# On the geometric trace of a generalized Selberg trace formula

András Biró  Alfréd Rényi Institute of Mathematics,
Budapest, Hungary
<EMAIL_ADDRESS>and  Dávid Tóth  Alfréd Rényi Institute of Mathematics,
Budapest, Hungary
and
Department of Computer Science and Information Theory
Budapest University of Technology and Economics<EMAIL_ADDRESS>
###### Abstract.

A certain generalization of the Selberg trace formula was proved by the first
named author in 1999. In this generalization instead of considering the
integral of $K(z,z)$ (where $K(z,w)$ is an automorphic kernel function) over
the fundamental domain, one considers the integral of $K(z,z)u(z)$, where
$u(z)$ is a fixed automorphic eigenfunction of the Laplace operator. This
formula was proved for discrete subgroups of $PSL(2,\mathbb{R})$, and just as
in the case of the classical Selberg trace formula it was obtained by
evaluating in two different ways ("geometrically" and "spectrally") the
integral of $K(z,z)u(z)$.

In the present paper we work out the geometric side of a further
generalization of this generalized trace formula: we consider the case of
discrete subgroups of $PSL(2,\mathbb{R})^{n}$ where $n>1$. Many new
difficulties arise in the case of these groups due to the fact that the
classification of conjugacy classes is much more complicated for $n>1$ than in
the case $n=1$.

The research reported in this paper is supported by the “TKP2020, National
Challenges Program” of the National Research, Development and Innovation
Office (BME NC TKP2020) and by the Ministry of Innovation and Technology and
the National Research, Development and Innovation Office within the Artificial
Intelligence National Laboratory of Hungary. It is also supported by the
MTA–RI Lendület ”Momentum” Analytic Number Theory and Representation Theory
Research Group, by the NKFIH (National Research, Development and Innovation
Office) grants FK 135218 (D.T.), K135885 and K 143876 (A.B.) and by the Rényi
Intézet Lendület Automorphic Research Group.

##  1\. Introduction

###  1.1. The Selberg trace formula and its generalizations

The Selberg trace formula (introduced by A. Selberg, see [10]) is a
particularly important tool in the theory of automorphic functions, it has
many applications in different branches of mathematics. Briefly speaking, it
is obtained by computing the integral

$\textrm{Tr}K=\int_{F}K(z,z)\,d\mu(z)$

in two different ways ("geometrically" and "spectrally"), where
$d\mu(z)=y^{-2}\,dx\,dy$ is the usual measure on the complex upper half-plane
$\mathbb{H}$, $F\subset\mathbb{H}$ is the fundamental domain of a finite
volume Fuchsian group $\Gamma\leq PSL(2,\mathbb{R})$ acting on $\mathbb{H}$,
and $K(z,w)$ is an appropriate automorphic kernel function (i.e. a function
that is invariant under the action of $\Gamma$).

A. Biró obtained a generalization of this formula in [1] by evaluating the
integral

(1) $\displaystyle\textrm{Tr}K=\int_{F}K(z,z)u(z)\,d\mu(z),$

where the weight $u(z)$ is a Maass form (i.e. an automorphic eigenfunction of
the Laplace operator). Biró applied his generalization to the the hyperbolic
circle problem (see [4]), and some ideas of [1] are applied for cycle integral
and triple product identities in [2] and [3].

Selberg’s trace formula was developed for a general family of groups, and our
aim is to work out the geometric side of Biró’s generalized formula for
discrete subgroups of $PSL(2,\mathbb{R})^{n}$ where $n>1$. For these, the
details of the Selberg trace formula are given in [6].

###  1.2. Discrete subgroups of $PSL(2,\mathbb{R})^{n}$

The main example of the discrete subgroups of $PSL(2,\mathbb{R})^{n}$ is the
Hilbert modular group. For any totally real finite extension $\mathbb{Q}\leq
K$ of degree $n>1$ it is defined as

$\Gamma_{K}:=\left\\{\left(\left[\begin{array}[]{cc}a^{(1)}&b^{(1)}\\\
c^{(1)}&d^{(1)}\end{array}\right],\dots,\left[\begin{array}[]{cc}a^{(n)}&b^{(n)}\\\
c^{(n)}&d^{(n)}\end{array}\right]\right):\,\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]\in PSL(2,\mathcal{O}_{K})\right\\},$

where $K^{(1)},\dots,K^{(n)}$ are the different embeddings of $K$ into
$\mathbb{R}$, and the images of an element $a\in K$ by these embeddings are
$a^{(1)},\dots,a^{(n)}$. As usually, $\mathcal{O}_{K}$ denotes the ring of
integers in $K$.

The generalized trace formula (derived from (1)) is fully worked out in the
PhD thesis [13] in the special case when $\Gamma$ is a Hilbert modular group
for a _quadratic field_ _of class number one_. Though a more general situation
is handled here, the Hilbert modular groups still play an important role in
the following. Before specifying this role, we give a short summary of the
main results about discrete subgroups of $PSL(2,\mathbb{R})^{n}$. Many of
these are proved in [7].

####  1.2.1. Action on the product of upper half-planes.

The group $PSL(2,\mathbb{R})$ acts on $\mathbb{H}$ in the usual way, if
$\gamma=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\in
PSL(2,\mathbb{R})$ and $z\in\mathbb{H}$, then

(2) $\displaystyle\gamma z=\frac{az+b}{cz+d}.$

This induces a coordinate-wise action of $PSL(2,\mathbb{R})^{n}$ on the
product space $\mathbb{H}^{n}$. For an element $z\in\mathbb{H}^{n}$ we will
use the notation $z=(z_{1},\dots,z_{n})$ where
$z_{k}=x_{k}+iy_{k}\in\mathbb{H}$ with $x_{k}\in\mathbb{R}$ and
$y_{k}\in\mathbb{R}^{+}$ ($k=1,\dots,n$). That is, if
$\gamma=(\gamma^{(1)},\dots,\gamma^{(n)})\in PSL(2,\mathbb{R})^{n}$ and
$z\in\mathbb{H}^{n}$, then $\gamma
z=(\gamma^{(1)}z_{1},\dots,\gamma^{(n)}z_{n})$. It is convenient to represent
an element $\gamma\in PSL(2,\mathbb{R})^{n}$ as in the one dimensional case,
i.e. as a matrix $\gamma=\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]$, whose elements are $n$ dimensional row vectors with
real coordinates, e.g. $a=(a^{(1)},\dots,a^{(n)})$. Then the action of
$PSL(2,\mathbb{R})^{n}$ on $\mathbb{H}^{n}$ can be written formally as in (2),
where the operations are meant to be performed coordinate-wise. It is known
that $\Gamma\leq PSL(2,\mathbb{R})^{n}$ is discrete if and only if it acts
discontinuously on $\mathbb{H}^{n}$ (see Proposition 2.1 in [7]).

####  1.2.2. Irreducible groups

The groups $\Gamma,\Gamma^{\prime}\leq PSL(2,\mathbb{R})^{n}$ are said to be
_strictly commensurable_ if $\Gamma\cap\Gamma^{\prime}$ has finite index in
both $\Gamma$ and $\Gamma^{\prime}$. They are said to be _commensurable_ if
$\Gamma$ is strictly commensurable with a conjugate of $\Gamma^{\prime}$. A
discrete subgroup $\Gamma\leq PSL(2,\mathbb{R})^{n}$ is said to be
_irreducible_ if $\Gamma$ is not commensurable with any direct product
$\Gamma^{\prime}\times\Gamma^{\prime\prime}$, where $\Gamma^{\prime}$ and
$\Gamma^{\prime\prime}$ are discrete subgroups of some non-trivial groups
$G^{\prime}$ and $G^{\prime\prime}$, respectively, for which
$PSL(2,\mathbb{R})^{n}=G^{\prime}\times G^{\prime\prime}$ holds. For a
discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R})^{n}$ we have the following
equivalent conditions for irreducibility (see the Corollary after Theorem 2 in
[11]):

  1. (i)

$\Gamma$ contains no element $\gamma=(\gamma^{(1)},\dots,\gamma^{(n)})$ such
that $\gamma^{(i)}=1$ holds for some $i$ while $\gamma^{(j)}\neq 1$ holds for
some $j$,

  2. (ii)

there exist no partial product $G^{\prime}$ of $PSL(2,\mathbb{R})^{n}$ such
that the projection of $\Gamma$ to $G^{\prime}$ is discrete,

  3. (iii)

for every $\gamma\in\Gamma\setminus\\{1\\}$ the centralizer of $\gamma$ in
$PSL(2,\mathbb{R})^{n}$ is commutative.

In the following $\Gamma$ always denotes an irreducible subgroup. Note that
property (i) above implies that the Hilbert modular group is irreducible.

####  1.2.3. Cusps.

The group $PSL(2,\mathbb{R})^{n}$ and hence its subgroup $\Gamma$ act on the
set $(\mathbb{R}\cup\\{\infty\\})^{n}$. This action is also given by (2)
(using the usual extended operations on the set $\mathbb{R}\cup\\{\infty\\}$).
Roughly speaking, an element of $(\mathbb{R}\cup\\{\infty\\})^{n}$ is a _cusp_
if its stabilizer in $\Gamma$ is in some sense as large as possible.

First, let us consider the element $\infty=(\infty,\dots,\infty)$. The
stabilizer $\Gamma_{\infty}$ of $\infty$ contains elements of the form

$\gamma=\left[\begin{array}[]{cc}a&b\\\ 0&d\end{array}\right],$

and among them there are those for which all the coordinates of the vectors
$a$ and $d$ are $1$, i.e. the translations. Let us define

$\mathbf{t}_{\infty}=\mathbf{t}_{\infty}(\Gamma)=\left\\{b\in\mathbb{R}^{n}:\left[\begin{array}[]{cc}1&b\\\
0&1\end{array}\right]\in\Gamma\right\\},$

then, since $\Gamma$ is discrete, $\mathbf{t}_{\infty}$ is a discrete subgroup
of $\mathbb{R}^{n}$, and hence it is isomorphic to $\mathbb{Z}^{m}$ for some
$1\leq m\leq n$. The important cases are those, for which $m=n$ holds (i.e.
$\mathbf{t}_{\infty}$ is a lattice).

For a general element $\gamma\in\Gamma_{\infty}$ we have
$d^{(k)}=(a^{(k)})^{-1}$ ($1\leq k\leq n$), and therefore $\gamma z=a^{2}z+ab$
holds for any $z\in\mathbb{H}^{n}$ (again, the operations are accomplished
coordinate-wise). Notice that the the coordinates of the vector $a^{2}$ are
all positive. The vectors with this property are called _totally positive_ and
they form the group $(\mathbb{R}^{+})^{n}$ w.r.t. the coordinate-wise
multiplication. An element $\varepsilon\in(\mathbb{R}^{+})^{n}$ is called a
_multiplier_ for $\Gamma$ if there is a $\gamma\in\Gamma$ for which $\gamma
z=\varepsilon z+b$ holds for some $b\in\mathbb{R}^{n}$. The multipliers form a
subgroup of $(\mathbb{R}^{+})^{n}$, it is denoted by
$\Lambda_{\infty}=\Lambda_{\infty}(\Gamma)$.

If $\mathbf{t}_{\infty}$ is a lattice, then $\Lambda_{\infty}$ is a discrete
subgroup of $(\mathbb{R}^{+})^{n}$ and for each
$\varepsilon\in\Lambda_{\infty}$ we have
$\varepsilon^{(1)}\cdots\varepsilon^{(n)}=1$ (see Remark 2.3 in [7]). Taking
the logarithm coordinate-wise, we obtain that $\log\Lambda_{\infty}$ is a
discrete subgroup of $\mathbb{R}^{n}$ contained in the hyperplane

(3) $\displaystyle V=\\{a\in\mathbb{R}^{n}:\,a^{(1)}+\dots+a^{(n)}=0\\},$

and hence cannot be a lattice. That is, if
$\mathbf{t}_{\infty}\cong\mathbb{Z}^{n}$, then
$\Lambda_{\infty}\cong\mathbb{Z}^{m}$ for some $0\leq m\leq n-1$. We say that
$\infty$ is a _cusp_ for $\Gamma$ if $\mathbf{t}_{\infty}\cong\mathbb{Z}^{n}$
and $\Lambda_{\infty}\cong\mathbb{Z}^{n-1}$.

Note that $\infty$ is a cusp for the Hilbert modular group. Indeed, in this
case the $n$ different embeddings of $K$ induce a map from $\mathcal{O}_{K}$
to $\mathbb{R}^{n}$ whose image is well-known to be a lattice that
$\mathbf{t}_{\infty}$ is isomorphic to. Also, $\Lambda_{\infty}$ consists of
the squares of the units of $\mathcal{O}_{K}$. By Dirichlet’s unit theorem we
have that $\mathcal{O}_{K}^{\times}\cong\mathbb{Z}^{n-1}\times\\{\pm 1\\}$,
hence $\Lambda_{\infty}\cong\mathbb{Z}^{n-1}$ and our claim follow.

We say that an element $\kappa\in(\mathbb{R}\cup\\{\infty\\})^{n}$ is a _cusp_
for the discrete group $\Gamma\leq PSL(2,\mathbb{R})^{n}$ if $\infty$ is a
cusp for the group $\sigma^{-1}\Gamma\sigma$ for some $\sigma\in
PSL(2,\mathbb{R})^{n}$ with $\sigma\infty=\kappa$. It is not hard to see that
in this case $\infty$ is a cusp for $\rho^{-1}\Gamma\rho$ for every element
$\rho\in PSL(2,\mathbb{R})^{n}$ with $\sigma\infty=\rho\infty=\kappa$.
Moreover, we have in fact
$\Lambda_{\infty}(\sigma^{-1}\Gamma\sigma)=\Lambda_{\infty}(\rho^{-1}\Gamma\rho)=:\Lambda_{\kappa}$,
hence this group is determined uniquely by the cusp $\kappa$ and it is called
the _multiplier group for $\kappa$_. Note that
$\mathbf{t}_{\infty}(\sigma^{-1}\Gamma\sigma)$ is determined only up to a
coordinate-wise scalar multiple. Having this in mind, we will use the notation
$\mathbf{t}_{\kappa}$ for such a lattice.

####  1.2.4. Fundamental domain

Let $X$ be a locally compact second-countable topological space, and assume
that $\Gamma$ is a group of topological (i.e. bijective, continuous and open)
maps of $X$ onto itself. The set $F\subset X$ is called a _fundamental set_
for $\Gamma$ if

(4) $\displaystyle X=\bigcup_{\gamma\in\Gamma}\gamma(F).$

For example the space $X$ itself is always a fundamental set. Of course we are
interested in fundamental sets that are in some sense as small as possible.

Also, we would like to consider measurable sets, so assume further that a
$\Gamma$-invariant Radon measure $\mu$ is given on $X$. Then a measurable set
$F\subset X$ is called a _fundamental domain_ for $\Gamma$ if (4) holds, and
there is a set $S\subset F$ such that $\mu(S)=0$ and different points of
$F\setminus S$ are not on the same $\Gamma$-orbit.

Every measurable fundamental set contains a fundamental domain (see Appendix
II in [7]), hence there exists a fundamental domain $F\subset\mathbb{H}^{n}$
for every irreducible discrete subgroup $\Gamma\leq PSL(2,\mathbb{R})^{n}$. In
the following we always assume that the volume of $F$ is finite. This holds of
course when the quotient space $\mathbb{H}^{n}\setminus\Gamma$ is compact.
However, this quotient is not compact once there are cusps for $\Gamma$ since
in that case $F$ contains parts that stretch out to the boundary of
$\mathbb{H}^{n}$. Now we are in the position to state the following important
theorem (see Theorem I.1.5 in [6]):

######  Theorem 1.1.

Assume that $n\geq 2$ and $\Gamma\leq PSL(2,\mathbb{R})^{n}$ is an irreducible
discrete subgroup whose fundamental domain has finite volume but the quotient
space $\mathbb{H}^{n}\setminus\Gamma$ is not compact. Then $\Gamma$ is
commensurable with a Hilbert modular group for a number field $K$ of degree
$n$.

In addition, the field $K$ in the theorem above is generated by the elements
of $\Lambda_{\kappa}$ for any cusp $\kappa$ (see the proof of Theorem 4 in
[11]). Note that if $\Gamma$ is commensurable with the Hilbert modular group,
then its fundamental domain is of finite volume.

From now on, $\Gamma$ always denotes a discrete irreducible subgroup of
$PSL(2,\mathbb{R})^{n}$ with a fundamental domain of finite volume, and we
will assume also that $\Gamma$ has at least one cusp. The group $\Gamma$ acts
on the set of its cusps. It is well known that the number of the equivalence
classes (orbits) of the cusps for a Hilbert modular group is the class number
of the corresponding field $K$ (see [12], Proposition 20 on p. 188). It
follows from this and from the theorem above, that the number of the
equivalence classes of the cusps for $\Gamma$ is finite.

To describe the fundamental domain of $\Gamma$ we first define the cusp
regions. We set

$U_{C}:=\\{z\in\mathbb{H}^{n}:\,Ny>C\\},$

where $C>0$ is a positive real number and $Na=\prod_{k=1}^{n}a^{(k)}$ for any
real vector $a=(a^{(1)},\dots,a^{(n)})^{T}$. If $\kappa$ is a cusp for
$\Gamma$ and $\sigma\in PSL(2,\mathbb{R})^{n}$ is a fixed element for which
$\sigma\infty=\kappa$ holds, then the sets of the form $\sigma(U_{C})$ are
called the neighbourhoods of $\kappa$. Note that the set of these
neighbourhoods does not depend on the choice of $\sigma$.

The stabilizer $\Gamma_{\kappa}$ of $\kappa$ acts on the sets of the form
$U_{\kappa}=\sigma(U_{C_{\kappa}})$ ($C_{\kappa}\in\mathbb{R}^{+}$,
$\sigma\infty=\kappa$). To construct a fundamental domain for
$\Gamma_{\kappa}$ in $U_{\kappa}$, it is sufficient to give a fundamental
domain $F^{\prime}_{\kappa}$ for
$\sigma^{-1}\Gamma_{\kappa}\sigma=(\sigma^{-1}\Gamma\sigma)_{\infty}$ in
$U_{C_{\kappa}}$, and then the set $F_{\kappa}:=\sigma(F_{\kappa}^{\prime})$
is a fundamental domain for $\Gamma_{\kappa}$ in $U_{\kappa}$. Recall that
$\mathbf{t_{\kappa}}$ is a lattice in $\mathbb{R}^{n}$ and that
$\log\Lambda_{\kappa}$ has rank $n-1$ and it is contained in the hyperplane
$V$ defined in (3). Let $P_{\kappa}$ be a fundamental parallelotope for
$\mathbf{t_{\kappa}}$ in $\mathbb{R}^{n}$ and let $Q_{\kappa}$ be a
fundamental parallelotope for $\log\Lambda_{\kappa}$ in the vector space $V$.
It is easy to see that

$F^{\prime}_{\kappa}=\left\\{z=x+iy\in U_{C_{\kappa}}:x\in
P_{\kappa},\,\log\frac{y}{(Ny)^{1/n}}\in Q_{\kappa}\right\\}$

is a fundamental domain for $\sigma^{-1}\Gamma_{\kappa}\sigma$ in
$U_{C_{\kappa}}$.

To express $F_{\kappa}$ in a simple way, we introduce the coordinates at the
cusp $\kappa$. For this we fix a _scaling element_ $\sigma_{\kappa}\in
PSL(2,\mathbb{R})^{n}$ such that $\sigma_{\kappa}\infty=\kappa$, a basis
$b_{1}^{\kappa},\dots,b_{n}^{\kappa}$ in $\mathbb{R}^{n}$ such that

$P_{\kappa}=\left\\{b\in\mathbb{R}^{n}:b=\sum_{k=1}^{n}t_{k}b_{k}^{\kappa},\,0\leq
t_{k}<1\textrm{ for }1\leq k\leq n\right\\}$

is a fundamental parallelotope for $\mathbf{t}_{\kappa}$, and a basis
$a_{1}^{\kappa},\dots,a_{n-1}^{\kappa}$ in $V$ such that

$Q_{\kappa}=\left\\{a\in V:a=\sum_{l=1}^{n-1}s_{l}a_{l}^{\kappa},\,0\leq
s_{l}<1\textrm{ for }1\leq l\leq n-1\right\\}$

is a fundamental parallelotope for $\log\Lambda_{\kappa}$. Note that
$a_{l}^{\kappa}=\log\varepsilon_{l}^{\kappa}$, where
$\varepsilon_{1}^{\kappa},\dots,\varepsilon_{n-1}^{\kappa}$ generate the group
$\Lambda_{\kappa}$. Then for any $z\in\mathbb{H}^{n}$ we write
$\sigma_{\kappa}^{-1}z=x^{\prime}+iy^{\prime}$ and

$\begin{array}[]{c}x^{\prime}=X_{1}^{\kappa}(z)b_{1}^{\kappa}+\dots+X_{n}^{\kappa}(z)b_{n}^{\kappa},\\\
Y_{0}^{\kappa}(z)=Ny^{\prime},\quad\log\dfrac{y^{\prime}}{(Ny^{\prime})^{1/n}}=Y_{1}^{\kappa}(z)\log\varepsilon_{1}^{\kappa}+\dots+Y_{n-1}^{\kappa}(z)\log\varepsilon_{n-1}^{\kappa}.\end{array}$

Once the element $\sigma_{\kappa}$ and the bases above are fixed, the numbers
$X_{k}^{\kappa}(z)$ and $Y_{l}^{\kappa}(z)$ are uniquely determined and called
the _coordinates of $z$ at the cusp $\kappa$_. We may simply write
$X_{k}^{\kappa}$ and $Y_{l}^{\kappa}$. Note that we also use the notation
$Y_{0}(z)=Ny$ or simply $Y_{0}$ (which is the same as $Y_{0}^{\infty}(z)$
above once $\infty$ is a cusp and we choose $\sigma_{\infty}=id$). The
fundamental domain of $\Gamma_{\kappa}$ in $U_{\kappa}$ can be expressed in a
simple form in terms of these coordinates:

$F_{\kappa}=\\{z\in U_{\kappa}:\,0\leq
X_{1}^{\kappa},\dots,X_{n}^{\kappa}<1,\,0\leq
Y_{1}^{\kappa},\dots,Y_{n-1}^{\kappa}<1\\}.$

If $\kappa$ and $\kappa^{\prime}$ are inequivalent cusps of $\Gamma$, then
there exists neighbourhoods $U$ and $U^{\prime}$ of $\kappa$ and
$\kappa^{\prime}$, respectively, such that $\gamma(U)\cap
U^{\prime}=\emptyset$ holds for any $\gamma\in\Gamma$ (see Lemma $2.9_{2}$ in
[7]). Hence, if we fix a maximal set $\mathcal{S}$ of $\Gamma$-inequivalent
cusps, then a real number $C>0$ can be chosen such that the sets
$U_{\kappa}=\sigma_{\kappa}(U_{C})$ are pairwise disjoint for the cusps in
$\mathcal{S}$, and the corresponding sets $F_{\kappa}$ contain at most $1$
point from every $\Gamma$ orbit. Finally, the fundamental domain for $\Gamma$
is given in the form

$F=F_{0}\cup\left(\bigcup_{\kappa\in\mathcal{S}}F_{\kappa}\right),$

where $F_{0}\subset\mathbb{H}^{n}$ is compact.

####  1.2.5. Classification of the elements of $\Gamma$

Recall that an element $\mathrm{id}\neq\gamma\in PSL(2,\mathbb{R})$ is called
elliptic, parabolic or hyperbolic, if $\left|\textrm{tr}\,\gamma\right|<2$,
$\left|\textrm{tr}\,\gamma\right|=2$ or $\left|\textrm{tr}\,\gamma\right|>2$,
respectively. An element of $\Gamma$ is called _totally elliptic_ or _totally
parabolic_ , if each of its components are elliptic or parabolic,
respectively. If there are elements of different types among the components,
then this element is called _mixed_. Note that if one component of an element
is parabolic, then so are the others by Theorem 1.1. Hence a mixed element
consists of elliptic and hyperbolic components.

Before we turn to the case when every component is hyperbolic we examine the
fixed points of the elements. A totally elliptic element has a single fixed
point $z\in\mathbb{H}^{n}$. Since $\Gamma$ acts discontinuously on
$\mathbb{H}^{n}$, $z$ has a neighborhood $U$ such that the set
$\\{\gamma\in\Gamma:\,\gamma U\cap U\neq\emptyset\\}$ is finite. This means
that a totally elliptic element must be of finite order. A totally parabolic
element fixes a single point in $(\mathbb{R}\cup\\{\infty\\})^{n}$. Since
$\Gamma$ is irreducible, the parabolic fixed points are exactly the cusps of
$\Gamma$ (see Theorem 3 in [11]). A mixed element with $1\leq m<n$ hyperbolic
components fixes $2^{m}$ points in
$(\mathbb{H}\cup\mathbb{R}\cup\\{\infty\\})^{n}$. If every component of
$\gamma\in\Gamma$ is hyperbolic, then $\gamma$ fixes $2^{n}$ points in
$(\mathbb{R}\cup\\{\infty\\})^{n}$. Such an element is called _hyperbolic-
parabolic_ if there is cusp among its fixed points. Otherwise it is called
_totally hyperbolic_.

###  1.3. Fourier expansion of automorphic forms

A function $f:\mathbb{H}^{n}\to\mathbb{C}$ is called an _automorphic function_
with respect to the group $\Gamma$ if it is invariant under the action of
$\Gamma$, that is, $f(\gamma z)=f(z)$ holds for every $z\in\mathbb{H}^{n}$ and
$\gamma\in\Gamma$. An _automorphic form_ $u$ is a smooth automorphic function
which is an eigenfunction of the Laplace operators

$\Delta_{k}=y_{k}^{2}\left(\frac{\partial^{2}}{\partial
x_{k}^{2}}+\frac{\partial^{2}}{\partial
y_{k}^{2}}\right),\qquad(k=1,\dots,n),$

that is, for which the equations $(\Delta_{k}+\lambda_{k})u=0$ hold with some
$\lambda_{k}\in\mathbb{C}$. We write these eigenvalues in the form
$\lambda_{k}=s_{k}(1-s_{k})$ for some $s_{k}\in\mathbb{C}$.

If $u$ is an automorphic form and $\kappa$ is a cusp, then
$u(\sigma_{\kappa}z)$ is invariant under the action of the translation
operator $T_{\alpha}u=u(z_{1}+\alpha_{1},\dots,z_{n}+\alpha_{n})$ for any
$\alpha\in\mathbf{t}_{\kappa}$, hence it has the Fourier expansion

$u(z)=\sum_{l\in\mathbf{t}_{\kappa}^{*}}\phi(y,l)e^{2\pi i<l,x>},$

where $x=(x_{1},\dots,x_{n})$, $y=(y_{1},\dots,y_{n})$ and
$\mathbf{t}_{\kappa}^{*}=\\{v\in\mathbb{R}^{n}:\left<v,w\right>\in\mathbb{Z}\,\textrm{
for any }w\in\mathbf{t}_{\kappa}\\}$ is the _dual lattice_ of
$\mathbf{t}_{\kappa}$. In general, the dual lattice is given in the following
way. If $L=A(\mathbb{Z}^{n})\subset\mathbb{R}^{n}$ is a lattice, where $A\in
GL(\mathbb{R}^{n})$, then its dual is given by
$L^{*}=(A^{-1})^{T}(\mathbb{Z}^{n})$. In our case the columns of $A$ are the
vectors $b_{1}^{\kappa},\dots,b_{n}^{\kappa}$.

For a vector $\alpha\in\mathbb{R}^{n}$ and a lattice $L\subset\mathbb{R}^{n}$
we define

$\alpha L=\\{(\alpha_{1}l_{1},\dots,\alpha_{n}l_{n}):\,(l_{1},\dots,l_{n})\in
L\\}.$

It is easy to see, that if $a\in\mathbf{t}_{\kappa}$ and
$\varepsilon\in\Lambda_{\kappa}$, then $\varepsilon a\in\mathbf{t}_{\kappa}$,
and as $\Lambda_{\kappa}$ is a group, we have in fact
$\varepsilon\mathbf{t}_{\kappa}=\mathbf{t}_{\kappa}$. If $E$ is the diagonal
matrix with the coordinates of $\varepsilon$ in its diagonal, then

(5) $\displaystyle\varepsilon\mathbf{t}_{\kappa}^{*}$
$\displaystyle=E(A^{-1})^{T}(\mathbb{Z}^{n})=E^{T}(A^{-1})^{T}(\mathbb{Z}^{n})$
$\displaystyle=(A^{-1}E)^{T}(\mathbb{Z}^{n})=((E^{-1}A)^{-1})^{T}(\mathbb{Z}^{n})=(\varepsilon^{-1}\mathbf{t}_{\kappa})^{*}=\mathbf{t}_{\kappa}^{*}.$

Since the Laplace operator commutes with the action of
$PSL(2,\mathbb{R})^{n}$, $u(\sigma_{\kappa}z)$ is still an eigenfunction of
the Laplacians, and its Fourier coefficients can be expressed by means of its
eigenvalues and the modified Bessel function of the second kind, denoted by
$K_{\nu}(z)$ (see Theorem 5.1 in [14]):

######  Theorem 1.2.

Let $u$ be an automorphic form that satisfies the growth condition
$u(\sigma_{\kappa}z)=o(e^{2\pi y_{k}})$ as $y_{k}\to\infty$ ($k=1,\dots,n$)
(where $\kappa$ is a cusp for $\Gamma$). Then $u(\sigma_{\kappa}z)$ admits a
Fourier expansion of the form

(6)
$u(\sigma_{\kappa}z)=\sum_{l\in\mathbf{t}_{\kappa}^{*}}a_{\kappa}(l,y)e^{2\pi
i<l,x>},$

where

$a_{\kappa}(l,y)=c_{\kappa}(l)\sqrt{y_{1}\dots
y_{n}}K_{s_{1}-1/2}(2\pi\left|l_{1}\right|y_{1})\dots
K_{s_{n}-1/2}(2\pi\left|l_{n}\right|y_{n})$

for $l\neq 0$, while $a_{\kappa}(0,y)=:a_{\kappa}(y)$ is the linear
combination of two terms of the form $y_{1}^{s_{1}}\dots y_{n}^{s_{n}}$ and
$y_{1}^{1-s_{1}}\dots y_{n}^{1-s_{n}}$, where the numbers $s_{k}\in\mathbb{C}$
are such that $(\Delta_{k}+s_{k}(1-s_{k}))u=0$.

In the following we always assume that an automorphic form $u$ satisfies the
growth condition in Theorem 1.2 and hence admits the Fourier expansion (6).
Since $u(\sigma_{\kappa}z)$ remains unchanged if we substitute
$z\mapsto\varepsilon z$ for any $\varepsilon\in\Lambda_{\kappa}$, comparing
the Fourier coefficients, using (5) and also that $N\varepsilon=1$ holds, we
obtain that $c_{\kappa}({\varepsilon l})=c_{\kappa}(l)$ for every cusp
$\kappa$ and for every $\varepsilon\in\Lambda_{\kappa}$,
$l\in\mathbf{t}_{\kappa}^{*}\setminus 0$. Also, well-known bounds for the
Bessel function $K_{\nu}(z)$ and the absolute convergence of the sum in (6)
easily imply the trivial bound $c_{\kappa}(l)\ll
e^{\delta\left|N(l)\right|^{1/n}}$ for any $\delta>0$, where the implied
constant depends on $\delta$. From this the exponential decay of
$u(\sigma_{\kappa}z)-a_{\kappa}(y)$ "near the cusp" can be derived. Though we
will not detail its (technical but straightforward) proof, the precise
statement is given in the following

######  Proposition 1.3.

Let $u$ be an automorphic form with respect to $\Gamma$ with Laplace
eigenvalues $s_{k}(1-s_{k})$ that satisfies $u(\sigma_{\kappa}z)=o(e^{2\pi
y_{k}})$ for any $1\leq k\leq n$. Assume that
$\log\left(\frac{y}{(Ny)^{1/n}}\right)$ is bounded, then
$u(\sigma_{\kappa}z)-a_{0}(y)=O(e^{-CN(y)^{1/n}})$ for some constant $C>0$ if
$N(y)$ is big enough. The implied constant depends on the bounds on $N(y)$ and
$\log\left(\frac{y}{(Ny)^{1/n}}\right)$.

Recall that above we fixed the generators
$\varepsilon_{1}^{\kappa},\dots,\varepsilon_{n-1}^{\kappa}$ of
$\Lambda_{\kappa}$. Their coordinates will be denoted by
$(\varepsilon_{j}^{\kappa})^{(k)}$ ($k=1,\dots,n$). If the zeroth Fourier
coefficient of $u(\sigma_{\kappa}z)$ is non-zero, then the comparison of them
on both sides of the equation
$u(\sigma_{\kappa}z)=u(\sigma_{\kappa}(\varepsilon_{j}^{\kappa}z))$ gives for
each $1\leq j\leq n-1$ (similarly as in section II.1 of [6]) that

(7)
$\displaystyle\prod_{k=1}^{n}\left[(\varepsilon_{j}^{\kappa})^{(k)}\right]^{s_{k}}=1.$

Let us define $s:=(s_{1}+\dots+s_{n})/n$, then by (7) we have

(8)
$(s_{1},\dots,s_{n})\left[\begin{array}[]{cccc}1&\log(\varepsilon_{1}^{\kappa})^{(1)}&\dots&\log(\varepsilon_{n-1}^{\kappa})^{(1)}\\\
1&\log(\varepsilon_{1}^{\kappa})^{(2)}&\dots&\log(\varepsilon_{n-1}^{\kappa})^{(2)}\\\
\vdots&\vdots&\ddots&\vdots\\\
1&\log(\varepsilon_{1}^{\kappa})^{(n)}&\dots&\log(\varepsilon_{n-1}^{\kappa})^{(n)}\end{array}\right]=(ns,2\pi
im_{u,\kappa}^{(1)},\dots,2\pi im_{u,\kappa}^{(n-1)})$

for some
$m_{u,\kappa}=(m_{u,\kappa}^{(1)},\dots,m_{u,\kappa}^{(n-1)})^{T}\in\mathbb{Z}^{n-1}$.
Let us denote the matrix above by $\mathcal{E}_{\kappa}$. Since the vectors
$\log\varepsilon_{i}^{\kappa}$ form a basis in the trace $0$ subspace of
$\mathbb{R}^{n}$ and the first column of $\mathcal{E}_{\kappa}$ is not in that
subspace, we get that $\mathcal{E}_{\kappa}$ is invertible. Its inverse is of
the form

$\mathcal{E}_{\kappa}^{-1}=\left[\begin{array}[]{cccc}1/n&1/n&\dots&1/n\\\
(e_{1}^{\kappa})^{(1)}&(e_{1}^{\kappa})^{(2)}&\dots&(e_{1}^{\kappa})^{(n)}\\\
\vdots&\vdots&\ddots&\vdots\\\
(e_{n-1}^{\kappa})^{(1)}&(e_{n-1}^{\kappa})^{(2)}&\dots&(e_{n-1}^{\kappa})^{(n)}\end{array}\right],$

and the values $s_{1},\dots,s_{n}$ are determined by $s$ and $m_{u,\kappa}$
through

$\displaystyle(s_{1},\dots,s_{n})=(ns,2\pi im_{u,\kappa}^{(1)},\dots,2\pi
im_{u,\kappa}^{(n-1)})\mathcal{E}_{\kappa}^{-1}.$

That is,

(9) $s_{k}=s+\sum_{j=1}^{n-1}2\pi
im_{u,\kappa}^{(j)}(e_{j}^{\kappa})^{(k)},\qquad
y_{k}^{s_{k}}=y_{k}^{s}\exp\left(\sum_{j=1}^{n-1}2\pi
im_{u,\kappa}^{(j)}(e_{j}^{\kappa})^{(k)}\log y_{k}\right).$

For a cusp $\kappa$ and for any $m\in\mathbb{Z}^{n-1}$ set

(10)
$\displaystyle\lambda_{m}^{\kappa}(y)=\exp\left(\sum_{k=1}^{n}\sum_{j=1}^{n-1}2\pi
im_{j}(e_{j}^{\kappa})^{(k)}\log
y_{k}\right)=\prod_{k=1}^{n}\prod_{j=1}^{n-1}y_{k}^{2\pi
im_{j}(e_{j}^{\kappa})^{(k)}}$

for every $y\in(\mathbb{R}^{+})^{n}$. With this notation we may write the
zeroth coefficient of $u(\sigma_{\kappa}z)$ in the following way:

$\displaystyle a_{\kappa}(y)=\eta_{\kappa}(y_{1}\dots
y_{n})^{s}\lambda_{m_{u,\kappa}}^{\kappa}(y)+\phi_{\kappa}(y_{1}\dots
y_{n})^{1-s}\lambda_{-m_{u,\kappa}}^{\kappa}(y).$

Later we will see that $m_{u,\kappa}$ can be assumed to be the same vector
$m_{u}$ for every cusp, at least if it can be defined. If however
$\eta_{\kappa}=\phi_{\kappa}=0$ holds for all $\kappa\in\mathcal{S}$ (i.e. $u$
is a cusp form), then we simply set $m_{u}=0$. Aside from the next paragraph,
in the following we always assume that $0<\mathrm{Re}\,s<1$ holds whenever the
number $s$ is associated with the form $u$.

Later we will make use of a specific family automorphic forms, namely the
Eisenstein series that are defined as follows. Let $\kappa\in\mathcal{S}$ be a
cusp. For an $s\in\mathbb{C}$ and $m\in\mathbb{Z}^{n-1}$ the Eisenstein series
belonging to $\kappa$ is given by

(11)
$E_{\kappa}(z,s,m)=\sum_{\gamma\in\Gamma_{\kappa}\setminus\Gamma}y(\gamma^{(1)}z_{1})^{s_{1}}\dots
y(\gamma^{(n)}z_{n})^{s_{n}}$

for any $z\in\mathbb{H}^{n}$, where the exponents $s_{1},\dots,s_{n}$ are
defined as in (9). This series converges absolutely and uniformly on compact
subsets for $\mathrm{Re}\,s>1$. Also, $E_{\kappa}(z,s,m)$ (as a function in
the variable $z$) is clearly a $\Gamma$-invariant eigenfunction of the
Laplacians and (as a function of $s$) it can be continued meromorphically to
the whole complex plane. Moreover, for a cusp $\kappa^{\prime}\in\mathcal{S}$
the coefficient $\eta_{\kappa^{\prime}}$ in the Fourier coefficient
$a_{\kappa^{\prime}}(y)$ is $1$ if $\kappa^{\prime}=\kappa$ and $0$ otherwise.
For the details see chapter II and also section III.4 of [6].

##  2\. The geometric trace

###  2.1. The automorphic kernel

In the following we fix a compactly supported smooth function $\psi\in
C^{\infty}_{0}(\mathbb{R}^{n})$ and define the _point pair invariant kernel_

$k_{\psi}(z,w)=k(z,w)=\psi\left(\frac{\left|z_{1}-w_{1}\right|^{2}}{\textrm{Im}\,z_{1}\cdot\textrm{Im}\,w_{1}},\dots,\frac{\left|z_{n}-w_{n}\right|^{2}}{\textrm{Im}\,z_{n}\cdot\textrm{Im}\,w_{n}}\right)=\psi\left(\frac{\left|z-w\right|^{2}}{\textrm{Im}\,z\cdot\textrm{Im}\,w}\right)$

for every $z,w\in\mathbb{H}^{n}$. Invariance means that $k(z,w)=k(\sigma
z,\sigma w)$ holds for every $z,w\in\mathbb{H}^{n}$ and $\sigma\in
PSL(2,\mathbb{R})^{n}$. To avoid long formulae we often use the latter compact
notation for $\psi$ and its transforms defined below. In these cases the
operations on vectors always indicate coordinate-wise operations. The
_automorphic kernel_ $K(z,w)$ is given by the sum

(12) $K(z,w)=\sum_{\gamma\in\Gamma}k(z,\gamma w)$

that clearly defines an automorphic function w.r.t. $\Gamma$.

The following transformations of $\psi$ often occur in computations:

$\displaystyle Q(w)=Q(w_{1},\dots,w_{n})$
$\displaystyle:=\int_{w_{n}}^{\infty}\dots\int_{w_{1}}^{\infty}\frac{\psi(t_{1},\dots,t_{n})}{\sqrt{t_{1}-w_{1}}\dots\sqrt{t_{n}-w_{n}}}\,dt_{1}\dots
dt_{n},$ (13) $\displaystyle g(u)=g(u_{1},\dots,u_{n})$
$\displaystyle:=Q(e^{u_{1}}+e^{-u_{1}}-2,\dots,e^{u_{n}}+e^{-u_{n}}-2),$
$\displaystyle h(r)=h(r_{1},\dots,r_{n})$
$\displaystyle:=\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}g(u_{1},\dots,u_{n})e^{i\sum_{k=1}^{n}r_{k}u_{k}}\,du_{1}\dots
du_{n}.$

Note that this is the multidimensional version of the Harish-Chandra
transform. Since $\psi$ is a compactly supported smooth function, $g$ is also
a smooth function with compact support and hence $h$ is rapidly decreasing.

The inverses of the transforms above are

$\displaystyle g(u_{1},\dots,u_{n})$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}h(r_{1},\dots,r_{n})e^{-i\sum_{k=1}^{n}r_{k}u_{k}}\,dr_{1}\dots
dr_{n},$ (14) $\displaystyle Q(w_{1},\dots,w_{n})$
$\displaystyle=g\left(2\log\left(\sqrt{\frac{w_{1}}{4}+1}+\sqrt{\frac{w_{1}}{4}}\right),\dots,2\log\left(\sqrt{\frac{w_{n}}{4}+1}+\sqrt{\frac{w_{n}}{4}}\right)\right),$
$\displaystyle\psi(t_{1},\dots,t_{n})$
$\displaystyle=\frac{(-1)^{n}}{\pi^{n}}\int_{t_{n}}^{\infty}\dots\int_{t_{1}}^{\infty}\frac{\frac{\partial^{n}Q}{\partial
w_{1}\dots\partial
w_{n}}(w_{1},\dots,w_{n})}{\sqrt{w_{1}-t_{1}}\dots\sqrt{w_{n}-t_{n}}}\,dw_{1}\dots
dw_{n}$

(see Proposition I.2.2 in [6]).

###  2.2. The geometric trace

Now we turn to the multidimensional version of the generalized Selberg trace
formula, more precisely, to the geometric trace that is computed by collecting
the terms of the conjugacy classes in the sum (12). As in [1], our starting
point is the integral

$\textrm{Tr}_{u}K=\int_{F}K(z,z)u(z)\,\mu(z),$

where $u$ is a fixed automorphic form that satisfies the growth condition
$u(z)=o(e^{2\pi y_{k}})$ for $k=1,\dots,n$, $F$ is the fundamental domain of
$\Gamma$, and $\mu$ is the product measure on $\mathbb{H}^{n}$ obtained from
the measure $y^{-2}\,dx\,dy$ on $\mathbb{H}$. Note that $\mathrm{Re}\,s_{k}<1$
is assumed for each $k$ (excluding the case $u(z)=1$, that would yield the
trace formula given in [6]). Since this integral is not necessarily
convergent, we work with the _truncated trace_ defined by

(15) $\textrm{Tr}_{u}^{A}K:=\int_{F_{A}}K(z,z)u(z)\,d\mu(z)$

for every $A>0$, where

$F_{A}=F_{0}\cup\left(\bigcup_{\kappa\in\mathcal{S}}F_{\kappa}^{A}\right)$

with $F_{\kappa}^{A}=\\{z\in F_{\kappa}:\,Y_{0}^{\kappa}(z)\leq A\\}$.

Substituting the definition of $K(z,w)$ into (15) and summing over the
conjugacy classes in $\Gamma$ we get

$\textrm{Tr}_{u}^{A}K=\sum_{\\{\gamma\\}}\sum_{\sigma\in\\{\gamma\\}}\int_{F_{A}}k(z,\sigma
z)u(z)\,d\mu(z),$

where $\\{\gamma\\}$ denotes the conjugacy class of an element
$\gamma\in\Gamma$. Note that the conjugacy class of the identity element
consists only of itself, and the term that belongs to it is a constant
multiple of the integral

$\int_{F_{A}}u(z)\,d\mu(z).$

This integral converges as $A\to\infty$ and the limit is zero since the
Laplacians are symmetric operators and the eigenvalues of $1$ and $u$ are
different.

Our aim is to give the contribution of the different types of classes in this
trace. The main result can be summarized in the form

$\textrm{Tr}_{u}^{A}K=\Sigma_{\mathrm{ell}}+\Sigma_{\mathrm{mix}}+\Sigma_{\mathrm{par}}+\Sigma_{\mathrm{hyp-
par}},$

where the four terms on the right hand side stand for the contribution of
totally elliptic, mixed (and totally hyperbolic), totally parabolic and
hyperbolic-parabolic classes, respectively. Note that the totally hyperbolic
classes can be handled in the same way as the mixed classes, hence they are
melted in a single term above. Since the individual terms are given by lengthy
and complicated formulae, we do not give the whole sum in one statement, but
split the main result into four theorems below instead. We begin with the
contribution of elliptic, mixed and totally hyperbolic classes, here the
corresponding results are similar to the ones in [1]. Our main focus is
therefore on the parabolic and hyperbolic-parabolic classes, that are handled
afterwards.

###  2.3. Contribution of totally elliptic, mixed and totally hyperbolic
classes

In these cases the sum

$\sum_{\sigma\in\\{\gamma\\}}\int_{F_{A}}k(z,\sigma z)u(z)\,d\mu(z)$

in $\textrm{Tr}_{u}^{A}K$ belonging to the class $\\{\gamma\\}$ actually
converges as $A\to\infty$ (one can see this by analysing the detailed
computations in the proofs of the next two theorems). We will also see that
there are only finitely many classes for which the sum above is non-zero,
hence we can integrate over $F$ instead of $F_{A}$ (by including an $o(A)$
term as well).

Since $\sigma_{1}^{-1}\gamma\sigma_{1}=\sigma_{2}^{-1}\gamma\sigma_{2}$ holds
if and only if $\sigma_{2}\sigma_{1}^{-1}$ is in the centralizer $C(\gamma)$
of $\gamma$, and this is equivalent to $\sigma_{2}\in C(\gamma)\sigma_{1}$, we
get that

(16) $T_{\gamma}:=\sum_{\sigma\in\\{\gamma\\}}\int_{F}k(z,\sigma
z)u(z)\,d\mu(z)=\sum_{\sigma\in
C(\gamma)\setminus\Gamma}\int_{F}k(z,\sigma^{-1}\gamma\sigma z)u(z)\,d\mu(z).$

As $k(\varrho z,\varrho w)=k(z,w)$ holds for every $\varrho\in
PSL(2,\mathbb{R})^{n}$ and $u$ is invariant under the action of $\Gamma$, this
last sum is

$\sum_{\sigma\in C(\gamma)\setminus\Gamma}\int_{F}k(\sigma z,\gamma\sigma
z)u(\sigma z)\,d\mu(z)=\int_{C(\gamma)\setminus\mathbb{H}^{n}}k(z,\gamma
z)u(z)\,d\mu(z),$

and for every $\varrho\in PSL(2,\mathbb{R})^{n}$ this can be written as

(17) $\int_{\varrho^{-1}(C(\gamma)\setminus\mathbb{H}^{n})}k(\varrho
z,\gamma\varrho z)u(\varrho
z)\,d\mu(z)=\int_{(\varrho^{-1}C(\gamma)\varrho)\setminus\mathbb{H}^{n}}k(z,\varrho^{-1}\gamma\varrho
z)u(\varrho z)\,d\mu(z)$

since the measure $\mu$ and the function $k$ are $PSL(2,\mathbb{R})^{n}$
invariant. Note that $(\varrho^{-1}C(\gamma)\varrho)\setminus\mathbb{H}^{n}$
is nothing else but the fundamental domain of the group
$\varrho^{-1}C(\gamma)\varrho$.

Now we turn to the contribution of totally elliptic classes. Let us first note
that by Corollary $2.14_{1}$ in [7] there are only finitely many such classes,
hence $\Sigma_{\mathrm{ell}}$ is a sum of finitely many terms of the form
(17).

Before giving the value $\Sigma_{\mathrm{ell}}$ let us fix the following
notations. Every elliptic element $\gamma\in\Gamma$ is conjugate in
$PSL(2,\mathbb{R})^{n}$ to an element of the form
$(R(\theta_{\gamma}^{(1)}),\dots,R(\theta_{\gamma}^{(n)}))$, where

(18) $R(\alpha)=\left[\begin{array}[]{cc}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{array}\right]$

and the vector
$(\theta_{\gamma}^{(1)},\dots,\theta_{\gamma}^{(n)})\in[0,\pi)^{n}$ depends
only on the conjugacy class of $\gamma$.

Besides, let $g_{\lambda}(r):[0,\infty)\to\mathbb{C}$ be the unique solution
of the differential equation

(19) $g^{\prime\prime}(r)+\frac{\cosh r}{\sinh r}g^{\prime}(r)=\lambda g(r)$

satisfying the initial condition $g(0)=1$.

######  Theorem 2.1.

The contribution of the totally elliptic classes in the truncated trace, i.e.,
the value of $\Sigma_{\mathrm{ell}}$ is

$\sum_{\\{\gamma\\}\textrm{
t.ell.}}\frac{(2\pi)^{n}}{m_{\gamma}}u(z_{\gamma})\int\limits_{0}^{\infty}\dots\int\limits_{0}^{\infty}\psi(S(r_{1},\theta_{\gamma}^{(1)}),\dots,S(r_{n},\theta_{\gamma}^{(n)}))\left(\prod_{k=1}^{n}g_{\lambda_{k}}(r_{k})\sinh
r_{k}\,dr_{k}\right)+o(A),$

where the sum runs over all totally elliptic classes and for every class
$\\{\gamma\\}$ the point $z_{\gamma}\in\mathbb{H}^{n}$ is the fixed point of
$\gamma$, $m_{\gamma}\in\mathbb{N}^{+}$ is the order of the centralizer of
$\gamma$, $S(r,\theta)=(2\sinh r\sin\theta)^{2}$ for any
$r,\vartheta\in\mathbb{R}$, $(\lambda_{1},\dots,\lambda_{n})$ is the Laplacian
eigenvalue vector of $u$, and the functions $g_{\lambda_{k}}$ and the vector
$(\theta_{\gamma}^{(1)},\dots,\theta_{\gamma}^{(n)})$ are defined above the
theorem. Moreover, there are only finitely many totally elliptic conjugacy
classes, hence the sum above is finite.

Next we handle the mixed and totally hyperbolic classes, i.e. the classes
whose elements have at least one hyperbolic coordinates. For simplicity, we
assume that the first $1\leq m\leq n$ coordinates of the element are
hyperbolic, while the following $n-m$ coordinates are elliptic. The results
below can easily be reformulated and proved for other distributions of
coordinates of different types. Note also that the case $m=n$ is the totally
hyperbolic case, at least if the fixed points of the element are not cusps,
which is assumed in this section.

Any mixed or totally hyperbolic element $\gamma$ is conjugate in
$PSL(2,\mathbb{R})^{n}$ to an element of the form

(20)
$\nu=(D(N_{\gamma}^{(1)}),\dots,D(N_{\gamma}^{(m)}),R(\theta_{\gamma}^{(m+1)}),\dots,R(\theta_{\gamma}^{(n)}))$

for some $N_{\gamma}^{(k)}>1$ and $\theta_{\gamma}^{(l)}\in[0,\pi)$, where

(21) $D(N)=\left[\begin{array}[]{cc}N^{1/2}&0\\\ 0&N^{-1/2}\end{array}\right]$

and $R(\theta)$ is defined in (18) above. It is not hard to see that all these
numbers are determined uniquely by the class of $\gamma$ (and hence the
notations $N_{\gamma}^{(k)}$ and $\theta_{\gamma}^{(l)}$ are justified). The
number $N_{\gamma}^{(k)}$ is called the _norm_ of $\gamma^{(k)}$. We also set

(22)
$N(\vartheta,\gamma^{(k)}):=\frac{N_{\gamma}^{(k)}+(N_{\gamma}^{(k)})^{-1}-2}{\cos^{2}\vartheta}$

for any $\vartheta\in(-\frac{\pi}{2};\frac{\pi}{2})$ and $1\leq k\leq m$.

Let $\varrho_{\gamma}\in PSL(2,\mathbb{R})^{n}$ be an element such that
$\nu=\varrho_{\gamma}^{-1}\gamma\varrho_{\gamma}$ holds. To give the
contribution of the class $\\{\gamma\\}$ we need to describe that centralizer
$C(\nu)$ of $\nu$ in $\varrho_{\gamma}^{-1}\Gamma\varrho_{\gamma}$. By the
results of section I.5 in [6] the centralizer $C(\gamma)$ of $\gamma$ is a
free abelian group of rank $m$. We fix a set of its generators denoted by
$\gamma_{1},\dots,\gamma_{m}$, then the centralizer
$C(\nu)\leq\varrho_{\gamma}^{-1}\Gamma\varrho_{\gamma}$ is
$\varrho_{\gamma}^{-1}C(\gamma)\varrho_{\gamma}$ and it is generated by the
elements $\nu_{i}=\varrho_{\gamma}^{-1}\gamma_{i}\varrho_{\gamma}$ for
$i=1,\dots,m$. As the $\gamma_{i}$’s have the same fixed points as $\gamma$
this is true also for the conjugates and therefore

$\nu_{i}=\left(D(N_{\gamma_{i}}^{(1)}),\dots,D(N_{\gamma_{i}}^{(m)}),R(\theta_{\gamma_{i}}^{(m+1)}),\dots,R(\theta_{\gamma_{i}}^{(n)})\right).$

This is a somewhat imprecise notation since $N_{\gamma_{i}}^{(k)}>1$ may not
be assured for all $k$. This means that $N_{\gamma_{i}}^{(k)}$ is not
necessarily the norm of $\nu_{i}^{(k)}$ in the above sense, but it is still
determined by the (fixed) generator $\nu_{i}$ and we keep using this notation.
The action of the first $m$ coordinates of the elements $\nu_{i}$ is simple:
for every $z=(z_{1},\dots,z_{n})\in\mathbb{H}^{n}$ and $k=1,\dots,m$ we have
$|\nu_{i}^{(k)}z_{k}|=N_{\gamma_{i}}^{(k)}\left|z_{k}\right|$ and
$\arg\nu_{i}^{(k)}z_{k}=\arg z_{k}$. The next statement follows now easily:

######  Proposition 2.2.

The fundamental domain of the centralizer
$C(\nu)=\varrho_{\gamma}^{-1}C(\gamma)\varrho_{\gamma}$ is

$F_{C(\nu)}=\\{z\in\mathbb{H}^{n}:\,(\log\left|z_{1}\right|,\dots,\log\left|z_{m}\right|)\in
P_{\gamma}\\}$

where $P_{\gamma}$ is the fundamental parallelepiped of the lattice in
$\mathbb{R}^{m}$ generated by the vectors

$(\log N_{\gamma_{i}}^{(1)},\dots,\log
N_{\gamma_{i}}^{(m)})\qquad(i=1,\dots,m).$

Before the next theorem we introduce one more notation. Let
$f_{\lambda}(\vartheta)$ be the unique solution of the differential equation

(23)
$F^{\prime\prime}(\vartheta)=\frac{\lambda}{\cos^{2}\vartheta}F(\vartheta)\qquad(\vartheta\in(-\pi/2,\pi/2))$

with the initial condition $f_{\lambda}(0)=1$ and $f^{\prime}_{\lambda}(0)=0$.
We are now ready to state

######  Theorem 2.3.

The contribution of the mixed and totally hyperbolic classes in the truncated
trace is

$\Sigma_{\mathrm{mix}}=\sum_{\\{\gamma\\}\textrm{ mixed or totally
hyperbolic}}T_{\gamma}+o(A),$

where for a class $\\{\gamma\\}$, for which the first $m$ coordinates of
$\gamma$ are hyperbolic and the rest are elliptic, the value of $T_{\gamma}$
is

$\displaystyle(2\pi)^{n-m}F_{\gamma}(0,\dots,0)\times$
$\displaystyle\quad\times\int\limits_{0}^{\infty}\dots\int\limits_{0}^{\infty}\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dots\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\psi(N(\vartheta_{1},\gamma^{(1)}),\dots,N(\vartheta_{m},\gamma^{(m)}),S(r_{m+1},\theta_{\gamma}^{(m+1)}),\dots,S(r_{n},\theta_{\gamma}^{(n)}))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\left(\prod_{k=1}^{m}f_{\lambda_{k}}(\vartheta_{k})\,\frac{d\vartheta_{k}}{\cos^{2}\vartheta_{k}}\right)\left(\prod_{k=m+1}^{n}g_{\lambda_{k}}(r_{k})\sinh
r_{k})\,dr_{k}\right),$

where $N(\vartheta,\gamma^{(k)})$ is defined in (22), $S(r,\theta)=(2\sinh
r\sin\theta)^{2}$ for any $r,\vartheta\in\mathbb{R}$,
$(\lambda_{1},\dots,\lambda_{n})$ is the Laplacian eigenvalue vector of $u$,
the functions $f_{\lambda_{k}}$ were defined before the theorem, the functions
$g_{\lambda_{k}}$ were defined before Theorem 2.1, the vector
$(\theta_{\gamma}^{(m+1)},\dots,\theta_{\gamma}^{(n)})$ is given by (20) and

$F_{\gamma}(0,\dots,0)=\int\limits_{(\log r_{1},\dots,\log r_{m})\in
P_{\gamma}}u(\varrho_{\gamma}^{(1)}(r_{1}i),\dots,\varrho_{\gamma}^{(m)}(r_{m}i),\varrho_{\gamma}^{(m+1)}i,\dots,\varrho_{\gamma}^{(n)}i)\prod_{k=1}^{m}\frac{dr_{k}}{r_{k}}.$

Here $\varrho_{\gamma}\in PSL(2,\mathbb{R})^{n}$ is an element for which
$\varrho_{\gamma}^{-1}\gamma\varrho$ is of the form (20) and the set
$P_{\gamma}$ is given in Proposition 2.2. An analogous formula gives the value
of $T_{\gamma}$ when the $m$ hyperbolic and $n-m$ elliptic coordinates are
distributed differently. Moreover, the value of $T_{\gamma}$ is zero except
for finitely many classes.

###  2.4. Contribution of hyperbolic-parabolic classes

We continue with the contribution of those classes whose elements have only
hyperbolic coordinates but also fix a cusp. Let
$\gamma=(\gamma^{(1)},\dots,\gamma^{(n)})$ be such an element and let
$x=(x_{1},\dots,x_{n})$ be a cusp fixed by $\gamma$. That is, $x_{i}$ is a
fixed point of the hyperbolic coordinate $\gamma^{(i)}$ and we denote its
other one by $x_{i}^{\prime}$. Then, by the results of §20 in [11] the fixed
point $(x_{1}^{\prime},\dots,x_{n}^{\prime})$ of $\gamma$ is also a cusp.

Every hyperbolic-parabolic class is represented by an element that fixes a
cusp $\kappa\in\mathcal{S}$. An element of this type is conjugated by the
scaling element $\sigma_{\kappa}\in PSL(2,\mathbb{R})^{n}$ to an element of
the form

$\gamma_{m,\alpha}^{\kappa}:=\left[\begin{array}[]{cc}(u^{\kappa}_{m})^{1/2}&\alpha(u^{\kappa}_{m})^{-1/2}\\\
0&(u^{\kappa}_{m})^{-1/2}\end{array}\right],$

where $\alpha\in\mathbf{t}_{\kappa}$,
$m=(m_{1},\dots,m_{n-1})\in\mathbb{Z}^{n-1}\setminus\\{0\\}$ and
$u^{\kappa}_{m}=(\varepsilon_{1}^{\kappa})^{m_{1}}\dots(\varepsilon_{n-1}^{\kappa})^{m_{n-1}}\in\Lambda_{\kappa}$.
The cusp in the notation $u_{m}^{\kappa}$ indicates that this unit depends
also on the multiplier group $\Lambda_{\kappa}$ and its generators. However,
as a byproduct of the proof of this section’s main result we also get the
following

######  Proposition 2.4.

The multiplier group $\Lambda_{\kappa}$ is the same for any
$\kappa\in\mathcal{S}$.

This fact allows us to drop the index from the notation of the multiplier
group and we simply write $\Lambda$ in the following. Also, we can and will
fix the same generators $\varepsilon_{1},\dots,\varepsilon_{n-1}$ for every
cusp and therefore it is legitimate to write $u_{m}$ instead of
$u_{m}^{\kappa}$. It follows also that the matrix $\mathcal{E}_{\kappa}$ and
consequently the integer vector $m_{u,\kappa}$ (defined in (8)) are
independent of $\kappa$ and will be denoted simply by $\mathcal{E}$ and
$m_{u}$, respectively. Note that the lattice $\mathbf{t}_{\kappa}$ does depend
on the cusp $\kappa$.

The element $\gamma_{m,\alpha}^{\kappa}$ fixes the points $\infty$ and
$q=\frac{\alpha}{1-u_{m}}$ and according to the first paragraph of this
section both points are cusps for $\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$.
We will denote by $\tilde{\kappa}_{m,\alpha}\in\mathcal{S}$ the cusp for
$\Gamma$ that can be taken (by an element of $\Gamma$) to $\sigma_{\kappa}q$.

The centralizer $C(\gamma_{m,\alpha}^{\kappa})$ of
$\gamma_{m,\alpha}^{\kappa}$ in $\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$ is
given in §20 of [11]:

######  Proposition 2.5.

The centralizer $C(\gamma_{m,\alpha}^{\kappa})$ of the element
$\gamma_{m,\alpha}^{\kappa}$ is a free abelian group of rank $n-1$ generated
by some elements $\gamma(l_{1}),\dots,\gamma(l_{n-1})$, where
$l_{j}\in\mathbb{Z}^{n-1}\setminus\\{0\\}$ for any $1\leq j\leq n-1$ and

$\gamma(l_{j})=\left[\begin{array}[]{cc}u_{l_{j}}^{1/2}&\frac{u_{l_{j}}-1}{u_{m}-1}\alpha
u_{l_{j}}^{-1/2}\\\\[5.69054pt] 0&u_{l_{j}}^{-1/2}\end{array}\right].$

In the following we fix a generating set of elements described in the
proposition above and define the $(n-1)\times(n-1)$ matrix

(24)
$L_{m,\alpha}^{\kappa}:=\left[\begin{array}[]{cccc}l_{1}^{(1)}&l_{2}^{(1)}&\dots&l_{n-1}^{(1)}\\\
l_{1}^{(2)}&l_{2}^{(2)}&\dots&l_{n-1}^{(2)}\\\ \vdots&\vdots&\ddots&\vdots\\\
l_{1}^{(n-1)}&l_{2}^{(n-1)}&\dots&l_{n-1}^{(n-1)}\end{array}\right].$

As before, we need to describe the fundamental domain
$F_{C(\gamma_{m,\alpha}^{\kappa})}$ of $C(\gamma_{m,\alpha}^{\kappa})$. One
shows by induction that $\gamma(l_{j})^{h}=\gamma(hl_{j})$ holds for any
$h\in\mathbb{Z}$. Let $C$ denote the group generated by the (clearly
independent) elements

(27) $\displaystyle\rho_{l_{j}}:=\left[\begin{array}[]{cc}u_{l_{j}}^{1/2}&0\\\
0&u_{l_{j}}^{-1/2}\end{array}\right]\qquad(1\leq j\leq n-1).$

We set $T=\left[\begin{array}[]{cc}1&-\frac{\alpha}{1-u_{m}}\\\
0&1\end{array}\right]$, then $C(\gamma_{\alpha,m}^{\kappa})=T^{-1}CT$ and
hence if $F_{C}$ is a fundamental domain for $C$, then

$F_{C(\gamma_{m,\alpha}^{\kappa})}=T^{-1}F_{C}=F_{C}+\frac{\alpha}{1-u_{m}}=F_{C}+q$

is a fundamental domain for $C(\gamma_{\alpha,m}^{\kappa})$.

It remains to describe the fundamental domain $F_{C}$. As in the case of mixed
and totally hyperbolic elements we use polar coordinates. That is, for a point
$z=(z_{1},\dots,z_{n})\in\mathbb{H}^{n}$ we write
$z_{k}=r_{k}e^{i(\pi/2+\vartheta_{k})}$ where $r_{k}\in\mathbb{R}^{+}$ and
$-\frac{\pi}{2}<\vartheta_{k}<\frac{\pi}{2}$ ($1\leq k\leq n$). Let
$\tilde{P}_{m,\alpha}^{\kappa}$ be the fundamental domain of the $n-1$
dimensional lattice in $V=\\{a\in\mathbb{R}^{n}:\,a_{1}+\dots+a_{n}=0\\}$
generated by the vectors
$v_{j}=l_{j}^{(1)}\log\varepsilon_{1}+\dots+l_{j}^{(n-1)}\log\varepsilon_{n-1}$
($1\leq j\leq n-1$). For later purposes we specify the choice of
$\tilde{P}_{m,\alpha}^{\kappa}$, namely we take the shifted image of the
parallelpiped spanned by $v_{1},\dots,v_{n-1}$ (in $V$) symmetric to the
origin. Let us fix the unit vector
$\mathbf{1}=(n^{-\frac{1}{2}},\dots,n^{-\frac{1}{2}})^{T}$, it spans the
subspace $V^{\perp}$. If
$P_{m,\alpha}^{\kappa}=\\{t\mathbf{1}+\tilde{P}_{m,\alpha}^{\kappa}:\,t\in\mathbb{R}\\}$,
then the fundamental domain $F_{C}$ is given by

$(\vartheta_{1},\dots,\vartheta_{n})\in(-\pi/2;\pi/2),\qquad(\log
r_{1},\dots,\log r_{n})\in P_{m,\alpha}^{\kappa}.$

The contribution of the class belonging to $\gamma_{m,\alpha}^{\kappa}$ in the
truncated trace can be divided into two parts. A main term (that diverges as
$A\to\infty$) comes from the zeroth Fourier coefficient of
$u(\sigma_{\kappa}(z+q))$ and the transformed zeroth coefficient of
$u(\sigma_{\tilde{\kappa}_{m,\alpha}}(z+q))$ while we obtain the remaining
convergent part by subtracting these from $u(\sigma_{\kappa}(z+q))$. Note that
here the argument is shifted since we will give the result in terms of the
fundamental domain $F_{C}$. For any cusp $\kappa^{\prime}\in\mathcal{S}$ we
set

$M_{\kappa^{\prime}}(z):=\eta_{\kappa^{\prime}}y_{1}^{s_{1}}\dots
y_{n}^{s_{n}}+\phi_{\kappa^{\prime}}y_{1}^{1-s_{1}}\dots y_{n}^{1-s_{n}},$

and subtract $M_{\kappa}(z+q)=M_{\kappa}(z)$ from $u(\sigma_{\kappa}(z+q))$,
while in the case of $\tilde{\kappa}_{m,\alpha}$ we first apply a
transformation that maps $q$ to $\infty$. This is performed by an element
$\sigma_{\tilde{\kappa}_{m,\alpha}}^{-1}\gamma\sigma_{\kappa}\in\sigma_{\tilde{\kappa}_{m,\alpha}}^{-1}\Gamma\sigma_{\kappa}$
having the matrix form $\left[\begin{array}[]{cc}e&f\\\
\frac{u_{m}-1}{\delta}&\frac{\alpha}{\delta}\end{array}\right]$ where
$\delta=e\alpha+f(1-u_{m})$. Using this notation we define

$\tilde{u}_{m,\alpha}^{\kappa}(z):=u\left(\sigma_{\kappa}(z+q)\right)-M_{\kappa}\left(z\right)-M_{\tilde{\kappa}_{m,\alpha}}\left(-\frac{\delta^{2}E_{m}^{-2}u_{m}^{-1}}{z}\right)$

where $E_{m}:=u_{m}^{-1/2}-u_{m}^{1/2}$. Note that it is convenient to work
with the quantity $E_{m}$ because of its skew-symmetry in $m$. We mention in
advance that though the vector $\delta$ depends on the choice of
$\sigma_{\tilde{\kappa}_{m,\alpha}}^{-1}\gamma\sigma_{\kappa}$, but the norm
of $\delta^{2}$ depends only on $m$ and $\alpha$. Note also that the
translation invariance of $M_{\tilde{\kappa}_{m,\alpha}}(z)$ was also used to
simplify the defining formula of $\tilde{u}_{m,\alpha}$.

Before the main statement of the section we define an equivalence relation on
the lattice $\mathbf{t}_{\kappa}$ for any $\kappa$: the elements
$\alpha,\beta\in\mathbf{t}_{\kappa}$ are said to be equivalent if
$\beta=(u_{m}-1)a+u_{l}\alpha$ holds for some $l\in\mathbb{Z}^{n-1}$ and
$a\in\mathbf{t}_{\kappa}$, that is, if and only if $\beta$ and $u_{l}\alpha$
represent the same element in the finite factor group
$\mathbf{t}_{\kappa}^{m}:=\mathbf{t}_{\kappa}/(u_{m}-1)\mathbf{t}_{\kappa}$.
These classes (represented as elements of $\mathbf{t}_{\kappa}^{m}/\Lambda$)
are used to list the hyperbolic-parabolic conjugacy classes in the next
result:

######  Theorem 2.6.

The contribution of the hyperbolic-parabolic classes in the truncated trace is

$\Sigma_{\mathrm{hyp-
par}}=\delta_{m_{u}}M(A)+\sum_{\kappa\in\mathcal{S}}\,\,\sum_{m\in\mathbb{Z}^{n-1}\setminus\\{0\\}}\,\,\sum_{\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda}C_{\kappa}(m,\alpha)+o(A),$

the main term $M(A)$ is given by

(28)
$\frac{\left|\det\mathcal{E}\right|}{n}\sum_{\kappa\in\mathcal{S}}\left(\frac{\eta_{\kappa}A^{s}}{s}+\frac{\phi_{\kappa}A^{1-s}}{1-s}\right)\sum_{m\in\mathbb{Z}^{n-1}\setminus\\{0\\}}g(\log
u_{m})$

where $\mathcal{E}$ and $g$ were defined in (8) and (2.1), respectively, and
the term $C_{\kappa}(m,\alpha)$ is

$\displaystyle\frac{1}{2}\int\limits_{\log r\in
P_{m,\alpha}^{\kappa}}\tilde{u}_{m,\alpha}^{\kappa}(ri)\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}$
$\displaystyle\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dots\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\psi\left(\frac{E_{m}^{2}}{\cos^{2}\vartheta}\right)\left(\prod_{k=1}^{n}\frac{f_{\lambda_{k}}(\vartheta_{k})\,d\vartheta_{k}}{\cos^{2}\vartheta_{k}}\right),$

where $(\lambda_{1},\dots,\lambda_{n})$ is the Laplacian eigenvalue vector of
$u$ and $f_{\lambda_{k}}(\vartheta)$ is the unique solution of the
differential equation (23) satisfying the initial condition
$f_{\lambda_{k}}(0)=1$ and $f^{\prime}_{\lambda_{k}}(0)=0$. Moreover, the
terms $C_{\kappa}(m,\alpha)$ and the terms in (28) are zero for any cusp
$\kappa$ for all but finitely many $m$.

###  2.5. The $\zeta$-functions at the cusps

In this section we introduce the $\zeta$-function belonging to the lattice
$\mathbf{t}_{\kappa}$ and the multiplier group $\Lambda$ where $\kappa$ is any
cusp in $\mathcal{S}$, these will be needed for the last part of our result in
the next section. In fact we define these objects in the following general
situation. Let $L\leq\mathbb{R}^{n}$ be a lattice of full rank for which the
following holds: if $l=(l^{(1)},\dots,l^{(n)})^{T}\in L$ and $l^{(k)}=0$ for
any $1\leq k\leq n$, then $l=0$. We define the _norm_ of $l$ by
$Nl=\prod_{k=1}^{n}l^{(k)}$. Using this terminology, we assume that any non-
zero element of $L$ has non-zero norm. In addition, let
$M\leq(\mathbb{R}^{+})^{n}$ be a discrete norm-$1$ multiplicative subgroup of
rank $n-1$ that acts on $\mathbb{R}^{n}$ by coordinate-wise multiplication so
that $L$ is invariant under this action. That is, let us assume that
$M\cong\mathbb{Z}^{n-1}$, for every $\varepsilon\in M$ $N\varepsilon=1$ holds,
and finally, for any $\varepsilon\in M$ and $l\in L$ we have $\varepsilon
l:=(\varepsilon^{(1)}l^{(1)},\dots,\varepsilon^{(n)}l^{(n)})^{T}\in L$. We
remark that $N(\varepsilon l)=Nl$ holds for every $\varepsilon\in M$ and
$l\in\mathbb{R}^{n}$ and hence the norm, restricted to the lattice $L$, is in
fact defined on $M$-orbits.

Though it is more convenient to give some required technical statements in the
above context, it is important to mention that this generality is illusory.
Namely, it only simplifies the notations and helps to focus on the important
properties of the underlying objects, but does not give a wider point of view
since the setting we talk about is basically the same as in our initial
situation above. More concretely, a slight modification of the proof of
Theorem 4 in [11] gives the following

######  Proposition 2.7.

Assume that $L$ and $M$ are as above. Then there exists a totally real number
field $\mathbb{Q}\leq K$ of degree $n$ with embeddings
$K^{(1)}\subset\mathbb{R},\dots,K^{(n)}\subset\mathbb{R}$ such that for each
$\varepsilon=(\varepsilon^{(1)},\dots,\varepsilon^{(n)})^{T}\in M$ the
coordinate $\varepsilon^{(k)}$ is a totally positive unit in $K^{(k)}$ for all
$1\leq k\leq n$ and the coordinates of $\varepsilon$ are conjugates of each
other. Moreover, there is a vector
$\nu=(\nu_{1},\dots,\nu_{n})^{T}\in\mathbb{R}^{n}$ with non-zero coordinates
such that for any element $\alpha\in\nu\cdot L$ (where the coordinate-wise
product is taken) we have $\alpha^{(k)}\in K^{(k)}$ for every $k$ and the
coordinates of $\alpha$ are conjugates of each other.

Let us fix the generators $\varepsilon_{1},\dots,\varepsilon_{n-1}$ of the
group $M$ and define the matrices $\mathcal{E}_{M}$ and $\mathcal{E}_{M}^{-1}$
analogously as $\mathcal{E}(=\mathcal{E}_{\kappa})$ and its inverse were
defined in Section 1.3. Then the corresponding Grössencharacter-type
exponential sum $\lambda_{M,m}$ is given for every $m\in\mathbb{Z}^{n-1}$ in
the same way as $\lambda^{\kappa}_{m}$ was in (10).

We are interested in the sum

(29) $\displaystyle Z_{L,M}(s,m):=\sum_{0\neq l\in
L/M}\frac{\lambda_{M,-m}(\left|l\right|)}{\left|Nl\right|^{s}},$

where $s\in\mathbb{C}$, $m\in\mathbb{Z}^{n-1}$ and $\left|l\right|$ is the
vector whose coordinates are the absolute values of the corresponding
coordinates of $l$. Since $\lambda_{M,-m}$ is a multiplicative function and it
is trivial on $M$, the function $Z_{L,M}(s,m)$ is well-defined and it will be
called the zeta function belonging to the lattice $L$ and the group $M$. For
$L=\mathbf{t}_{\kappa}$ and $M=\Lambda$ we simply write $Z_{\kappa}(s,m)$ and
it will be called the _zeta function of $\Gamma$ belonging to the cusp_
$\kappa$. Note that the index of $\lambda$ in (29) is $-m$ in order to obtain
the equivalent form

$Z_{L,M}(s,m)=\sum_{0\neq l\in
L/M}\frac{1}{\left|l_{1}\right|^{s_{1}}\dots\left|l_{n}\right|^{s_{n}}},$

where $s_{1},\dots,s_{n}$ are defined as in (9).

We will show a few properties of these functions, they are summarized in the
following lemma:

######  Lemma 2.8.

The sum in (29) converges absolutely and locally uniformly for
$\mathrm{Re}\,s>1$, hence it defines an analytic function on this half-plane.
It can be continued meromorphically to the whole plane $\mathbb{C}$ and has no
poles on $\mathbb{C}$ except for the case $m=0$ when $s=1$ and $s=0$ are the
only poles of $Z_{L,M}(s,0)$, they are simple and

$\mathrm{Res}_{s=1}Z_{L,M}(s,0)=\frac{2^{n}\left|\det\mathcal{E}_{M}\right|}{n\cdot\mathrm{vol}(\mathbb{R}^{n}/L)}.$

Moreover, the completed function

$\Xi_{L,M}(s,m):=\pi^{-\frac{ns}{2}}\left(\prod_{k=1}^{n}\Gamma\left(\frac{s_{k}}{2}\right)\right)Z_{L,M}(s,m)$

satisfies the functional equation

$\mathrm{vol}(\mathbb{R}^{n}/L)^{1/2}\Xi_{L,M}(s,m)=\mathrm{vol}(\mathbb{R}^{n}/L^{*})^{1/2}\Xi_{L^{*},M}(1-s,-m)$

where $L^{*}$ is the dual lattice of $L$. The convexity bound

$Z_{L,M}(s,m)\ll_{\varepsilon,m}\left|\textrm{Im}\,s\right|^{n(1-\mathrm{Re}\,s)/2+\varepsilon}$

holds for any $\varepsilon>0$ if $0\leq\mathrm{Re}\,s\leq 1$ and
$\left|\textrm{Im}\,s\right|$ is bounded from below by some positive constant.
Also,
$Z_{L,M}(s,m)\ll_{\varepsilon,m}\left|\textrm{Im}\,s\right|^{\varepsilon}$
holds if $\mathrm{Re}\,s>1$ and $\left|\textrm{Im}\,s\right|$ is bounded away
from zero.

###  2.6. Contribution of totally parabolic classes

At last we give the contribution of the totally parabolic classes in the
geometric trace. In advance of that we introduce some notations. If $\kappa$
is a cusp of $\Gamma$ and $\sigma_{\kappa}$ is the corresponding scaling
element, then the zeroth Fourier coefficient of $u(\sigma_{\kappa}z)$ is
$a_{\kappa}(y)=\eta_{\kappa}y_{1}^{s_{1}}\dots
y_{n}^{s_{n}}+\phi_{\kappa}y_{1}^{1-s_{1}}\dots y_{n}^{1-s_{n}}$. Recall that
if at least one of $\eta_{\kappa}$ and $\phi_{\kappa}$ is non-zero (i.e. when
$u$ does not vanish at $\kappa$) then the zeroth coefficient $a_{\kappa}(y)$
can be written in the form $\eta_{\kappa}(y_{1}\dots
y_{n})^{s}\lambda_{m_{u}}(y)+\phi_{\kappa}(y_{1}\dots
y_{n})^{1-s}\lambda_{-m_{u}}(y)$, where $s=\frac{s_{1}+\dots+s_{n}}{n}$ and
$m_{u}\in\mathbb{Z}^{n-1}$. If $\eta_{\kappa}=\phi_{\kappa}=0$ for all
$\kappa$, then $m_{u}=0$ holds by definition. Note that in (10) the function
$\lambda_{m}=\lambda^{\kappa}_{m}$ was defined in terms of the entries of
$\mathcal{E}_{\kappa}=\mathcal{E}$ and hence by Proposition 2.4 and its
subsequent paragraph $\lambda_{m}$ is independent of the cusp that (from now
on) will not be included in our notation. Now we are in the position to give
$\Sigma_{\mathrm{par}}$ explicitly:

######  Theorem 2.9.

The contribution of the totally parabolic classes in the truncated trace,
i.e., the value of $\Sigma_{\mathrm{par}}$ is

$\displaystyle\sum_{\kappa\in\mathcal{S}}\delta_{m_{u}}\frac{\left|\det\mathcal{E}\right|}{n}\left(\frac{\eta_{\kappa}A^{s}}{s}+\frac{\phi_{\kappa}A^{1-s}}{1-s}\right)g\left(0\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\mathrm{vol}(\mathbb{R}^{n}/\mathbf{t}_{\kappa})(\eta_{\kappa}Z_{\kappa}(1-s,-m_{u})F(0)+\phi_{\kappa}Z_{\kappa}(s,m_{u})\tilde{F}(0))+o(1)$

as $A\to\infty$, where

$\displaystyle
F(S)=\int\limits_{(\mathbb{R}^{+})^{n}}\psi\left(t^{2}\right)\prod_{k=1}^{n}t_{k}^{S-s_{k}}\,dt_{k}\quad\textrm{and}\quad\tilde{F}(S)=\int\limits_{(\mathbb{R}^{+})^{n}}\psi\left(t^{2}\right)\prod_{k=1}^{n}t_{k}^{S+s_{k}-1}\,dt_{k}.$

The values $F(0)$ and $\tilde{F}(0)$ can be expressed in terms of the function
$h$ defined in (2.1):

$F(0)=\left(\frac{i}{2^{2-s}\pi^{2}}\right)^{n}\left(\prod_{k=1}^{n}\Gamma\left(\frac{1-s_{k}}{2}\right)^{2}\right)\int\limits_{(\mathbb{R}^{+})^{n}}h(r)\prod_{k=1}^{n}\frac{\Gamma(\frac{s_{k}}{2}+ir_{k})}{\Gamma(\frac{2-s_{k}}{2}+ir_{k})}r_{k}\,dr_{k},$

and

$\tilde{F}(0)=\left(\frac{i}{2^{s+1}\pi^{2}}\right)^{n}\left(\prod_{k=1}^{n}\Gamma\left(\frac{s_{k}}{2}\right)^{2}\right)\int\limits_{(\mathbb{R}^{+})^{n}}h(r)\prod_{k=1}^{n}\frac{\Gamma(\frac{1-s_{k}}{2}+ir_{k})}{\Gamma(\frac{s_{k}+1}{2}+ir_{k})}r_{k}\,dr_{k}.$

##  3\. Proofs of the theorems

###  3.1. Proof in the totally elliptic case

We first prove Theorem 2.1. As it was already mentioned, there are only
finitely many elliptic conjugacy classes by Corollary $2.14_{1}$ in [7]),
hence it remains to show the formula for $T_{\gamma}$ (defined in (16)). We
prove by induction. Since our argument is very similar to the one in [1], we
only sketch the induction step.

Let $\gamma\in\Gamma$ be a totally elliptic element with the elliptic fixed
point $z_{\gamma}\in\mathbb{H}^{n}$. The centralizer $C(\gamma)$ consists of
the elements in $\Gamma$ which leave the point $z_{\gamma}$ fixed (see [11],
p. 37) and the stabilizer $\Gamma_{z_{\gamma}}$ of $z_{\gamma}$ in $\Gamma$ is
a finite cyclic group (see Remark 2.14 in [7]). Let us denote by $m_{\gamma}$
the order of $C(\gamma)$. Every elliptic element in $PSL(2,\mathbb{R})$ is
conjugate to an element of the form
$\left[\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right]$, hence the generator $\gamma_{0}$ of
$C(\gamma)$ can be chosen so that it is conjugate in $PSL(2,\mathbb{R})^{n}$
to the element

$\gamma^{\prime}_{0}=\left(\left[\begin{array}[]{cc}\displaystyle{\cos\displaystyle{\frac{\pi}{m_{\gamma}}}}&\displaystyle{\sin\displaystyle{\frac{\pi}{m_{\gamma}}}}\\\\[8.53581pt]
\displaystyle{-\sin\displaystyle{\frac{\pi}{m_{\gamma}}}}&\displaystyle{\cos\displaystyle{\frac{\pi}{m_{\gamma}}}}\end{array}\right],\left[\begin{array}[]{cc}\displaystyle{\cos\displaystyle{\frac{l_{2}\pi}{m_{\gamma}}}}&\displaystyle{\sin\displaystyle{\frac{l_{2}\pi}{m_{\gamma}}}}\\\\[8.53581pt]
\displaystyle{-\sin\displaystyle{\frac{l_{2}\pi}{m_{\gamma}}}}&\displaystyle{\cos\displaystyle{\frac{l_{2}\pi}{m_{\gamma}}}}\end{array}\right],\dots,\left[\begin{array}[]{cc}\displaystyle{\cos\displaystyle{\frac{l_{n}\pi}{m_{\gamma}}}}&\displaystyle{\sin\displaystyle{\frac{l_{n}\pi}{m_{\gamma}}}}\\\\[8.53581pt]
\displaystyle{-\sin\displaystyle{\frac{l_{n}\pi}{m_{\gamma}}}}&\displaystyle{\cos\displaystyle{\frac{l_{n}\pi}{m_{\gamma}}}}\end{array}\right]\right),$

where $l_{k}\in\mathbb{Z}$ with $\gcd(l_{k},m_{\gamma})=1$ for every
$k=2,\dots,n$. Let us write $\gamma^{\prime}=\varrho^{-1}\gamma\varrho$, we
give the fundamental domain $F_{C(\gamma^{\prime})}$ of
$C(\gamma^{\prime})=\left<\gamma^{\prime}_{0}\right>\leq\varrho^{-1}\Gamma\varrho$.
The first coordinate of $\gamma^{\prime}_{0}$ is a rotation around the point
$i\in\mathbb{H}$ by the angle $2\pi/m_{\gamma}$, therefore every
$C(\gamma^{\prime})$-orbit has exactly one point in the set
$F_{0}\times\mathbb{H}^{n-1}$, where $F_{0}\subset\mathbb{H}$ is a sector
enclosed by two half-lines with endpoint $i$ and angle $2\pi/m_{\gamma}$. Note
that in fact each coordinate is a rotation around $i$ which means that
$\varrho$ takes the point $(i,\dots,i)$ to the fixed point $z_{\gamma}$ of
$\gamma$. Now by (17) we have

$T_{\gamma}=\int_{F_{C(\gamma^{\prime})}}k(z,\gamma^{\prime}z)u(\varrho
z)\,d\mu(z)=\frac{1}{m_{\gamma}}\int_{\mathbb{H}^{n}}k(z,\gamma^{\prime}z)u(\varrho
z)\,d\mu(z),$

where we used the $PSL(2,\mathbb{R})^{n}$-invariance of the function $k$ and
the measure $\mu$, the $\Gamma$-invariance of $u$ and that $\gamma^{\prime}$
and $\gamma^{\prime}_{0}$ commute. Writing $z=(z_{1},\dots,z_{n})$ we have

(30) $\int_{\mathbb{H}^{n}}k(z,\gamma^{\prime}z)u(\varrho
z)\,d\mu(z)=\int_{\mathbb{H}}\dots\int_{\mathbb{H}}k(z,\gamma^{\prime}z)u(\varrho
z)\,d\mu(z_{1})\dots\,d\mu(z_{n}),$

where $\mu(z_{k})$ denotes the measure $y_{k}^{-2}dx_{k}\,dy_{k}$. In the
inner integral above the coordinates $z_{2},\dots,z_{n}$ are fixed, and the
function $u(\varrho z)$ can be regarded as a function of $z_{1}$. It is the
eigenfunction of the Laplace operator $\Delta_{1}$ (because the operator
commutes with the group action), furthermore, the value of
$k(z,\gamma^{\prime}z)$ depends only on the hyperbolic distance of $z_{1}$ and
$\gamma^{\prime(1)}z_{1}$. To simplify the notation we write $u(\varrho
z)=u_{1}(z_{1})$ and $k(z,w)=k_{1}(z_{1},w_{1})$. Furthermore, as
$\gamma^{\prime}$ is fixed we can simply write
$k(z,\gamma^{\prime}z)=k_{1}(z_{1},\gamma^{\prime(1)}z_{1})$. With this
notation the inner integral becomes

$T_{1}:=\int_{\mathbb{H}}k(z,\gamma^{\prime}z)u(\varrho
z)\,d\mu(z_{1})=\int_{\mathbb{H}}k_{1}(z_{1},\gamma^{\prime(1)}z_{1})u_{1}(z_{1})\,d\mu(z_{1}).$

Recall that in (18) we introduced the notation

$R(\alpha)=\left[\begin{array}[]{cc}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{array}\right],$

for any $\alpha\in\mathbb{R}$. For a vector
$\varphi=(\varphi^{(1)},\dots,\varphi^{(n)})\in\mathbb{R}^{n}$ we set
$R(\varphi)=(R(\varphi^{(1)}),\dots,R(\varphi^{(n)}))$. The elements of the
centralizer $C(\gamma^{\prime})$ are of the form $R(\varphi)$, in particular
$\gamma^{\prime}=R(\theta_{\gamma})$ for some
$\theta_{\gamma}=(\theta_{\gamma}^{(1)},\dots,\theta_{\gamma}^{(n)})$ where
$\theta_{\gamma}^{(k)}\in[0,\pi)$. As it was mentioned in Section 2.3 it is
not hard to see that the vector $\theta_{\gamma}$ is determined by the class
of $\gamma$, i.e. it is independent of the choice of $\varrho$ (at least if
every coordinate is chosen from the interval $[0,\pi)$). Since
$\gamma^{\prime}\in C(\gamma^{\prime})$ we have in fact
$\theta_{\gamma}^{(k)}=l_{k}\pi/m_{\gamma}$ for some integer
$0<l_{k}<m_{\gamma}$ ($k=1,\dots,n$).

Next we use geodesic polar coordinates (see [8], section 1.3), i.e. we make
the substitution $z_{1}=R(\varphi_{1})e^{-r_{1}}i$ where $r_{1}\in(0,\infty)$
is the hyperbolic distance of $i$ and $z_{1}$ and $\varphi_{1}\in[0,\pi)$.
Then we have $d\mu(z_{1})=(2\sinh r_{1})\,dr_{1}\,d\varphi_{1}$ and

$T_{1}=\int_{0}^{\infty}\int_{0}^{\pi}k_{1}(R(\varphi_{1})e^{-r_{1}}i,R(\theta_{\gamma}^{(1)})R(\varphi_{1})e^{-r_{1}}i))u_{1}(R(\varphi_{1})e^{-r_{1}}i)2\sinh
r_{1}\,d\varphi_{1}\,dr_{1}.$

As the elements $R(\theta_{\gamma}^{(1)})$ and $R(\varphi_{1})$ commute and
$k_{1}$ depends only on the hyperbolic distance of the variables, we get that

$T_{1}=\int_{0}^{\infty}k_{1}(e^{-r_{1}}i,R(\theta_{\gamma}^{(1)})e^{-r_{1}}i))\left(\int_{0}^{\pi}u_{1}(R(\varphi_{1})e^{-r_{1}}i)\,d\varphi_{1}\right)(2\sinh
r_{1})\,dr_{1}.$

Recall that
$k_{1}(z_{1},w_{1})=\psi\left(\rho(z_{1},w_{1}),\dots,\rho(z_{n},w_{n})\right)=:\psi_{1}(\rho(z_{1},w_{1}))$,
where

$\rho(z_{k},w_{k})=\frac{\left|z_{k}-w_{k}\right|^{2}}{\textrm{Im}\,z_{k}\,\textrm{Im}\,w_{k}}$

for $k=1,\dots,n$. One gets by a computation that

$\displaystyle\rho(e^{-r_{k}}i,R(\theta_{\gamma}^{(k)})e^{-r_{k}}i)=\frac{\left|-e^{-2r_{k}}+1\right|^{2}\sin^{2}\theta_{\gamma}^{(k)}}{e^{-2r_{k}}}=(2\sinh
r_{k}\sin\theta_{\gamma}^{(k)})^{2},$

hence

(31) $T_{1}=\int_{0}^{\infty}\psi_{1}((2\sinh
r_{1}\sin\theta_{\gamma}^{(1)})^{2})\left(\int_{0}^{\pi}u_{1}(R(\varphi_{1})e^{-r_{1}}i)\,d\varphi_{1}\right)(2\sinh
r_{1})\,dr_{1}.$

Let us define the function

$G_{1}(w)=\frac{1}{\pi}\int_{0}^{\pi}u_{1}(R(\varphi_{1})w)\,d\varphi_{1},$

where $w\in\mathbb{H}$. By Lemma 1.10 in [8] the value of $G_{1}$ depends only
on the hyperbolic distance $r$ of $w$ and $i$. Moreover, $G_{1}$ is the
eigenfunction of the (one dimensional) Laplace operator $\Delta$ with
eigenvalue $\lambda_{1}$, where $\lambda_{1}$ is the first coordinate of the
eigenvalue vector of $u$. Now by Lemma 1.12 of [8] this function is unique up
to a constant factor. Furthermore

$\Delta=\frac{\partial^{2}}{\partial r^{2}}+\frac{\cosh r}{\sinh
r}\frac{\partial}{\partial
r}+\frac{1}{4\sinh^{2}r}\frac{\partial^{2}}{\partial\varphi^{2}}$

hence the function $G_{1}$ (as a function of $r$) satisfies the differential
equation (19) with the constant $\lambda=\lambda_{1}$, and consequently

$G_{1}(w)=g_{\lambda_{1}}(r)u_{1}(i)=g_{\lambda_{1}}(r)u(\varrho^{(1)}i,\varrho^{(2)}z_{2},\dots,\varrho^{(n)}z_{n}),$

where $g_{\lambda_{1}}(r):[0,\infty)\to\mathbb{C}$ is the solution of (19)
with $\lambda=\lambda_{1}$ satisfying the initial condition
$g_{\lambda_{1}}(0)=1$. By substituting this in (31), then interchanging the
integrals in (30) and proceeding by induction one gets the statement of the
theorem.

###  3.2. Proof in the mixed and totally hyperbolic cases

We continue with the proof of Theorem 2.3. Let $\gamma\in\Gamma$ a mixed or a
totally hyperbolic element. We assume that the first $1\leq m\leq n$
coordinate of $\gamma$ are hyperbolic while the following $n-m$ coordinates
are elliptic, the proof of the statement is similar in the other cases. We
have seen in Section 2.3 that such an element is conjugated by an element
$\varrho\in PSL(2,\mathbb{R})^{n}$ to an element of the form
$\nu=\left(D(N_{\gamma}^{(1)}),\dots,D(N_{\gamma}^{(m)}),R(\theta_{\gamma}^{(m+1)}),\dots,R(\theta_{\gamma}^{(n)})\right)$
for some $N_{\gamma}^{(k)}>1$ and $\theta_{\gamma}^{(l)}\in[0;2\pi)$, where
$D(N)$ and $R(\theta)$ were defined in (21) and (18), respectively. An easy
computation shows that these numbers are uniquely defined by the class
$\\{\gamma\\}$.

As in the totally elliptic case, by (17) we need to consider the integral

$T_{\gamma}=\int_{F_{C(\nu)}}k(z,\nu z)u(\varrho z)\,d\mu(z),$

where $F_{C(\nu)}$ is the fundamental domain for the centralizer
$C(\nu)\leq\varrho^{-1}\Gamma\varrho$ of $\nu=\varrho^{-1}\gamma\varrho$. This
domain was described in Proposition 2.2 and the notations introduced there
will be used in the following.

For the first $m$ coordinates of $z$ we change to polar coordinates, i.e. make
the substitution $z_{k}=r_{k}e^{i(\pi/2+\vartheta_{k})}$ where
$r_{k}\in(0,\infty)$ and $\vartheta_{k}\in(-\frac{\pi}{2},\frac{\pi}{2})$
($k=1,\dots,m$), while for the last $n-m$ coordinates we change to geodesic
polar coordinates as in the previous proof. A simple computation shows that
$\rho(z_{k},\nu^{(k)}z_{k})=N(\vartheta_{k},\gamma^{(k)})$ for $k=1,\dots,m$
(where $N(\vartheta_{k},\gamma^{(k)})$ was defined in (22)), and by
$y_{k}^{-2}\,dx_{k}\,dy_{k}=(r_{k}\cos^{2}\vartheta_{k})^{-1}\,dr_{k}\,d\vartheta_{k}$
and the results of the previous proof we obtain that $T_{\gamma}$ is

$\displaystyle(2\pi)^{n-m}$
$\displaystyle\int\limits_{0}^{\infty}\dots\int\limits_{0}^{\infty}\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dots\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\psi(N(\vartheta_{1},\gamma^{(1)}),\dots,N(\vartheta_{m},\gamma^{(m)}),S(r_{m+1},\theta_{\gamma}^{(m+1)}),\dots,S(r_{n},\theta_{\gamma}^{(n)}))$
$\displaystyle\qquad\,\,\times
F(e^{i(\frac{\pi}{2}+\vartheta_{1})},\dots,e^{i(\frac{\pi}{2}+\vartheta_{m})})\,\frac{d\vartheta_{1}}{\cos^{2}\vartheta_{1}}\dots\frac{d\vartheta_{m}}{\cos^{2}\vartheta_{m}}\left(\prod_{k=m+1}^{n}g_{\lambda_{k}}(r_{k})\sinh
r_{k}\,dr_{k}\right),$

where $S(r,\theta)=(2\sinh r\sin\theta)^{2}$ and

$F(z)=\int\limits_{(\log r_{1},\dots,\log r_{m})\in
P_{\gamma}}u(\varrho^{(1)}(r_{1}z_{1}),\dots,\varrho^{(m)}(r_{m}z_{m}),\varrho^{(m+1)}i,\dots,\varrho^{(n)}i)\prod_{k=1}^{m}\frac{dr_{k}}{r_{k}}$

for any $z\in\mathbb{H}^{m}$. Since $u(\varrho z)$ is invariant under the
action of the centralizer $C(\nu)$, one sees easily that the function $F$ is
invariant under is coordinate-wise scalar multiplication, i.e.
$F(R_{1}z_{1},\dots,R_{m}z_{m})=F(z_{1},\dots,z_{m})$ holds for any
$R_{1},\dots,R_{m}\in(0,\infty)$ and $z_{1},\dots,z_{m}\in\mathbb{H}$. This
means that $F$ depends only on the vector
$(\vartheta_{1},\dots,\vartheta_{m})$ (where
$z_{k}=r_{k}e^{i(\frac{\pi}{2}+\vartheta_{k})}$).

Moreover, since $u$ is the eigenfunction of every $\Delta_{k}$ with eigenvalue
$\lambda_{k}$ and these operators commute with the group action, we infer that
$F(z)$ is also an eigenfunction of the Laplacians
$\Delta_{1},\dots,\Delta_{m}$ with the same corresponding eigenvalues. As

$\Delta_{k}=(r_{k}\cos\vartheta_{k})^{2}\left(\frac{\partial^{2}}{\partial
r_{k}^{2}}+r_{k}^{-1}\frac{\partial}{\partial
r_{k}}+r_{k}^{-2}\frac{\partial^{2}}{\partial\vartheta_{k}^{2}}\right),$

we obtain the differential equations

(32)
$\frac{\partial^{2}F}{\partial\vartheta_{k}^{2}}(\vartheta_{1},\dots,\vartheta_{m})=\frac{\lambda_{k}}{\cos^{2}\vartheta_{k}}F(\vartheta_{1},\dots,\vartheta_{m})\qquad(\vartheta_{k}\in(-\pi/2,\pi/2),\,k=1,\dots,m).$

Let $f_{\lambda_{k}}(\vartheta)$ be the unique solution of the differential
equation (23) with $\lambda=\lambda_{k}$ and the initial conditions
$f_{\lambda_{k}}(0)=1$ and $f^{\prime}_{\lambda_{k}}(0)=0$, and
$\tilde{f}_{\lambda_{k}}(\vartheta)$ the one with
$\tilde{f}_{\lambda_{k}}(0)=0$ and $\tilde{f}^{\prime}_{\lambda_{k}}(0)=1$.
Note that $f_{\lambda_{k}}(-\vartheta)$ satisfies (23) and the initial
conditions of $f_{\lambda_{k}}(\vartheta)$ and hence they agree, i.e.
$f_{\lambda_{k}}$ is an even function. Similarly, $\tilde{f}_{\lambda_{k}}$ is
an odd function.

The equation

$F(\vartheta_{1},\dots,\vartheta_{m})=F(0,\vartheta_{2},\dots,\vartheta_{m})f_{\lambda_{1}}(\vartheta_{1})+\frac{\partial
F}{\partial\vartheta_{1}}(0,\vartheta_{2},\dots,\vartheta_{m})\tilde{f}_{\lambda_{1}}(\vartheta_{1})$

holds by (32) for every fixed $\vartheta_{2},\dots,\vartheta_{m}$, hence the
inner integral in $T_{\gamma}$ is

$\displaystyle\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\psi(N(\vartheta_{1},\gamma^{(1)}),\dots,N(\vartheta_{m},\gamma^{(m)}),S(r_{m+1},\theta_{\gamma}^{(m+1)}),\dots,S(r_{n},\theta_{\gamma}^{(n)}))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\left(F(0,\vartheta_{2},\dots,\vartheta_{m})f_{\lambda_{1}}(\vartheta_{1})+\frac{\partial
F}{\partial\vartheta_{1}}(0,\vartheta_{2},\dots,\vartheta_{m})\tilde{f}_{\lambda_{1}}(\vartheta_{1})\right)\frac{d\vartheta_{1}}{\cos^{2}\vartheta_{1}}$
$\displaystyle=\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\psi(N(\vartheta_{1},\gamma^{(1)}),\dots,N(\vartheta_{m},\gamma^{(m)}),S(r_{m+1},\theta_{\gamma}^{(m+1)}),\dots,S(r_{n},\theta_{\gamma}^{(n)}))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
F(0,\vartheta_{2},\dots,\vartheta_{m})f_{\lambda_{1}}(\vartheta_{1})\frac{d\vartheta_{1}}{\cos^{2}\vartheta_{1}}$

because $N(\vartheta_{1},\gamma^{(1)})$ and $\cos^{-2}(\vartheta_{1})$ are
even and $\tilde{f}_{\lambda_{1}}(\vartheta_{1})$ is an odd function. Then, by
induction (using also that $f_{\lambda_{k}}$ is even) we infer

$\displaystyle T_{\gamma}=$ $\displaystyle(2\pi)^{n-m}F(0,\dots,0)\times$
$\displaystyle\times\int\limits_{0}^{\infty}\dots\int\limits_{0}^{\infty}\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dots\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\psi(N(\vartheta_{1},\gamma^{(1)}),\dots,N(\vartheta_{m},\gamma^{(m)}),S(r_{m+1},\theta_{\gamma}^{(m+1)}),\dots,S(r_{n},\theta_{\gamma}^{(n)}))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\left(\prod_{k=1}^{m}f_{\lambda_{k}}(\vartheta_{k})\,\frac{d\vartheta_{k}}{\cos^{2}\vartheta_{k}}\right)\left(\prod_{k=m+1}^{n}g_{\lambda_{k}}(r_{k})\sinh
r_{k})\,dr_{k}\right),$

where

$F(0,\dots,0)=\int\limits_{(\log r_{1},\dots,\log r_{m})\in
P_{\gamma}}u(\varrho^{(1)}(r_{1}i),\dots,\varrho^{(2)}(r_{m}i),\varrho^{(m+1)}i,\dots,\varrho^{(n)}i)\prod_{k=1}^{m}\frac{dr_{k}}{r_{k}}.$

It remains to show that there are only finitely many mixed or totally
hyperbolic equivalence classes for which $T_{\gamma}$ is non-zero. Note that
since $\psi$ has compact support and

$N(\vartheta_{k},\gamma^{(k)})=\frac{N_{\gamma}^{(k)}+(N_{\gamma}^{(k)})^{-1}-2}{\cos^{2}\vartheta_{k}}\geq
N_{\gamma}^{(k)}+(N_{\gamma}^{(k)})^{-1}-2=\left|\textrm{tr}\,[\gamma^{(k)}]\right|-2,$

we get $T_{\gamma}=0$ once $\left|\textrm{tr}\,[\gamma^{(k)}]\right|$ is big
enough for some $k$. Hence it is enough to show that there are only finitely
many classes whose representatives have hyperbolic coordinates of bounded norm
(and trace).

By Theorem 1.1 $\Gamma$ is commensurable with a Hilbert modular group
$\Gamma_{K}$, so there is an $M_{\gamma}\in\mathbb{N}^{+}$ such that
$\gamma^{M_{\gamma}}$ is conjugate to an element in $\Gamma_{K}$, moreover,
the exponent $M_{\gamma}$ is bounded by a constant depending on $\Gamma$. It
follows that it is enough to show that there are only finitely many mixed or
totally hyperbolic classes in $\Gamma_{K}$ with coordinates of bounded trace.

Note that the trace of $(\gamma^{M_{\gamma}})^{(k)}$ is in
$\mathcal{O}_{K^{(k)}}$ for every $1\leq k\leq n$, and these values are
conjugates of each other. Hence if each of them is bounded, then the norm of
them is bounded as well, so there are finitely many possibilities for the
values of these traces. Finally, by Proposition I.7.1 and the paragraph after
Definition I.7.2 in [6], there are only finitely many totally hyperbolic
conjugacy classes with fixed traces, and this completes the proof of Theorem
2.3.

###  3.3. Proof in the hyperbolic-parabolic case

In this section we prove Theorem 2.6. In the course of the following proof the
statement of Proposition 2.4 will also be verified, but until that point of
our argument we always indicate any possible or evident dependence on a cusp.
Accordingly, we temporarily use the notations $\Lambda_{\kappa}$ for the
multiplier group and $\varepsilon_{j}^{\kappa}$ for its generators.

Recall that every hyperbolic-parabolic class is represented by an element that
fixes a cusp $\kappa\in\mathcal{S}$ and such an element is conjugated by the
scaling element $\sigma_{\kappa}\in PSL(2,\mathbb{R})^{n}$ to an element of
the form

(33)
$\gamma_{m,\alpha}^{\kappa}=\left[\begin{array}[]{cc}(u_{m}^{\kappa})^{1/2}&\alpha(u_{m}^{\kappa})^{-1/2}\\\
0&(u_{m}^{\kappa})^{-1/2}\end{array}\right],$

where $\alpha\in\mathbf{t}_{\kappa}$,
$m=(m_{1},\dots,m_{n-1})\in\mathbb{Z}^{n-1}\setminus\\{0\\}$ and
$u_{m}^{\kappa}=(\varepsilon_{1}^{\kappa})^{m_{1}}\dots(\varepsilon_{n-1}^{\kappa})^{m_{n-1}}\in\Lambda_{\kappa}$.
To simplify the notation, we often write $u_{m}$ instead of $u_{m}^{\kappa}$
in the following, at least when $\kappa$ is fixed. Note that the action of
$\gamma_{m,\alpha}^{\kappa}$ on a point $z\in\mathbb{H}^{n}$ can be written as
$\gamma_{m,\alpha}^{\kappa}z=u_{m}^{\kappa}z+\alpha$ and hence
$q=\frac{\alpha}{1-u_{m}^{\kappa}}$ is the real fixed vector of
$\gamma_{m,\alpha}^{\kappa}$ that is also a cusp for
$\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$ (as it was already mentioned in
Section 2.4).

A simple computation shows that if elements $\gamma_{m,\alpha}$ and
$\gamma_{{m^{\prime}},\beta}$ of the form (33) are conjugate in
$\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$, then $m^{\prime}=m$ or
$m^{\prime}=-m$ holds. Assume first that the elements $\gamma_{m,\alpha}$ and
$\gamma_{m,\beta}$ are conjugate to each other. Again, it follows by a
straightforward calculation that

$\beta=(u_{m}^{\kappa}-1)a+(u_{l}^{\kappa})^{-1}\alpha$

holds for some $l\in\mathbb{Z}^{n-1}$ and $a\in\mathbf{t}_{\kappa}$ in this
case. This means exactly that $\beta$ represents the same element in the
finite factor group
$\mathbf{t}_{\kappa}^{m}:=\mathbf{t}_{\kappa}/(u_{m}^{\kappa}-1)\mathbf{t}_{\kappa}$
as $(u_{l}^{\kappa})^{-1}\alpha$, and hence for a fixed $m$ (and $\kappa$) the
hyperbolic-parabolic classes are represented by the equivalence classes of
$\mathbf{t}_{\kappa}^{m}/\Lambda_{\kappa}$.

Now assume that the elements $\gamma_{m,\alpha}$ and $\gamma_{-m,\beta}$ are
conjugate for some $m\in\mathbb{Z}^{n-1}\setminus\\{0\\}$ and
$\alpha,\beta\in\mathbf{t}_{\kappa}$, i.e.
$\tau^{-1}\gamma_{m,\alpha}\tau=\gamma_{-m,\beta}$ for some
$\tau\in\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$. Since
$\tau^{-1}\gamma_{m,\alpha}\tau$ fixes $\tau^{-1}\infty$ and also
$\tau^{-1}q$, one of these points must be $\infty$. If
$\tau^{-1}\infty=\infty$ was true, then the conjugate would be of the form
$\gamma_{m,\beta}$, which is impossible (since $m\neq 0$). It follows that
$\tau^{-1}$ takes $q$ to $\infty$, hence these cusps are equivalent in
$\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$. Similarly, if these cusps are
equivalent then $\gamma_{m,\alpha}$ is conjugate to an element
$\gamma_{-m,\beta}$.

Based on this the contribution of the hyperbolic-parabolic classes in the
trace can be written as

(36)
$\displaystyle\frac{1}{2}\sum_{\kappa\in\mathcal{S}}\,\sum_{m\in\mathbb{Z}^{n-1}\setminus
0}\,\,\sum_{\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda_{\kappa}}\sum_{\begin{array}[]{c}\scriptstyle{\sigma\in
C(\gamma)\setminus\Gamma}\\\
\scriptstyle{\gamma\sim\gamma_{m,\alpha}^{\kappa}}\end{array}}\int_{F_{A}}k(z,\sigma^{-1}\gamma\sigma
z)u(z)\,d\mu(z)=$ (37)
$\displaystyle\quad=\frac{1}{2}\sum_{\kappa\in\mathcal{S}}\,\sum_{m\in\mathbb{Z}^{n-1}\setminus
0}\,\,\sum_{\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda_{\kappa}}\,\,\sum_{\sigma\in
C(\gamma_{m,\alpha}^{\kappa})\setminus\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}}\int_{\sigma(\sigma_{\kappa}^{-1}F_{A})}k(z,\gamma_{m,\alpha}^{\kappa}z)u(\sigma_{\kappa}z)\,d\mu(z),$

where $C(\gamma_{m,\alpha}^{\kappa})$ is the centralizer of
$\gamma_{m,\alpha}^{\kappa}$ in $\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$.
We multiply the whole sum by $1/2$ since every class is taken into account for
both fixed cusps of their elements except for those whose fixed points are
equivalent in $\Gamma$. But in the latter case we count these classes twice
for an $m$ and $-m$ as well.

Let us focus on the inner sum

(38) $\sum_{\sigma\in
C(\gamma_{m,\alpha}^{\kappa})\setminus\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}}\int_{\sigma(\sigma_{\kappa}^{-1}F_{A})}k(z,\gamma_{m,\alpha}^{\kappa}z)u(\sigma_{\kappa}z)\,d\mu(z).$

First, note that $k(z,\gamma_{m,\alpha}^{\kappa}z)$ can be written as

$\displaystyle\psi\left(\frac{(E_{m}^{(1)}x_{1}-\alpha_{1}(u_{m}^{(1)})^{-1/2})^{2}}{y_{1}^{2}}+(E_{m}^{(1)})^{2},\dots,\frac{(E_{m}^{(n)}x_{n}-\alpha_{n}(u_{m}^{(n)})^{-1/2})^{2}}{y_{n}^{2}}+(E_{m}^{(n)})^{2}\right),$

where $E_{m}=E_{m}^{\kappa}=u_{m}^{-1/2}-u_{m}^{1/2}$. Since $\psi$ is
compactly supported, it follows immediately that (38) is zero for all but
finitely many $m$, and hence the sum in (36) is finite.

The union of the sets $\sigma(\sigma_{\kappa}^{-1}F_{A})$ in the integrals
above, where $\sigma$ runs through the right cosets of the centralizer
$C(\gamma_{m,\alpha}^{\kappa})$, makes up the fundamental domain
$F_{C(\gamma_{m,\alpha}^{\kappa})}$ of $C(\gamma_{m,\alpha}^{\kappa})$ except
for the images of the part $F\setminus F_{A}=:F^{*}_{A}$. We will now show
that for some cosets it is unnecessary to omit the images of $F_{A}^{*}$,
since there we integrate only the zero function. For this we write the kernel
function in the form

(39)
$k(z,\gamma_{m,\alpha}^{\kappa}z)=\psi\left(\frac{\left|z_{1}-q_{1}\right|^{2}}{(E_{m}^{(1)})^{-2}y_{1}^{2}},\dots,\frac{\left|z_{n}-q_{n}\right|^{2}}{(E_{m}^{(n)})^{-2}y_{n}^{2}}\right).$

The part $\sigma(\sigma_{\kappa}^{-1}F^{*}_{A})$ is the same as

$\bigcup_{\kappa^{\prime}}\\{z\in
F_{C({\gamma_{m,\alpha}^{\kappa}})}:\,\sigma_{\kappa}\sigma^{-1}z\in
F,\,Y_{0}^{\kappa^{\prime}}(\sigma_{\kappa}\sigma^{-1}z)\geq A\\}.$

The condition $Y_{0}^{\kappa^{\prime}}(\sigma_{\kappa}\sigma^{-1}z)\geq A$
means that $z\in\sigma\sigma_{\kappa}^{-1}\sigma_{\kappa^{\prime}}U_{A}$, that
is, there is a $w\in U_{A}$ such that
$z=\sigma\sigma_{\kappa}^{-1}\sigma_{\kappa^{\prime}}w$. Let us define
$\nu=\sigma\sigma_{\kappa}^{-1}\sigma_{\kappa^{\prime}}=\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]$. If $w=u+iv$, then the product of the expressions in
the arguments on the right hand side of (39) is

$\displaystyle\prod_{k=1}^{n}\frac{\left|\nu_{k}w_{k}-q_{k}\right|^{2}}{(E_{m}^{(k)})^{-2}y_{k}(\nu
w)^{2}}$ $\displaystyle=\frac{N(E_{m}^{2})}{Y_{0}(\nu
w)^{2}}\prod_{k=1}^{n}\left|\nu_{k}w_{k}-\frac{\alpha_{k}}{1-u_{m}^{(k)}}\right|^{2}$
$\displaystyle=\frac{N((1-u_{m})^{2})}{Y_{0}(\nu
w)^{2}}\prod_{k=1}^{n}\left|\frac{a_{k}w_{k}+b_{k}}{c_{k}w_{k}+d_{k}}-\frac{\alpha_{k}}{1-u_{m}^{(k)}}\right|^{2}$
$\displaystyle\geq\frac{N((1-u_{m})^{2})}{Y_{0}(\nu
w)^{2}}\prod_{k=1}^{n}\frac{((1-u_{m}^{(k)})a_{k}-\alpha_{k}c_{k})^{2}v_{k}^{2}}{(1-u_{m}^{(k)})^{2}\left|c_{k}w_{k}+d_{k}\right|^{2}}$
$\displaystyle=N((1-u_{m})a-\alpha
c)^{2}\prod_{k=1}^{n}\left|c_{k}w_{k}+d_{k}\right|^{2}$ $\displaystyle\geq
N((1-u_{m})a-\alpha c)^{2}N(c)^{2}Y_{0}(w)^{2}.$

The first two factors of the last product are bounded away from zero by Lemma
$2.9_{1}$ in [7], at least if they are non-zero. This follows for the second
factor easily but requires some explanation in the case of the first factor.
The point $q$ is a cusp for $\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$, hence
$q=\sigma_{\kappa}^{-1}\beta$ for some cusp $\beta$ of $\Gamma$. Let
$\lambda\in\mathcal{S}$ denote the base element of the $\Gamma$-equivalence
class of $\beta$, then $q=\sigma_{\kappa}^{-1}\gamma_{0}\lambda$ for some
$\gamma_{0}\in\Gamma$, i.e.
$\sigma_{\lambda}^{-1}\gamma_{0}^{-1}\sigma_{\kappa}\in\sigma_{\lambda}^{-1}\Gamma\sigma_{\kappa}$
takes $q$ to $\infty$, hence it is of the form
$\left[\begin{array}[]{cc}e&f\\\
\frac{u_{m}-1}{\delta}&\frac{\alpha}{\delta}\end{array}\right]$, where
$\delta=e\alpha+f(1-u_{m})$.

We first show that $N(\delta^{2})$ depends only on $q$, i.e. on $m$ and
$\alpha$. Assume that $\sigma_{\lambda}^{-1}\gamma^{\prime}\sigma_{\kappa}$
also takes $q$ to $\infty$ for some $\gamma^{\prime}\in\Gamma$, and therefore
it is of the form $\left[\begin{array}[]{cc}e^{\prime}&f^{\prime}\\\
\frac{u_{m}-1}{\delta^{\prime}}&\frac{\alpha}{\delta^{\prime}}\end{array}\right]$.
Now

$(\sigma_{\lambda}^{-1}\gamma^{\prime}\sigma_{\kappa})(\sigma_{\lambda}^{-1}\gamma_{0}^{-1}\sigma_{\kappa})^{-1}=\sigma_{\lambda}^{-1}\gamma^{\prime}\gamma_{0}\sigma_{\lambda}\in\sigma_{\lambda}^{-1}\Gamma\sigma_{\lambda}$

fixes $\infty$ (which is a cusp for
$\sigma_{\lambda}^{-1}\Gamma\sigma_{\lambda}$), and then

$\left[\begin{array}[]{cc}e^{\prime}&f^{\prime}\\\
\frac{u_{m}-1}{\delta^{\prime}}&\frac{\alpha}{\delta^{\prime}}\end{array}\right]\left[\begin{array}[]{cc}e&f\\\
\frac{u_{m}-1}{\delta}&\frac{\alpha}{\delta}\end{array}\right]^{-1}=\left[\begin{array}[]{cc}e^{\prime}&f^{\prime}\\\
\frac{u_{m}-1}{\delta^{\prime}}&\frac{\alpha}{\delta^{\prime}}\end{array}\right]\left[\begin{array}[]{cc}\frac{\alpha}{\delta}&-f\\\
\frac{1-u_{m}}{\delta}&e\end{array}\right]=\left[\begin{array}[]{cc}\pm
u^{\frac{1}{2}}&t\\\ 0&\pm u^{-\frac{1}{2}}\end{array}\right]$

holds for some $u\in\Lambda_{\lambda}$ and $t\in\mathbf{t}_{\lambda}$. It
follows that $\delta^{\prime}\,{}^{2}=u\delta^{2}$, hence
$N(\delta^{2})=N(\delta^{\prime 2})$.

Now

$\sigma_{\lambda}^{-1}\gamma_{0}^{-1}\sigma_{\kappa}\nu=\left[\begin{array}[]{cc}*&*\\\
-\frac{(1-u_{m})a-\alpha
c}{\delta}&*\end{array}\right]\in\sigma_{\lambda}^{-1}\Gamma\sigma_{\kappa^{\prime}},$

and choosing $\Delta:=\sigma_{\lambda}^{-1}\Gamma\sigma_{\kappa^{\prime}}$ in
Lemma $2.9_{1}$ of [7] we get that $N((1-u_{m})a-\alpha c)^{2}\geq
N(\delta^{2})C_{0}$ for some positive constant $C_{0}>0$ (assuming that
$(1-u_{m})a-\alpha c\neq 0$).

Next we handle the cases where one of the factors $N(c)^{2}$ and
$N((1-u_{m})a-\alpha c)^{2}$ is $0$. Since
$\sigma=\sigma_{\kappa}^{-1}\gamma\sigma_{\kappa}$ for some $\gamma\in\Gamma$,
hence $c=0$ holds if and only if

$\infty=\nu\infty=\sigma\sigma_{\kappa}^{-1}\sigma_{\kappa^{\prime}}\infty=\sigma_{\kappa}^{-1}\gamma\kappa^{\prime},$

that is, $\kappa=\sigma_{\kappa}\infty=\gamma\kappa^{\prime}$. But $\Gamma$
permutes the elements of the class of $\kappa^{\prime}$, so $c=0$ can hold
only if $\kappa=\kappa^{\prime}$, and then $\nu=\sigma$, and therefore
$\infty=\sigma\infty$ holds. Consequently, the condition
$Y_{0}^{\kappa^{\prime}}(\sigma_{\kappa}\sigma^{-1}z)\geq A$ reduces to
$Y_{0}(\sigma^{-1}z)=Y_{0}(z)\geq A$ in this case.

Assume now that $(1-u_{m})a-\alpha c=0$, that is, $\frac{a}{c}=q$ holds. This
means that

$\sigma_{\kappa}^{-1}\gamma_{0}\lambda=q=\nu\infty=\sigma_{\kappa}^{-1}\gamma\kappa^{\prime},$

hence $\kappa^{\prime}=\lambda$ must hold. This means that the exceptional set
$\sigma(\sigma_{\kappa}^{-1}F^{*}_{A})$ can be reduced to

$\\{z\in
F_{C({\gamma_{m,\alpha}^{\kappa}})}:\,Y_{0}(\sigma_{\lambda}^{-1}\sigma_{\kappa}\sigma^{-1}z)\geq
A\\}.$

The element $\sigma_{\lambda}^{-1}\sigma_{\kappa}\sigma^{-1}$ takes $q$ to
$\infty$, hence - as we have already seen above - it is of the form
$\left[\begin{array}[]{cc}e&f\\\
\frac{u_{m}-1}{\delta}&\frac{\alpha}{\delta}\end{array}\right]$, where
$\delta=e\alpha+f(1-u_{m})$, and then

$Y_{0}(\sigma_{\lambda}^{-1}\sigma_{\kappa}\sigma^{-1}z)=\frac{N(\delta^{2})Y_{0}(z)}{\prod_{k=1}^{n}\left|(u_{m}^{(k)}-1)z_{k}+\alpha_{k}\right|^{2}}=\frac{N(\delta
E_{m}^{-1})^{2}Y_{0}(z)}{\prod_{k=1}^{n}\left|z_{k}-q_{k}\right|^{2}}.$

All this shows that the expression in (38) can be written as

$\int_{S_{A}}k(z,\gamma_{m,\alpha}^{\kappa}z)u(\sigma_{\kappa}z)\,d\mu(z),$

where

$S_{A}=\left\\{z\in F_{C(\gamma_{m,\alpha}^{\kappa})}:\,Y_{0}(z)\leq
A,\,\frac{N(\delta
E_{m}^{-1})^{2}Y_{0}(z)}{\prod_{k=1}^{n}\left|z_{k}-q_{k}\right|^{2}}\leq
A\right\\},$

at least when $A$ is big enough. Recall that the centralizer
$C(\gamma_{m,\alpha}^{\kappa})$ and its fundamental domain was described in
Section 2.4. In the following we also use some relating notations defined
there. A direct calculation shows now that the last integral above is

$\displaystyle\int_{S_{A}}\psi$
$\displaystyle\left(\frac{\left|z_{1}-q_{1}\right|^{2}}{(E_{m}^{(1)})^{-2}y_{1}^{2}},\dots,\frac{\left|z_{n}-q_{n}\right|^{2}}{(E_{m}^{(n)})^{-2}y_{n}^{2}}\right)u(\sigma_{\kappa}z)\,d\mu(z)=$
$\displaystyle\qquad\qquad\qquad=\int\limits_{S_{A}-q}\psi\left(\frac{|E_{m}^{(1)}z_{1}|^{2}}{y_{1}^{2}},\dots,\frac{|E_{m}^{(n)}z_{n}|^{2}}{y_{n}^{2}}\right)u(\sigma_{\kappa}(z+q))\,d\mu(z),$

where

$S_{A}-q=\left\\{z\in F_{C}:\,Y_{0}(z)\leq A,\,\frac{N(\delta
E_{m}^{-1})^{2}Y_{0}(z)}{\prod_{k=1}^{n}\left|z_{k}\right|^{2}}\leq
A\right\\}.$

The two inequalities above can be written in terms of the polar coordinates as
follows:

$A_{\vartheta}:=\frac{N(\delta
E_{m}^{-1})^{2}\prod_{k=1}^{n}\cos\vartheta_{k}}{A}\leq\prod_{k=1}^{n}r_{k}\leq\frac{A}{\prod_{k=1}^{n}\cos\vartheta_{k}}=:A^{\vartheta}.$

Since
$\displaystyle{\frac{dx\,dy}{y^{2}}=\frac{dr\,d\vartheta}{r\cos^{2}\vartheta}}$,
the integral above becomes after the change of variables

(40)
$\int\limits_{-\pi/2}^{\pi/2}\dots\int\limits_{-\pi/2}^{\pi/2}\psi\left(\frac{(E_{m}^{(1)})^{2}}{\cos^{2}\vartheta_{1}},\dots,\frac{(E_{m}^{(n)})^{2}}{\cos^{2}\vartheta_{n}}\right)I_{u}(A,\vartheta,m,\alpha)\prod_{k=1}^{n}\frac{d\vartheta_{k}}{\cos^{2}\vartheta_{k}}$

where

$I_{u}(A,\vartheta,m,\alpha)=\int\limits_{\begin{array}[]{c}\scriptstyle{\log
r\in P_{m,\alpha}^{\kappa}}\\\ \scriptstyle{A_{\vartheta}\leq Nr\leq
A^{\vartheta}}\end{array}}u\left(\sigma_{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)}+q)\right)\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}.$

We handle the zeroth term and the remaining terms of the Fourier expansion of
the function $u(\sigma_{\kappa}(z+q))$ separately. To this end, for any cusp
$\kappa^{\prime}$ we write

$u(\sigma_{\kappa^{\prime}}z)=\eta_{\kappa^{\prime}}y_{1}^{s_{1}}\dots
y_{n}^{s_{n}}+\phi_{\kappa^{\prime}}y_{1}^{1-s_{1}}\dots
y_{n}^{1-s_{n}}+R_{\kappa^{\prime}}(z)=M_{\kappa^{\prime}}(z)+R_{\kappa^{\prime}}(z).$

Subtracting the contribution of the zeroth term
$M_{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)}+q)$ from the integral
$I_{u}(A,\vartheta,m,\alpha)$ one obtains

$\int\limits_{\begin{array}[]{c}\scriptstyle{\log r\in
P_{m,\alpha}^{\kappa}}\\\ \scriptstyle{A_{\vartheta}\leq Nr\leq
A^{\vartheta}}\end{array}}R_{\kappa}\left(re^{i(\frac{\pi}{2}+\vartheta)}+q\right)\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}.$

However, this integral does not converge as $A\to\infty$, but Proposition 1.3
gives that it does converge if one integrates only over $\\{\log r\in
P_{m,\alpha}^{\kappa}:1\leq Nr\\}$. Note that by the compact support of $\psi$
the coordinates of the vector $\cos\vartheta$ can be assumed to be bounded
away from zero and hence $R_{\kappa}$ in the above integral can be bounded
uniformly exponentially using Proposition 1.3. To ensure convergence on the
other half of the set $\\{\log r\in P_{m,\alpha}^{\kappa}\\}$ we will subtract
the main term of $u$ at $q$. By the $\Gamma$-invariance of $u$ we have

$u(\sigma_{\kappa}(z+q))=u(\sigma_{\lambda}(\sigma_{\lambda}^{-1}\gamma_{0}^{-1}\sigma_{\kappa}(z+q)))=u\left(\sigma_{\lambda}\left(-e(eq+f)-\frac{(eq+f)^{2}}{z}\right)\right),$

hence (after the substitution $r\mapsto\frac{1}{r}$) the integral of
$u\left(\sigma_{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)}+q)\right)$ over the
set $\\{\log r\in P_{m,\alpha}^{\kappa}:A_{\vartheta}\leq Nr\leq 1\\}$ becomes

$\displaystyle\int\limits_{\begin{array}[]{c}\scriptstyle{\log r\in
P_{m,\alpha}^{\kappa}}\\\ \scriptstyle{1\leq Nr\leq
A_{\vartheta}^{-1}}\end{array}}\\!\\!\\!\\!\\!\\!u\left(\sigma_{\lambda}\left(-e(eq+f)+(eq+f)^{2}re^{i(\frac{\pi}{2}-\vartheta)}\right)\right)\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}=$
$\displaystyle\qquad=\int\limits_{\begin{array}[]{c}\scriptstyle{\log r\in
P_{m,\alpha}^{\kappa}}\\\ \scriptstyle{1\leq Nr\leq
A_{\vartheta}^{-1}}\end{array}}\\!\\!\\!\\!\\!\\!\\!M_{\lambda}\left((eq+f)^{2}re^{i(\frac{\pi}{2}-\vartheta)}\right)+R_{\lambda}\left((eq+f)^{2}re^{i(\frac{\pi}{2}-\vartheta)}-e(eq+f)\right)\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}.$

Here we also used the translation invariance of $M_{\lambda}$. It is now clear
that the integral of the second term above converges, i.e.

$u\left(\sigma_{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)}+q)\right)-M_{\lambda}\left(-\frac{(eq+f)^{2}}{re^{i(\frac{\pi}{2}+\vartheta)}}\right)=u\left(\sigma_{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)}+q)\right)-M_{\lambda}\left(-\frac{\delta^{2}E_{m}^{-2}u_{m}^{-1}}{re^{i(\frac{\pi}{2}+\vartheta)}}\right)$

is integrable over $\\{\log r\in P_{m,\alpha}^{\kappa}:\,Nr\leq 1\\}$. A
straightforward computation (detailed below) shows the same for
$M_{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)})$ and then consequently for the
function

$\tilde{u}_{m,\alpha}^{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)})=u\left(\sigma_{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)}+q)\right)-M_{\kappa}\left(re^{i(\frac{\pi}{2}+\vartheta)}\right)-M_{\lambda}\left(-\frac{\delta^{2}E_{m}^{-2}u_{m}^{-1}}{re^{i(\frac{\pi}{2}+\vartheta)}}\right).$

A similar argument gives that
$\tilde{u}_{m,\alpha}^{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)})$ is integrable
over $\\{\log r\in P_{m,\alpha}^{\kappa}:1\leq Nr\\}$ and hence over the whole
set $\\{\log r\in P_{m,\alpha}^{\kappa}\\}$.

Now $I_{u}(A,\vartheta,m,\alpha)$ can be written as

(41) $\int\limits_{\begin{array}[]{c}\scriptstyle{\log r\in
P_{m,\alpha}^{\kappa}}\\\ \scriptstyle{A_{\vartheta}\leq Nr\leq
A^{\vartheta}}\end{array}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[M_{\kappa}\\!\left(re^{i(\frac{\pi}{2}+\vartheta)}\right)+M_{\lambda}\\!\left(-\frac{\delta^{2}E_{m}^{-2}u_{m}^{-1}}{re^{i(\frac{\pi}{2}+\vartheta)}}\right)\right]\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}+\\!\\!\int\limits_{\log
r\in
P_{m,\alpha}^{\kappa}}\\!\\!\\!\\!\\!\\!\\!\\!\tilde{u}_{m,\alpha}^{\kappa}(re^{i(\frac{\pi}{2}+\vartheta)})\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}+o(A).$

First we turn to the second integral above. Observe that the function
$U(z):=u(\sigma_{\kappa}(z+q))$ is invariant under the action of
$\rho_{l_{j}}$ (defined in (27)):

$\displaystyle U(\rho_{l_{j}}z)$
$\displaystyle=u(\sigma_{\kappa}(u_{l_{j}}z+q))=u(\sigma_{\kappa}(u_{l_{j}}(z+q)+(1-u_{l_{j}})q))$
$\displaystyle=u\left(\sigma_{\kappa}\gamma(l_{j})(z+q)\right)=u(\sigma_{\kappa}(z+q))=U(z),$

because the element $\gamma(l_{j})$ is in the centralizer
$C(\gamma_{\alpha,m}^{\kappa})\leq\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$,
and $u$ is invariant under the action of $\Gamma$. The same invariance holds
for $M_{\kappa}(z)$ and
$M_{\lambda}\left(-\frac{\delta^{2}E_{m}^{-2}u_{m}^{-1}}{z}\right)$ as well,
hence the invariance of $\tilde{u}_{m,\alpha}^{\kappa}(z)$ under the action of
$\rho_{lj}$ follows.

Let us define the function

$F^{\kappa}_{m,\alpha}(z)=\int\limits_{\log r\in
P_{m,\alpha}^{\kappa}}\tilde{u}_{m,\alpha}^{\kappa}(rz)\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}.$

By the observation of the last paragraph we can see as in the case of mixed
elements that $F^{\kappa}_{m,\alpha}$ is invariant under coordinate-wise
scalar multiplication, i.e.
$F^{\kappa}_{m,\alpha}(z)=F^{\kappa}_{m,\alpha}(\vartheta)$ depends only on
$\vartheta$ where $z=re^{i(\frac{\pi}{2}+\vartheta)}$.

Since the Laplacian $\Delta_{k}$ is an invariant operator (i.e. it commutes
with the action of $PSL(2,\mathbb{R})^{n}$), we get that $U(z)$ is an
eigenfunction of it with the eigenvalue $\lambda_{k}$. But the same is true
$M_{\kappa}(z)$ and
$M_{\lambda}\left(-\frac{\delta^{2}E_{m}^{-2}u_{m}^{-1}}{z}\right)$, and
therefore $\tilde{u}_{m,\alpha}^{\kappa}(z)$ and also
$F^{\kappa}_{m,\alpha}(z)$ are eigenfunctions of $\Delta_{k}$ with the
eigenvalue $\lambda_{k}$. In the same way as in the case of mixed elements we
conclude that the contribution of the second integral of (41) in (40) is

$\displaystyle F^{\kappa}_{m,\alpha}(0,\dots,0)$
$\displaystyle\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dots\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\psi\left(\frac{(E_{m}^{(1)})^{2}}{\cos^{2}\vartheta_{1}},\dots,\frac{(E_{m}^{(n)})^{2}}{\cos^{2}\vartheta_{n}}\right)\left(\prod_{k=1}^{n}\frac{f_{\lambda_{k}}(\vartheta_{k})\,d\vartheta_{k}}{\cos^{2}\vartheta_{k}}\right),$

where $f_{\lambda_{k}}(\vartheta)$ is the unique solution of the differential
equation (23) with the initial condition $f_{\lambda_{k}}(0)=1$ and
$f^{\prime}_{\lambda_{k}}(0)=0$ and

$F^{\kappa}_{m,\alpha}(0,\dots,0)=\int\limits_{\log r\in
P_{m,\alpha}^{\kappa}}\tilde{u}(r_{1}i,\dots,r_{n}i)\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}.$

Finally we calculate the first integral in (41). The map $\,r\mapsto(\log
r_{1},\dots,\log r_{n})$ maps the set we integrate on to

$\mathcal{F}_{A}:=\bigcup_{n^{-\frac{1}{2}}\log A_{\vartheta}\leq t\leq
n^{-\frac{1}{2}}\log
A^{\vartheta}}\\{t\mathbf{1}\\}\times\tilde{P}_{m,\alpha}^{\kappa},$

and the determinant of its Jacobian is $\prod_{k=1}^{n}r_{k}^{-1}$, so the
integral is

(42)
$\displaystyle\int\limits_{\mathcal{F}_{A}}M_{\kappa}\left(e^{x+i(\frac{\pi}{2}+\vartheta)}\right)+M_{\lambda}\left(-\frac{\delta^{2}E_{m}^{-2}u_{m}^{-1}}{e^{x+i(\frac{\pi}{2}+\vartheta)}}\right)\,\prod_{k=1}^{n}dx_{k}.$

The first term of the zeroth Fourier coefficient $M_{\kappa}$ gives the
following contribution to the integral above:

(43)
$\displaystyle\eta_{\kappa}\prod_{k=1}^{n}\left(\cos\vartheta_{k}\right)^{s_{k}}\int\limits_{\mathcal{F}_{A}}\exp\left(\sum_{k=1}^{n}s_{k}x_{k}\right)\,\prod_{k=1}^{n}\,dx_{k}.$

Let $\mathcal{L}$ be the linear map that maps the standard basis of
$\mathbb{R}^{n}$ to $\mathbf{1},v_{1}^{\kappa},\dots,v_{n-1}^{\kappa}$ (see
Section 2.4 for the definitions) so that $\mathcal{L}([n^{-\frac{1}{2}}\log
A_{\vartheta};n^{-\frac{1}{2}}\log
A^{\vartheta}]\times[-\frac{1}{2};\frac{1}{2}]^{n-1})=\mathcal{F}_{A}$. The
matrix of this map w.r.t. the standard basis is
$[\mathcal{L}]=\mathcal{E}_{\kappa}\left[\begin{array}[]{cc}n^{-\frac{1}{2}}&\\\
&L_{m,\alpha}^{\kappa}\end{array}\right]$, where $\mathcal{E}_{\kappa}$ is
defined in (8) and $L_{m,\alpha}^{\kappa}$ is defined in (24). Hence, after a
change of variables, the integral in (43) becomes

$\displaystyle\left|\det[\mathcal{L}]\right|$
$\displaystyle\int\limits_{n^{-\frac{1}{2}}\log
A_{\vartheta}}^{n^{-\frac{1}{2}}\log
A^{\vartheta}}e^{n^{-\frac{1}{2}}t_{0}\sum_{k=1}^{n}s_{k}}\,dt_{0}\prod_{q=1}^{n-1}\int\limits_{-1/2}^{1/2}\exp\left(t_{q}\sum_{j=1}^{n-1}l_{q}^{(j)}\sum_{k=1}^{n}s_{k}\log\varepsilon_{j}^{\kappa}\,{}^{(k)}\right)\,dt_{q}=$
(44)
$\displaystyle=\left|\det[\mathcal{L}]\right|\int\limits_{n^{-\frac{1}{2}}\log
A_{\vartheta}}^{n^{-\frac{1}{2}}\log
A^{\vartheta}}e^{t_{0}n^{\frac{1}{2}}s}\,dt_{0}\prod_{q=1}^{n-1}\int\limits_{-1/2}^{1/2}\exp\left(2\pi
it_{q}\sum_{j=1}^{n-1}l_{q}^{(j)}m_{u,\kappa}^{(j)}\right)\,dt_{q}$

by (8). This expression is zero unless
$(L_{m,\alpha}^{\kappa})^{T}m_{u,\kappa}=0$ holds. Since

$0\neq\det[\mathcal{L}]=n^{-\frac{1}{2}}\det\mathcal{E}_{\kappa}\det
L_{m,\alpha}^{\kappa},$

i.e. $\det L_{m,\alpha}^{\kappa}\neq 0$, this can hold only if
$m_{u,\kappa}=0$. In the latter case (43) becomes

$\frac{\eta_{\kappa}\left|\det\mathcal{E}_{\kappa}\right|\left|\det
L_{m,\alpha}^{\kappa}\right|}{ns}\left[A^{s}-\frac{N(\delta
E_{m}^{-1})^{2s}\prod_{k=1}^{n}(\cos\vartheta_{k})^{2s}}{A^{s}}\right]=\frac{\eta_{\kappa}\left|\det\mathcal{E}_{\kappa}\right|\left|\det
L_{m,\alpha}^{\kappa}\right|A^{s}}{ns}+o(A).$

Here we used that since $\psi$ has a compact support, the values
$\cos\vartheta_{k}$ are bounded away from zero by a constant depending on
$\psi$ and $\Gamma$. The same argument gives that the contribution of the
second term of the zeroth Fourier coefficient $M_{\kappa}$ in
$I_{u}(A,\vartheta,m,\alpha)$ is

(45) $\frac{\phi_{\kappa}\left|\det\mathcal{E}_{\kappa}\right|\left|\det
L_{m,\alpha}^{\kappa}\right|A^{1-s}}{n(1-s)}+o(A).$

Now we turn to the integral of the second term in (42). The first term of this
Fourier coefficient contributes

$\displaystyle\eta_{\lambda}\prod_{k=1}^{n}\left((\delta^{(k)})^{2}(E_{m}^{(k)})^{-2}(u_{m}^{(k)})^{-1}\cos\vartheta_{k}\right)^{s_{k}}\int\limits_{\mathcal{F}_{A}}\exp\left(-\sum_{k=1}^{n}s_{k}x_{k}\right)\,\prod_{k=1}^{n}\,dx_{k}.$

The same computation as above shows that this is zero unless $m_{u,\kappa}=0$,
in which case one gets

$\displaystyle\eta_{\lambda}\left|\det[\mathcal{L}]\right|N(\delta^{2}E_{m}^{-2})^{s}$
$\displaystyle\prod_{k=1}^{n}(\cos\vartheta_{k})^{s}\int\limits_{n^{-\frac{1}{2}}\log(A^{\vartheta})^{-1}}^{n^{-\frac{1}{2}}\log
A_{\vartheta}^{-1}}\\!\\!\\!\\!\\!\\!\\!e^{t_{0}n^{\frac{1}{2}}s}\,dt_{0}=\frac{\eta_{\lambda}\left|\det\mathcal{E}_{\kappa}\right|\left|\det
L_{m,\alpha}\right|}{ns}A^{s}+o(A).$

Finally, we get one more term which is of the form as the one in (45),
replacing $\phi_{\kappa}$ by $\phi_{\lambda}$.

We conclude that (42) contributes in (40) (aside from an $o(A)$ term) as

$\left(\frac{(\eta_{\kappa}+\eta_{\lambda})A^{s}}{s}+\frac{(\phi_{\kappa}+\phi_{\lambda})A^{1-s}}{1-s}\right)\frac{\left|\det
L_{m,\alpha}^{\kappa}\right|\left|\det\mathcal{E}_{\kappa}\right|}{n}\\!\\!\\!\int\limits_{-\pi/2}^{\pi/2}\\!\\!\\!\\!\dots\\!\\!\\!\\!\int\limits_{-\pi/2}^{\pi/2}\\!\\!\psi\left(\frac{E_{m}^{2}}{\cos^{2}\vartheta}\right)\prod_{k=1}^{n}\frac{d\vartheta_{k}}{\cos^{2}\vartheta_{k}},$

where the last integral is

$\displaystyle
2^{n}\int\limits_{0}^{\frac{\pi}{2}}\dots\int\limits_{0}^{\frac{\pi}{2}}\psi\left(\frac{(E_{m}^{(1)})^{2}}{\cos^{2}\vartheta_{1}},\dots,\frac{(E_{m}^{(n)})^{2}}{\cos^{2}\vartheta_{n}}\right)\prod_{k=1}^{n}\frac{d\vartheta_{k}}{\cos^{2}\vartheta_{k}}=$
$\displaystyle=\frac{1}{N(\left|E_{m}\right|)}\int\limits_{(E_{m}^{(1)})^{2}}^{\infty}\dots\int\limits_{(E_{m}^{(n)})^{2}}^{\infty}\frac{\psi(t_{1},\dots,t_{n})}{\prod\limits_{k=1}^{n}\sqrt{t_{k}-(E_{m}^{(k)})^{2}}}\,\prod_{k=1}^{n}\,dt_{k}=\frac{1}{N(\left|E_{m}\right|)}g(\log
u_{m}^{(1)},\dots,\log u_{m}^{(n)}).$

We summarize the results obtained so far. The contribution of the hyperbolic-
parabolic classes in the truncated trace is

(46) $\Sigma_{\mathrm{hyp-
par}}=\frac{1}{2}\sum_{\kappa\in\mathcal{S}}\,\,\sum_{m\in\mathbb{Z}^{n-1}\setminus\\{0\\}}\,\,\sum_{\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda_{\kappa}}\,\left[\delta_{m_{u,\kappa}}M_{\kappa}(m,\alpha,A)+C_{\kappa}(m,\alpha)\right]+o(A),$

where the main term $M_{\kappa}(m,\alpha,A)$ is given by

$\left(\frac{(\eta_{\kappa}+\eta_{\tilde{\kappa}_{m,\alpha}})A^{s}}{s}+\frac{(\phi_{\kappa}+\phi_{\tilde{\kappa}_{m,\alpha}})A^{1-s}}{1-s}\right)\frac{\left|\det
L_{m,\alpha}^{\kappa}\right|\left|\det\mathcal{E}_{\kappa}\right|}{nN(\left|E_{m}^{\kappa}\right|)}g(\log
u_{m}^{\kappa}),$

where $\tilde{\kappa}_{m,\alpha}\in\mathcal{S}$ is the cusp for $\Gamma$ that
can be taken (by an element of $\Gamma$) to
$\sigma_{\kappa}\frac{\alpha}{1-u_{m}^{\kappa}}$, and the term
$C_{\kappa}(m,\alpha)$ is

(47) $\displaystyle\int\limits_{\log r\in
P_{m,\alpha}^{\kappa}}\tilde{u}_{m,\alpha}^{\kappa}(ri)\prod_{k=1}^{n}\frac{dr_{k}}{r_{k}}$
$\displaystyle\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dots\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\psi\left(\frac{(E_{m}^{\kappa})^{2}}{\cos^{2}\vartheta}\right)\left(\prod_{k=1}^{n}\frac{f_{\lambda_{k}}(\vartheta_{k})\,d\vartheta_{k}}{\cos^{2}\vartheta_{k}}\right),$

Moreover, the terms above are zero for any cusp $\kappa$ for all but finitely
many $m$.

Let us fix a $\kappa\in\mathcal{S}$, an $m\in\mathbb{Z}^{n-1}\setminus\\{0\\}$
and an $\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda_{\kappa}$ in the sum (46)
above. The corresponding term is counted twice, it occurs also in the case of
the cusp $\kappa^{\prime}=\tilde{\kappa}_{m,\alpha}$ for an appropriate
$m^{\prime}\in\mathbb{Z}^{n-1}\setminus\\{0\\}$ and a class $\beta$. It
follows that the main terms $\delta_{m_{u,\kappa}}M_{\kappa}(m,\alpha,A)$ and
$\delta_{m_{u,\kappa^{\prime}}}M_{\kappa^{\prime}}(m^{\prime},\beta,A)$ are
equal.

At this point we specify the function $u(z)$, namely, we work with the
Eisenstein series $E_{\kappa}(z,s,0)$ for some fixed $\frac{1}{2}<s<1$
(defined in (11)). It is not a cusp form and $s_{1}=\ldots=s_{n}$ hold for its
eigenvalues, and therefore $m_{u,\kappa}=m_{u,\kappa^{\prime}}=0$ must hold by
(8) which implies
$M_{\kappa}(m,\alpha,A)=M_{\kappa^{\prime}}(m^{\prime},\beta,A)$. Also, since
$\eta_{\kappa}=1$ and $\eta_{\kappa^{\prime}}=0$, the first factor of each of
these main terms is non-zero.

Assume that $u_{m}^{\kappa}\neq u_{m^{\prime}}^{\tilde{\kappa}_{m,\alpha}}$
holds, then we can choose the function $g$ so that exactly one of $g(\log
u_{m}^{\kappa})$ and $g(\log u_{m}^{\tilde{\kappa}_{m,\alpha}})$ is zero. This
yields that exactly one of $M_{\kappa}(m,\alpha,A)$ and
$M_{\tilde{\kappa}_{m,\alpha}}(m^{\prime},\beta,A)$ is zero, which is
impossible, and hence
$u_{m}^{\kappa}=u_{m^{\prime}}^{\tilde{\kappa}_{m,\alpha}}$ must hold.

Let us choose the vector $m=\mathbf{e}_{j}=(0,\dots,0,1,0,\dots,0)^{T}$ whose
the $j$th coordinate is $1$ and the others are zero ($1\leq j\leq n-1$). Then
$u_{\mathbf{e}_{j}}^{\kappa}=\varepsilon_{j}^{\kappa}$, and it follows that
$\varepsilon_{j}^{\kappa}\in\Lambda_{\tilde{\kappa}_{m,\alpha}}$, i.e.
$\Lambda_{\kappa}\subset\Lambda_{\tilde{\kappa}_{m,\alpha}}$. Changing the
role of $\kappa$ and $\tilde{\kappa}_{m,\alpha}$ in the previous argument and
specifying $m^{\prime}$ instead of $m$ we infer that
$\Lambda_{\tilde{\kappa}_{m,\alpha}}\subset\Lambda_{\kappa}$ hence these
groups are identical. Hence to conclude the proof of Proposition 2.4., that
is, to show that the multiplier group $\Lambda_{\kappa}$ is independent of
$\kappa$ it is enough to prove the following:

######  Lemma 3.1.

For any two different cusps $\kappa$, $\kappa^{\prime}$ there is a hyperbolic-
parabolic element in $\Gamma$ with fixed points $\kappa$ and
$\kappa^{\prime}$.

###### Proof.

In the first step we show the analogous statement for the Hilbert modular
group $\Gamma_{K}$ and any two different cusps $\kappa$ and $\kappa^{\prime}$
for $\Gamma_{K}$. These cusps can and will be represented by an element of the
field $K$ and the corresponding vector is obtained via the different
embeddings of $K$ into $\mathbb{R}$. It is well-known that the number of the
equivalence classes of cusps for $\Gamma_{K}$ is the class number $h=h(K)$ of
$K$ (see Proposition 20 on page 188 in [12]). These classes are represented by
a fixed set of integer ideals
$\mathfrak{a}_{1},\dots,\mathfrak{a}_{h}\subset\mathcal{O}_{K}$ such that the
corresponding cusps are written in the form $\lambda_{j}=\rho_{j}/\sigma_{j}$
where $\rho_{j},\sigma_{j}\in\mathcal{O}_{K}$ and
$\mathfrak{a}_{j}=(\rho_{j},\sigma_{j})$.

Assume first that $\kappa=\lambda_{j}$ for some $1\leq j\leq h$. Let us fix
the elements $\eta_{j},\xi_{j}\in\mathfrak{a}_{j}^{-1}$ such that
$\rho_{j}\eta_{j}-\xi_{j}\sigma_{j}=1$ holds, then the matrix

$\displaystyle A_{j}=\left[\begin{array}[]{cc}\rho_{j}&\xi_{j}\\\
\sigma_{j}&\eta_{j}\end{array}\right]\in SL(2,K)$

takes $\infty$ to $\lambda_{j}$, hence $\infty$ is a cusp of
$A_{j}^{-1}\Gamma_{K}A_{j}$. The stabilizer of $\infty$ in this group consists
of elements of the form

$\left[\begin{array}[]{cc}u&\zeta u^{-1}\\\ 0&u^{-1}\end{array}\right]$

with $u\in\mathcal{O}_{K}^{\times}$ and $\zeta\in\mathfrak{a}_{j}^{-2}$ (see
[12]). For a $\kappa^{\prime}\in(K\cup\\{\infty\\})\setminus\\{\kappa\\}$, let
$c=A_{j}^{-1}\kappa^{\prime}$ be a cusp of $A_{j}^{-1}\Gamma_{K}A_{j}$
different from $\infty$. We show that the latter matrix above can be chosen so
that its other fixed cusp is $c$. For this, it is enough to choose the unit
$u$ so that $c(1-u^{2})\in\mathfrak{a}_{j}^{-2}$ holds. But this can be
reached since for an arbitrary integral ideal $\mathfrak{a}$ one can choose
$u$ so that $\mathfrak{a}\mid(1-u^{2})\Longleftrightarrow
1-u^{2}\in\mathfrak{a}$ holds.

It follows that there is a hyperbolic-parabolic element in $\Gamma$ with fixed
points $\lambda_{j}$ and $\kappa^{\prime}$. But any cusp $\kappa$ can be
written as $\gamma_{\kappa}^{-1}\lambda_{j}$ for some $j$ and
$\gamma_{\kappa}^{-1}\in\Gamma$, so if $\gamma\in\Gamma$ is a hyperbolic-
parabolic element that fixes $\lambda_{j}$ and
$\gamma_{\kappa}\kappa^{\prime}$, then
$\gamma_{\kappa}^{-1}\gamma\gamma_{\kappa}$ fixes $\kappa$ and
$\kappa^{\prime}$. Since the cusps are the same for any finite index subgroup
of $\Gamma_{K}$ (though their equivalence classes are not), the claim of the
lemma follows now from Theorem 1.1. ∎

From now on, we drop the index in the notation of the multiplier group and
assume that the generating set
$\varepsilon_{1}^{\kappa},\dots,\varepsilon_{n-1}^{\kappa}$ is the same for
any $\kappa\in\mathcal{S}$. Hence the matrices $\mathcal{E}_{\kappa}$ are
identical for any $\kappa$, so we omit the indices here as well. Note that the
integer vectors $m_{u,\kappa}$ are also defined in terms of $\mathcal{E}$ and
therefore their common value will be denoted by $m_{u}$.

Returning to the main terms $M_{\kappa}(m,\alpha,A)$ and
$M_{\tilde{\kappa}_{m,\alpha}}(m^{\prime},\beta,A)$ in our argument, from
$u_{m}^{\kappa}=u_{m^{\prime}}^{\tilde{\kappa}_{m,\alpha}}$ we infer that
$m=m^{\prime}$ and we simply write $u_{m}$ and $E_{m}$ in the following.
Finally, the equality of the main terms implies that $|\det
L_{m,\alpha}^{\kappa}|=|\det L_{m,\beta}^{\tilde{\kappa}_{m,\alpha}}|$.

Now we return to a general form $u$, and we only assume that it is not a cusp
form and $m_{u}=0$ holds (otherwise there are no main terms in (46)). We split
each main term $M_{\kappa}(m,\alpha,A)$ into two parts:

$\displaystyle\left(\frac{\eta_{\kappa}A^{s}}{s}+\frac{\phi_{\kappa}A^{1-s}}{1-s}\right)\frac{\left|\det
L_{m,\alpha}^{\kappa}\right|\left|\det\mathcal{E}\right|}{nN(\left|E_{m}\right|)}g(\log
u_{m})+$
$\displaystyle\qquad\qquad+\left(\frac{\eta_{\tilde{\kappa}_{m,\alpha}}A^{s}}{s}+\frac{\phi_{\tilde{\kappa}_{m,\alpha}}A^{1-s}}{1-s}\right)\frac{\left|\det
L_{m,\alpha}^{\kappa}\right|\left|\det\mathcal{E}\right|}{nN(\left|E_{m}\right|)}g(\log
u_{m}).$

This main term has the (equal) pair
$M_{\tilde{\kappa}_{m,\alpha}}(m,\beta,A)$, and if
$\lambda=\tilde{\kappa}_{m,\alpha}$ then clearly
$\tilde{\lambda}_{m,\beta}=\kappa$ and the pair of $M_{\lambda}(m,\beta,A)$ is
$M_{\kappa}(m,\alpha,A)$. That is, this pairing gives a bijection for every
$m$ on the set of pairs $(\kappa,\alpha)$, where $\kappa\in\mathcal{S}$ and
$\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda$. Moreover, the term
$M_{\tilde{\kappa}_{m,\alpha}}(m,\beta,A)$ has the split form

$\displaystyle\left(\frac{\eta_{\tilde{\kappa}_{m,\alpha}}A^{s}}{s}+\frac{\phi_{\tilde{\kappa}_{m,\alpha}}A^{1-s}}{1-s}\right)\frac{|\det
L_{m,\beta}^{\tilde{\kappa}_{m,\alpha}}|\left|\det\mathcal{E}\right|}{nN(\left|E_{m}\right|)}g(\log
u_{m})+$
$\displaystyle\qquad\qquad+\left(\frac{\eta_{\kappa}A^{s}}{s}+\frac{\phi_{\kappa}A^{1-s}}{1-s}\right)\frac{|\det
L_{m,\beta}^{\tilde{\kappa}_{m,\alpha}}|\left|\det\mathcal{E}\right|}{nN(\left|E_{m}\right|)}g(\log
u_{m}).$

By the last remark of the previous paragraph we have that the first term of
$M_{\kappa}(m,\alpha,A)$ is the second term of
$M_{\tilde{\kappa}_{m,\alpha}}(m,\beta,A)$ and vice versa. These simple
observations imply immediately that if we sum the main terms in (46) obtaining

$\displaystyle\frac{\delta_{m_{u}}\left|\det\mathcal{E}\right|}{2n}$
$\displaystyle\left[\sum_{\kappa\in\mathcal{S}}\left(\frac{\eta_{\kappa}A^{s}}{s}+\frac{\phi_{\kappa}A^{1-s}}{1-s}\right)\sum_{m\in\mathbb{Z}^{n-1}\setminus\\{0\\}}\frac{g(\log
u_{m})}{N(\left|E_{m}\right|)}\sum_{\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda}\,\,\left|\det
L_{m,\alpha}^{\kappa}\right|\right.$
$\displaystyle\quad+\left.\sum_{\kappa\in\mathcal{S}}\,\,\sum_{m\in\mathbb{Z}^{n-1}\setminus\\{0\\}}\frac{g(\log
u_{m})}{N(\left|E_{m}\right|)}\sum_{\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda}\left(\frac{\eta_{\tilde{\kappa}_{m,\alpha}}A^{s}}{s}+\frac{\phi_{\tilde{\kappa}_{m,\alpha}}A^{1-s}}{1-s}\right)\left|\det
L_{m,\alpha}^{\kappa}\right|\right],$

then the two triple sums above are equal, hence this expression simply becomes

(48)
$\frac{\delta_{m_{u}}\left|\det\mathcal{E}\right|}{n}\left[\sum_{\kappa\in\mathcal{S}}\left(\frac{\eta_{\kappa}A^{s}}{s}+\frac{\phi_{\kappa}A^{1-s}}{1-s}\right)\sum_{m\in\mathbb{Z}^{n-1}\setminus\\{0\\}}\frac{g(\log
u_{m})}{N(\left|E_{m}\right|)}\sum_{\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda}\,\,\left|\det
L_{m,\alpha}^{\kappa}\right|\right].$

Next we give group theoretic interpretations of the quantities $N(|E_{m}|)$
and $|\det L_{m,\alpha}^{\kappa}|$. The sublattice
$(u_{m}-1)\mathbf{t}_{\kappa}$ of $\mathbf{t}_{\kappa}$ is obtained by
coordinate-wise multiplication, i.e. via multiplication by a diagonal matrix
with entries $(u_{m}-1)^{(k)}$ in its diagonal. It is well-known that the
index of this sublattice in $\mathbf{t}_{\kappa}$, i.e. the order of the
factor group
$\mathbf{t}_{\kappa}^{m}=\mathbf{t}_{\kappa}/(u_{m}-1)\mathbf{t}_{\kappa}$ is
the absolute value of the determinant of this matrix, that is simply
$|N(u_{m}-1)|=N(|E_{m}|)$.

Now we consider the $\Lambda$-equivalent elements of
$\mathbf{t}_{\kappa}^{m}$. Assume that for an
$\alpha\in\mathbf{t}_{\kappa}^{m}$ we have $u_{l}\alpha=\alpha$ (in
$\mathbf{t}_{\kappa}^{m}$) for some $l\in\mathbb{Z}^{n-1}$. This means exactly
that $\frac{u_{l}-1}{u_{m}-1}\alpha\in\mathbf{t}_{\kappa}$. A simple
computation shows that in this case the element

$\gamma(l,m,\alpha):=\left[\begin{array}[]{cc}u_{l}^{1/2}&\frac{u_{l}-1}{u_{m}-1}\alpha
u_{l}^{-1/2}\\\ 0&u_{l}^{-1/2}\end{array}\right]\in
C(\gamma^{\kappa}_{m,\alpha}).$

Every element of the centralizer $C(\gamma^{\kappa}_{m,\alpha})$ has this form
by Proposition 2.5, and hence $u_{l}\alpha=\alpha$ is equivalent to
$\gamma(l,m,\alpha)\in C(\gamma^{\kappa}_{m,\alpha})$. Again, by Proposition
2.5 this holds if and only if $\log u_{l}$ is in the lattice spanned by
$v_{1}^{\kappa},\dots,v_{n-1}^{\kappa}$ in the subspace
$V=\\{a\in\mathbb{R}^{n}:a_{1}+\dots+a_{n}=0\\}$, where
$v_{j}^{\kappa}=l_{j}^{(1)}\log\varepsilon_{1}+\dots+l_{j}^{(n-1)}\log\varepsilon_{n-1}$
and the integer vectors $l_{j}\in\mathbb{Z}^{n-1}$ are defined in Proposition
2.5. Hence, for a fixed $\alpha$, the number of inequivalent points
$u_{l}\alpha\in\mathbf{t}_{\kappa}^{m}$ is exactly the index of the before-
mentioned sublattice in the lattice generated by
$\log\varepsilon_{1},\dots,\log\varepsilon_{n-1}$ in $V$, and this is exactly
$|\det L_{m,\alpha}^{\kappa}|$. It follows that

$\sum_{\alpha\in\mathbf{t}_{\kappa}^{m}/\Lambda}|\det
L_{m,\alpha}^{\kappa}|=|\mathbf{t}_{\kappa}^{m}|=N(|E_{m}|),$

and hence (48) becomes

$\frac{\delta_{m_{u}}\left|\det\mathcal{E}\right|}{n}\sum_{\kappa\in\mathcal{S}}\left(\frac{\eta_{\kappa}A^{s}}{s}+\frac{\phi_{\kappa}A^{1-s}}{1-s}\right)\sum_{m\in\mathbb{Z}^{n-1}\setminus\\{0\\}}g(\log
u_{m})$

and this (together with (46) and (47)) completes the proof of Theorem 2.6.

###  3.4. Extension of $\zeta$-functions corresponding to lattices

In this section we prove Lemma 2.8. We use the notations of Section 2.5 and
note that the following argument is a standard one in analytic number theory
(we basically copy the proof of Theorem 1.7.2 in [5]) and hence some details
will be omitted.

It is easy to see that the sum in (29) converges absolutely and locally
uniformly for $\mathrm{Re}\,s>1$, and this latter condition will be assumed in
the first part of the proof. We write the terms of the sum in (29) as follows:
let $0\neq l\in L$, $l=(l^{(1)},\dots,l^{(n)})$, then

$\int_{0}^{\infty}e^{-\pi
x(l^{(k)})^{2}}x^{\frac{s_{k}}{2}}\,\frac{dx}{x}=\frac{1}{\pi^{\frac{s_{k}}{2}}(l^{(k)})^{s_{k}}}\int_{0}^{\infty}e^{-u}u^{\frac{s_{k}}{2}}\,\frac{du}{u}=\frac{\Gamma\left(\frac{s_{k}}{2}\right)}{\pi^{\frac{s_{k}}{2}}\left|l^{(k)}\right|^{s_{k}}}$

and hence

$\int_{(\mathbb{R}^{+})^{n}}e^{-\pi\textrm{tr}\,(\mathbf{x}l^{2})}\prod_{k=1}^{n}x_{k}^{\frac{s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}=\pi^{-\frac{ns}{2}}\frac{\lambda_{M,-m}(\left|l\right|)}{\left|Nl\right|^{s}}\prod_{k=1}^{n}\Gamma\left(\frac{s_{k}}{2}\right),$

where $\mathbf{x}=(x_{1},\dots,x_{n})^{T}$, $\mathbf{x}l^{2}$ is the
coordinate-wise product of $\mathbf{x}$ and $l^{2}$, and the trace
$\textrm{tr}\,(\cdot)$ of a vector is the sum of its coordinates. Then

$\displaystyle\pi^{-\frac{ns}{2}}\left(\prod_{k=1}^{n}\Gamma\left(\frac{s_{k}}{2}\right)\right)Z_{L,M}(s,m)$
$\displaystyle=\sum_{0\neq l\in
L/M}\int_{(\mathbb{R}^{+})^{n}}e^{-\pi\textrm{tr}\,(\mathbf{x}l^{2})}\prod_{k=1}^{n}x_{k}^{\frac{s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}$
$\displaystyle=\sum_{\varepsilon\in
M^{2}}\int_{(\mathbb{R}^{+})^{n}/M^{2}}\left(\sum_{0\neq l\in
L/M}e^{-\pi\textrm{tr}\,(\varepsilon\mathbf{x}l^{2})}\right)\prod_{k=1}^{n}(\varepsilon^{(k)}x_{k})^{\frac{s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}$
$\displaystyle=\int_{(\mathbb{R}^{+})^{n}/M^{2}}\left(\sum_{0\neq l\in
L}e^{-\pi\textrm{tr}\,(\mathbf{x}l^{2})}\right)\prod_{k=1}^{n}x_{k}^{\frac{s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}.$

Let us define the following theta function for the lattice $L$:

$\Theta_{L}(\mathbf{x}):=\sum_{l\in L}e^{-\pi\textrm{tr}\,(\mathbf{x}l^{2})}.$

Using this notation we have

(49)
$\displaystyle\Xi_{L,M}(s,m):=\pi^{-\frac{ns}{2}}\left(\prod_{k=1}^{n}\Gamma\left(\frac{s_{k}}{2}\right)\right)Z_{L,M}(s,m)=\int_{(\mathbb{R}^{+})^{n}/M^{2}}(\Theta_{L}(\mathbf{x})-1)\prod_{k=1}^{n}x_{k}^{\frac{s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}.$

For a fixed $\mathbf{x}\in(\mathbb{R}^{+})^{n}$ we set
$f_{\mathbf{x}}(\mathbf{y})=e^{-\pi\textrm{tr}\,(\mathbf{x}\mathbf{y}^{2})}$,
its Fourier transform is

$\displaystyle\hat{f}_{\mathbf{x}}(\xi)$
$\displaystyle=\int_{\mathbb{R}^{n}}f_{\mathbf{x}}(\mathbf{y})e^{-2\pi
i\left<\mathbf{y},\mathbf{\xi}\right>}\,\mathrm{d}\mathbf{y}=\prod_{k=1}^{n}\frac{1}{\sqrt{x_{k}}}e^{-\pi\xi_{k}^{2}/x_{k}}.$

By the Poisson summation formula we have

(50) $\displaystyle\Theta_{L}(\mathbf{x})$ $\displaystyle=\sum_{l\in
L}f_{\mathbf{x}}(l)=\frac{1}{\mathrm{vol}(\mathbb{R}^{n}/L)}\sum_{\beta\in
L^{*}}\hat{f_{\mathbf{x}}}(\beta)=\frac{1}{\mathrm{vol}(\mathbb{R}^{n}/L)(N\mathbf{x})^{1/2}}\Theta_{L^{*}}(1/\mathbf{x}).$

Now we split the integral in (49) into two parts depending on $N\mathbf{x}$.
If $N\mathbf{x}<1$, we use (50) and then substitute $1/\mathbf{x}$ to obtain
that $\Xi_{L,M}(s,m)$ is

$\displaystyle\int\limits_{\begin{array}[]{c}\scriptstyle{(\mathbb{R}^{+})^{n}/M^{2}}\\\
\scriptstyle{N\mathbf{x}>1}\end{array}}(\Theta_{L}(\mathbf{x})-1)\prod_{k=1}^{n}x_{k}^{\frac{s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}+\dfrac{1}{\mathrm{vol}(\mathbb{R}^{n}/L)}\int\limits_{\begin{array}[]{c}\scriptstyle{(\mathbb{R}^{+})^{n}/M^{2}}\\\
\scriptstyle{N\mathbf{x}>1}\end{array}}(\Theta_{L^{*}}(\mathbf{x})-1)\prod_{k=1}^{n}x_{k}^{\frac{1-s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}$
$\displaystyle\qquad-\int\limits_{\begin{array}[]{c}\scriptstyle{(\mathbb{R}^{+})^{n}/M^{2}}\\\
\scriptstyle{N\mathbf{x}<1}\end{array}}\prod_{k=1}^{n}x_{k}^{\frac{s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}+\dfrac{1}{\mathrm{vol}(\mathbb{R}^{n}/L)}\int\limits_{\begin{array}[]{c}\scriptstyle{(\mathbb{R}^{+})^{n}/M^{2}}\\\
\scriptstyle{N\mathbf{x}>1}\end{array}}\prod_{k=1}^{n}x_{k}^{\frac{1-s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}.$

A straightforward computation (similar to the one that led to (3.3) in the
previous proof) shows that

$\displaystyle\int\limits_{\begin{array}[]{c}\scriptstyle{(\mathbb{R}^{+})^{n}/M^{2}}\\\
\scriptstyle{N\mathbf{x}<1}\end{array}}\prod_{k=1}^{n}x_{k}^{\frac{s_{k}}{2}}$
$\displaystyle\,\frac{dx_{k}}{x_{k}}=2^{n-1}\left|\det\mathcal{E}_{M}\right|\int_{-\infty}^{0}e^{nsy_{0}/2}\,dy_{0}\prod_{j=1}^{n-1}\int_{0}^{1}e^{2\pi
m_{j}iy_{j}}\,dy_{j}.$

This expression is $0$ unless all coordinates of $m$ are zero, in which case
it is $2^{n}\left|\det\mathcal{E}_{M}\right|/(ns)$. Similarly,

$\displaystyle\int\limits_{\begin{array}[]{c}\scriptstyle{(\mathbb{R}^{+})^{n}/M^{2}}\\\
\scriptstyle{N\mathbf{x}>1}\end{array}}\prod_{k=1}^{n}$ $\displaystyle
x_{k}^{\frac{1-s_{k}}{2}}\,\frac{dx_{k}}{x_{k}}=2^{n-1}\left|\det\mathcal{E}_{M}\right|\int_{0}^{\infty}e^{n(1-s)y_{0}/2}\,dy_{0}\prod_{j=1}^{n-1}\int_{0}^{1}e^{-2\pi
m_{j}iy_{j}}\,dy_{j},$

and as above, this is $0$ if $m\neq 0$ and otherwise we get
$\frac{2^{n}\left|\det\mathcal{E}_{M}\right|}{n(s-1)}$ (note that
$\mathrm{Re}\,s>1$ is still assumed here).

Therefore if $m\neq 0$, then $\Xi_{L,M}(s,m)$ is entire and

(51)
$\mathrm{vol}(\mathbb{R}^{n}/L)^{1/2}\Xi_{L,M}(s,m)=\mathrm{vol}(\mathbb{R}^{n}/L^{*})^{1/2}\Xi_{L^{*},M}(1-s,-m)$

holds. If $m=0$, then $\Xi_{L,M}(s,m)$ is holomorphic except for $s=1$ and
$s=0$, where it has simple poles with residues
$\frac{2^{n}\left|\det\mathcal{E}_{M}\right|}{n\cdot\mathrm{vol}(\mathbb{R}^{n}/L)}$
and $-\frac{2^{n}\left|\det\mathcal{E}_{M}\right|}{n}$, respectively. The
functional equation (51) holds also in this case for any $s\neq 0,1$.

One can reorder the equation (51) asymmetrically:

$Z_{L,M}(s,m)=\mathrm{vol}(\mathbb{R}^{n}/L^{*})\pi^{ns-\frac{n}{2}}\left(\prod_{k=1}^{n}\frac{\Gamma\left(\frac{1-s_{k}}{2}\right)}{\Gamma\left(\frac{s_{k}}{2}\right)}\right)Z_{L^{*},M}(1-s,-m).$

Here

$\frac{\Gamma\left(\frac{1-s_{k}}{2}\right)}{\Gamma\left(\frac{s_{k}}{2}\right)}=\Gamma\left(\frac{1-s_{k}}{2}\right)\Gamma\left(1-\frac{s_{k}}{2}\right)\frac{\sin\frac{\pi
s_{k}}{2}}{\pi}=2^{s_{k}}\Gamma\left(1-s_{k}\right)\frac{\sin\frac{\pi
s_{k}}{2}}{\pi^{\frac{1}{2}}},$

i.e.

$Z_{L,M}(s,m)=\mathrm{vol}(\mathbb{R}^{n}/L^{*})2^{ns}\pi^{n(s-1)}\left(\prod_{k=1}^{n}\Gamma\left(1-s_{k}\right)\sin\frac{\pi
s_{k}}{2}\right)Z_{L^{*},M}(1-s,-m).$

If $\mathrm{Re}\,s$ is bounded and $\left|\textrm{Im}\,s\right|\geq t_{0}$ for
some big enough $t_{0}>0$ then by Stirling’s formula

$\left|\Gamma(1-s_{k})\sin\frac{\pi
s_{k}}{2}\right|\asymp\left|\textrm{Im}\,s\right|^{1/2-\mathrm{Re}\,s}$

and hence
$\left|Z_{L,M}(s,m)\right|\asymp(\textrm{Im}\,s)^{n\left(\frac{1}{2}-\mathrm{Re}\,s\right)}\left|Z_{L^{*},M}(1-s,-m)\right|$.
By the Phragmén-Lindelöf principle, it follows from this and from the trivial
bound $\left|Z_{L,M}(s,m)\right|\leq\left|Z_{L,M}(\mathrm{Re}\,s,0)\right|$
for $\mathrm{Re}\,s>1$ that

$Z_{L,M}(s,m)\ll_{\varepsilon,m}\left|\textrm{Im}\,s\right|^{n(1-\mathrm{Re}\,s)/2+\varepsilon}$

holds for $0\leq\mathrm{Re}\,s\leq 1$ and $\left|\textrm{Im}\,s\right|\geq
t_{0}>0$. Similarly, one can bound $Z_{L,M}(s,m)$ by
$\left|\textrm{Im}\,s\right|^{\varepsilon}$ once $\mathrm{Re}\,s>1$ and
$\left|\textrm{Im}\,s\right|\geq t_{0}$ (or by a constant if
$\mathrm{Re}\,s>1+\delta$ for some $\delta>0$). This completes the proof of
Lemma 2.8.

###  3.5. Proof in the totally parabolic case

We proceed by calculating the part of the trace where we sum over parabolic
classes. Every such class is represented by an element that fixes a cusp
$\kappa\in\mathcal{S}$. An element of this type is conjugated by
$\sigma_{\kappa}\in PSL(2,\mathbb{R})^{n}$ to an element of the form

$\gamma_{\alpha}^{\kappa}:=\left[\begin{array}[]{cc}1&\alpha\\\
0&1\end{array}\right],$

where $0\neq\alpha\in\mathbf{t}_{\kappa}$. A simple computation shows that two
such elements $\gamma_{\alpha}^{\kappa}$ and $\gamma_{\beta}^{\kappa}$ are
conjugate in $\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$ if and only if
$\alpha=\varepsilon\beta$ for some $\varepsilon\in\Lambda$ (see also [6],
section III.2). Hence summation over parabolic classes means a double
summation over the elements of $\mathcal{S}$ and the non-zero elements of
$\mathbf{t}_{\kappa}/\Lambda$. Therefore the contribution of the parabolic
classes in the trace can be written as

(54)
$\displaystyle\sum_{\kappa\in\mathcal{S}}\,\sum_{0\neq\alpha\in\mathbf{t}_{\kappa}/\Lambda}\sum_{\begin{array}[]{c}\scriptstyle{\sigma\in
C(\gamma)\setminus\Gamma}\\\
\scriptstyle{\gamma\sim\gamma_{\alpha}^{\kappa}}\end{array}}\int_{F_{A}}k(z,\sigma^{-1}\gamma\sigma
z)u(z)\,d\mu(z)=$ (55)
$\displaystyle\qquad=\sum_{\kappa\in\mathcal{S}}\,\sum_{0\neq\alpha\in\mathbf{t}_{\kappa}/\Lambda}\,\sum_{\sigma\in
C(\gamma_{\alpha}^{\kappa})\setminus\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}}\int_{\sigma(\sigma_{\kappa}^{-1}F_{A})}k(z,\gamma_{\alpha}^{\kappa}z)u(\sigma_{\kappa}z)\,d\mu(z)$

where $C(\gamma_{\alpha}^{\kappa})$ is the centralizer of
$\gamma_{\alpha}^{\kappa}$ in $\sigma_{\kappa}^{-1}\Gamma\sigma_{\kappa}$
given by

$C(\gamma_{\alpha}^{\kappa}):=\left\\{\left[\begin{array}[]{cc}1&\beta\\\
0&1\end{array}\right]\in
PSL(2,\mathbb{R})^{n}:\,\beta\in\mathbf{t}_{\kappa}\right\\},$

and its fundamental domain is
$F_{C(\gamma_{\alpha}^{\kappa})}=\left\\{z\in\mathbb{H}^{n}:\,0\leq
X_{1}^{\kappa}(z),\dots,X_{n}^{\kappa}(z)<1\right\\}$.

The union of the sets $\sigma(\sigma_{\kappa}^{-1}F_{A})$ in (54) makes up the
set $F_{C({\gamma_{\alpha}^{\kappa}})}$ except for the images of the part
$F\setminus F_{A}=F^{*}_{A}$. As in the hyperbolic-parabolic case, for some