})\backslash
S\simeq\mathbb{U}^{n}/\\{\pm 1_{2n}\\}$ and $\mathbb{U}^{n}=S\cap
Sp_{2n}(\mathbb{R})$ is isomorphic to the unitary group of order $n$ through
the map

$\matrixquantity(A&B\\\ -B&A)\mapsto A+iB.$

The restriction of the action defined in (2.5) to the arithmetic subgroup
$Sp_{2n}(\mathbb{Z})$ will be of special importance to us. Note that from the
identification (34) we see that the action of $\Gamma_{n}$ on
$\mathbb{H}_{n}^{+}$ is proper since $\Gamma_{n}=Sp_{2n}(\mathbb{Z})$ is
discrete and the subgroup $K$ in (2.5) is compact. The action of
$Sp_{2n}(\mathbb{Z})$ on $\mathbb{H}_{n}^{+}$ admits a fundamental domain and
thanks to a result by Siegel [4], it has the following concrete description
[21]. Let $U_{n}$ be the subset of matrices
$\tau=\tau_{1}+i\tau_{2}\in\mathbb{H}_{n}^{+}$ satisfying the following
conditions:

  1. (1)

$\absolutevalue{\det(C\tau+D)}\geq 1$ for every $\matrixquantity(A&B\\\
C&D)\in Sp_{2n}(\mathbb{Z});$

  2. (2)

$\tau_{2}=(\tau_{2})_{ij}$ is Minkowski reduced, i.e

$a^{t}\tau_{2}a\geq(\tau_{2})_{kk},\,\,\,1\leq k\leq n\quad\text{for all}\quad
a=\matrixquantity(a_{1}\\\ \\\ \vdots\\\
a_{n})\in\mathbb{Z}^{n}\,\,\,\text{where}\,\,\,(a_{1},\dots,a_{n})=1;$

  3. (3)

$\absolutevalue{(\tau_{1})_{ij}}\leq 1/2.$

Then $U_{n}$ is a fundamental domain of $Sp_{2n}(\mathbb{Z})$ on
$\mathbb{H}_{n}^{+}$ of finite volume with the respect to the element of
volume (33):

$\textmd{vol}(U_{n})=2\prod_{i=1}^{n}\pi^{-k}\Gamma(i)\zeta(2i),$

where $\Gamma(s)$ denotes the Gamma functions and $\zeta(s)$ is the Riemann
zeta function.

###  2.6. Bost-Connes-Marcolli system: the $GSp_{2n}$-case

We consider the connected Shimura datum $(GSp_{2n}^{+},\mathbb{H}^{+}_{n})$
together with the BCM pair

$\displaystyle G^{+}=GSp_{2n}^{+},\quad X^{+}=\mathbb{H}^{+}_{n},\quad
V=\mathbb{Q}^{2n},\quad M=MSp_{2n}(\hat{\mathbb{Z}}),$ $\displaystyle
L=\mathbb{Z}^{2n},\quad K=GSp_{2n}(\hat{\mathbb{Z}}),\quad
K_{M}=MSp_{2n}(\hat{\mathbb{Z}}).$

Let $M\in GSp_{2n}^{+}(\mathbb{Q})\cap GSp_{2n}(\hat{\mathbb{Z}})$. Since
$\hat{\mathbb{Z}}\,\cap\,\mathbb{Q}=\mathbb{Z}$ it is clear that $M\in
MSp_{2n}(\mathbb{Z})$ and there exists $M^{\prime}\in
MSp_{2n}(\mathbb{Z})^{+}$ such that $MM^{\prime}=1$ so that $\lambda(M)=1$,
that is $\Gamma_{+}=GSp_{2n}^{+}(\mathbb{Q})\cap
GSp_{2n}(\hat{\mathbb{Z}})=\Gamma_{n}$. Corollary 2.10 tells us that
$h(G,K)=1$ so by Proposition 1.3 it is enough to work with the space
$\mathfrak{Z}_{\mathcal{D},\mathcal{L}}^{\textmd{+}}$. As in the case of the
$GL_{2}$ system, the first difficulty that arises is the presence of points in
$\mathbb{H}_{n}^{+}$ with nontriival stabilizers:

######  Proposition 2.11.

The groupoid structure on
$\mathfrak{U}_{\mathcal{D},\mathcal{L}}^{\textmd{+}}$ does not pass to the
quotient by the action of $\Gamma_{n}\times\Gamma_{n}$.

###### Proof.

Let $g=\matrixquantity(1_{n}&0_{n}\\\ 0_{n}&\frac{1}{2}1_{n})\in
GSp_{2n}^{+}(\mathbb{Q})$ and assume the groupoid composition is defined when
we pass to the quotient. Since $g\cdot\frac{1}{2}i1_{2n}=i1_{2n}$ we obtain
that

$(g,0,i1_{2n})(g,0,\frac{1}{2}i1_{2n})=(g^{2},0,\frac{1}{2}i1_{2n}),$

where the equality holds in the quotient. On the other hand, let
$\gamma=\matrixquantity(0_{n}&-1_{n}\\\ 1_{n}&0_{n})$ so that
$g\gamma^{-1}g=-\frac{1}{2}\gamma$ and $\gamma\cdot
i1_{2n}=g\cdot\frac{1}{2}i1_{2n}$. We then have the following equality in the
quotient:

$(g^{2},0,\frac{1}{2}i1_{2n})=(g\gamma^{-1},0,\gamma\cdot
i1_{2n})(g,0,\frac{1}{2}i1_{2n})=(g\gamma^{-1}g,0,\frac{1}{2}i1_{2n}).$

Hence there exist $\gamma_{1},\gamma_{2}\in\Gamma_{n}=Sp_{2n}(\mathbb{Z})$
satisfying the following two conditions:

$\gamma_{2}\cdot i1_{2n}=i1_{2n},$
$\gamma_{1}g^{2}\gamma_{2}^{-1}=\frac{1}{2}\matrixquantity(0_{n}&1_{n}\\\
-1_{n}&0_{n}).$

The first condition implies that $\gamma_{2}$ is of the form
$\gamma_{2}=\matrixquantity(A&B\\\ -B&A)$ while the second condition gives

$\gamma_{1}=\matrixquantity(\frac{1}{2}B&-2A\\\ \frac{1}{2}A&2B).$

Note that since $\gamma_{1}\in Sp_{2n}(\mathbb{Z})$ we get

$\matrixquantity(\frac{1}{4}(BA-AB)&A^{2}+B^{2}\\\ -B^{2}-A^{2}&4(BA-
AB))=\Omega,$

that is $I=4(A^{\prime 2}+B^{\prime 2})$ for some
$A^{\prime},B^{\prime}\in\textrm{Mat}_{n}(\mathbb{Z})$, which is a
contradiction. ∎

###  2.7. The $GSp_{4,\mathbb{Q}}$-system

We restrict our attention to the case $n=2$ and fix the following notation. We
let $Y=\mathbb{H}^{+}_{2}\times MSp_{4}(\hat{\mathbb{Z}})$ so that
$X=GSp_{4}(\mathbb{Q})^{+}Y=\mathbb{H}^{+}_{2}\times
MSp_{4}(\mathbb{A}_{f,\mathbb{Q}})$ and $\Gamma_{2}=Sp_{4}(\mathbb{Z})$. As
observed above the action of $\Gamma_{2}$ on $Y$ is not free and it turns out
that the set of points in $Y$ with non-trivial stabilizers strictly contains
$\mathbb{H}_{2}^{+}\times\\{0_{4}\\}$. Let

$F_{Y}=\\{h\in MSp_{4}(\hat{\mathbb{Z}})\mid\rank_{\mathbb{Q}_{p}}(h_{p})\leq
2\quad\textmd{for all primes $p$}\\}.$

Then the action of $\Gamma_{2}$ on
$\tilde{Y}=Y\backslash(\mathbb{H}_{2}^{+}\times F_{Y})$ is free. To see this,
we suppose that for some $\gamma\in\Gamma_{2}$ we have $\gamma\cdot\tau=\tau$
and $\gamma h_{p}=h_{p}$ for some $\tau\in\mathbb{H}_{2}^{+}$ and $h_{p}\in
MSp_{4}(\mathbb{Q}_{p})$ with $\rank_{\mathbb{Q}_{p}}(h_{p})>2$ for some prime
$p$. Then we can find $T\in GL_{4}(\mathbb{Q}_{p})$ such that

$T\gamma T^{-1}=\matrixquantity(1&0&0&x_{1}\\\ 0&1&0&x_{2}\\\ 0&0&1&x_{3}\\\
0&0&0&x_{4}),$

for some $x_{1}\dots,x_{4}\in\mathbb{Q}_{p}$. Since the entries of $\gamma$
are in $\mathbb{Z}$ we see that $x_{4}\in\mathbb{Q}$ and thus
$C_{\mathbb{Q},\gamma}(x)=(x-1)^{3}(x-x_{4})$. On the other hand, since
$\gamma$ fixes a point in $\mathbb{H}_{2}^{+}$, then there exists $P\in
Sp_{4}(\mathbb{R})$ such that

$P\gamma P^{-1}=\matrixquantity(a_{1}&b_{1}\\\
-b_{1}&a_{1})\odot\matrixquantity(a_{2}&b_{2}\\\ -b_{2}&a_{2}),\quad
a_{i},b_{i}\in\mathbb{R}\quad a_{1}^{2}+b_{1}^{2}=a_{2}^{2}+b_{2}^{2}=1,$

So
$C_{\mathbb{C},\gamma}(x)=(x-\lambda_{1})(x-\overline{\lambda}_{1})(x-\lambda_{2})(x-\overline{\lambda}_{2})$
where $\lambda_{1}=a_{1}+ib_{1}$ and $\lambda_{2}=a_{2}+ib_{2}$. Hence
$\lambda_{1}=\lambda_{2}=1$ and $\gamma=1$. It is easy to see that the set of
points in $Y$ with non-trivial stabilizers is strictly larger than
$\mathbb{H}_{2}^{+}\times\mathbf{\\{}0\\}$.

The fact that $F_{Y}$ is not invariant under scalar matrices in
$GSp_{4}^{+}(\mathbb{Q})$ creates a new difficulty that was not present in the
$GL_{2}$-system. For this reason, and for the purpose of
$\textmd{KMS}_{\beta}$ analysis, instead of working with the quotient
$\mathbb{H}_{2}^{+}=GSp_{4}^{+}(\mathbb{R})/K$, we consider first the quotient
$PGSp_{4}^{+}(\mathbb{R})=GSp_{4}^{+}(\mathbb{R})/Z(\mathbb{R})$, where
$Z(\mathbb{R})$ is the center of the group $GSp_{4}^{+}(\mathbb{R})$. From now
on we refer to this system as the $GSp_{4}$-system and we call the original
dynamical system (corresponding to the Shimura datum
$(GSp_{4}^{+},\mathbb{H}_{2}^{+})$) the Connes-Marcolli $GSp_{4}$-system. We
will show later that the two systems have the same thermodynamical properties.
Since now $PGSp_{4}^{+}(\mathbb{R})$ is a group, we get the following:

######  Proposition 2.12.

For $\beta\neq 0$, there exists a correspondence between
$\textmd{KMS}_{\beta}$ states on the $GSp_{4}$-system and
$\Gamma_{2}$-invariant measures $\mu$ on $PGSp_{4}^{+}(\mathbb{R})\times
MSp_{4}(\mathbb{A}_{f,\mathbb{Q}})$ such that

$\nu(\Gamma_{2}\backslash PGSp_{4}^{+}(\mathbb{R})\times
MSp_{4}(\hat{\mathbb{Z}}))=1,\quad\mu(gB)=\lambda(g)^{-\beta}\mu(B)$ (35)

for any $g\in GSp_{4}^{+}(\mathbb{Q})$ and Borel compact sublet $B\subset
PGSp_{4}^{+}(\mathbb{R})\times MSp_{4}(\mathbb{A}_{f,\mathbb{Q}})$. Here $\nu$
denotes the measure on $\Gamma_{2}\backslash PGSp_{4}^{+}(\mathbb{R})\times
MSp_{4}(\mathbb{A}_{f,\mathbb{Q}})$ corresponding to $\mu$.

###### Proof.

Since the action of $\Gamma_{2}$ on $PGSp_{4}^{+}(\mathbb{R})$ is free,
Proposition 1.2 applied to the group $G=GSp_{4}^{+}(\mathbb{Q})$ and the
spaces $X=PGSp_{4}^{+}(\mathbb{R})\times MSp_{4}(\mathbb{A}_{f,\mathbb{Q}})$
and $Y=PGSp_{4}^{+}(\mathbb{R})\times MSp_{4}(\hat{\mathbb{Z}})$ gives a one-
to-one correspondence between the set of $\textmd{KMS}_{\beta}$ states on
$(\mathcal{A},\sigma)$ and $\Gamma_{2}$-invariant measures $\mu$ on $Y$ such
that $\mu(gB)=\lambda(g)^{-\beta}\mu(B)$ if $gZ$ and $Z$ are measurable
subsets of $Y$. The equality
$MSp_{4}(\mathbb{A}_{f})=GSp_{4}^{+}(\mathbb{Q})MSp_{4}(\hat{\mathbb{Z}})$
allows us to extend ([18, Lemma 2.2]) this measure to a Radon measure on $X$
such that $\mu(gB)=\lambda(g)^{-\beta}\mu(B)$ for every Borel subset
$B\subseteq X$. Since the algebra $\mathcal{A}$ is not unital, from the
normalization condition (1) and equation (7) we obtain that $\nu$ is a
probability measure on $\Gamma_{2}\backslash Y$. ∎

From now on, we let $Y=PGSp_{4}^{+}(\mathbb{R})\times
MSp_{4}(\hat{\mathbb{Z}})$ and $X=PGSp_{4}^{+}(\mathbb{R})\times
MSp_{4}(\mathbb{A}_{f,\mathbb{Q}})$. For $\beta>0$, we denote by
$\mathcal{E}_{\beta}$ the set of Radon measures on $X$ satisfying the
properties in Proposition 2.12. Note that the extremal $\textmd{KMS}_{\beta}$
states correspond to point mass measures.

##  3\. $\textmd{KMS}_{\beta}$ states analysis

###  3.1. High temperature region

We begin the $\textmd{KMS}_{\beta}$ of the $GSp_{4}$-system analysis by first
considering the high temperature region $0<\beta<3$. Our first goal is to show
that the $GSp_{4}$-system constructed above does not admit a
$\textmd{KMS}_{\beta}$ state for $0<\beta<3$ with $\beta\notin\\{1,2\\}$. We
first show some useful lemmas.

######  Lemma 3.1.

Let $F$ be a finite set of prime numbers and $g=(g_{p})_{p\in F}\in\prod_{p\in
F}MSp_{4}{(\mathbb{Z}_{p})}\subset\prod_{p\in F}MSp_{4}{(\mathbb{Q}_{p})}$
with $\lambda(g_{p})\neq 0$ for all $p\in F$. Then there exist $g_{1}\in
S_{F}$ and $g_{2}\in\prod_{p\in F}GSp_{4}{(\mathbb{Z}_{p})}$ such that
$g=g_{1}g_{2}$.

###### Proof.

It follows from Corollary 2.10 that we can find $g_{1}\in
GSp_{4}^{+}(\mathbb{Q})$ and $g_{2}\in GSp_{4}(\mathbb{Z}_{p})$ such that
$g=g_{1}g_{2}$ with $g_{1}\in GSp_{4}(\mathbb{Z}_{q})$, $q\neq p$ and
$g_{1}\in MSp_{4}(\mathbb{Z}_{p})$ and $\lambda(g_{1})\in\mathbb{N}_{F}$, that
is $g_{1}\in S_{F}$. ∎

For $k_{0},k_{1},k_{2}\in\mathbb{Z}$, we set

$P_{k_{0}}:=\matrixquantity(0&0\\\ 0&p^{k_{0}})\odot 0_{2},\quad
P_{k_{1},k_{2}}:=\matrixquantity(0&0\\\
0&p^{k_{1}})\odot\matrixquantity(0&0\\\ 0&p^{k_{2}}).$ $\displaystyle
Z_{k_{0}}^{(0)}:=Sp_{4}(\mathbb{Z}_{p})\,P_{k_{0}}\,GSp_{4}(\mathbb{Z}_{p}),\quad
Z_{k_{1},k_{2}}^{(1)}:=Sp_{4}(\mathbb{Z}_{p})\,P_{k_{1},k_{2}}\,GSp_{4}(\mathbb{Z}_{p}).$
(36)

######  Lemma 3.2.

The sets $Z_{k_{0}}^{(0)}$ and $Z_{k_{1},k_{2}}^{(1)}$,
$k_{0},k_{1},k_{2}\in\mathbb{Z}$ are pairwise disjoint. Moreover, given any
nonzero matrix $a\in MSp_{4}(\mathbb{Q}_{p})$ with $\lambda(a)=0$, then $a\in
Z_{k_{0}}^{(0)}\cup Z_{k_{1},k_{2}}^{(1)}$ for some
$k_{0},k_{1},k_{2}\in\mathbb{Z}$.

###### Proof.

We first fix some notations. For $1\leq i,j\leq 4$, let $E_{ij}$ be the
elementary matrix with coefficient $1$ at the position $(i,j)$ and $0$
otherwise. For $U\in GL_{2}(\mathbb{Z}_{p})$ and
$S\in\textrm{Sym}_{2}(\mathbb{Z}_{p})$, we put

$J(U)=\matrixquantity(U^{t}&0_{2}\\\ 0_{2}&U^{-1}),\quad
J(S)=\matrixquantity(1_{2}&S\\\ 0_{2}&1_{2})$

Consider $g\in MSp_{4}(\mathbb{Q}_{p})$ with $\mu(g)=0$. Let $g_{0}$ be any
entry of $g$ with maximal $p$-adic valuation and we write
$g_{0}=a_{0}p^{k_{0}}$, where $a_{0}\in\mathbb{Z}_{p}^{\times}$. Using the
matrices $\Omega$ and $J_{1}(P)$ (where $P$ is a permutation matrix), we may
assume that $g_{11}=g_{0}$. If $a$ is an entry of the matrix $g$, we set

$U_{a}=1_{2}-g_{0}^{-1}aE_{21},\quad
S_{a}=-g_{0}^{-1}aE_{11},\quad\tilde{S}_{a}=-g_{0}^{-1}a(E_{12}+E_{21})$

Observe that by maximality, these matrices are in $Sp_{4}(\mathbb{Z}_{p})$. We
multiply $g$ from the right by

$J(U_{g_{{}_{12}}})J(S_{g_{{}_{13}}+g_{{}_{11}}g_{{}_{12}}g_{{}_{14}}})J(\tilde{S}_{g_{{}_{14}}}).$

to obtain a matrix whose first row is $g_{0}\mathbf{e_{1}}$. Taking the
transpose and repeating this process, we obtain a matrix whose first column is
equal to $g_{0}\mathbf{e_{1}}^{t}$. The symplectic relations 16 and 17 imply
that this matrix has the following form:

$\matrixquantity(g_{0}&0\\\ 0&0)\odot M,\quad
M\in\textrm{Mat}_{2}(\mathbb{Q}_{p}),\quad\det(M)=0.$

If $M=0$, then one has

$\matrixquantity(0&-1\\\ 1&0)\odot 1_{2}\cdot\matrixquantity(g_{0}&0\\\
0&0)\odot M\cdot\matrixquantity(0&a_{0}^{-1}\\\
1&0)\odot\matrixquantity(0&a_{0}^{-1}\\\ 1&0)=\matrixquantity(0&0\\\
0&p^{k_{0}})\odot 0_{2}\in P_{k_{0}}.$

Otherwise, we can use right and left multiplication to find $\gamma_{1}\in
SL_{2}(\mathbb{Z}_{p})$ and $\gamma_{2}\in GL_{2}(\mathbb{Z}_{p})$ such that
$\gamma_{1}M\gamma_{2}=\matrixquantity(0&0\\\ 0&p^{k_{1}})$, for some
$k_{1}\in\mathbb{Z}$. Then the matrix

$\matrixquantity(0&-1\\\ 1&0)\odot\gamma_{1}\cdot\matrixquantity(g_{0}&0\\\
0&0)\odot M\cdot\matrixquantity(0&a_{0}^{-1}\\\
-a_{0}\det(\gamma_{2})&0)\odot\gamma_{2}=\matrixquantity(0&0\\\
0&p^{k_{0}})\odot\matrixquantity(0&0\\\ 0&p^{k_{1}})\in P_{k_{0},k_{1}}$

has the desired form. One can easily check that this decomposition is unique.
∎

######  Lemma 3.3.

Let $p$ be a prime number and $g\in MSp_{4}(\mathbb{Z}_{p})$ such that
$\absolutevalue{\lambda(g)}_{p}=p^{-k},k\in\mathbb{N}$. Then there exist
$\gamma_{1},\gamma_{2}\in Sp_{4}(\mathbb{Z}_{p})$ such that
$\gamma_{1}g\gamma_{2}$ is of the form

$\textmd{diag}(a_{1},a_{2},d_{1},d_{2}),\quad\absolutevalue{a_{1}}_{p}\geq\absolutevalue{a_{2}}_{p}\geq\absolutevalue{d_{2}}_{p}\geq\absolutevalue{d_{1}}_{p}$

###### Proof.

The proof is similar to [16, Theorem 2.2, Chapter V]. ∎

######  Lemma 3.4.

Let $p$ be a prime and we put

$g_{1,p}:=\textrm{diag}(1,1,p,p),\quad g_{2,p}:=\textrm{diag}(p,p,p,p),\quad
g_{3,p}:=\textrm{diag}(1,p,p^{2},p).$

A set of representatives of the right cosets relative to $\Gamma_{2}$ in
$\Gamma_{2}g_{1,p}\Gamma_{2}$ is given by the matrices

$\matrixquantity(p&0&0&0\\\ 0&p&0&0\\\ 0&0&1&0\\\
0&0&0&1),\quad\matrixquantity(p&0&0&0\\\ 0&1&0&k_{1}\\\ 0&0&1&0\\\
0&0&0&p),\quad\matrixquantity(1&-k_{2}&k_{3}&0\\\ 0&p&0&0\\\ 0&0&p&0\\\
0&0&k_{2}&1),\quad\matrixquantity(1&0&k_{4}&k_{5}\\\ 0&1&k_{5}&k_{6}\\\
0&0&p&0\\\ 0&0&0&p),\quad$

where $0\leq k_{1},k_{2},\dots,k_{6}<p$.

A set of representatives of the right cosets relative to $\Gamma_{2}$ in
$\Gamma_{2}g_{3,p}\Gamma_{2}$ is given by the matrices

In this section, we study the equilibrium states of the $GSp_{4}$-system
within the low temperature region. More specifically, we give an explicit
construction of the extremal $\textmd{KMS}_{\beta}$ states for $\beta>4$,
which of course provides a complete description of the set of
$\textmd{KMS}_{\beta}$ states.

######  Theorem 3.9.

For $\beta>4$, the extremal KMS states of the GSp4-system are given by the
Gibbs states

$\phi_{\beta}(f)=\frac{\zeta(2\beta-2)\textmd{Tr}(\pi_{y}(f)e^{-\beta
H_{y}})}{\zeta(\beta)\zeta(\beta-1)\zeta(\beta-2)\zeta(\beta-3)},$ (45)

where $y\in PGSp_{4}^{+}(\mathbb{R})\times GSp_{4}(\hat{\mathbb{Z}})$.

###### Proof.

Let $F$ be an arbitrary finite set of primes and denote by $G_{F}$ the group
generated by $G_{p}$ for $p\in F$. We denote by
$MSp_{4}^{+}(\mathbb{Z})=GSp_{4}^{+}(\mathbb{Q})\cap MSp_{4}(\mathbb{Z})$ and
we put

$S_{F}:=\\{m\in MSp_{4}^{+}(\mathbb{Z})\mid\lambda(m)\in\mathbb{N}(F)\\},$

and

$Y_{F}=PGSp_{4}^{+}(\mathbb{R})\times\prod_{p\in
F}GSp_{4}(\mathbb{Z}_{p})\times\prod_{q\notin F}MSp_{4}(\mathbb{Z}_{q}).$

Similarly to the proof of Theorem LABEL:label_68 (we replace $Y_{p}$ by
$Y_{F}$ and $S_{2,p}$ by $S_{F}$), we get

$1=\nu_{\beta}(\Gamma_{2}\backslash
Y_{F})\zeta_{S_{F},\Gamma_{2}}(\beta)=\nu_{\beta}(\Gamma_{2}\backslash
Y_{F})\prod_{p\in F}\zeta_{S_{2,p},\Gamma_{2}}$

Note that $Y_{F}\subseteq Y_{F^{\prime}}$ for $F^{\prime}\subseteq F$ and the
intersection of $Y_{F}$ over all finite primes is the set
$PGSp_{4}^{+}(\mathbb{R})\times GSp_{4}(\hat{\mathbb{Z}})$. Hence for
$\beta>4$, we get

$\nu_{\beta}(\Gamma_{2}\backslash(PGSp_{4}^{+}(\mathbb{R})\times
GSp_{4}(\hat{\mathbb{Z}}))=\zeta_{MSp_{4}^{+}(\mathbb{Z}),\Gamma_{2}}(\beta)^{-1},$

where

$\zeta_{MSp_{4}^{+}(\mathbb{Z}),\Gamma_{2}}(\beta)=\prod_{p\in\mathcal{P}}\zeta_{S_{2,p},\Gamma_{2}}=\frac{\zeta(\beta)\zeta(\beta-1)\zeta(\beta-2)\zeta(\beta-3)}{\zeta(2\beta-2)}.$

On the other hand, the sets $\Gamma_{2}s(PGSp_{4}^{+}(\mathbb{R})\times
GSp_{4}(\hat{\mathbb{Z}}))$ are disjoints for $s\in\Gamma_{2}\backslash
MSp_{4}^{+}(\mathbb{Z})/\Gamma_{2}$. We thus obtain

$\displaystyle\nu_{\beta}(\Gamma_{2}\backslash
MSp_{4}^{+}(\mathbb{Z})(PGSp_{4}^{+}(\mathbb{R})\times
GSp_{4}(\hat{\mathbb{Z}})))=\sum_{s\in\Gamma_{2}\backslash
MSp_{4}^{+}(\mathbb{Z})/\Gamma_{2}}\nu_{\beta}(\Gamma_{2}\backslash\Gamma_{2}s(PGSp_{4}^{+}(\mathbb{R})\times
GSp_{4}(\hat{\mathbb{Z}})))$
$\displaystyle=\zeta_{MSp_{4}^{+}(\mathbb{Z}),\Gamma_{2}}(\beta)\nu_{\beta}(\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})\times GSp_{4}(\hat{\mathbb{Z}}))=1,$

Hence $MSp_{4}^{+}(\mathbb{Z})(PGS_{4}^{+}(\mathbb{R})\times
GSp_{4}(\hat{\mathbb{Z}}))$ has full measure in $PGS_{4}^{+}(\mathbb{R})\times
MSp_{4}(\hat{\mathbb{Z}})$ and by Corollary 2.9 the subset
$PGSp_{4}^{+}(\mathbb{R})\times GSp_{4}(\mathbb{A}_{f})$ has full measure in
$PGSp_{4}^{+}(\mathbb{R})\times MSp_{4}(\mathbb{A}_{f})$. Conversely, any
probability $\Gamma_{2}$-invariant measure on $PGSp_{4}^{+}(\mathbb{R})\times
GSp_{4}(\hat{\mathbb{Z}})$ extends [18, Lemma 2.4] uniquely to a measure on
$PGSp_{4}^{+}(\mathbb{R})\times GSp_{4}(\mathbb{A}_{f})$ satisfying condition
35.

Suppose now that $\mu_{\beta}\in\mathcal{E}_{\beta}$ is a Dirac measure
centered on $y\in PGSp_{4}^{+}(\mathbb{R})\times GSP_{4}(\hat{\mathbb{Z}})$.
Then

$\displaystyle\phi(f)$ $\displaystyle=\sum_{s\in\Gamma_{2}\backslash
MSp_{4}^{+}(\mathbb{Z})/\Gamma_{2}}\int_{\Gamma_{2}\backslash\Gamma_{2}s(PGSp_{4}^{+}(\mathbb{R})\times
GSp_{4}(\hat{\mathbb{Z}}))}f(1,\omega)d\nu_{\beta}(\omega)$
$\displaystyle=\zeta_{MSp_{4}^{+}(\mathbb{Z}),\Gamma_{2}}^{-1}\sum_{s\in\Gamma_{2}\backslash
MSp_{4}^{+}(\mathbb{Z})/\Gamma_{2}}\lambda(s)^{-\beta}\sum_{h\in\Gamma_{2}\backslash\Gamma_{2}s\Gamma_{2}}f(1,hy)$
$\displaystyle=\zeta_{MSp_{4}^{+}(\mathbb{Z}),\Gamma_{2}}(\beta)^{-1}\sum_{h\in\Gamma_{2}\backslash
MSp_{4}^{+}(\mathbb{Z})}\lambda(h)^{-\beta}f(1,hy)$
$\displaystyle=\frac{\textmd{Tr}(\pi_{y}(f)e^{-\beta
H_{y}})}{\textmd{Tr}(e^{-\beta H_{y}})},$

since the operator $H_{y}$ is positive and $G_{y}=MSp_{4}^{+}(\mathbb{Z})$ for
$y\in GSp_{4}^{+}(\mathbb{R})\times GSp_{4}(\hat{\mathbb{Z}})$.

∎

###  3.3. The critical region

We denote by $\hat{\mathcal{E}}_{\beta}$ the subset of right
$GSp_{4}(\hat{\mathbb{Z}})$-invariant measures in $\mathcal{E}_{\beta}$. The
next proposition shows that this set is not empty for $3\leq\beta<4$.

######  Proposition 3.10.

For each $\beta\in(3,4]$, the $\textmd{GSp}_{4}$-system
$\\{\mathcal{A},(\sigma_{t})_{t\in\mathbb{R}}\\}$ admits at least one
$\textrm{KMS}_{\beta}$-state.

###### Proof.

We generalize the construction in [18]. By the correspondence in Proposition
2.12, it is enough to construct a measure $\mu_{\beta}$ on
$\mathbb{H}_{2}\times MSp_{4}(\mathbb{A}_{f})$ such that
$\mu_{\beta}\in\mathcal{E}_{\beta}$. For each prime $p$ and $3<\beta\leq 4$,
we consider the normalized Haar measure on $GSp_{4}(\mathbb{Z}_{p})$ so that
the total volume is $\zeta_{S_{2,p},\Gamma_{2}}(\beta)^{-1}$ (we denote this
measure by $\textmd{meas}_{\beta,p}$). Observe that
$GSp_{4}(\mathbb{Q}_{p})=G_{p}GSp_{4}(\mathbb{Z}_{p})$ and hence by [18, lemma
2.4] we can uniquely extend this measure to a measure $\mu_{\beta,p}$ on
$GSp_{4}(\mathbb{Q}_{p})$ such that if $Z$ is a compact measurable subset in
$GSp_{4}(\mathbb{Q}_{p})$, then

$\mu_{\beta,p}(Z)=\sum_{g\in
G_{p}}|\lambda(g)|_{p}^{-\beta}\textmd{meas}_{\beta,p}(gZ\cap
GSp_{4}(\mathbb{Z}_{p})),$

where $|a|_{p}$ denotes the $p$-adic valuation of $a$. Since
$\mu_{\beta,p}(hZ)=|\lambda(h)|_{p}^{\beta}\mu_{\beta,p}(Z)$ for $g\in
GSp_{4}(\mathbb{Q}_{p})$, it is clear that $\mu_{\beta,p}$ is left
$GSp_{4}(\mathbb{Z}_{p})$-invariant. It is also right
$GSp_{4}(\mathbb{Z}_{p})$-invariant since the Haar measure
$\textmd{meas}_{\beta,p}$ is right translation invariant. We extend the
measure $\mu_{\beta,p}(Z)$ to a measure on $MSp_{4}(\mathbb{Q}_{p})$ by
setting $\mu_{\beta,p}(Z):=\mu_{\beta,p}(Z\cap GSp_{4}(\mathbb{Q}_{p}))$ for
Borel $Z\subseteq MSp_{4}(\mathbb{Q}_{p})$. To extend this measure to
$MSp_{4}(\mathbb{A}_{f})$ we first check that
$\mu_{\beta,p}(MSp_{4}(\mathbb{Z}_{p}))=1$. The proof of Lemma
(LABEL:label_47) applied to the space $MSp_{4}(\mathbb{Q}_{p})$ shows that the
set $MSp_{4}(\mathbb{Z}_{p})\cap GSp_{4}(\mathbb{Q}_{p})$ has full measure.
Since this set is precisely $S_{2,p}GSp_{4}(\mathbb{Z}_{p})$, then similarly
to the calculation in the proof of Theorem LABEL:label_68 we get

$\mu_{\beta,p}(MSp_{4}(\mathbb{Z}_{p}))=\sum_{g\in\Gamma_{2}\backslash
S_{2,p}/\Gamma_{2}}\lambda(g)^{-\beta}\deg_{\Gamma_{2}}(g)\mu_{\beta,p}(GSp_{4}(\mathbb{Z}_{p}))=1.$

We thus define a measure on $MSp_{4}(\mathbb{A}_{f})$ by
$\mu_{\beta,f}=\prod_{p\in\mathcal{P}}\mu_{\beta,p}$. Then for $g\in
GSp_{4}(\mathbb{Q})$ and measurable subset $Z\subseteq
MSp_{4}(\mathbb{A}_{f})$, we get

$\mu_{\beta,f}(gZ)=\Big{(}\prod_{p\in\mathcal{P}}|\lambda(g)|_{p}^{\beta}\Big{)}\mu_{\beta,f}(Z)=\lambda(g)^{-\beta}\mu_{\beta,f}(Z).$

If we denote by $\mu_{\beta,PGSp_{4}^{+}(\mathbb{R})}$ the normalized Haar
measure on $PGSp_{4}^{+}(\mathbb{R})$ such that
$\nu_{\beta,PGSp_{4}^{+}(\mathbb{R})}$ is a probability measure on
$\Gamma_{2}\backslash PGSp_{4}^{+}(\mathbb{R})$, then it is clear that the
measure defined by
$\mu_{\beta}:=\mu_{\beta,PGSp_{4}^{+}(\mathbb{R})}\times\mu_{\beta,f}$ is an
element of $\mathcal{E}_{\beta}$. By construction, $\mu_{\beta}$ is also right
$GSp_{4}(\hat{\mathbb{Z}})$-invariant; that is
$\mu_{\beta}\in\hat{\mathcal{E}}_{\beta}$. ∎

Our next goal is to show that for $3<\beta\leq 4$, the $\textmd{KMS}_{\beta}$
constructed in Proposition 3.10 is the unique equilibrium state. We first
recall the definition of an ergodic action.

######  Definition 5.

If $\mu\in\mathcal{E}_{\beta}$, the action of $G$ on the measure space
$(X,\mu)$ is ergodic if the following holds: If $A$ is any $G$-invariant Borel
subset of $X$, then $\mu(A)=0$ or $\mu(A^{c})=0$.

Recall that if $W$ be a locally compact group then a character of $W$ is a
continuous homomorphisms $\chi:W\rightarrow\mathbb{T}$.

######  Lemma 3.11.

For $n\in\mathbb{N}$, we let
$G_{n}=1+p^{n}\mathbb{Z}_{p}^{\times}\subseteq\mathbb{Z}_{p}$ and $\chi$ be
any character of $\mathbb{Z}_{p}^{\times}$. Then $G_{k}\subseteq\ker(\chi)$
for some $k\in\mathbb{N}$.

###### Proof.

Consider the open subset of $\mathbb{T}$ given by

$V=\\{z\in\mathbb{T}\mid\textrm{Re}(z)>0\\}.$ (46)

Observe first that the only subgroup of $V$ is $\\{1\\}$ and a fundamental
system of neighborhood of the neutral element of $\mathbb{Z}_{p}^{\times}$ is
given by the subgroups

$G_{1}\supset G_{2}\supset\dots\supset G_{n}\dots$

Consider now the open subset of $\mathbb{T}$ given in (46). Since the
character $\chi$ is continuous, there exists an integer $k\geq 1$ such that
$\chi(G_{k})\subseteq V$. Now $\chi$ is homomorphism and therefore the subset
$\chi(G_{k})$ is a subgroup of $V$, that is $\chi(G_{k})=1$. ∎

######  Lemma 3.12.

Let $m$ be an integer and $B$ a finite set of prime numbers. Then the set

$\\{(n,\dots,n)\in\prod_{p\in B}\mathbb{Z}_{p}^{\times}\mid
n\in\mathbb{Z}\,\,\textmd{and}\,\,(n,p)=1\quad\forall p\in B\\}$ (47)

is dense in $\prod_{p\in B}\mathbb{Z}_{p}^{\times}$.

###### Proof.

Any $a\in\mathbb{Z}_{p}^{\times}$ admits a $p$-adic expansion of the form

$a=\sum_{i\geq 0}c_{i}p^{i},\quad 0<c_{i}<p.$

Hence, given $\epsilon>0$ and $x=(x_{1},\dots,x_{|B|})\in\prod_{p\in
B}\mathbb{Z}_{p}^{\times}$, we choose $a_{k}\in\mathbb{Z}$ and
$e_{k}\in\mathbb{N}$ such that

$|x_{k}-a_{k}|_{p_{k}}<\frac{\epsilon}{2},\quad(a_{k},p_{k})=1,\quad
p_{k}^{-e_{k}}<\frac{\epsilon}{2}$

for $k=1,\dots,|B|$. By the Chinese remainder theorem, the congruence system

$n\equiv a_{k}\mod p_{k}^{e_{k}},\quad 1\leq k\leq|B|$

has a solution $n\in\mathbb{Z}$. The condition $(a_{k},p_{k})$ implies that
$(n,p_{k})=1$ for all $1\leq k\leq|B|$. Hence

$|n-x_{k}|_{p_{k}}\leq|n-a_{k}|_{p_{k}}+|a_{k}-x_{k}|_{p_{k}}\leq
p^{e_{k}}+\frac{\epsilon}{2}\leq\epsilon,$

as desired.

∎

Given $m\in\mathbb{N}$, a Dirichlet character modulo $m$ is a function
$\chi_{m}:\mathbb{Z}\rightarrow\mathbb{C}$ obtained by extending a character
of $(\mathbb{Z}/m\mathbb{Z})^{\times}$ to $0$ on $\mathbb{Z}/m\mathbb{Z}$ and
lifted to $\mathbb{Z}$ by composition. The corresponding Dirichlet
$L$-function is defined by

$L(s,\chi_{m})=\sum_{n=1}^{\infty}\chi(n)/n^{s},\quad s\in\mathbb{C}.$

######  Theorem 3.13.

Let $\hat{\mu}_{\beta}\in\hat{\mathcal{E}}_{\beta}$ and
$A=L^{\infty}(PGSp^{+}_{4}(\mathbb{R})\times
MSp_{4}(\mathbb{A}_{f}),\hat{\mu}_{\beta})$. We have that

$A^{PGSp_{4}^{+}(\mathbb{R})\times GSp_{4}^{+}(\mathbb{Q})}=\mathbb{C}.$ (48)

###### Proof.

It is enough to show that the action of $GSp_{4}^{+}(\mathbb{Q})$ on
$(MSp_{4}{(\mathbb{A}_{f}),\hat{\mu}_{\beta,f}})$ (where $\hat{\mu}_{\beta,f}$
is the measure on $MSp_{4}(\mathbb{A}_{f})$ obtained by projecting onto the
second factor) is ergodic. The strategy is similar to [22] and [18]. Since
every $GSp_{4}^{+}(\mathbb{Q})$-invariant subset is completely determined by
its intersection with $MSp_{4}(\hat{\mathbb{Z}})$, it is enough to show that
the closed subspace

$H=\\{f\in
L^{2}(MSp_{4}(\hat{\mathbb{Z}})),\mbox{$\,\mathrm{d}$}\hat{\mu}_{\beta,f}\mid
V_{m}f=f,\forall m\in MSp_{4}^{+}(\mathbb{Z})\\},\quad(V_{m}f)(x):=f(mx),$

consists of constant functions. Denote by $P$ the orthogonal projection onto
$H$. Since every function in $H$ is $\Gamma_{2}$-invariant, it is enough to
show that $P$ maps $\Gamma_{2}$-invariants functions to constants.

Consider the subspace

$H_{F}=\\{f\in
L^{2}(MSp_{4}(\hat{\mathbb{Z}})),\mbox{$\,\mathrm{d}$}\hat{\mu}_{\beta,f})\mid
V_{s}f=f,\forall s\in S_{F}\\},$

and denote by $P_{F}$ the orthogonal projection onto $H_{F}$. Consider the
subset

$W_{F}=\prod_{p\in F}GSp_{4}(\mathbb{Z}_{p})\times\prod_{q\notin
F}MSp_{4}(\mathbb{Z}_{q}).$

We obtain by Lemma 3.1 and Corollary 3.12 that the disjoint union
$\cup_{s\in\Gamma_{2}\backslash S_{F}/\Gamma_{2}}sW_{F}$ is a subset of full
measure. Hence given any $\Gamma_{2}$-invariant function $f\in
L^{2}(MSp_{4}(\hat{\mathbb{Z}})),\mbox{$\,\mathrm{d}$}\hat{\mu}_{\beta,f})$,
we deduce from [18, Lemma 2.9] that

$P_{F}f=\zeta_{S_{F},\Gamma_{2}}(\beta)^{-1}\sum_{s\in\Gamma_{2}\backslash
S_{F}/\Gamma_{2}}\lambda(s)^{-\beta}R_{\Gamma_{2}}(s)T_{s}f.$ (49)

We fix a finite set of primes $B$ such that $B\cap F=\emptyset$ and consider
the functions in
$L^{2}(MSp_{4}(\hat{\mathbb{Z}}),\mbox{$\,\mathrm{d}$}\mu_{\beta,f})$ of the
form

$\chi_{B}(x)=\begin{cases}\chi(\lambda((x_{p})_{p\in
B}))\quad&\text{if}\,\,\,x\in W_{B}\\\ 0\quad&\text{otherwise}\end{cases},$

where $\chi$ is a character of the compact abelian group $\prod_{p\in
B}\mathbb{Z}_{p}^{\times}$.

We will first show that $P\chi_{B}(x)$ is constant a.e. This is easy to prove
if the character $\chi$ is trivial. Indeed, in this cases
$\chi_{B}=\mathbb{1}_{B}$ and the projection formula (49) we get that
$P_{B}\chi_{B}=\zeta_{S_{B},\Gamma_{2}}(\beta)^{-1}$. Since $P=PP_{B}$ the
result follows. We now consider the case where $\chi$ is non-trivial. We first
write $\chi=\prod_{p\in B}\chi_{p}$, where $\chi_{p}$ is a character of
$\mathbb{Z}_{p}^{\times}$ given by

$\chi_{p}(a)=\chi(1,\dots,a,\dots,1).$

Then by Lemma 3.11 for each prime $p\in B$ there exists an integer
$k_{p}\in\mathbb{N}$ such that $G_{k_{p}}\subseteq\ker(\chi_{p})$. Let
$m=\prod_{p\in B}p^{k_{p}}$ and define
$\chi_{m}:\mathbb{Z}\rightarrow\mathbb{C}$ by

$\chi_{m}(n)=\begin{cases}\chi((n,\dots,n))\quad&\text{if}\,\,\,(n,m)=1\\\
0\quad&\text{otherwise}\end{cases}$

It is clear that $\chi_{m}$ is multiplicative and $\chi_{m}(n+m)=\chi_{m}(n)$
if $(n,m)\neq 1$. Suppose that $(n,m)=1$ so that
$n\in\mathbb{Z}_{p}^{\times}$. Hence

$\chi_{m}(n+m)=\prod_{p\in B}\chi_{p}(n+m)=\prod_{p\in
B}\chi_{p}(n)\chi_{p}(1+mn^{-1})=\prod_{p\in B}\chi_{p}(n)=\chi_{m}(n),$

since $p^{k_{p}}$ divides $m$. Hence $\chi_{m}$ is a Dirichlet character
modulo $m$. We claim that $\chi_{m}$ is nontrivial. Indeed, let
$a\in\prod_{p\in B}\mathbb{Z}_{p}^{\times}$ such that $\chi(a)\neq 1$. By
Lemma LABEL:label_56 the set

$\\{(n,\dots,n)\in\prod_{p\in B}\mathbb{Z}_{p}^{\times}\mid
n\in\mathbb{Z}\,\,\,\text{and}\,\,\,(n,m)=1\\}$

is dense in $\prod_{p\in B}\mathbb{Z}_{p}^{\times}$. Since $\chi$ is
continuous there exists $n_{0}\in\mathbb{Z}$ with $(n_{0},m)=1$ and
$1\neq\chi(n_{0},\dots,n_{0})=\chi_{m}(n_{0})$ as desired.

Since $F\cap B=\emptyset$, for $s\in S_{F}$ we can write

$(T_{s}\chi_{B})(x)=\begin{cases}\chi(\lambda((x_{p})_{p\in
B}))\chi_{m}(\lambda(s))\quad&\text{if}\,\,\,x\in W_{B}\\\
0\quad&\text{otherwise}.\end{cases}$

This allows us to obtain an explicit upper bound of the $L^{2}$-norm of
$P\chi_{B}$ as follows. Since $P=PP_{F}$, from the projection formula (49) we
get

$\displaystyle\norm{P\chi_{B}}$
$\displaystyle=\norm{PP_{F}\chi_{B}}\leq\norm{P_{F}\chi_{B}}\leq\zeta_{S_{F},\Gamma_{2}}(\beta)^{-1}\absolutevalue{\sum_{s\in\Gamma\
S_{F}/\Gamma}\lambda(s)^{-\beta}R_{\Gamma_{2}}(s)\chi_{m}(\lambda(s))}$
$\displaystyle=\zeta_{S_{F},\Gamma_{2}}(\beta)^{-1}\absolutevalue{\sum_{n\in\mathbb{N}(F)}n^{-\beta}R_{\Gamma_{2}}(n)\chi_{m}(n)}$
$\displaystyle=\zeta_{S_{F},\Gamma_{2}}(\beta)^{-1}\absolutevalue{\prod_{p\in
F}\sum_{l=0}^{\infty}p^{-l\beta}R_{\Gamma_{2}}(p^{l})\chi_{m}(p^{l})},$

since the function $R_{\Gamma_{2}}(n)$ is multiplicative by Lemma 2.5. As in
the proof of Proposition LABEL:label_36 we get

$\displaystyle\norm{P\chi_{B}}$
$\displaystyle\leq\zeta_{F_{S},\Gamma_{2}}(\beta)^{-1}\prod_{p\in
F}\absolutevalue{(\frac{1}{1-\chi_{m}(p)p^{-\beta}}-\frac{p+p^{2}+p^{3}}{1-\chi_{m}(p)p^{2-\beta}}+\frac{p^{2}+p^{4}}{1-\chi_{m}(p)p^{3-\beta}})}$
$\displaystyle=\prod_{p\in
F}\absolutevalue{\frac{(1+p^{1-\beta}(\chi_{m}(p)-1))-p^{2-2\beta}\chi_{m}(p))(1-p^{3-\beta})(1-p^{2-\beta})(1-p^{-\beta})}{(1-\chi_{m}(p)p^{3-\beta})(1-\chi_{m}(p)p^{2-\beta})(1-\chi_{m}(p)p^{1-\beta})(1-\chi_{m}(p)p^{-\beta})(1-p^{2-2\beta})}}$
$\displaystyle=\absolutevalue{\frac{(\sum_{n\in\mathbb{N}(F)}\frac{\chi_{m}(n)}{n^{\beta-3}})(\sum_{n\in\mathbb{N}(F)}\frac{\chi_{m}(n)}{n^{\beta-2}})(\sum_{n\in\mathbb{N}(F)}\frac{\chi_{m}(n)}{n^{\beta-1}})}{\zeta_{\mathbb{N}(F)}(\beta-3)\zeta_{\mathbb{N}(F)}(\beta-2)\zeta_{\mathbb{N}(F)}(\beta-1)}}\prod_{p\in
F}\absolutevalue{\frac{1+\chi_{m}(p)p^{1-\beta}}{1+p^{1-\beta}}}$
$\displaystyle\leq\absolutevalue{\frac{(\sum_{n\in\mathbb{N}(F)}\frac{\chi_{m}(n)}{n^{\beta-3}})(\sum_{n\in\mathbb{N}(F)}\frac{\chi_{m}(n)}{n^{\beta-2}})(\sum_{n\in\mathbb{N}(F)}\frac{\chi_{m}(n)}{n^{\beta-1}})}{\zeta_{\mathbb{N}(F)}(\beta-3)\zeta_{\mathbb{N}(F)}(\beta-2)\zeta_{\mathbb{N}(F)}(\beta-1)}}$
$\displaystyle\leq\frac{L(\beta-3,\chi_{m})L(\beta-2,\chi_{m})L(\beta-1,\chi_{m})}{\zeta_{\mathbb{N}(F)}(\beta-3)\zeta_{\mathbb{N}(F)}(\beta-2)\zeta_{\mathbb{N}(F)}(\beta-1)}.$

Since the character $\chi_{m}$ is non-trivial and $3<\beta\leq 4$, it follows
from [30, Proposition 12, Chapter VI] that the the right hand side above can
be made arbitrary small as $F\nearrow\mathcal{P}$ (with $F\cap B=\emptyset$).
This shows that $P\chi_{B}=0$, in particular $P\chi_{B}$ is again a constant
function when $\chi$ is a nontrivial character.

Consider now the functions $F_{B}\in
L^{2}(MSp_{4}(\hat{\mathbb{Z}}),\mbox{$\,\mathrm{d}$}\mu_{\beta,f})$ of the
form

$F_{B}(x)=\begin{cases}f((x_{p})_{p\in B})\quad&\text{if}\,\,\,x\in W_{B}\\\
0\quad&\text{otherwise}.\end{cases}$

where $f\in L^{2}(\prod_{p\in
B}GSp_{4}(\mathbb{Z}_{p}),\mbox{$\,\mathrm{d}$}\mu_{B})$ and
$\mu_{B}=(\pi_{B})_{*}(\mu_{\beta,f})$, where $\pi_{B}$ is the projection
$\pi_{B}:MSp_{4}(\hat{\mathbb{Z}})\rightarrow\prod_{p\in
B}MSp_{4}(\mathbb{Z}_{p})$. We first show that it is enough to assume that $f$
is $\Gamma_{2}$-invariant. Indeed, when this is not the case, we denote by $Q$
the projection onto the space of $Sp_{4}(\hat{\mathbb{Z}})$-invariant
functions. Since $\hat{\mu}_{\beta,f}$ is $Sp_{4}(\hat{\mathbb{Z}})$-invariant
we have that

$QF_{B}(x)=\int_{Sp_{4}(\hat{\mathbb{Z}})}F_{B}(gx)\mbox{$\,\mathrm{d}$}g=\begin{cases}\int_{\prod_{p\in
B}Sp_{4}(\mathbb{Z}_{p})}f(gx)\mbox{$\,\mathrm{d}$}g_{B}\quad&\text{if}\,\,\,x\in
W_{B}\\\ 0\quad&\text{otherwise},\end{cases}$

Since the Haar measure on $Sp_{4}(\hat{\mathbb{Z}})$ is left-translation-
invariant and $PQ=P$ (since $\Gamma_{2}$ is dense in
$Sp_{4}(\hat{\mathbb{Z}})$), this shows that WLOG we can assume that $f$ is
$\Gamma_{2}$-invariant. Then by the density of $\Gamma_{2}$ in $\prod_{p\in
B}Sp_{4}(\mathbb{Z}_{p})$, we see that the function $f$ depends only on
$\lambda((x_{p})_{p\in B})$, i.e

$F_{B}(x)=\begin{cases}f^{\prime}(\lambda(x_{p})_{p\in
B})\quad&\text{if}\,\,\,x\in W_{B}\\\ 0\quad&\text{otherwise},\end{cases}$

where $f^{\prime}$ is a square integrable function in $\prod_{p\in
B}\mathbb{Z}^{\times}_{p}$ (with the natural pushforward measure). The linear
span of $\chi$ ( where $\chi$ is a character of $\prod_{p\in
B}\mathbb{Z}^{\times}_{p}$) form a dense subspace of such square integrable
functions and we have shown that $P\chi_{B}$ is constant. It follows that
$PF_{B}$ is constant.

We can easily verify that the adjoint of the the operator $V_{s}$ (where $s\in
S_{B}$) is given by

$(V^{*}_{s}h)(x)=\begin{cases}\lambda(s)^{\beta}h(s^{-1}x)\quad&\text{if}\,\,\,x\in
sMSp_{4}(\hat{\mathbb{Z}})\\\ 0&\quad\text{otherwise}.\end{cases}$

Hence the map $\lambda(s)^{-\beta/2}V^{*}_{s}$ maps isometrically the
functions of the form $F_{B}$ to functions in
$L^{2}(MSp_{4}(\hat{\mathbb{Z}}),\mbox{$\,\mathrm{d}$}\mu_{\beta,f})$ of the
form

$G_{B}(x)=\begin{cases}g((x_{p})_{p\in B})\quad&\text{if}\,\,\,x\in sW_{B}\\\
0&\quad\text{otherwise},\end{cases}$ (50)

where $g\in L^{2}(s\prod_{p\in
B}GSp_{4}(\mathbb{Z}_{p}),\mbox{$\,\mathrm{d}$}\mu_{B})$. It follows then from
$P=PV_{s}^{*}$ that the projection $P$ maps every function of the form (50) to
a constant function. The set $\cup_{s\in\Gamma_{2}\backslash
S_{B}/\Gamma_{2}}sW_{B}$ has full measure in $MSp_{4}(\hat{\mathbb{Z}})$ and
thus $P$ maps functions depending only on $(x_{p})_{p\in B}$ to constants.
Finally observe that as $B\nearrow\mathcal{P}$, the union of
$L^{2}(\prod_{p\in B}MSp_{4}(\mathbb{Z}_{p}),\mbox{$\,\mathrm{d}$}\mu_{B})$
over all finite sets of primes is a dense subspace of $L^{2}(\prod_{p\in
B}MSp_{4}(\mathbb{Z}_{p}),\mbox{$\,\mathrm{d}$}\hat{\mu}_{\beta,f})$ (this
follows from weak convergence of measures). Since $P$ maps this dense subspace
to constant functions, this finishes the proof.

∎

######  Lemma 3.14.

Let $f$ be a smooth function on $\Gamma\backslash PGSp_{4}^{+}(\mathbb{R})$
with a compact support and let $\Omega$ be any compact subset of
$\Gamma\backslash PGSp_{4}^{+}(\mathbb{R})$. If we denote by
$\mbox{$\,\mathrm{d}$}\mu$ the normalized Haar measure on $\Gamma\backslash
PGSp_{4}^{+}(\mathbb{R})$, then for all $\epsilon>0$ there exist
$\kappa_{1}(\epsilon)>0$, $\kappa_{2}>0$ and $M_{2}>0$ (depending on $f$) such
that the inequality

$\frac{1}{R_{\Gamma_{2}}(m)}\sum_{a\in\Gamma_{2}\backslash
S_{m}/\Gamma_{2}}\absolutevalue{(T_{a}f)(\tau)-\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}fd\mu}\deg(a)\leq\kappa_{1}m^{(2\epsilon-1)}\prod_{i=1}^{i=l}(1+\kappa_{2}p_{i}^{-1})\frac{1}{(1-p^{-2\epsilon-1})^{2}},$

holds for all $\tau\in\Omega$ and every integer of the form
$m=\prod_{i=1}^{i=l}p_{i}^{l_{i}}$ where $\min\\{p_{1},\dots,p_{l}\\}>M_{2}$.

###### Proof.

Let $p$ be a given prime and $l\in\mathbb{N}$. The formula of
$R_{\Gamma_{2}}(p^{l})$ in the proof of Proposition LABEL:label_36 gives

$\displaystyle(R_{\Gamma_{2}}(p^{l})-p^{3l})(1-p)^{2}(1+p+p^{2})$
$\displaystyle=1-p^{3l}-p^{1+2l}-p^{2+2l}-p^{2+2l}+p^{1+3l}+p^{3+3l}$
$\displaystyle\geq 1+3p^{3l}-p^{2+2l}$ $\displaystyle\geq
p^{3l}(1-p^{2-l})\geq 0,\quad\forall l\geq 2$

A simple verification for the case $l=1$ shows that

$(R_{\Gamma_{2}}(p^{l})-p^{3l})(1-p)^{2}(1+p+p^{2})\geq 0,\quad\forall
l\in\mathbb{N},$

that is

$R_{\Gamma_{2}}(p^{l})p^{-3l}\geq 1.$ (51)

We suppose now that $m\in\mathbb{N}$ has the form
$m=\prod_{i=1}^{i=r}p_{i}^{l_{i}}$. Then by [6, Theorem 1.7 and section 4.7]
together with the calculations for $G=GSp_{2n}$, $n\geq 2$ in [6, pages 22-23]
and the degree formula in $\eqref{label 38}$ applied to $n=2$ (recall that the
big $O$ depends only on $n$), we can find $\kappa_{1}>0$, $\kappa_{2}>0$ and
$M_{2}>0$ (depending on $f$) such that

$\displaystyle\frac{1}{R_{\Gamma_{2}}(m)}\sum_{a\in\Gamma_{2}\backslash
S_{m}/\Gamma_{2}}\absolutevalue{(T_{a}f)(\tau)-\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}fd\mu}\deg(a)$
$\displaystyle\leq\frac{\kappa_{1}}{R_{\Gamma_{2}}(m)}\sum_{k_{ij}\leq
m_{ij}\leq[l_{i}/2]}\prod_{i=1}^{i=r}(p^{2l_{i}-2k_{ij}-2m_{ij}})^{\epsilon-\frac{1}{2}}p_{i}^{3l_{i}-4k_{ij}-2m_{ij}}(1+O(p_{i}^{-1}))$
$\displaystyle\leq\frac{\kappa_{1}}{R_{\Gamma_{2}}(m)p^{-3l_{i}}}\prod_{i=1}^{i=l}\sum_{k_{ij}\leq
m_{ij}\leq[l_{i}/2]}p^{l_{i}(2\epsilon-1)}p_{i}^{(-3-2\epsilon)k_{ij}+(-1-2\epsilon)m_{ij}}(1+\kappa_{2}p_{i}^{-1})$
$\displaystyle\leq\kappa_{1}m^{2\epsilon-1}\prod_{i=1}^{i=l}\sum_{k_{ij}\leq
m_{ij}\leq[l_{i}/2]}p^{-(3+2\epsilon)k_{ij}-(1+2\epsilon)m_{ij}}(1+\kappa_{2}p_{i}^{-1})$
$\displaystyle\leq\kappa_{1}m^{2\epsilon-1}\prod_{i=1}^{i=l}(1+\kappa_{2}p_{i}^{-1})\sum_{m_{ij}=0}^{\infty}\sum_{k_{ij}=0}^{\infty}p^{-(3+2\epsilon)k_{ij}-(1+2\epsilon)m_{ij}}$
$\displaystyle\leq\kappa_{1}m^{2\epsilon-1}\prod_{i=1}^{i=l}(1+\kappa_{2}p_{i}^{-1})\frac{1}{(1-p^{-1-2\epsilon})^{2}},$

holds for $\min\\{p_{1},\dots,p_{l}\\}>M_{2}$ as desired.

∎

######  Proposition 3.15.

Let $J$ be any nonempty finite set of prime numbers, $3<\beta\leq 4$, $f$ a
smooth function on $\Gamma\backslash GSp_{4}^{+}(\mathbb{R})$ with a compact
support and $\Omega$ any compact subset of $\Gamma\backslash
GSp_{4}^{+}(\mathbb{R})$. Then for any $\delta>0$, there exists a sequence
$\\{F_{n}\\}_{n\geq 1}$ consisting of finite set of prime numbers that are
disjoint from $J$ and such that for all $\tau\in\Omega$ we have

$\absolutevalue{(T_{F_{n}}f)(\tau)-\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}fd\mu}<\delta.$

###### Proof.

Given $3<\beta\leq 4$, we fix $0<\epsilon<\frac{\beta-3}{2}\leq\frac{1}{2}$
and choose $\kappa_{1},\kappa_{2}$ and $M_{2}$ as in Lemma 3.14. Let $M_{1}>0$
be such that

$x^{\beta}(1-x^{2\epsilon-1}-\kappa_{2}x^{2\epsilon-2})>\kappa_{2}\quad\forall
x>M_{1},$ (52)

and we set $M:=\max\\{M_{1},M_{2}\\}$. Let $F$ be a finite set of prime
numbers with $F\cap J=\emptyset$ and $\min\\{p\in F\\}>M$. Then by Lemma 3.14,
we have

$\displaystyle\absolutevalue{(T_{F}f)(\tau)-\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}fd\mu}$
$\displaystyle\leq\kappa_{1}\xi_{S_{F},\Gamma}(\beta)^{-1}(\sum_{m\in\mathbb{N}(F)}m^{(2\epsilon-1)}\prod_{i=1}^{i=l}(1+\kappa_{2}p_{i}^{-1})\frac{1}{(1-p^{-2\epsilon-1})^{2}}R_{\Gamma}(m)m^{-\beta})$
$\displaystyle\leq\kappa_{1}\xi_{S_{F},\Gamma}(\beta)^{-1}(\prod_{p\in
F}\sum_{l=0}^{\infty}p^{l(2\epsilon-1)}R_{\Gamma}(p^{l})p^{-l\beta}(1+\kappa_{2}p^{-1})\frac{1}{(1-p^{-2\epsilon-1})^{2}})$
$\displaystyle\leq\kappa_{1}\xi_{S_{F},\Gamma}(\beta)^{-1}(\prod_{p\in
F}(1+\kappa_{2}p^{-1})\prod_{p\in F}\frac{1}{(1-p^{-2\epsilon-1})^{2}}$
$\displaystyle\prod_{p\in
F}\frac{1+p^{2\epsilon-1}p^{\beta-1}}{1+p^{\beta-1}}\frac{\xi_{\mathbb{N}(F)}(\beta-2-2\epsilon)\xi_{\mathbb{N}(F)}(\beta-1-2\epsilon)\xi_{\mathbb{N}(F)}(\beta-2\epsilon)}{\xi_{\mathbb{N}(F)}(\beta-3)\xi_{\mathbb{N}(F)}(\beta-2)\zeta_{\mathbb{N}(F)}(\beta-1)}.$

Notice that by our choice of $M$ and $F$, Equation (52) gives

$(1+\kappa_{2}p^{-1})\frac{1+p^{2\epsilon-2+\beta}}{1+p^{\beta-1}}<1,\quad\forall
p\in F.$

Hence

$\displaystyle\absolutevalue{(T_{F}f)(\tau)-\int_{\Gamma\backslash
PGSp_{4}^{+}(\mathbb{R})}fd\mu}\leq\kappa_{1}\prod_{p\in
F}\frac{\zeta_{\mathbb{N}(F)}(\beta-2-2\epsilon)\zeta_{\mathbb{N}(F)}(\beta-1-2\epsilon)\zeta_{\mathbb{N}(F)}(\beta-2\epsilon)}{(1-p^{(-2\epsilon-1)})\zeta_{\mathbb{N}(F)}(\beta-3)\zeta_{\mathbb{N}(F)}(\beta-2)\zeta_{\mathbb{N}(F)}(\beta-1)}.$

As $F\nearrow\mathcal{P}$ with $F\cap J=\emptyset$, the right hand side can be
made arbitrary small since $3<\beta\leq 4$, $\beta-2-2\epsilon>1$ and
$\epsilon>0$, ∎

######  Theorem 3.16.

Let $\hat{\mu}_{\beta}\in\hat{\mathcal{E}}_{\beta}$ and
$A=L^{\infty}(PGSp^{+}_{4}(\mathbb{R})\times
MSp_{4}(\mathbb{A}_{f}),\hat{\mu}_{\beta})$. We have that

$A^{GSp_{4}^{+}(\mathbb{Q})\times GSp_{4}(\hat{\mathbb{Z}})}=\mathbb{C}.$ (53)

###### Proof.

Consider the space $\mathcal{H}=L^{2}(GSp_{4}^{+}(\mathbb{R})\times
MSp_{4}(\hat{\mathbb{Z}}),\mbox{$\,\mathrm{d}$}\hat{\mu}_{\beta})$. Observe
that any $GSp_{4}^{+}(\mathbb{Q})\times GSp_{4}(\hat{\mathbb{Z}})$-invariant
subset of $GSp^{+}_{4}(\mathbb{R})\times MSp_{4}(\mathbb{A}_{f})$ is
completely determined by its intersection with $GSp^{+}_{4}(\mathbb{R})\times
MSp_{4}(\mathbb{\hat{\mathbb{Z}}})$, hence it is enough to show that any
$MSp_{4}^{+}(\mathbb{Z})\times GSp_{4}(\hat{\mathbb{Z}})$-invariant function
in $\mathcal{H}$ is constant. We denote by $H$ the closed subspace of
$MSp_{4}^{+}(\mathbb{Z})\times GSp_{4}(\hat{\mathbb{Z}})$-invariant functions
in $\mathcal{H}$ and denote by $P$ the orthogonal projection onto $H$. We will
show that the image under $P$ of a dense subspace consists of constant
functions. Given any non-empty finite sets of primes $F$ and $J$, we denote by
$H_{F}$ the closed subspace of $S_{F}$-invariant functions in $\mathcal{H}$
and by $P_{F}$ the orthogonal projection onto $H_{F}$. Let $\mathcal{H}_{J}$
be the subspace of $\Gamma_{2}\times\prod_{p\in
J}GSp_{4}(\mathbb{Z}_{p})$-invariant functions depending only on
$GSp_{4}^{+}(\mathbb{R})\times\prod_{p\in J}MSp_{4}(\mathbb{Z}_{p})$.

Recall that $S_{J}Y_{J}$ is a subset of full measure, whence by [18, Lemma
2.9] given any function $f$ in $\mathcal{H}_{J}$ we get

$P_{J}f=\zeta_{S_{F},\Gamma_{2}}(\beta)\sum_{\Gamma_{2}\backslash
S_{J}/\Gamma_{2}}\lambda(s)^{-\beta}\deg_{\Gamma_{2}}(s)T_{s}f.$ (54)

It follows that the value $P_{J}f(\tau,m)$ depends only on $\tau\in
GSP_{4}^{+}(\mathbb{R})$. We can then write

$P_{J}f(x)=\begin{cases}\tilde{f}(\tau)\quad&\text{if}\,\,\,x=(\tau,m)\in
Y_{J}\\\ 0\quad&\text{otherwise}\end{cases}$

We put $\tilde{f}_{J}:=P_{J}f$. Observe that since $\tilde{f}_{J}$ is
$S_{J}$-invariant we get that $\tilde{f}$ is $\Gamma_{2}$-invariant and
therefore can view it as a square integrable function on $\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})$. We first suppose that $\tilde{f}$ is smooth with
compact support $\Omega$. For $F\cap J=\emptyset$, the projection formula (54)
gives

$P_{F}\tilde{f}_{J}(x)=\begin{cases}T_{F}\tilde{f}(\tau)\quad&\text{if}\,\,\,x=(\tau,m)\in
Y_{J}\\\ 0\quad&\text{otherwise}.\end{cases}$

We put $(T_{F}\tilde{f})_{J}:=P_{F}\tilde{f}_{J}(x)$. Given $\epsilon>0$, by
Proposition 3.15 there exists some finite set of primes $F$ disjoint from $J$
such that

$\absolutevalue{T_{F}f(\tau)-\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}\tilde{f}\mbox{$\,\mathrm{d}$}\mu}<\epsilon,\quad\forall\tau\in\Omega.$

Since $PP_{J}=PP_{F}=P$, we get

$\displaystyle\norm{Pf-P\mathbb{1}_{Y_{J}}\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}\tilde{f}\mbox{$\,\mathrm{d}$}\mu}_{2}$
$\displaystyle=\norm{PP_{J}f-P\mathbb{1}_{Y_{J}}\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}\tilde{f}\mbox{$\,\mathrm{d}$}\mu}_{2}$
$\displaystyle=\norm{PP_{F}P_{J}f-P\mathbb{1}_{Y_{J}}\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}\tilde{f}\mbox{$\,\mathrm{d}$}\mu}_{2}$
$\displaystyle\leq\norm{P_{F}P_{J}f-\mathbb{1}_{Y_{J}}\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}\tilde{f}\mbox{$\,\mathrm{d}$}\mu}_{2}$
$\displaystyle\leq\norm{(T_{F}\tilde{f})_{J}-\mathbb{1}_{Y_{J}}\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}\tilde{f}\mbox{$\,\mathrm{d}$}\mu}+2<\epsilon.$

Hence using the projection formula 54 with $f=\mathbb{1}_{Y_{J}}$ we get

$Pf=P\mathbb{1}_{Y_{J}}\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}\tilde{f}\mbox{$\,\mathrm{d}$}\mu=PP_{J}1_{Y_{J}}\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}\tilde{f}\mbox{$\,\mathrm{d}$}\mu=\zeta_{S_{J},\Gamma_{2}}(\beta)^{-1}\int_{\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})}\tilde{f}\mbox{$\,\mathrm{d}$}\mu,$

which is constant. Any integrable functions on $\Gamma_{2}\backslash
PGSp_{4}^{+}(\mathbb{R})$ can be approximated by a compactly supported smooth
function and hence $Pf$ is constant for all $\Gamma_{2}$-invariant functions
in $\mathcal{H}_{J}$. The results follows since the union of $\mathcal{H}_{J}$
over all finite set of primes is dense in the space of square integrable
$\Gamma_{2}\times GSp_{4}(\mathbb{\hat{\mathbb{Z}}})$-invariant functions.

∎

######  Theorem 3.17.

For $3<\beta\leq 4$, The $GSp_{4}$-system admits a unique
$\textmd{KMS}_{\beta}$ state.

###### Proof.

We will show that the set $\mathcal{E}_{\beta}$ consists of a single point. We
first use [18, Propositon 4.6] together with Theorem 3.13 and Theorem 3.16 to
conclude that $A^{GSp_{4}^{+}(\mathbb{Q})}=\mathbb{C}$ where
$A=L^{\infty}(PGSp_{4}^{+}(\mathbb{R})\times
MSp_{4}(\mathbb{A}_{f,\mathbb{Q}}),\hat{\mu}_{\beta})$ for any
$\hat{\mu}_{\beta}\in\hat{\mathcal{E}}_{\beta}$ and $3<\beta\leq 4$, in other
words there exists a unique right $GSp_{4}(\hat{\mathbb{Z}})$-invariant
measure $\mbox{$\,\mathrm{d}$}\hat{\mu}_{\beta}$ in $\mathcal{E}_{\beta}$.
Suppose now that $\upsilon_{\beta}$ is any other point of
$\mathcal{E}_{\beta}$. Then the measure defined by

$\omega=\int_{GSp_{4}(\hat{\mathbb{Z}})}g\cdot\upsilon_{\beta}\mbox{$\,\mathrm{d}$}g$

is an element of $\mathcal{E}_{\beta}$ and by unicity we get that
$\omega=\hat{\mu}_{\beta,f}$. Since the point $\hat{\mu}_{\beta,f}$ is
extremal we conclude that $\hat{\mu}_{\beta,f}=\upsilon_{\beta}$. This
completes the proof. ∎

######  Remark 1.

We have studied the $GSp_{4}$-system in the region $\beta>0$ with
$\beta\notin\\{1,2,3\\}$. Let us now consider the cases where the inverse
temperature is a pole of the Dirichlet series (LABEL:label_94). If $\beta=2$,
it is possible to construct explicit measures $\mu_{2}\in\mathcal{E}_{2}$. We
consider the normalized Haar measure on $\mathbb{A}_{f,\mathbb{Q}}$ such that
$\textmd{meas}(\hat{\mathbb{Z}})=1$ and

$\textmd{meas}(aE)=\prod_{p}|a_{p}|_{p}\,\textmd{meas}(E).$ (55)

for any $a\in\mathbb{A}_{f,\mathbb{Q}}^{\times}$ and measurable subset
$E\subseteq\mathbb{A}_{f,\mathbb{Q}}$. Let $\mu_{f}$ be the product measure on
$\mathbb{A}_{f,\mathbb{Q}}^{2}$. Since

$\matrixquantity(0&0&0&0&x_{1}\\\ 0&0&0&0&x_{2}\\\ 0&0&0&0&x_{3}\\\
0&0&0&0&x_{4}\\\ )\in MSp_{4}(\mathbb{A}_{f,\mathbb{Q}}),\quad
x_{1},\dots,x_{4}\in A_{f,\mathbb{Q}}$

we may consider $\mu_{f}$ as a measure on $MSp_{4}(A_{f,\mathbb{Q}})$ such
that $\mu_{f}(MSp_{4}(\hat{\mathbb{Z}}))=1$. We claim that
$\mu_{2}=\mu_{\infty}\times\mu_{f}\in\mathcal{E}_{2}$. By construction it is
enough to show that $\mu_{2}$ satisfies the scaling condition (35). Given
$g\in GSp_{4}^{+}(\mathbb{Q})$, we can find
$\gamma_{1},\gamma_{2}\in\Gamma_{2}$ and a diagonal matrix $D\in
GSp_{4}^{+}(\mathbb{Q})$ such that $g=\gamma_{1}D\gamma_{2}$. Since
$\gamma\hat{\mathbb{Z}}^{4}=\hat{\mathbb{Z}}^{4}$ for any
$\gamma\in\Gamma_{2}$ and the Haar measure is translation invariant we
conclude that $\mu_{1}(gB)=\lambda^{-2}\mu(B)$ for any Borel subset of
$MSp_{4}(\mathbb{A}_{f,\mathbb{Q}})$.

One would expect to use a similar construction for $\beta=1$ and $\beta=3$.
However, since the only subspace of $\mathbb{A}_{f,\mathbb{Q}}^{4}$ stable
under the action of $GSp_{4}^{+}(\mathbb{Q})$ is
$\mathbb{A}_{f,\mathbb{Q}}^{4}$ itself, this argument fails in the case
$\beta=1$ or $\beta=3$. We conjecture that the $GSp_{4}$-system does not admit
any $\textmd{KMS}_{\beta}$ state in these two cases.

######  Remark 2.

The results we prove in this paper completely classify the
$\textmd{KMS}_{\beta}$ states on the Bost-Connes-Marcolli $GSp_{4}$-system. In
fact, we will show that given $\beta>0$ with $\beta\notin\\{1,2,3\\}$, there
exists a one-to-one correspondence between $\textmd{KMS}_{\beta}$ on the
Connes-Marcolli $GSp_{4}$-system and the $GSp_{4}$-system
$(\mathcal{A},\sigma_{t})$. Recall the set

$F_{Y}=\\{h\in
MSp_{4}(\hat{\mathbb{Z}})\mid\textmd{rank}_{\mathbb{Q}_{p}}(h_{p})\leq
2\,\,\,\text{for all}\,\,\,p\in\mathcal{P}\\},$

and consider the dynamical system $I=C^{*}_{r}(\Gamma_{2}\backslash
GSp_{4}^{+}(\mathbb{Q})\boxtimes_{\Gamma_{2}}(\mathbb{H}_{2}^{+}\times
F_{Y}))$. We claim that $I$ can not have any $\textmd{KMS}_{\beta}$ states. To
see this, observe that any $KMS_{\beta}$ state on $I$ gives rise to a regular
$\Gamma_{2}$-invariant measure $\mu_{\beta}$ on the space
$\mathbb{H}_{2}^{+}\times F_{Y}$ (note that unlike the case where the
underlying space is an $r$-discreet principal groupoid, the support of this
measure is not necessarily contained in $\mathbb{H}_{2}^{+}\times F_{Y}$.) By
the $\textmd{KMS}_{\beta}$ condition, this measure still satisfies the scaling
property 2.12. Now since $\mathbb{H}_{2}^{+}=(\mathbb{U}^{2}/\\{\pm
1_{4}\\})\backslash PGSp_{4}^{+}(\mathbb{R})$, we can define a measure on
$PGSp_{4}^{+}(\mathbb{R})\times F_{Y}$ by the formula

$\int_{PGSp_{4}^{+}(\mathbb{R})\times
F_{Y}}f(x)d\tilde{\mu}_{\beta}(x)=\int_{\mathbb{H}_{2}^{+}\times
F_{Y}}\Big{(}\int_{\mathbb{U}^{2}/\\{\pm
1_{4}\\}}f(xg)\mbox{$\,\mathrm{d}$}g\Big{)}\mbox{$\,\mathrm{d}$}\mu_{\beta}(x).$

The measure $\tilde{\mu}_{\beta}$ satisfies the condition 2.12. There is a
canonical extension of this measure to a $\Gamma_{2}$-invariant measure
$\mu_{\beta}\in\mathcal{E}_{\beta}$ on the space
$PGp_{4}^{+}(\mathbb{R})\times MSp_{4}(\mathbb{A}_{f})$. This leads to a
contradiction since the set $PGSp_{4}^{+}(\mathbb{R})\times F_{Y}$ has measure
zero by Corollary LABEL:label_56. This shows that the set
$\mathbb{H}_{2}^{+}\times F_{Y}$ can be ignored in the the analysis of
$\textmd{KMS}_{\beta}$ states for $\beta>0$ and $\beta\notin\\{1,2,3\\}$ and
if we let $\tilde{Y}=Y\backslash F_{Y}$, it is clear that different
$\textmd{KMS}_{\beta}$ state on $C_{r}^{*}(\Gamma_{2}\backslash
GSp_{4}^{+}(\mathbb{Q})\boxtimes_{\Gamma_{2}}\tilde{Y})$ give rise to
different $\textmd{KMS}_{\beta}$ state on the $GSp_{4}$-system
$(\mathcal{A},\sigma_{t})$.

## References

  * [1] Jean-Benoit Bost and Alain Connes.  Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory.  Selecta Mathematica, 1(3):411–457, 1995.
  * [2] O. Bratteli and D.W. Robinson.  Operator algebras and quantum statistical mechanics. Vol. 1.  Springer-Verlag, New York, 1979.
  * [3] O. Bratteli and D.W. Robinson.  Operator algebras and quantum statistical mechanics. Vol. 2.  Springer-Verlag, New York., 1981.
  * [4] Siegel C.F.  Symplectic Geometry.  Academic Press, New York, 1964.
  * [5] Johannes Christensen.  The structure of KMS weights on étale groupoid $C^{*}$-algebras.  arXiv:2005.01792, 2020.
  * [6] Laurent Clozel, Hee Oh, and Emmanuel Ullmo.  Hecke operators and equidistribution of Hecke points.  Inventiones mathematicae, 144(2):327–380, 2001.
  * [7] F Combes.  Poids associé à une algèbre hilbertienne à gauche.  Compositio Mathematica, 23(1):49–77, 1971.
  * [8] Alain Connes and Matilde Marcolli.  From physics to number theory via noncommutative geometry. Part I: Quantum statistical mechanics of Q-lattices.  arXiv preprint math/0404128, 2004.
  * [9] Alain Connes, Matilde Marcolli, and Niranjan Ramachandran.  KMS states and complex multiplication.  Selecta Mathematica, 11(3):325–347, 2006.
  * [10] Benedict H Gross.  On the Satake isomorphism.  London Mathematical Society Lecture Note Series, pages 223–238, 1998.
  * [11] E Ha, F Paugam, and A Bost.  Connes-Marcolli system for Shimura varieties.  IMRP, 5:237–286, 2005.
  * [12] Eugene Ha and Frédéric Paugam.  Bost-Connes-Marcolli systems for Shimura varieties. Part I. Definitions and formal analytic properties.  International mathematics research papers, 2005(5):237–286, 2005\.
  * [13] Rudolf Haag, Nicolaas Marinus Hugenholtz, and Marinus Winnink.  On the equilibrium states in quantum statistical mechanics.  Communications in Mathematical Physics, 5(3):215–236, 1967.
  * [14] B Julia.  Statistical theory of numbers.  In Number theory and physics, pages 276–293. Springer, 1990.
  * [15] Martin Kneser.  Starke Approximation in algebraischen Gruppen. I.  Journal für die reine und angewandte Mathematik, 1965.
  * [16] Aloys Krieg.  Hecke algebras, volume 87.  American Mathematical Soc., 1990.
  * [17] Johan Kustermans.  KMS-weights on C*-algebras.  arXiv preprint funct-an/9704008, 1997.
  * [18] Marcelo Laca, Nadia S Larsen, and Sergey Neshveyev.  Phase transition in the Connes–Marcolli GL 2-system.  Journal of Noncommutative Geometry, 1(4):397–430, 2007.
  * [19] Gregori A Margulis.  Discrete subgroups of semisimple Lie groups, volume 17.  Springer Science & Business Media, 1991.
  * [20] Gregory A Margulis.  Cobounded subgroups of algebraic groups over local fields.  Functional Analysis and Its Applications, 11(2):119–128, 1977.
  * [21] Yukihiko Namikawa.  Toroidal compactification of Siegel spaces, volume 812.  Springer, 2006.
  * [22] Sergey Neshveyev.  Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem.  Proceedings of the American Mathematical Society, 130(10):2999–3003, 2002.
  * [23] M. Newman and J.R. Smart.  Symplectic Modulary Groups.  Acta Arithmetica, 9:83–89, 1964.
  * [24] Onorato Timothy O’Meara.  Symplectic groups, volume 16.  American Mathematical Soc., 1978.
  * [25] Ameya Pitale.  Siegel Modular Forms.  Lecture Notes in Mathematics, 2240, 2019.
  * [26] Vladimir P Platonov.  The problem of strong approximation and the Kneser-Tits conjecture for algebraic groups.  Mathematics of the USSR-Izvestiya, 3(6):1139, 1969.
  * [27] Gopal Prasad.  Strong approximation for semi-simple groups over function fields.  Annals of Mathematics, 105(3):553–572, 1977.
  * [28] Jean Renault.  A groupoid approach to C*-algebras, volume 793.  Springer, 2006.
  * [29] Haag Rudolf.  Local Quantum Physics.  Springer, Berlin, 1992.
  * [30] Jean-Pierre Serre.  Cours d’arithmétique, volume 2.  Presses universitaires de France, 1970.
  * [31] Marko Tadić.  Representations of $p$-adic symplectic groups.  Compositio Mathematica, 90(2):123–181, 1994.