# The Eisenstein homology and cohomology groups for Imaginary quadratic fields

Debargha Banerjee  and  Pranjal Vishwakarma  INDIAN INSTITUTE OF SCIENCE
EDUCATION AND RESEARCH, PUNE, INDIA

###### Abstract.

We explicitly write down the Eisenstein cycles in the first homology groups of
quotients of the hyperbolic $3$-spaces as linear combinations of Manin symbols
for the imaginary quadratic field. They generate the Eisenstein part of the
homology groups. We also study the Eisenstein part of the dual space (the
cohomology groups) and give a formula for the trace of complex conjugation on
the Eisenstein part of cohomology groups for the congruence subgroups of the
form $\Gamma_{1}(N)$ inside the Bianchi groups.

###### Key words and phrases:

Eisenstein series, Modular symbols, Special values of $L$-functions

###### 2010 Mathematics Subject Classification:

Primary: 11F67, Secondary: 11F41, 11F20, 11F30

The first named author was partially supported by the SERB grant
CRG/2020/000223, and the second named author by the PMRF fellowship. Authors
are greatly indebted to Professor Haluk Sengün for several email
correspondences. The article is dedicated to the fond memory of Professor Yuri
Manin.

##  1\. Introduction

###  1.1. Introduction

Let $\mathcal{O}$ be the ring of integers of an imaginary quadratic field $K$
and $\Gamma$ be a subgroup of finite index of the full Bianchi modular group
$G={\mathrm{SL}}_{2}(\mathcal{O})$. Let $M_{k}(\Gamma)$ be the complex vector
space of modular forms of scalar weight $k$ for the subgroup $\Gamma$ with
$S_{k}(\Gamma)$ be the subspace of cusp forms. There is a natural complement
$E_{k}(\Gamma)$ of $S_{k}(\Gamma)$ inside $M_{k}(\Gamma)$, Eisenstein subspace
of $M_{k}(\Gamma)$. In other words, we have a decomposition
$M_{k}(\Gamma)=S_{k}(\Gamma)\oplus E_{k}(\Gamma)$. To study arithmetic
properties like explicit Fourier coefficients of modular forms in
$M_{k}(\Gamma)$, it is natural to study the dual space, the space of modular
symbols $\mathcal{M}_{k}(\Gamma)$ (basically the homology groups or the dual
cohomology groups of the modular curve associated to $\Gamma$). Denote the
corresponding space of the cuspidal and Eisenstein part by
$\mathcal{S}_{k}(\Gamma)$ and $\mathcal{E}_{k}(\Gamma)$. The whole space of
modular symbols $\mathcal{M}_{k}(\Gamma)$ also breaks into
$\mathcal{M}_{k}(\Gamma)=\mathcal{S}_{k}(\Gamma)\oplus\mathcal{E}_{k}(\Gamma)$.
The aim of this article is to study the space $\mathcal{E}_{k}(\Gamma)$.

Eisenstein cycles also known as Eisenstein elements, are elements in the
Eisenstein homology groups of $\Gamma$. Merel invented Eisenstein cycles in
order to give an expression for elements in the Eisenstein ideals [24] and
have recently been applied to the study of the rank of Eisenstein ideals [21].
This is related to certain conjectures of Mazur [23]. These Eisenstein cycles
are determined by periods of the Eisenstein series. Much remains to be known
about extensions of these results to the Bianchi modular forms.

Banerjee-Merel [1] studied the Eisenstein cycles as modular symbols for the
congruence subgroups in the classical case of elliptic modular forms. They
write down the Eisenstein cycles in the first homology group of modular curves
of level $N$ as $\overline{\mathbb{Q}}$ linear combinations of Manin symbols
for any odd integer $N$. They generate the Eisenstein part of the space of
modular symbols. In [2], they extended the result to the subgroups of finite
indices of the full modular group. Of course, in general these Eisenstein
cycles are not necessarily $\overline{\mathbb{Q}}$ valued. In continuation, we
extend the computation to the Bianchi modular forms (cf. § 2.3 for the
definition of the same). The main novelty in this section is the extensive use
of $\star$ delta operators. In our setting, differential forms are harmonic
rather than holomorphic.

John Cremona and his collaborators initiated the study of modular symbols
(homology groups of quotients of three-dimensional hyperbolic space) for the
imaginary quadratic fields. They are mostly interested in the cuspidal part of
the homology group. Over imaginary quadratic fields, Ito [18] showed that the
periods are again described by Sczech’s Dedekind sums [27] for
$\Gamma={\mathrm{SL}}_{2}(\mathcal{O})$. In the case of congruence subgroups,
however the analog of elliptic Dedekind sums are not defined yet.

The study of Eisenstein part of the cohomology groups started with the pioneer
work of [16]. Note that there is a duality between the homology and cohomology
groups [5, p. 12] (see also [6], [7]). In [12], Sengün and his co-authors
studied the dimension of the cohomology of Bianchi groups using Sczech
cocycles for the principal congruence subgroups. It is natural to ask if the
computation can be generalized to other important congruence subgroups like
$\Gamma_{0}(N)$ or $\Gamma_{1}(N)$.

In this paper, we extend the study of Eisenstein elements to subgroups of the
Bianchi groups of the form $\Gamma_{1}(N)$. Similar to loc. cit, we use Sczech
cocycles [28] to study the Eisenstein part of the cohomology. Note that Sczech
cocycles determine the periods of the Eisenstein series. The main novelty of
our work for the subgroup $\Gamma_{1}(N)$ stems from the explicit
understanding of cusps for these subgroups in Proposition 14. We also give
reasons why the same computation may not be extended for the subgroups of the
form $\Gamma_{0}(N)$. Although, we understand the cuspidal subgroup for
$\Gamma_{0}(N)$ but it is not possible to compute the trace of complex
conjugation on the Eisenstein part of the cohomology groups.

We now proceed to describe our results. Let
$\mathbb{H}_{3}\mathrel{\mathop{\ordinarycolon}}=\\{(z,t)\mathrel{\mathop{\ordinarycolon}}z\in\mathbb{C},t\in\mathbb{R}^{+}\\}$
be the hyperbolic $3$-space and let
$\Gamma\subset{\mathrm{SL}}_{2}(\mathcal{O})\mathrel{\mathop{\ordinarycolon}}=G$
(For example
${\mathrm{SL}}_{2}(\mathbb{Z}[i])\mathrel{\mathop{\ordinarycolon}}=G$) be a
subgroup of the finite index with no elements of finite order. We use the
Baily-Borel compactification of hyperbolic three-space
$\overline{\mathbb{H}}_{3}=\mathbb{H}_{3}\cup{\mathbb{P}}^{1}(K)$. Consider
the set of cusps
$\partial_{\Gamma}\mathrel{\mathop{\ordinarycolon}}=\Gamma\backslash{\mathbb{P}}^{1}(K)$
and denote
$X_{\Gamma}\mathrel{\mathop{\ordinarycolon}}=\Gamma\backslash\overline{\mathbb{H}}_{3}$.
In the present setting, results are difficult to get because $X_{\Gamma}$ do
not carry the same algebraic geometric structure as in the cases over
$\mathbb{Q}$ or totally real extensions of $\mathbb{Q}$.

We then have an exact sequence of relative homology groups.

$0=\mathrm{H}_{1}(\partial_{\Gamma};\mathbb{Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma};\mathbb{Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};\mathbb{Z})\xrightarrow[]{\delta}\mathbb{Z}[\partial_{\Gamma}]\rightarrow\mathbb{Z}\rightarrow
0,$

where the map
$\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};\mathbb{Z})\xrightarrow[]{\delta}\mathbb{Z}[\partial_{\Gamma}]$
is obtained from the boundary map

$\delta\mathrel{\mathop{\ordinarycolon}}\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};\mathbb{Z})\rightarrow\mathrm{H}_{0}(\partial_{\Gamma};\mathbb{Z}).$

The space of $1$-differential forms on $\mathbb{H}_{3}$ is given by the basis

$\beta=(\beta_{0},\beta_{1},\beta_{2})=(-\frac{dz}{t},\frac{dt}{t},\frac{d\bar{z}}{t}).$

######  Definition 1.

The Eisenstein classes corresponding to a cusp $\xi$ is the modular symbol
${\mathcal{E}}_{\xi}\in\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma},\mathbb{R})$
such that

$\int_{{\mathcal{E}}_{\xi}}F\cdot\beta=0$

for all cusp forms $F$ (cf. Definition 5). The Eisenstein cycles are then
paths in the Eisenstein classes corresponding to the cusp $\xi$.

We explicitly write an Eisenstein cycle corresponding to $\xi$. Cremona [11]
showed that (see also [10, thm 4.3.2]) the map

$\eta\mathrel{\mathop{\ordinarycolon}}(\gamma)=\\{\gamma 0,\gamma\infty\\}$

is a surjective map from $\Gamma\backslash{\mathrm{SL}}_{2}(\mathcal{O})$ onto
$\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};\mathbb{Z})$. The kernel of the
map can be computed as in loc. cit.

For the Eisenstein series $E_{\xi}$, denote the differential $1$-form
associated to $\xi$ as $\omega_{\xi}$ and consider the differential form
$\star\omega_{\xi}$ with $\star$ adjoint operators acting on differential
forms. Let $B_{0}=\\{0,P_{2},P_{3}\\}$ be the face of the octahedron in [10,
Diagram 4.2]. Define:

$F_{\xi}(g)\mathrel{\mathop{\ordinarycolon}}=\int_{B_{0}}*\omega_{\xi}[g].$

We now state our first theorem.

######  Theorem 2.

The modular symbol

${\mathcal{E}}_{\xi}=\sum_{g\in\Gamma\backslash G}F_{\xi}(g)\eta(g)$

is the Eisenstein element corresponding to the Eisenstein series $E_{\xi}\in
E_{2}(\Gamma)$.

Similar to the classical case [2], we can’t determine if these Eisenstein
cycles are $\overline{\mathbb{Q}}$ valued. Of course, we expect these
Eisenstein cycles to be $\overline{\mathbb{Q}}$ valued only for the Eisenstein
series for the congruence subgroups. We expect that the values of the integral
should be related to Eisenstein cocycles of [8] in a subtle manner. For every
cusp, we produce an explicit element of the Eisenstein part of the modular
symbols. They generate the Eisenstein part of the modular symbols.

The above calculations are for the homology group. We know that cohomology and
homology groups are dual to each other. It makes sense to study the Eisenstein
part of cohomology groups. Let $\mathfrak{a}$ be an ideal in $\mathcal{O}$.
Define the congruence subgroup of level $\mathfrak{a}$,

$\Gamma_{1}(\mathfrak{a})\mathrel{\mathop{\ordinarycolon}}=\Big{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in
SL_{2}(\mathcal{O})\mathrel{\mathop{\ordinarycolon}}a-1,d-1,c\in\mathfrak{a}\Big{\\}}.$

We will be interested in the subgroups of the form $\Gamma_{1}(\mathfrak{a})$.
We first compute the dimension of the Eisenstein cohomology groups and then
calculate traces of complex conjugation for the Eisenstein cohomology groups
for the congruence subgroups of the form $\Gamma_{1}(N)$ for a principal ideal
$(N)$. Sengün-Türkelli [12] computes traces of complex conjugations on the
Eisenstein cohomology groups for the principal congruence subgroups that we
generalize for the congruence subgroups of the form $\Gamma_{1}(N)$.

######  Theorem 3.

Let $K$ be an imaginary quadratic field and $t$ be the number of rational
primes ramifying in $K$. Let $N=p_{1}^{n_{1}}\ldots p_{\tau}^{n_{r}}$ be a
positive odd number whose prime divisors $p_{i}$ are unramified in $K$ and
$\Gamma_{1}(N)$ be the congruence subgroup of the Bianchi group
${\mathrm{SL}}_{2}(\mathcal{O})$ of level ideal $(N)$. Let $\sigma$ be the
involutions induced on the Eisenstein cohomology of $\Gamma_{1}(N)$ by
nontrivial automorphisms of $K$. Then

$\operatorname{tr}\left(\sigma\mid\mathrm{H}_{\text{Eis
}}^{2}\left(\Gamma_{1}(N),E_{k,k}\right)\right)=\alpha^{\rho}.-2^{t-\tau-1}\cdot\prod_{i=1}^{\tau}\left(p_{i}^{2n_{i}}-p_{i}^{2n_{i}-2}\right)+\delta(0,k),$

where $\alpha=\\#(\Gamma_{1}(N)/\Gamma(N))$ and $\delta$ is the Kronecker
$\delta$-function.

In particular, the trace of $\sigma_{\text{Eis }}^{2}$ on
$\mathrm{H}^{2}\left(\mathrm{SL}_{2}(\mathcal{O}),E_{k,k}\right)$ is
$-2^{t-1}+\delta(0,k)$. For $N\in\mathbb{N}$, let $\mathcal{C}_{N}$ be the set
of cusps of the congruence subgroups of the form $\Gamma_{1}(N)$.

######  Theorem 4.

Assume that the class number of $K$ is one and let $p$ be a rational prime
that is inert in $K$. Let $\sigma$ denote involution induced on the Eisenstein
cohomology of $\Gamma_{1}(N)$ by the nontrivial an automorphism of $K$. Then
we have

$\operatorname{tr}\left(\sigma\mid\mathrm{H}_{\text{Eis}}^{1}\left(\Gamma_{1}\left(p^{n}\right),\mathbb{C}\right)\right)=\begin{cases}-2,&\text{
if }n=1\\\ -\\#\mathcal{C}_{p^{n}}.\left(\frac{-2}{p^{2}-1}\right).&\text{ if
}n>1.\end{cases}$

Using these, we calculate the asymptotic lower bounds for the dimension of the
cuspidal cohomology in Proposition 16 for the congruence subgroup of the form
$\Gamma_{1}(N)$ of the Bianchi modular groups similar to Sengün-Türkelli [12].
In future work, we wish to relate the Eisenstein cycles in the homology groups
constructed in this paper and the Szech cocycles for the congruence subgroups
of the form $\Gamma_{1}(N)$ that we use in the proof of our two theorems on
cohomology groups.

##  2\. Bianchi modular symbols and Manin symbols

###  2.1. Fundamental domains

Recall that

$\mathbb{H}_{3}=\\{(z,t)\mathrel{\mathop{\ordinarycolon}}z\in\mathbb{C},t\in\mathbb{R}^{+}\\}\simeq{\mathrm{GL}}_{2}(\mathbb{C})/Z\cdot\mathrm{SU}(2)$

is the hyperbolic $3$ space, where $Z$ denotes the diagonal matrices in
${\mathrm{GL}}_{2}(\mathbb{C})$. An invariant metric for the action of
${\mathrm{GL}}_{2}(\mathbb{C})$ on $\mathbb{H}_{3}$ is then given by

$ds=\frac{(dzd\bar{z}+(dt)^{2})}{t^{2}}.$

We say that a subgroup $\Gamma\subset{\mathrm{SL}}_{2}(\mathbb{C})$ acts
discontinuously on $\mathbb{H}_{3}$ if every compact subset of
$\mathbb{H}_{3}$ meets only a finite number of its images under $G$.

A subgroup $\Gamma$ acts discontinuously if and only if it is discrete (in the
matrix topology), and we define a fundamental domain or region for $\Gamma$ to
be an open subset $D\subset\mathbb{H}_{3}$ such that every orbit of $\Gamma$
in $\mathbb{H}_{3}$ meets $D$ at most once, and meets the closure of $D$ at
least once.

Choose a fundamental region $D$ defined in [10, §2.3] for the action of
$\Gamma$ on $\bar{\mathbb{H}}_{3}$ with $\\{0,j\infty\\}$ as one of its edges.
Let $\Gamma_{P}$ be the stabilizer of a vertex $P$ of $D$. Form a larger basic
polyhedron by taking the union of the transforms of $D$ by a finite subgroup
$\Gamma_{P}$.

Recall that we are interested in the case
$G={\mathrm{SL}}_{2}(\mathbb{Z}[i])$. This subgroup is generated by

$S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},\ T=\begin{pmatrix}1&1\\\
0&1\end{pmatrix},\ U=\begin{pmatrix}1&i\\\ 0&1\end{pmatrix},\
R=\begin{pmatrix}0&i\\\ i&0\end{pmatrix},$

with the relations $TU=UT,S^{2}=R^{2}=(RS)^{2}=(URS)^{2}=(TS)^{3}=(UR)^{3}=1$.
Consider three points on the unit sphere
$P_{1}=(\frac{1}{2},0,\frac{1}{2}\sqrt{3}),P_{2}=(\frac{1}{2},\frac{1}{2},\frac{1}{2}\sqrt{2}),$
and $P_{3}=(0,\frac{1}{2},\frac{1}{2}\sqrt{3})$. A fundamental domain for the
action of $\Gamma$ is given by vertices at $0,\infty$, and $P_{1},P_{2}$ and
$P_{3}$. The stabilizer of $P_{2}$ is $\Gamma_{P}=\langle TS,UR\rangle,$ which
is of order 12 [10, p.59].

Call the union of the $12$ translates of $D$ by $\Gamma_{P}$ the basic
polyhedron, denoted $B$. It is a hyperbolic octahedron, and its $12$ edges are
precisely the images of $\\{0,\infty\\}$ under the action of $\Gamma_{P}$. We
refer to loc. cit. for the diagram depicting the region $D$ and the octahedron
$B$. Note that $B$ has a triangular face whose edges are the transforms of
$\\{0,\infty\\}$ under $I,TS,$ and $(TS)^{2}$; while the images of
$\\{0,\infty\\}$ under $UR,(UR)^{2}$ are $\\{\infty,i\\}$ and $\\{i,0\\}$
respectively.

###  2.2. Differential forms

Recall that the space of real $1$-differential forms on $\mathbb{H}_{3}$ is
given by the basis

$\beta=(\beta_{0},\beta_{1},\beta_{2})=(-\frac{dz}{t},\frac{dt}{t},\frac{d\bar{z}}{t}),$

and denote the pullback of each $\beta_{i}$ to ${\mathrm{GL}}_{2}(\mathbb{C})$
by $\omega_{i}$. A differential form on ${\mathrm{GL}}_{2}(\mathbb{C})$ is the
inverse image of a differential form on $\mathbb{H}_{3}$ if and only if it can
be written as
$\varphi_{0}\omega_{0}+\varphi_{1}\omega_{1}+\varphi_{2}\omega_{2}$. Here,
$\Phi=(\varphi_{0},\varphi_{1},\varphi_{2})$ satisfies
$\Phi(gkz)=\Phi(g)\rho(kz)$ for every $g\in{\mathrm{GL}}_{2}(\mathbb{C}),k\in
SU(2),z\in Z$, and $\rho$ is a fixed representation of $SU(2)$.

Let $F=(F_{0},F_{1},F_{2})$ be a function on $\mathbb{H}_{3}$, and
$F\cdot\beta=F_{0}\beta_{0}+F_{1}\beta_{1}+F_{2}\beta_{2}$ a $1$-form on
$\mathbb{H}_{3}$. Recall that $*$ is the usual adjoint operator of
differential forms. By definition, one has
$*(F\cdot\beta)=-\frac{1}{2}i\bar{F}_{1}(\beta_{0}\wedge\beta_{2})+i\bar{F}_{0}(\beta_{1}\wedge\beta_{2})+i\bar{F}_{2}(\beta_{0}\wedge\beta_{1})$.
The differential form is harmonic if and only if $F\cdot\beta$ and
$*(F\cdot\beta)$ are closed forms.

We say a function
$F\mathrel{\mathop{\ordinarycolon}}\mathbb{H}_{3}\to\mathbb{C}^{3}$ is
harmonic if $F\cdot\beta$ is a harmonic differential form and is slowly
increasing in the sense that there exists $N\geq 0$ such that

$F\Big{(}\begin{pmatrix}x&0\\\ 0&1\end{pmatrix}(z,t)\Big{)}=O(|x|^{N}),\quad
x\in\mathbb{R}$

as $x\to\infty$ uniformly over compact sets in $\mathbb{H}_{3}$.

###  2.3. Bianchi modular forms

Let $\Delta=d\circ\delta+\delta\circ d$ is the usual Laplacian with $d$ being
the exterior derivative and $\delta$ the co-differential operator on
$\mathbb{H}_{3}$. Recall first a differential form $\omega$ is harmonic if
$\Delta\omega=0$. We now recall the weights of Bianchi modular forms. The
weights of Bianchi modular forms can be scalar-valued [10] or more generally
vector-valued [33].

Now, we define scalar-valued Bianchi modular form of weight $2$. We wish to
define a slashing operator such that $F\cdot\beta$ is an invariant
differential under the action of a discrete subgroup $\Gamma$ of
${\mathrm{SL}}(2,\mathcal{O})$. The complex modular group
${\mathrm{SL}}_{2}(\mathbb{C})$ acts on the space of differential $1$ -forms
as $\beta^{\prime}=\mathrm{J}(\gamma;(z,t))\beta$; with
$\mathrm{J}(\gamma;(z,t))=\frac{1}{\left(|r|^{2}+|s|^{2}\right)}\left(\begin{array}[]{ccc}r^{2}&-2rs&s^{2}\\\
\bar{r}\bar{s}&(r\bar{r}-s\bar{s})&-\bar{r}s\\\
\bar{s}{}^{2}&\overline{2rs}&\bar{r}{}^{2}\end{array}\right)$,
$\gamma=\left(\begin{array}[]{ll}a&b\\\ c&d\end{array}\right)$,
$r=\overline{cz+d}$ and $s=\bar{c}t$.

Let $F\mathrel{\mathop{\ordinarycolon}}\mathbb{H}_{3}\to\mathbb{C}^{3}$ be a
harmonic function. Define the slash operator

($F\mid_{\gamma})(z,t)=F(g(z,t)$) $\mathrm{J}(\gamma;(z,t)$).

We now define vector-valued Bianchi modular form of weight $2$. Let
$F\mathrel{\mathop{\ordinarycolon}}\mathbb{H}_{3}\to\mathbb{C}^{3}$ be a
harmonic function. Define the slash operator:

$\left(\left.F\right|_{\gamma}\right)(w)\mathrel{\mathop{\ordinarycolon}}=Sym^{2}\left(j(\gamma,w)^{-1}\right)F(\gamma
w),$

where

$j(\gamma,w)=\begin{pmatrix}cz+d&-ct\\\
\bar{c}t&\overline{cz+t}\end{pmatrix},\qquad\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix};$

and $Sym^{2}$ is the symmetric $2$-nd power of the standard representation of
${\mathrm{SL}}_{2}(\mathbb{C})$ on $\mathbb{C}^{2}$.

Denote by $S_{2}(\Gamma)$ the space of weight $2$ Bianchi cusp forms on
$\Gamma$ and the set of cusps of $\Gamma$ (for both Satake and Bailey-Borel
compactification) can be identified with the orbit space
$\Gamma\backslash{\mathbb{P}}^{1}(K)$.

We have
$F\mathrel{\mathop{\ordinarycolon}}\mathbb{H}_{3}\rightarrow\mathbb{C}^{3}$
and

$(F|_{\gamma})(z)=\frac{1}{|r|^{2}+|s|^{2}}\left(\begin{array}[]{ccc}{r}^{2}&2r\bar{s}&\bar{s}^{2}\\\
-{r}{s}&|r|^{2}-|s|^{2}&\bar{r}\bar{s}\\\
{s}^{2}&-2\bar{r}{s}&\bar{r}^{2}\end{array}\right)F(\gamma z)$

where $\gamma=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$ and
$r=\overline{cz+d}$ and $s=\bar{c}t$. The $1$-forms
$\beta_{0}\mathrel{\mathop{\ordinarycolon}}=-\frac{dz}{t},\beta_{1}\mathrel{\mathop{\ordinarycolon}}=\frac{dt}{t},\beta_{2}\mathrel{\mathop{\ordinarycolon}}=\frac{d\bar{z}}{t}$
form a basis of differential $1$-forms on $\mathbb{H}_{3}$.

The modular group ${\mathrm{SL}}_{2}(\mathbb{C})$ acts on the space of
differential $1$ -forms as

$\gamma\cdot{}^{t}\left(\beta_{0},\beta_{1},\beta_{2}\right)=Sym^{2}(j(\gamma,z))^{t}\left(\beta_{0},\beta_{1},\beta_{2}\right).$

Here $Sym^{2}$ is the symmetric $2$-nd power representation of
${\mathrm{SL}}_{2}(\mathbb{C})$ on $\mathbb{C}^{2}$.

With both the definition of weights, the differential $F\cdot\beta$ is
$\Gamma$ invariant. We can also generalize vector-valued Bianchi modular form
to arbitrary weight $k$ by changing $Sym^{2}$ to $Sym^{k}$.

######  Definition 5.

A weight $2$ cuspidal Bianchi modular form for a congruence subgroup
$\Gamma\subset{\mathrm{SL}}_{2}(\mathcal{O})$ is a real analytic function
$F\mathrel{\mathop{\ordinarycolon}}\mathbb{H}_{3}\rightarrow\mathbb{C}^{3}$
with the following properties:

  1. (1)

$F\cdot\beta$ is a harmonic differential 1-form on $\mathbb{H}_{3}$ that is
$\Gamma$ -invariant,

  2. (2)

$\int_{\mathbb{C}/\mathcal{O}}(F\mid\gamma)(z,t)dz=0$ for every
$\gamma\in{\mathrm{SL}}_{2}(\mathcal{O})$.

The last condition is equivalent to saying that the constant coefficient in
the Fourier Bessel expansion of $F\mid\gamma$ is equal to zero for every
$\gamma\in{\mathrm{SL}}_{2}(\mathcal{O})$.

The $\Gamma$ invariance implies that $F$ has a Fourier-Bessel expansion of the
form

$F(z,t)={\sum_{\alpha\in\mathcal{O},\alpha\neq
0}c(\alpha)t^{2}\mathbf{K}\left(\frac{4\pi|\alpha|t}{\sqrt{d_{K}}}\right)\psi\left(\frac{\alpha
z}{\sqrt{d_{K}}}\right)};$

where $\psi(z)=e^{2\pi(z+\bar{z})}$ and
$\mathbf{K}(y)=\left(-\frac{i}{2}K_{1}(y),K_{0}(y),\frac{i}{2}K_{1}(y)\right)$
with $K_{0},K_{1}$ being the modified Bessel functions satisfying the
differential equation

$\frac{dK_{j}}{dy^{2}}+\frac{1}{y}\frac{dK_{j}}{dt}-\left(1+\frac{1}{y^{2j}}\right)K_{j}=0,\quad
j=0,1.$

Note that both the functions $K_{0}$ and $K_{1}$ are rapidly decreasing at
infinity. Suppose that $\Gamma$ has no points of finite order. Then
$Y_{\Gamma}\mathrel{\mathop{\ordinarycolon}}=\Gamma\backslash\mathbb{H}_{3}$
is a real-analytic manifold of dimension three. Denote the closure by
$X_{\Gamma}=\Gamma\backslash\bar{\mathbb{H}}_{3}$. Then an automorphic form
$F$ is called a cusp form or cuspidal if it vanishes on all the cusps of
$X_{\Gamma}$. Conversely, we say $F$ is not a cusp form if $F_{i}$ does not
vanish at the cusp for some $i$.

###  2.4. Ito’s Eisenstein differential forms [18]

Let the quaternion number
$u=z+jt\in\mathbb{H}_{3}\left(j^{2}=-1,ij=-ji\right)$ with $z\in\mathbb{C}$
and $t>0$. Write $z(u)=z$ and $t(u)=t$ if $u=z+jt$. Every element
$M=\left(\begin{array}[]{ll}a&b\\\
c&d\end{array}\right)\in{\mathrm{SL}}_{2}(\mathbb{C})$ acts on
$\mathbb{H}_{3}$ by

$Mu\mathrel{\mathop{\ordinarycolon}}=(au+b)(cu+d)^{-1},$

where the right-hand side is taken in the skew field of quaternions. For two
complex numbers $(m,n)\neq(0,0)$, take a matrix
$M=\left(\begin{array}[]{cc}*&*\\\
m&n\end{array}\right)\in{\mathrm{SL}}_{2}(\mathbb{C})$ and let

$\displaystyle t(m,n;u)=t(Mu),$ $\displaystyle
J(m,n;u)=\left(\frac{\partial}{\partial z},\frac{\partial}{\partial
t},\frac{\partial}{\partial\bar{z}}\right)z(Mu).$

Fix a lattice $L$ in $\mathbb{C}$ and put
$\mathcal{O}_{L}=\\{m\in\mathbb{C}\mid mL\subset L\\}$. The Eisenstein series

$E(s;u)\mathrel{\mathop{\ordinarycolon}}=\sum_{m,n\in
L}^{\prime}J(m,n;u)t(m,n;u)^{s},\quad\operatorname{Re}(s)>1$

can be continued analytically to the whole $s$-plane. Here the prime on the
summation symbol means to omit the meaningless terms, i.e., the term
corresponding to $m=n=0$ in this case. Define the complex-valued differential
form $\omega$ on $\mathbb{H}_{3}$ by

$\omega=E(u,0)\left(\begin{array}[]{l}dz\\\ dt\\\ d\bar{z}\end{array}\right).$

This differential form is closed and invariant with respect to
${\mathrm{SL}}_{2}(\mathcal{O}_{L})$. An integral of $\omega$ can be written
in the form of the Fourier expansion as

$H(u)=G_{2}(0)(z-\bar{z})-\frac{4\pi}{D(L)}t\sum_{m\in L,n\in
L^{\prime}}^{\prime}\frac{\bar{m}n}{|mn|}K_{1}(4\pi|mn|t)e(mnz).$

Here, for each non-negative integer $k$,

$G_{k}(x)\mathrel{\mathop{\ordinarycolon}}=\left.\sum_{w\in
L}^{\prime}(w+x)^{-k}|w+x|^{-s}\right|_{s=0},$

where the value at $s=0$ is to be understood in the sense of analytic
continuation,
$D(L)\mathrel{\mathop{\ordinarycolon}}=w_{1}\bar{w}_{2}-\bar{w}_{1}w_{2}$ if
$L=\mathbb{Z}w_{1}+\mathbb{Z}w_{2}$ with
$\operatorname{Im}\left(w_{1}/w_{2}\right)>0$, and
$L^{\prime}=D(L)^{-1}\bar{L}$. We take the prime on the summation symbol to
omit the terms corresponding to $mn=0$. Furthermore,
$e(z)\mathrel{\mathop{\ordinarycolon}}=\exp(2\pi i(z+\bar{z}))$ and $K_{v}(z)$
denotes the modified Bessel function of the second kind.

The function $H$ is harmonic with respect to the Riemannian structure of
$\mathbb{H}_{3}$ given by the ${\mathrm{SL}}_{2}(\mathbb{C})$-invariant
Riemannian metric $\frac{1}{t^{2}}\left(dx^{2}+dy^{2}+dt^{2}\right)$
$\quad(u=z+jt\in\mathbb{H},z=x+iy)$.

######  Theorem 1.

Define the map
$\Phi\mathrel{\mathop{\ordinarycolon}}{\mathrm{SL}}_{2}(\mathcal{O}_{L})\rightarrow\mathbb{C}$
by

$\Phi\left(\begin{array}[]{ll}a&b\\\
c&d\end{array}\right)=\begin{cases}G_{2}(0)I\left(\begin{array}[]{c}a+d\\\
c\end{array}\right)-D(a,c),&c\neq 0,\\\
G_{2}(0)I\left(\frac{b}{d}\right),&c=0\end{cases}$

with $I(z)\mathrel{\mathop{\ordinarycolon}}=z-\bar{z}$ and

$D(a,c)\mathrel{\mathop{\ordinarycolon}}=\frac{1}{c}\sum_{r\in
L/cL}G_{1}\left(\frac{ar}{c}\right)G_{1}\left(\frac{r}{c}\right)$

Then

$H(Au)=H(u)+\Phi(A)$

for every $A$ in ${\mathrm{SL}}_{2}(\mathcal{O}_{L})$.

##  3\. Inner product formula for Bianchi modular forms

For this section, we work over the field $K=\mathbb{Q}(i)$ for simplicity. The
analysis that we carry out is exactly the same for any Euclidean imaginary
quadratic field, is slightly more involved for non-Euclidean $K$ but still
with class number one. To extend our results to these cases, one simply works
with the respective fundamental domains described in [11]. When the class
number is greater than one, the fundamental domain of the action of
${\mathrm{SL}}_{2}(\mathcal{O})$ on the hyperbolic three space
$\mathbb{H}_{3}$ becomes much more complicated to describe. It will be really
intriguing to find the relation of our work with group theoretic results about
homology groups of $3$-manifolds of Bartel[4], [3] to remove the dependence on
the explicit fundamental domain.

###  3.1. Quasi-periods

Consider the field $K=\mathbb{Q}[i]$ and let
$\Gamma\subset{\mathrm{SL}}_{2}(\mathbb{Z}[i])$ be a subgroup of the finite
index and let $M_{2}(\Gamma)$ be the space of all Bianchi modular forms of
weight two for the subgroup $\Gamma\subset{\mathrm{SL}}_{2}(\mathbb{Z}[i])$.
For this section, we choose the basis $\widetilde{\beta}=(-dz,dt,d\bar{z})$
and note that $\beta=\frac{1}{t}\widetilde{\beta}$. We know that both $\beta$
and $\widetilde{\beta}$ are the basis of the space of differential $1$ forms
on $\mathbb{H}_{3}$.

######  Definition 6.

(Quasi-periods of Bianchi modular forms) For a Bianchi modular form
$F=(F_{0},F_{1},F_{2})\mathrel{\mathop{\ordinarycolon}}\mathbb{H}_{3}\rightarrow\mathbb{C}^{3}$
and
$\widetilde{\beta}=(\widetilde{\beta}_{0},\widetilde{\beta}_{1},\widetilde{\beta}_{2})$
and $u\mathrel{\mathop{\ordinarycolon}}=z+tj\in\mathbb{H}_{3}$, we define the
quasi-period to be the $0$-form:

$\pi_{F_{0}}(z)\mathrel{\mathop{\ordinarycolon}}=\int_{1/2}^{z}{F_{0}(u)}\widetilde{\beta}_{0};$
$\pi_{F_{1}}(t)\mathrel{\mathop{\ordinarycolon}}=\int_{\frac{\sqrt{3}}{2}}^{t}{F_{2}(u)}\widetilde{\beta}_{1};\pi_{F_{2}}(\bar{z})\mathrel{\mathop{\ordinarycolon}}=\int_{\frac{1}{2}}^{\bar{z}}{F_{2}(u)}\widetilde{\beta}_{2};$

and

$\pi_{F}\mathrel{\mathop{\ordinarycolon}}=\pi_{F_{0}}(z)+\pi_{F_{1}}(t)+\pi_{F_{2}}(\bar{z}).$

Then, we have

$d\pi_{F}=d\pi_{F_{0}}(z)+d\pi_{F_{1}}(t)+d\pi_{F_{2}}(\bar{z});$
$d\pi_{F}={F_{0}}(u)dz+F_{1}(u)dt+F_{2}(u)d\bar{z}$

with $P_{1}$, $P_{2}$ and $P_{3}$ are points in fundamental domain defined in
§ 2.1.

By the Leibniz integral rule for differentiation under the integral sign, we
have

$d(\pi_{F_{i}})={F_{i}(u)}\widetilde{\beta}_{i}.$

We prove the following formula that works for any arbitrary subgroup
$\Gamma\subset
G\mathrel{\mathop{\ordinarycolon}}={\mathrm{SL}}_{2}(\mathbb{Z}[i])$ of finite
index. This is a generalization of the inner product formula of Banerjee-Merel
[2] for imaginary quadratic fields.

######  Proposition 7.

Let $F,S\in M_{2}(\Gamma)$ be two Bianchi modular forms, with at least one of
them being a cusp form. The inner product of these two modular forms is given
by $<F,S>=I$ with

$\displaystyle I=$
$\displaystyle\frac{1}{12[G\mathrel{\mathop{\ordinarycolon}}\Gamma]}\sum_{g\in\Gamma\backslash
G}\left(-\frac{1}{2}\int_{0}^{\infty}{tF_{1{|_{gh}}}\beta_{1}}\int_{B_{0}}t\bar{S}_{1}(\beta_{0}\wedge\beta_{2})+\int_{0}^{\infty}{tF_{0{|_{gh}}}\beta_{0}}\int_{B_{0}}t\bar{S}_{0}(\beta_{1}\wedge\beta_{2})\right)$
$\displaystyle+\sum_{g\in\Gamma\backslash
G}\left(\int_{0}^{\infty}{tF_{2{|_{gh}}}\beta_{2}}\int_{B_{0}}t\bar{S}_{2}(\beta_{0}\wedge\beta_{1})\right)$

Where $B_{0}$ is the face of the octahedron $\\{0,P_{2},P_{3}\\}$.

###### Proof.

Let $F=(F_{0},F_{1},F_{2})$ and $S=(S_{0},S_{1},S_{2})$. By [20, p. 549], the
inner product is given by

$\displaystyle<F,S>$ $\displaystyle\mathrel{\mathop{\ordinarycolon}}=$
$\displaystyle\frac{1}{12i[G\mathrel{\mathop{\ordinarycolon}}\Gamma]}\int_{\Gamma\backslash\mathbb{H}_{3}^{*}}F\cdot\widetilde{\beta}\wedge*(S\cdot\widetilde{\beta})$
$\displaystyle=$
$\displaystyle\frac{1}{12i[G\mathrel{\mathop{\ordinarycolon}}\Gamma]}\sum_{g\in\Gamma\backslash
G}\int_{\mathfrak{F}}F|_{g}\cdot\widetilde{\beta}\wedge*(S|_{g}\cdot\widetilde{\beta})$

In the above equations, $\mathfrak{F}$ is the fundamental domain as in [10,
diagram 4.2]. Now write $\\{P_{1},j\infty\\}$ as the translate
$h\\{0,\infty\\}$ with $h\in\Gamma$ as in loc. cit.

Let us consider the integral
$\int_{\mathfrak{F}}F|_{g}\cdot\widetilde{\beta}\wedge*(S|_{g}\cdot\widetilde{\beta})$.
The other two can be calculated similarly. Choose a quasi-period such that
$d(\pi_{F})=F\cdot\widetilde{\beta}$.

Note that
$d(\pi_{F}.(*(S.\widetilde{\beta}))=d(\pi_{F})\wedge*(S\cdot\widetilde{\beta})+\pi_{F}.d(*(S\cdot\widetilde{\beta}))$.
We know that $d(*(S\cdot\widetilde{\beta}))=0$ because $S$ is harmonic
function. Observe that

$\displaystyle d(\pi_{F}.(*(S.\widetilde{\beta}))$
$\displaystyle=d(\pi_{F})\wedge*(S\cdot\widetilde{\beta})+\pi_{F}.d(*(S\cdot\widetilde{\beta}))$
$\displaystyle d(\pi_{F}.(*(S.\widetilde{\beta}))$
$\displaystyle=d(\pi_{F})\wedge*(S\cdot\widetilde{\beta})+0$ $\displaystyle
d(\pi_{F}.(*(S.\widetilde{\beta}))$
$\displaystyle=F\cdot\widetilde{\beta}\wedge*(S\cdot\widetilde{\beta})$

The last equality follows from the fact $dz\wedge dz=dt\wedge dt=0$.

Let $B_{0}$ be the face of the octahedron. Note that all the sides are
translations of $\\{0,j\infty\\}$ by the group $\Gamma_{P_{1}}$. For the face
$B_{0}=\\{0,P_{2},P_{3}\\}$, we get integrals. By Stokes theorem, we have

$\displaystyle\int_{\Gamma\backslash\mathbb{H}_{3}^{*}}F\cdot\widetilde{\beta}\wedge*(S\cdot\widetilde{\beta})=\int_{\Gamma\backslash\mathbb{H}_{3}^{*}}d(\pi_{F}.(*(S.\widetilde{\beta}))$
$\displaystyle=\sum_{g\in\Gamma\backslash
G}\int_{\mathfrak{F}}d(\pi_{F_{|_{g}}}.(*(S|_{g}.\widetilde{\beta}))$
$\displaystyle=\sum_{g\in\Gamma\backslash
G}\int_{\partial\mathfrak{F}}(\pi_{F_{|_{g}}}.(*(S|_{g}.\widetilde{\beta})).$

Hence, we deduce that

$\displaystyle\sum_{g\in\Gamma\backslash
G}\int_{\partial\mathfrak{F}}(\pi_{F_{|_{g}}}.(*(S|_{g}.\widetilde{\beta}))$
$\displaystyle=\sum_{g\in\Gamma\backslash
G}\int_{\partial\mathfrak{F}}\left(\pi_{F_{|_{g}}}.(-\frac{1}{2}it\bar{S}_{1}(\beta_{0}\wedge\beta_{2})+it\bar{S}_{0}(\beta_{1}\wedge\beta_{2})+it\bar{S}_{2}(\beta_{0}\wedge\beta_{1})\right)$
$\displaystyle=\sum_{g\in\Gamma\backslash
G}\int_{\partial\mathfrak{F}}\left(-\frac{1}{2}i\pi_{F_{1{|_{g}}}}t\bar{S}_{1}(\beta_{0}\wedge\beta_{2})+i\pi_{F_{0{|_{g}}}}t\bar{S}_{0}(\beta_{1}\wedge\beta_{2})+i\pi_{F_{2{|_{g}}}}t\bar{S}_{2}(\beta_{0}\wedge\beta_{1})\right)$
$\displaystyle=$ $\displaystyle\sum_{g\in\Gamma\backslash
G}\left(-\frac{1}{2}i\int_{\frac{\sqrt{3}}{2}}^{\infty}{F_{1{|_{g}}}\widetilde{\beta}_{1}}\int_{B_{0}}t\bar{S}_{1}(\beta_{0}\wedge\beta_{2})+i\int_{\frac{1}{2}}^{\infty}{F_{0{|_{g}}}}\widetilde{\beta}_{0}\int_{B_{0}}t\bar{S}_{0}(\beta_{1}\wedge\beta_{2})\right)$
$\displaystyle+\sum_{g\in\Gamma\backslash
G}\left(i\int_{\frac{1}{2}}^{\infty}{F_{2{|_{g}}}\widetilde{\beta}_{2}}\int_{B_{0}}t\bar{S}_{2}(\beta_{0}\wedge\beta_{1})\right)$
$\displaystyle=$ $\displaystyle\sum_{g\in\Gamma\backslash
G}\left(-\frac{1}{2}i\int_{0}^{\infty}{F_{1{|_{gh}}}\widetilde{\beta}_{1}}\int_{B_{0}}t\bar{S}_{1}(\beta_{0}\wedge\beta_{2})+i\int_{0}^{\infty}{F_{0{|_{gh}}}\widetilde{\beta}_{0}}\int_{B_{0}}t\bar{S}_{0}(\beta_{1}\wedge\beta_{2})\right)$
$\displaystyle+\sum_{g\in\Gamma\backslash
G}\left(i\int_{0}^{\infty}{F_{2{|_{gh}}}\widetilde{\beta}_{2}}\int_{B_{0}}t\bar{S}_{2}(\beta_{0}\wedge\beta_{1})\right)$

for some $h\in\Gamma$.

∎

We now prove our Theorem 2.

###### Proof.

$\displaystyle<F,G>$ $\displaystyle\mathrel{\mathop{\ordinarycolon}}=$
$\displaystyle\int_{\Gamma\backslash\mathbb{H}_{3}^{*}}F\cdot\widetilde{\beta}\wedge*(G\cdot\widetilde{\beta})$
$\displaystyle=$
$\displaystyle\int_{\Gamma\backslash\mathbb{H}_{3}^{*}}(F_{0}\widetilde{\beta}_{0}+F_{1}\beta_{1}+F_{2}\widetilde{\beta}_{2})\wedge(-\frac{i}{2}\overline{G_{1}}\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{2}+i\overline{G_{0}}\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{2}+i\overline{G_{2}}\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{1})$
$\displaystyle=$
$\displaystyle\int_{\Gamma\backslash\mathbb{H}_{3}^{*}}(iF_{0}\overline{G_{0}}\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{2}-\frac{i}{2}F_{1}\overline{G_{1}}\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{2}+iF_{2}\overline{G_{2}}\widetilde{\beta}_{2}\wedge\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{1})$
$\displaystyle=$
$\displaystyle\int_{\Gamma\backslash\mathbb{H}_{3}^{*}}(iF_{0}\overline{G_{0}}\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{2}+\frac{i}{2}F_{1}\overline{G_{1}}\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{2}+iF_{2}\overline{G_{2}}\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{2})$
$\displaystyle=$
$\displaystyle\int_{\Gamma\backslash\mathbb{H}_{3}^{*}}[iF_{0}\overline{G_{0}}+\frac{i}{2}F_{1}\overline{G_{1}}+iF_{2}\overline{G_{2}}]\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{2}$
$\displaystyle=$
$\displaystyle\int_{\Gamma\backslash\mathbb{H}_{3}^{*}}H\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{2}$
$\displaystyle=$ $\displaystyle\sum_{g\in\Gamma\backslash
G}\int_{\mathfrak{F}}H|_{g}\widetilde{\beta}_{0}\wedge\widetilde{\beta}_{1}\wedge\widetilde{\beta}_{2}.$

Consider the function

$H=iF_{0}\overline{G_{0}}+\frac{i}{2}F_{1}\overline{G_{1}}+iF_{2}\overline{G_{2}}.$

By [11, theorem 1], we write the modular symbol
${\mathcal{E}}_{\xi}\in\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma},\mathbb{C})$
as

${\mathcal{E}}_{\xi}=\sum_{g\in\Gamma\backslash
G}\widetilde{F_{\xi}}(g)\\{g0,g\infty\\};$

for some arbitrary $\widetilde{F_{\xi}}(g)\in\mathbb{C}$ and for all
$g\in\Gamma\backslash G$. For $F=(F_{0},F_{1},F_{2})$, we write

$I=\int_{{\mathcal{E}}_{\xi}}F\cdot\beta=\sum_{g\in\Gamma\backslash
G}\widetilde{F_{\xi}}(g)\int_{\\{g0,g\infty\\}}F\cdot\beta.$

To prove the statement, we endeavor to find $\widetilde{F_{\xi}}(g)$ so that
we have an equality

$\widetilde{F_{\xi}}(g)\int_{\\{g0,g\infty\\}}F\cdot\beta=\int_{\mathfrak{F}}(H|_{g})\beta_{0}\wedge\beta_{1}\wedge\beta_{2}.$

We then have

$\displaystyle\int_{{\mathcal{E}}_{\xi}}F\cdot\beta$ $\displaystyle=$
$\displaystyle\sum_{g\in\Gamma\backslash
G}\widetilde{F_{\xi}}(g)\int_{g(0)}^{g(\infty)}F\cdot\beta$ $\displaystyle=$
$\displaystyle\sum_{g\in\Gamma\backslash
G}\int_{\mathfrak{F}}H|_{g}\beta_{0}\wedge\beta_{1}\wedge\beta_{2}.$

By Theorem 7, the right-hand side is equal to

$-12i[G\mathrel{\mathop{\ordinarycolon}}\Gamma]\langle E_{\xi},F\rangle=0.$

Recall that $\mathfrak{F}$ is the fundamental domain or octahedron. Apriori,
we define

$F_{\xi}(g)\mathrel{\mathop{\ordinarycolon}}=\frac{\int_{\mathfrak{F}}H|_{g}\beta_{0}\wedge\beta_{1}\wedge\beta_{2}}{\int_{g(0)}^{g(\infty)}F\cdot\beta}.$

We need to ensure this integral is independent of the choice of the cusp form
$F$. By changing of variable, we write

$\int_{\mathfrak{F}}H|_{g}\beta_{0}\wedge\beta_{1}\wedge\beta_{2}=(\int_{g(0)}^{g(j\infty)}F\cdot\beta)(\int_{B_{0}}\star(E_{\xi}\cdot\beta).$

Once we separate the variable, we now define the Eisenstein element. For the
Eisenstein series $E_{\xi}$, define

$\widetilde{F_{\xi}}(g)=F_{\xi}(g)\mathrel{\mathop{\ordinarycolon}}=\int_{B_{0}}*(E_{\xi}\cdot\beta)[g].$

This particular $F_{\xi}$ produces the modular symbol with the required
property as in the theorem. ∎

##  4\. Eisenstein Cohomology groups for $\Gamma_{1}(N)$

###  4.1. Dimension of Eisenstein cohomology

Recall the definition of Eisenstein cohomology groups following Sengün-
Türkelli [12]. The Borel-Serre compactification[29, appendix] $X_{\Gamma}$ of
$Y_{\Gamma}$ is a compact three manifold with boundary $\partial X_{\Gamma}$
whose interior is homeomorphic to $Y_{\Gamma}$.

Given $n\geqslant 0$, let the space of homogeneous polynomials of degree $n$
on the variables $x,y$ with complex coefficients be denoted by
$\mathbb{C}[x,y]_{n}$. The modular group ${\mathrm{SL}}_{2}(\mathbb{C})$ acts
on this space in the obvious way permitted by the two variables. Consider the
$\mathrm{SL}_{2}(\mathbb{C})$-module

$E_{n}\mathrel{\mathop{\ordinarycolon}}=\mathbb{C}[x,y]_{n}\otimes_{\mathbb{C}}\overline{\mathbb{C}[x,y]_{n}};$

where the overline on the second factor is to indicate that the action on this
factor is twisted with complex conjugation. Considered as a module, $E_{n}$
gives rise to a locally constant sheaf $\mathcal{E}_{n}$ on $Y_{\Gamma}$ whose
stalks are isomorphic to $E_{n}$. Consider the long exact sequence

$\ldots\rightarrow\mathrm{H}_{c}^{i}\left(Y_{\Gamma};\mathcal{E}_{n}\right)\rightarrow\mathrm{H}^{i}\left(X_{\Gamma};\overline{\mathcal{E}}_{n}\right)\rightarrow\mathrm{H}^{i}\left(\partial
X_{\Gamma};\overline{\mathcal{E}}_{n}\right)\rightarrow\ldots$

where $\mathrm{H}_{c}^{i}$ denotes the compactly supported cohomology and
$\overline{\mathcal{E}}_{n}$ is a certain natural extension of
$\mathcal{E}_{n}$ to $X_{\Gamma}$.

######  Definition 8.

(Eisenstein cohomology groups) The image of the compactly supported cohomology
is known as the cuspidal cohomology $\mathrm{H}_{\text{cusp}}^{i}$. The
complement of the cuspidal cohomology inside $\mathrm{H}^{i}$ is the
Eisenstein cohomology $\mathrm{H}_{\text{Eis}}^{i}$. This is isomorphic to the
image of the restriction map inside the cohomology of the boundary.

Recall that the Hecke action is defined as usual via correspondences on the
$3$-manifold $X_{\Gamma}$. The decomposition
$\mathrm{H}^{i}=\mathrm{H}_{cusp}^{i}\oplus\mathrm{H}_{Eis}^{i}$ respects the
Hecke action. By construction, the embedding $Y_{\Gamma}\hookrightarrow
X_{\Gamma}$ is a homotopy invariance. We have the following isomorphisms

$\mathrm{H}^{i}\left(X_{\Gamma};\overline{\mathcal{E}}_{n}\right)\simeq\mathrm{H}^{i}\left(Y_{\Gamma};\mathcal{E}_{n}\right)\simeq\mathrm{H}^{i}\left(\Gamma;E_{n}\right).$

Note that the last one is the group cohomology. Via the above isomorphisms, we
define the cuspidal and Eisenstein parts of the cohomology groups
$\mathrm{H}^{i}\left(\Gamma;E_{n}\right)$.

  * •

The boundary $\partial X_{\Gamma}$ is a disjoint union of $2$-tori, each
corresponds to a cusp of $Y_{\Gamma}$ (except for $\mathbb{Q}(i)$ or
$\mathbb{Q}(\sqrt{-3})$). The set of cusps of $\Gamma$ can be identified with
the orbit space $\Gamma\backslash\mathbb{P}^{1}(K)$.

  * •

The boundary $\partial X_{\Gamma}$ is made up of a disjoint union of $2$-tori,
each of which corresponds to a cusp of $Y_{\Gamma}$. The set of cusps of
$\Gamma$ can be represented by the orbit space
$\Gamma\backslash\mathbb{P}^{1}(K)$. It is well known that when $\Gamma$ is
the full Bianchi group, the number of cusps is $h(K)$, the class number of
$K$.

The fundamental group of a $2$-torus is a free abelian group on two
generators. It is easy to compute the size of the cohomology groups.

######  Proposition 9.

Let $\Gamma$ be a congruence subgroup of a Bianchi group and let
$\mathcal{C}_{\Gamma}$ be the set of cusps of $\Gamma$. Then

$\begin{gathered}\operatorname{dim}\mathrm{H}^{0}\left(\partial
X_{\Gamma},\overline{\mathcal{E}}_{k}\right)=\operatorname{dim}\mathrm{H}^{2}\left(\partial
X_{\Gamma},\overline{\mathcal{E}}_{k}\right)=\\#\mathcal{C}_{\Gamma}\\\
\operatorname{dim}\mathrm{H}^{1}\left(\partial
X_{\Gamma},\overline{\mathcal{E}}_{k}\right)=2\cdot\\#\mathcal{C}_{\Gamma}.\end{gathered}$

The long exact sequence associated with the pair $\left(X_{\Gamma},\partial
X_{\Gamma}\right)$ is compatible with the action of the involution $\tau$. It
follows from algebraic topology that for $k>0$, the image of the restriction
map.

$\mathrm{H}^{i}\left(X_{\Gamma},\overline{\mathcal{E}}_{k}\right)\rightarrow\mathrm{H}^{i}\left(\partial
X_{\Gamma},\overline{\mathcal{E}}_{k}\right)$

is onto when $i=2$ and its image has half the rank of the target space when
$i=1$. Hence we have the following result:

######  Proposition 10.

Let $k>0$ and $\Gamma$ as above. Then

  1. (1)

$\operatorname{dim}\mathrm{H}_{\text{Eis
}}^{0}\left(X_{\Gamma},\mathcal{E}_{k}\right)=0$.

  2. (2)

$\operatorname{dim}\mathrm{H}_{\text{Eis
}}^{i}\left(X_{\Gamma},\mathcal{E}_{k}\right)=\\#\mathcal{C}_{\Gamma}$ for
$i=1,2$.

###### Proof.

Consider the long exact sequence associated with $(X_{\Gamma},\partial
X_{\Gamma})$:

$\displaystyle...\rightarrow\mathrm{H}^{1}(\partial
X_{\Gamma},\mathcal{E}_{k})\rightarrow\mathrm{H}^{2}_{c}(X_{\Gamma},\mathcal{E}_{k})\rightarrow\mathrm{H}^{2}(X_{\Gamma},\mathcal{E}_{k})\rightarrow\mathrm{H}^{2}(\partial
X_{\Gamma},\mathcal{E}_{k})$
$\displaystyle\rightarrow\mathrm{H}^{3}_{c}(X_{\Gamma},\mathcal{E}_{k})\rightarrow\mathrm{H}^{3}(X_{\Gamma},\mathcal{E}_{k})=0.$

The subscript $``_{c}"$ denotes the compactly supported cohomology. If
$\mathcal{E}_{k}=\mathbb{C}$ is trivial, then
$\mathrm{H}^{3}_{c}(X_{\Gamma},\mathcal{E}_{k})=\mathbb{C}$, but if
$\mathcal{E}_{k}$ is not trivial, then
$\mathrm{H}^{3}_{c}(X_{\Gamma},\mathcal{E}_{k})$ vanishes. When
$\mathcal{E}_{k}$ is trivial, the image of the restriction map

$\mathrm{H}^{2}(X_{\Gamma},\mathcal{E}_{k})\rightarrow\mathrm{H}^{2}(\partial
X_{\Gamma},\mathcal{E}_{k})$

has one dimensional cokernel, and when $\mathcal{E}_{k}$ is non-trivial, it is
surjective. For $k>0$, we analyze the image of the restriction maps following
Harder [17, proposition 4.7.1]. From loc. cit., the image of the restriction
map

$\mathrm{H}^{i}\left(X_{\Gamma},{\mathcal{E}}_{k}\right)\rightarrow\mathrm{H}^{i}\left(\partial
X_{\Gamma},{\mathcal{E}}_{k}\right)$

is onto when $i=2$ and its image has half the rank of the target space when
$i=1$. So we have

$\operatorname{dim}\mathrm{H}_{\text{Eis
}}^{1}\left(X_{\Gamma},\mathcal{E}_{k}\right)=\frac{1}{2}\operatorname{dim}\mathrm{H}^{1}\left(\partial
X_{\Gamma},{\mathcal{E}}_{k}\right)$

$\operatorname{dim}\mathrm{H}_{\text{Eis
}}^{2}\left(X_{\Gamma},\mathcal{E}_{k}\right)=\operatorname{dim}\mathrm{H}^{2}\left(\partial
X_{\Gamma},{\mathcal{E}}_{k}\right)$

$\quad\operatorname{dim}\mathrm{H}_{\text{Eis
}}^{i}\left(X_{\Gamma},\mathcal{E}_{k}\right)=\\#\mathcal{C}_{\Gamma}\quad$
for $i=1,2$.

∎

##  5\. Lefschetz numbers for complex conjugations

Recall the study Lefschetz fixed point theorem following Sengün-Türkelli [12].
Let $\sigma$ be the complex conjugation. The action of $\sigma$ on
$\mathbb{H}_{3}$ is defined by $(z,t)\mapsto(\bar{z},t)$. It also acts on
${\mathrm{SL}}_{2}(\mathbb{C})$ by acting on the entries of a matrix in an
obvious way. If $M\in\mathrm{SL}_{2}(\mathbb{C})$, then we write
${}^{\sigma}M$, or simply $\bar{M}$, for the image of $M$ under the action of
$\sigma$.

In this section, we also consider the twisted complex conjugation and denote
the same by $\tau$. The action of $\tau$ on $\mathbb{H}_{3}$ is given by
$(z,t)\mapsto(-\bar{z},t)$ where $\bar{z}$ denotes the complex conjugate of
$z$. Its action on $\mathrm{SL}_{2}(\mathbb{C})$ is defined as
$\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\mapsto\left(\begin{array}[]{cc}\bar{a}&-\bar{b}\\\
-\bar{c}&\bar{d}\end{array}\right)$ where the bar in the notation denotes the
complex conjugation. It is convenient to regard $\tau$ as the composition
$\alpha\circ\sigma=\sigma\circ\alpha$ where

$\alpha\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)=\left(\begin{array}[]{cc}a&-b\\\
-c&d\end{array}\right)=\beta\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\beta$

where $\beta\mathrel{\mathop{\ordinarycolon}}=\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right)$, for every $\left(\begin{array}[]{ll}a&b\\\
c&d\end{array}\right)\in\mathrm{SL}_{2}(\mathbb{C})$ and $\alpha(z,t)=(-z,t)$
for every $(z,t)\in\mathbb{H}_{3}$.

Both $\sigma$ and $\tau$ are orientation-reversing, and they can be extended
to the Borel-Serre compactification naturally (see [26, section 1.4]). The
action of $\sigma$ on $E_{k,k}$ can be described as follows: $\sigma(P\otimes
Q)=Q\otimes P$. Similarly, we have $\tau(P\otimes
Q)=\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right)Q\otimes\left(\begin{array}[]{rr}-1&0\\\
0&1\end{array}\right)P$. These actions are compatible with those on
$\mathrm{SL}_{2}(\mathbb{C})$.

In this section, we discuss the Lefschetz numbers for these two involutions.
We use the symbol $\rho$ when we want to state results that are true for both
of them. We start with a useful lemma (see [9, I.4.3] for proof).

######  Lemma 11.

Let $\gamma\in\Gamma$ and $x=(\gamma\rho)^{2}$. Then
$\operatorname{tr}\left(\gamma\rho\mid
E_{k,k}\right)=\operatorname{tr}\left(x\mid E_{k}\right)$.

###  5.1. Lefschetz number of $\sigma$ for congruence subgroups

Let $\Gamma=\Gamma_{1}(N)\subseteq{\mathrm{SL}}_{2}(\mathcal{O}_{K})$ be a
congruence subgroup of level ($N$). Denote its image in
$\mathrm{PSL_{2}}(\mathcal{O})$ by $\bar{\Gamma}$. Then for $N>2,\Gamma$ (and
so $\bar{\Gamma}$) is torsion-free. In [31, §5.1], it is shown that for
sufficiently large N (for example if $N>3$), the congruence subgroup
$\Gamma_{1}(N)$ is torsion-free. In this section, we use [12, Theorem 2.1] to
calculate the Lefschetz numbers $L\left(\sigma,\Gamma_{1}(N),E_{k,k}\right)$.

First, we need to analyze the fixed point set $Y_{\Gamma}^{\sigma}$. Let
$H(1)$ be the subset of $H^{1}(\sigma,\bar{\Gamma})$ consisting of the
cocycles $\gamma\in\bar{\Gamma}$ with
$\operatorname{det}\left(\gamma^{\sigma}{\gamma}\right)=1$. Let $H(2)$ be the
subset consisting of the cocycles $\gamma$ with
$\operatorname{det}\left(\gamma^{\sigma}{\gamma}\right)=-1$. We have
$H^{1}(\sigma,\bar{\Gamma})=H(1)\cup H(2)$. If $\Gamma$ is torsion free, then
$H^{1}(\sigma,\bar{\Gamma})=H(1)$. Let
$\gamma_{1}=\left(\begin{array}[]{ll}1&0\\\
0&1\end{array}\right),\gamma_{1}^{\prime}=\left(\begin{array}[]{cc}1&\sqrt{d}\\\
0&1\end{array}\right)$ and $\gamma_{2}=\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)$. Consider now the matrix

$\gamma_{2}^{\prime}=\begin{cases}\left(\begin{array}[]{cc}1+\sqrt{d}&(2-d)/2\\\
-2&-1+\sqrt{d}\end{array}\right)&\text{if $d\equiv 2\bmod 4$}\\\
\left(\begin{array}[]{cc}\sqrt{d}&(d-1)/2\\\
2&\sqrt{d}\end{array}\right)&\text{if $d\equiv 1\bmod 4$.}\end{cases}$

Notice that $\gamma_{1},\gamma_{1}^{\prime}\in H(1)$ and
$\gamma_{2},\gamma_{2}^{\prime}\in H(2)$. The locally symmetric space
$F(\gamma)$ is a surface if $\gamma\in H(1)$ and is a point if $\gamma\in
H(2)$. In [25], Rohlfs gives the number of translations of the surfaces
corresponding to $\gamma_{1},\gamma_{1}^{\prime}$ and the number of
translations of the points corresponding to $\gamma_{2},\gamma_{2}^{\prime}$.

######  Theorem 2.

(Rohlfs, [25, Theorem 4.1]) . Let $D$ be the discriminant of $K/\mathbb{Q}$
and $t$ be the number of distinct prime divisors of $D$. Let $(N)=$
$\prod_{p\mid D}p_{p}^{j_{p}}\prod_{p\nmid D}(p)^{j_{p}}$ be an ideal with
$N>2$, and let $\Gamma=\Gamma_{1}(N)$ be the congruence subgroup of level
$(N)$. Let $s=\\#\\{p$ prime $|p|D,p\neq 2$ and $\left.j_{p}\neq 0\right\\}.$

Then $Y_{\Gamma}^{\sigma}$ consists of translations of surfaces
$F\left(\gamma_{1}\right)$ and $F\left(\gamma_{1}^{\prime}\right)$ and the
number of translations of these surfaces are denoted by $A$ and $B$
respectively in the table below.

$d$ | $j_{2}$ | $A$ | $B$
---|---|---|---
$d\equiv 1(4)$ | $\geq 0$ | $2^{t-s}$ | 0
$d\equiv 2(4)$ | 0 | $2^{t-s}$ | $2^{t-s-1}$
| 1 | $2^{t-s}$ | $2^{t-s-1}$
| 2 | $8\cdot 2^{t-s}$ | 0
| $\geq 3$ | $8\cdot 2^{t-s-1}$ | 0
$d\equiv 3(4)$ | 0 | $2^{t-s}$ | $2^{t-s-1}$
| 1 | $2^{t-s}$ | 0
| 2 | $8\cdot 2^{t-s}$ | 0
| $j_{2}=2n+1\geq 3$ | $2^{t-s-1}$ | 0
| $j_{2}=2n\geq 4$ | $8\cdot 2^{t-s-1}$ | 0

Now, using [[12],theorem $2.1$] and Theorem 2, we calculate the Lefschetz
numbers for $\Gamma_{1}(N)$. Consider the subgroup
$\Gamma(u,v)=\\{A\in{\mathrm{SL}}_{2}(\mathcal{O})|(u,v)A=(u,v)\\}$. Let
$\mathfrak{P}$ be a prime ideal of $\mathcal{O}$ and recall that
$\Gamma_{0}(\mathfrak{P})\mathrel{\mathop{\ordinarycolon}}=\Big{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in{\mathrm{SL}}_{2}(\mathcal{O})\mathrel{\mathop{\ordinarycolon}}c\in\mathfrak{P}\Big{\\}}$.

######  Lemma 12.

The congruence subgroup $\Gamma_{0}(\mathfrak{P})$ is not a subset of
$\Gamma(u,v)$ for all non-zero $(u,v)\in(\mathbb{C}/K)^{2}$.

###### Proof.

Let $A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma_{0}(\mathfrak{P})$ and
assume $A\in\Gamma(u,v)$ for $0\neq(u,v)\in(\mathbb{C}/K)^{2}$. We then have
$c\equiv 0\pmod{\mathfrak{P}}$.

$(u,v)A$ = $(u,v)\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\begin{pmatrix}au+cv,bu+dv\end{pmatrix}$.

$\begin{pmatrix}au+cv,bu+dv\end{pmatrix}=\begin{pmatrix}u,v\end{pmatrix}$.

$\begin{pmatrix}(a-1)u+cv,bu+(d-1)v\end{pmatrix}=(0,0)$.

$\implies(a-1)u+cv=0,bu+(d-1)v=0$ and we know that $c\equiv
0\pmod{\mathfrak{P}}$

$\implies(a-1)u=0$ and $bu+(d-1)v$=0 $(\pmod{\mathfrak{P}})$

$(a-1)u=0\implies u=0$ and $bu+(d-1)v=0\implies v=0$.

We do not get any non-zero solution. We deduce that
$\Gamma_{0}({\mathfrak{P}})\not\subset\Gamma(u,v)$ for all non-zero
$(u,v)\in(\mathbb{C}/K)^{2}$. ∎

Because of the above lemma, we will be interested in the congruence subgroups
of the form $\Gamma_{1}(N)$ in this article. It will be really interesting to
see similar results for the congruence subgroups of the form $\Gamma_{0}(N)$.

######  Proposition 13.

Let $\Gamma_{1}(N),A,B$ be as in the theorem above. Then

$L\left(\sigma,\Gamma_{1}(N),E_{k,k}\right)=\begin{cases}(A+(\frac{N+2}{2})B).\frac{-N^{2}}{12}\prod_{p\mid
N}\left(1-p^{-2}\right)\cdot(k+1)&\text{ if }N\text{ is even }\\\
(A+(\frac{N+1}{2})B).\frac{-N^{2}}{12}\prod_{p\mid
N}\left(1-p^{-2}\right)\cdot(k+1)&\text{ if }N\text{ is odd. }\end{cases}$

###### Proof.

For each $\gamma\in H(1)$ (cf. Lemma 11), we have
$\operatorname{tr}\left(\gamma\sigma\mid
E_{k,k}\right)=\operatorname{tr}\left(\mathbf{1}\mid E_{k,k}\right)=$ $(k+1)$.
By [12, Theorem $2.1$], we need to calculate the Euler Poincaré
characteristics $\chi\left(\Gamma_{1}(N)^{\gamma\sigma}\right)$ with
$\gamma\in\\{\gamma_{1},\gamma_{1}^{\prime}\\}$.

Let $\Gamma^{e}_{1}(N)$ be the congruence subgroup of the elliptic modular
group $\mathrm{SL}_{2}(\mathbb{Z})$ of level $N$. An easy calculation shows
that $\Gamma_{1}(N)^{\gamma_{1}\sigma}=\Gamma^{e}_{1}(N)$. Let $Y_{1}(N)$
denotes the hyperbolic surface associated to $\Gamma^{e}_{1}(N)$. The
compactification $X_{1}(N)$ is obtained from $Y_{1}(N)$ by adding the cusps.
It is well-known that $Y_{1}(N)$ has $\frac{1}{2}\sum_{d\mid
N}\phi(d)\cdot\phi(\frac{N}{d})$ cusps. By [30, 1.6 .4], we have

$\chi\left(X_{1}(N)\right)=(-1/12)N^{2}\prod_{p\mid
N}\left(1-p^{-2}\right)+\frac{1}{2}\sum_{d\mid
N}\phi(d)\cdot\phi(\frac{N}{d}).$

Therefore

$\displaystyle\chi\left(\Gamma^{e}_{1}(N)\right)\mathrel{\mathop{\ordinarycolon}}=\chi\left(Y_{1}(N)\right)=$
$\displaystyle\chi\left(X_{1}(N)\right)-\\#\left\\{\right.\text{cusps
of}\left.Y_{1}(N)\right\\}=(-1/12)N^{2}\prod_{p\mid N}\left(1-p^{-2}\right).$

If $d\equiv 1\pmod{4}$, the number $B$ of translations of the surfaces
$F\left(\gamma_{1}^{\prime}\right)$ is 0 as noted in the Theorem 2 above.

We now consider the cases $d\equiv 2,3\pmod{4}$ and calculate the Euler-
Poincaré characteristics $\chi\left(\Gamma_{1}(N)^{\gamma\sigma}\right)$ for
$\gamma\in\\{\gamma_{1},\gamma_{1}^{\prime}\\}$. Let
$h=\left(\begin{array}[]{cc}1&\sqrt{d}\\\ 0&2\end{array}\right)$ and a matrix
calculation shows that

$\displaystyle\Gamma^{\gamma_{1}^{\prime}\sigma}=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}x+z\sqrt{d}&y+\frac{w-x}{2}\sqrt{d}\\\
2z&w-z\sqrt{d}\end{array}\right)\in\mathrm{SL}_{2}(\mathcal{O})\mid\right.$
$\displaystyle x-1\equiv w-1\equiv y\equiv z\equiv 0\quad(\bmod N)\text{ and
}w\equiv x\quad(\bmod\ 2N)\\}.$

In this case, the condition that $w\equiv x\pmod{2N}$ is redundant as it
automatically follows from the condition that the determinant is $1$ for a
matrix in $\mathrm{SL}_{2}(\mathcal{O})$. Therefore, we have the following
description

$\Gamma_{1}(N)^{\gamma_{1}^{\prime}\sigma}=\left\\{\left(\begin{array}[]{cc}x+z\sqrt{d}&y+\frac{w-x}{2}\sqrt{d}\\\
2z&w-z\sqrt{d}\end{array}\right)\in\mathrm{SL}_{2}(\mathcal{O})\mid x\equiv
w\equiv 1,y\equiv z\equiv 0\quad(\bmod N)\right\\}.$

An easy calculation shows that

$h^{-1}\Gamma_{1}(N)^{\gamma_{1}^{\prime}\sigma}h=\left\\{\left(\begin{array}[]{cc}x&2y+zd\\\
z&w\end{array}\right)\in\mathrm{SL}_{2}(\mathbb{Z})\mid x\equiv w\equiv
1,\quad y\equiv z\equiv 0\quad(\bmod N)\right\\}.$

Thus, we have a relation:

$h^{-1}\Gamma_{1}(N)^{\gamma^{\prime}_{1}\sigma}h=\Gamma^{e}_{1}(N)\cap\Gamma^{\prime}.$

Here, $\Gamma^{{}^{\prime}}\subset\mathrm{SL}_{2}(\mathbb{Z})$ is the group of
matrices that are of type $\left\\{\left(\begin{array}[]{cc}x&2y\\\
z&w\end{array}\right)\in\mathrm{SL}_{2}(\mathcal{\mathbb{Z}})\right\\}$. A
small check shows that the index is given by

$\left[\Gamma^{e}_{1}(N)\mathrel{\mathop{\ordinarycolon}}\Gamma^{e}_{1}(N)\cap\Gamma^{{}^{\prime}}\right]=\begin{cases}\frac{N+1}{2}&\text{
if }N\text{ is odd }\\\ \frac{N+2}{2}&\text{ if }N\text{ is even}\end{cases}.$

If $N$ is odd, then the coset representatives of quotient spaces are
represented by the matrices

$\left(\begin{array}[]{ll}1&0\\\
0&1\end{array}\right),\left(\begin{array}[]{ll}1&1\\\
0&1\end{array}\right),.....,\left(\begin{array}[]{cc}1&N-4\\\
0&1\end{array}\right),\left(\begin{array}[]{cc}1&N-2\\\
0&1\end{array}\right).$

On the other hand for even $N$, they are given by

$\left(\begin{array}[]{ll}1&0\\\
0&1\end{array}\right),\left(\begin{array}[]{ll}1&1\\\
0&1\end{array}\right),.....,\left(\begin{array}[]{cc}1&N-3\\\
0&1\end{array}\right),\left(\begin{array}[]{cc}1&N-1\\\
0&1\end{array}\right).$

Hence, when $d\equiv 2,3\pmod{4}$, the index
$\left[\Gamma^{e}_{1}(N)\mathrel{\mathop{\ordinarycolon}}h^{-1}\Gamma_{1}(N)^{\gamma_{1}^{\prime}\sigma}h\right]$
is $\frac{N+2}{2}$ if $N$ is even, and it is $\frac{N+1}{2}$ if $N$ is odd.
This implies that

$\chi\left(\Gamma_{1}(N)^{\gamma_{1}^{\prime}\sigma}\right)=\begin{cases}(\frac{N+2}{2})\chi\left(\Gamma^{e}_{1}(N)\right)&\text{
if }N\text{ is even }\\\
(\frac{N+1}{2})\chi\left(\Gamma^{e}_{1}(N)\right)&\text{ if }N\text{ is odd
}\end{cases}.$

This completes the proof of the proposition. ∎

######  Corollary 2.1.

Let $p$ be an odd rational prime that is unramified over $K.$ Let $t$ be the
number of distinct prime divisors of $D$. Then, for $n>0$ we have

$L\left(\sigma,\Gamma_{1}\left(p^{n}\right),E_{k,k}\right)=\begin{cases}-2^{t}\cdot\frac{p^{2n}-p^{2n-2}}{12}\cdot(k+1)&\text{
if }d\equiv 1(\bmod 4),\\\
-2^{t-2}\cdot(p^{n}+5)\frac{p^{2n}-p^{2n-2}}{12}\cdot(k+1)&\text{ if }d\equiv
2,3(\bmod 4)\end{cases}$

###### Proof.

Since $p$ is odd, we have $j_{2}$ = 0. Recall that $p$ is the only prime
divisor in the level. Since the prime is unramified, we have $s=0$. If
$d\equiv 1\pmod{4}$, we have $A+(\frac{N+1}{2})B=2^{t}$. If $d\equiv
2,3\pmod{4}$, we have

$\displaystyle A+\left(\frac{N+1}{2}\right)B$
$\displaystyle=2^{t}+\left(\frac{N+1}{2}\right)\cdot 2^{t-1}$
$\displaystyle=2\cdot 2^{t-1}+\left(\frac{N+1}{2}\right)\cdot 2^{t-1}$
$\displaystyle=\left(\frac{N+5}{2}\right)\cdot
2^{t-1}=\left(\frac{p^{n}+5}{2}\right)\cdot 2^{t-1}$

∎

##  6\. Traces on the Eisenstein cohomology for $\Gamma_{1}(N)$

We compute the trace of complex conjugation on the Eisenstein cohomology
following Sengün-Türkelli [12] for the subgroups of the form $\Gamma_{1}(N)$
inside the Bianchi modular groups. We now prove our first theorem on the
cohomology groups (cf. Theorem 3).

###### Proof.

Assume that $k>0$ and $\Gamma=\Gamma_{1}(N)$. In this case, the restriction
map
$\mathrm{H}^{2}\left(X_{\Gamma},E_{k,k}\right)\rightarrow\mathrm{H}^{2}\left(\partial
X_{\Gamma},E_{k,k}\right)$ is onto, and thus it suffices to compute the trace
of $\rho^{2}$ on $\mathrm{H}^{2}\left(\partial X_{\Gamma},E_{k,k}\right)$.
Poincaré duality together with the fact that $\rho$ reverses the orientation
reduces the problem to computing the trace of $\rho^{0}$ on
$\mathrm{H}^{0}\left(\partial X_{\Gamma},E_{k,k}\right)$ instead.

The cohomology of the boundary can be expressed as a direct sum of the
cohomology of the boundary components $X_{c}$, which are 2-tori;

$\mathrm{H}^{0}\left(\partial
X_{\Gamma},E_{k,k}\right)\simeq\bigoplus_{c\in\mathcal{C}_{\Gamma}}\mathrm{H}^{0}\left(X_{c},E_{k,k}\right)\simeq\bigoplus_{c\in\mathcal{C}_{\Gamma}}\mathrm{H}^{0}\left(\Gamma_{c},E_{k,k}\right).$

If $c$ is a cusp, then $\rho$ takes $\Gamma_{c}$ to $\Gamma_{\rho(c)}$. If
$c\neq\rho(c)$, then

$\mathrm{H}^{0}\left(\Gamma_{c},E_{k,k}\right)\oplus\mathrm{H}^{0}\left(\Gamma_{\rho(c)},E_{k,k}\right)$

is a $\rho^{0}$-invariant subspace of
$\bigoplus_{c\in\mathcal{C}_{\Gamma}}\mathrm{H}^{0}\left(\Gamma_{c},E_{k,k}\right)$.
As $\rho^{0}$ takes the basis of the first summand to the basis of the second
summand, the trace of $\rho^{0}$ on this subspace is $0$.

If $c=\rho(c)$, then $\rho^{0}$ acts on the one-dimensional space
$\mathrm{H}^{0}\left(\Gamma_{c},E_{k,k}\right)$ and hence $\rho^{0}$ has trace
$1$. Thus

$\operatorname{tr}\left(\sigma\mid\mathrm{H}^{0}\left(\partial
X_{\Gamma},E_{k,k}\right)\right)=\\#\left(\mathcal{C}_{\Gamma}\right)^{\rho},$

that is, the trace of $\rho^{0}$ is equal to the number of cusps of $\Gamma$
that are invariant under the action of $\rho$.

We know that
$\mathcal{C}_{\Gamma}=\bigsqcup_{x\in\mathcal{C}_{G}}\Gamma\backslash Gx$.
Clearly a cusp $c=$ $gx\Gamma\in\mathcal{C}_{\Gamma}$ is $\rho$-invariant only
if $x\in\mathcal{C}_{G}$ is so. So we have

$\left(\mathcal{C}_{\Gamma}\right)^{\rho}=\bigsqcup_{x\in\left(\mathcal{C}_{G}\right)^{\rho}}(\Gamma\backslash
Gx)^{\rho}.$

From [[12], corollary $4.4$], we deduce that
$\\#\left(\mathcal{C}_{G}\right)^{\rho}=2^{t-1}$. By [[12], lemma $4.5$] , it
is enough to compute #( $\Gamma\backslash G\infty)^{\rho}$. For the rest of
the proof, we put

$\displaystyle R\mathrel{\mathop{\ordinarycolon}}=\mathcal{O}/(N)\quad
U^{+}(R)\mathrel{\mathop{\ordinarycolon}}=\left(\begin{array}[]{cc}\pm 1&R\\\
0&\pm
1\end{array}\right),U(R)\mathrel{\mathop{\ordinarycolon}}=\left(\begin{array}[]{ll}1&R\\\
0&1\end{array}\right).$

Following our discussion on the cusps, it is enough to compute

$\displaystyle\\#((\Gamma/\Gamma(N))\cdot(\Gamma(N)\backslash
G\infty))^{\rho}=$
$\displaystyle\\#(\Gamma/\Gamma(N))^{\rho}\cdot(\Gamma(N)\backslash
G\infty)^{\rho}=\alpha^{\rho}\cdot\\#\left(U^{+}(R)\backslash\mathrm{SL}_{2}(R)\right)^{\rho}.$

Following Sengün-Türkelli [12], it is enough to compute
$\\#\left(U^{+}(R)\backslash\mathrm{SL}_{2}(R)\right)^{\rho}$. Observe that
$\sigma$ fixes the ideal $(N)$. Hence, the action of $\sigma$ on $\mathcal{O}$
descends to action on $R$. The action of $\sigma$ and $\tau$ on
$\mathrm{SL}_{2}(R)$ are defined as follows

$\sigma\left(\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\right)=\left(\begin{array}[]{ll}\sigma_{a}&\sigma_{b}\\\
\sigma_{c}&\sigma_{d}\end{array}\right),\quad\tau\left(\left(\begin{array}[]{ll}a&b\\\
c&d\end{array}\right)\right)=\left(\begin{array}[]{cc}\sigma_{a}&-\sigma_{b}\\\
-\sigma_{c}&\sigma_{d}\end{array}\right).$

where $\sigma_{a}$ (respectively $\tau_{a}$) is action of $\sigma$
(respectively $\tau$) on $a$.

We first treat the special situation when $N=p^{n}$ for a prime $p$. Consider
the bijections of sets of matrices:

$U^{+}(R)\backslash\mathrm{SL}_{2}(R)\simeq U^{+}(R)\backslash B(R)\times
B(R)\backslash\mathrm{SL}_{2}(R);$

Where $B(R)$ is the subgroup of upper-triangular matrices, there are well-
known bijections.

$B(R)\backslash\mathrm{SL}_{2}(R)\leftrightarrow\mathbb{P}^{1}(R),\quad\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]\mapsto(a\mathrel{\mathop{\ordinarycolon}}c).$

Let $\mathbb{P}^{1}(R)$ denotes the projective line over $R$, and

$U(R)\backslash B(R)\leftrightarrow R^{*},\quad\left[\begin{array}[]{cc}a&b\\\
0&a^{-1}\end{array}\right]\mapsto a.$

These bijections lead to the identification:

$U^{+}(R)\backslash\mathrm{SL}_{2}(R)\simeq R^{*}/\\{\pm
1\\}\times\mathbb{P}^{1}(R).$

It is straightforward to transfer the action of $\sigma$ and $\tau$ to the
right-hand side. We immediately see that:

$\left(U^{+}(R)\backslash\mathrm{SL}_{2}(R)\right)^{\rho}\simeq\left(R^{*}/\\{\pm
1\\}\right)^{\rho}\times\mathbb{P}^{1}(R)^{\rho}.$

Start with computing $\\#\mathbb{P}^{1}(R)^{\rho}$. It can be seen that
$\mathbb{P}^{1}\left(R^{\sigma}\right)\hookrightarrow$ $\mathbb{P}^{1}(R)$ and
in fact $\mathbb{P}^{1}\left(R^{\sigma}\right)=\mathbb{P}^{1}(R)^{\sigma}$.
Note that
$\mathbb{P}^{1}\left(R^{\sigma}\right)\simeq\mathbb{P}^{1}\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)$
and thus has cardinality $p^{n}+p^{n-1}$. Computation shows that
$\mathbb{P}^{1}(R)^{\tau}$ has the same number of elements.

Let us now consider $\\#(R^{*}){}^{\rho}$. The action of $\sigma$ and $\tau$
are same on $R^{*}$. Clearly we have $\\#\left(R^{*}/\\{\pm
1\\}\right)^{\rho}=(1/2)\cdot\\#\left(R^{*}\right)^{\rho}$.

  * •

Let $p$ be a prime that splits in $K$. We then have

$\left(R\right)^{\rho}=\left(\mathcal{\mathcal{O}}/p^{n}\mathcal{\mathcal{O}}\right)^{\rho}=\left(\mathbb{Z}\sqrt{-d}/p^{n}\mathbb{Z}\sqrt{-d}\right)^{\rho}=\left(\mathbb{Z}\sqrt{-d}\right)^{\rho}/\left(p^{n}\mathbb{Z}\sqrt{-d}\right)^{\rho}.$

    1. (1)

When $\rho=\sigma$, the automorphism $\sigma$ acts on $\mathbb{Z}\sqrt{-d}$ by
$\sigma(a+b\sqrt{-d})=a-b\sqrt{-d}$. Let
$(a+b\sqrt{-d})\in\left(\mathbb{Z}\sqrt{-d}\right)^{\sigma}$ then
$(a+b\sqrt{-d})=(a-b\sqrt{-d})$, this implies $b=0$ and $a\in\mathbb{Z}$.

    2. (2)

On the other hand if $\rho=\tau$, the automorphism $\tau$ acts on
$\mathbb{Z}\sqrt{-d}$ by $\tau(a+b\sqrt{-d})=-a+b\sqrt{-d}$. Consider now
$(a+b\sqrt{-d})\in\left(\mathbb{Z}\sqrt{-d}\right)^{\tau}$ then
$(a+b\sqrt{-d})=(-a+b\sqrt{-d})$, this implies $a=0$ and $b\in\mathbb{Z}$. We
have $R\simeq\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)^{2}$ and hence
$\\#\left(R^{*}\right)^{\rho}=\\#\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)^{*}=p^{n}-p^{n-1}$.

  * •

When $p$ is inert in $K$, we can view $R$ as the quadratic extension
$\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)[\omega]$ of the ring
$\mathbb{Z}/p^{n}\mathbb{Z}$. It follows that
$R^{*}=\\{a+b\cdot\omega\in\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)^{*}[\omega]\\}$.
We calculate $\left(R\right)^{\rho}$ when $p$ is split in $K$. We deduce that
$\left(R^{*}\right)^{\rho}$ is given by

$\\{a+b\cdot\omega\in\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)[\omega]\mid
p\nmid a,b=0\ \text{for}\ \rho=\sigma\ \text{or}\ p\nmid b,a=0\ \text{for}\
\rho=\tau\\}.$

The cardinality of this set is $p^{n}-p^{n-1}$.

In both inert and split cases, we get the quantity
$\\#\left(U^{+}(R)\backslash\mathrm{SL}_{2}(R)\right)^{\rho}=(1/2)\cdot\left(p^{2n}-p^{2(n-1)}\right).$

To finish the proof, let us assume that $N=p_{1}^{n_{1}}\ldots
p_{r}^{n_{r}}\in\mathbb{N}$ whose prime divisors $p_{i}$ are unramified in
$K$. The result in this general case follows from the simple fact that

${\mathrm{SL}}_{2}(\mathcal{O}/(N))\simeq\mathrm{SL}_{2}\left(\mathcal{O}/\left(p_{1}\right)^{n_{1}}\right)\times\ldots\mathrm{SL}_{2}\left(\mathcal{O}/\left(p_{r}\right)^{n_{r}}\right)$

∎

###  6.1. Trace on $\mathrm{H}_{Eis}^{1}(\Gamma_{1}(N),\mathbb{C})$ and Szech
Cocyles

In the next subsection, we compute the trace of $\sigma$ on
$\mathrm{H}_{Eis}^{1}(\Gamma_{1}(N),\mathbb{C})$. We use the explicit
$1$-cocycles defined by Sczech in [28] to produce a basis for
$\mathrm{H}_{\text{Eis }}^{1}(\Gamma_{1}(N),\mathbb{C})$. Consider
$\mathcal{O}$ as a lattice in $\mathbb{C}$. For $k\in\\{0,1,2\\}$ and
$u\in\mathbb{C}$, put

$E_{k}(u)=E_{k}(u,\mathcal{O})=\left.\sum_{\begin{subarray}{c}w\in\mathcal{O}\\\
w\neq-u\end{subarray}}(w+u)^{-k}|w+u|^{-s}\right|_{s=0}$

where $..\left.\right|_{s=0}$ means that the value is defined by analytic
continuation to $s=0$. Define, $E(u)$ by setting

$2E(u)=\begin{cases}2E_{2}(0),&u\in\mathcal{O}\\\
\wp(u)-E_{1}(u)^{2},&u\notin\mathcal{O}\end{cases};$

where $\wp(u)$ denotes the Weierstrass $\wp$-function. Given
$u,v\in\frac{1}{N}\mathcal{O}$, Sczech defined certain homomorphisms

$\Psi(u,v)\mathrel{\mathop{\ordinarycolon}}\Gamma_{1}(N)\rightarrow\mathbb{C}$

dependent only on the classes of
$(u,v)\in(\frac{1}{N}\mathcal{O}/\mathcal{O})^{2}$. For
$A=\left(\begin{array}[]{cc}a&b\\\ 0&d\end{array}\right)\in\Gamma_{1}(N)$, we
have the simple description

$\Psi(u,v)(A)=-\left(\frac{\bar{b}}{d}\right)E(u)-\frac{b}{d}E_{0}(u)E_{2}(v)$

where

$\left(\frac{t}{s}\right)=-1+\\#\left\\{y\bmod s\mathcal{O}\mid y^{2}\equiv
t\quad\bmod s\mathcal{O}\right\\}$

is the Legendre symbol. For non-parabolic $A\in\Gamma_{1}(N)$, there is a
similar but more complicated description involving the Eisenstein series
$E_{k}$ ’s. This in turn generalizes the classical Dedekind sums. Note that
$E(-u)=E(u)$ and $E_{k}(-u)=E_{k}(u)$ for even $k$. So,
$\Psi(-u,-v)=\Psi(u,v)$ on parabolic elements. In fact, using the definition
in [28, section 4] one can see that $\Psi(-u,-v)=\Psi(u,v)$. Note that $\Psi$
and $\Phi$ are equivalent. However $\Phi$ is defined for the full group in
Theorem 1 whereas $\Psi$ is for the congruence subgroup $\Gamma_{1}(N)$.

Sczech shows that the collection $\Psi(u,v)$ with
$(u,v)\in\left(\frac{1}{N}\mathcal{O}/\mathcal{O}\right)^{2}$ live in the
Eisenstein part of the cohomology and that the number of linearly independent
such homomorphisms is equal to the number of cusps of $\Gamma$. Thus they
generate $\mathrm{H}_{\text{Eis }}^{1}(\Gamma_{1}(N),\mathbb{C})$. We now list
some properties of the Eisenstein series.

  1. (1)

Ito [18] showed that the cocycle given by the integral of this closed
differential 1-form differs from the cocycle $\Psi(u,v)$ of Sczech by a
coboundary for the full Bianchi group (see also Weselmann [32]).

  2. (2)

The Sczech cocycles are integrals of closed harmonic differential forms. These
differential forms are given by certain Eisenstein series defined on the
hyperbolic space $\mathbb{H}_{3}$.

  3. (3)

Following Ito, we form an Eisenstein series $E_{(u,v)}(\tau,s)$ for
$(\tau,s)\in\mathbb{H}_{3}\times\mathbb{C}$ with values in $\mathbb{C}^{3}$
associated to each cusp of $\Gamma_{1}(N)$. As a function of
$s,E_{(u,v)}(\tau,s)$ can be analytically continued to all of $\mathbb{C}$.

  4. (4)

Harder [15] shows that the differential $1$-form on the hyperbolic 3-space
induced by $E_{(u,v)}(\tau,s)$ is closed for $s=0$.

Note that the above Eisenstein series associated with different cusps are
linearly independent because they are non-vanishing only at their associated
cusp. This implies that the cohomology classes of Sczech cocycles, which are
associated with the cusps of $\Gamma_{1}(N)$ form a basis of
$\mathrm{H}_{Eis}^{1}(\Gamma_{1}(N),\mathbb{C})$.

Let
$P\mathrel{\mathop{\ordinarycolon}}=\left\\{\pm\left[\begin{array}[]{cc}1&j\\\
0&1\end{array}\right]\mathrel{\mathop{\ordinarycolon}}j\in\mathcal{O}\right\\}$
be the parabolic subgroup of $\mathrm{SL}_{2}(\mathcal{O})$. Note that
$\bar{P}=\left\\{\pm\left[\begin{array}[]{ll}1&j\\\
0&1\end{array}\right]\mathrel{\mathop{\ordinarycolon}}j\in\mathcal{O}/N\mathcal{O}\right\\}$
and $\bar{P}_{+}=\left\\{\left[\begin{array}[]{ll}1&j\\\
0&1\end{array}\right]\mathrel{\mathop{\ordinarycolon}}j\in\mathcal{O}\right\\}$
is the ”positive” half of $P$. Here, the overbar signifies reduction modulo
$(N)$.

######  Proposition 14.

$\\#\mathcal{C}_{N}=|\bar{P}_{+}\backslash\mathrm{SL}_{2}(\mathcal{O}/N\mathcal{O})/\bar{P}|$

###### Proof.

Recall that $P=\mathrm{SL}_{2}(\mathcal{O})_{\infty}$ is the subgroup of
$\mathrm{SL}_{2}(\mathcal{O})$ fixing the cusp $\infty$. Then the map

$\Gamma_{1}(N)\backslash\mathrm{SL}_{2}(\mathcal{O})/P\longrightarrow\\{\text{
cusps of }\Gamma_{1}(N)\\}$

given by

$\Gamma_{1}(N)\alpha P\xrightarrow{}\Gamma_{1}(N)\alpha(\infty)$

is a bijection [13] (the proof works for imaginary quadratic fields also). We
know that $\mathrm{SL}_{2}(\mathcal{O})/P$ identifies with
$K\cup\\{\infty\\}$, so that the double coset space
$\Gamma_{1}(N)\backslash\mathrm{SL}_{2}(\mathcal{O})/P$ identifies with the
cusps $\Gamma_{1}(N)\backslash({K}\cup\\{\infty\\})$.

The double coset space $\Gamma(N)\backslash\mathrm{SL}_{2}(\mathcal{O})/P$ is
naturally viewed as $\mathrm{SL}_{2}(\mathcal{O}/N\mathcal{O})/\bar{P}$ where
$\bar{P}$ denotes the projected image of $P$ in
$\mathrm{SL}_{2}(\mathcal{O}/N\mathcal{O})$, that is

$\bar{P}=\left\\{\pm\left[\begin{array}[]{ll}1&j\\\
0&1\end{array}\right]\mathrel{\mathop{\ordinarycolon}}j\in\mathcal{O}/N\mathcal{O}\right\\}.$

We have a decomposition $\Gamma_{1}(N)$ =
$\bigcup_{j}\left[\begin{array}[]{ll}1&j\\\ 0&1\end{array}\right]\Gamma(N)$,
the double coset space is naturally viewed as
$\bar{P}_{+}\backslash\mathrm{SL}_{2}(\mathcal{O}/N\mathcal{O})/\bar{P}$ where
$\bar{P}_{+}=\left\\{\left[\begin{array}[]{ll}1&j\\\
0&1\end{array}\right]\mathrel{\mathop{\ordinarycolon}}j\in\mathcal{O}\right\\}$
is the ”positive” half of $P$ and again the overbar signifies reduction modulo
$(N)$.

We deduce that

$|\bar{P}_{+}\backslash\mathrm{SL}_{2}(\mathcal{O}/N\mathcal{O})/\bar{P}|=\\#\mathcal{C}_{N}.$

∎

######  Corollary 15.

$\\#\mathcal{C}_{p}=p^{2}-1$

###### Proof.

We know $\mathcal{O}/p\mathcal{O}$ is isomorphic to
$\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}$ and number of non-zero
choices of $y\in\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}$ is
$p^{2}-1$ then $gcd(y,p)=1$. Possibilities of $x$ is $x\ \text{modulo}\
gcd(y,p)$ and $gcd(y,p)=1$ then $x\ \text{modulo}\ 1$. The number of choices
of $x$ is $1$, and the number of choices for $(x,y)$ is $p^{2}-1$. ∎

We now prove our Theorem 4.

###### Proof.

By Ito [19], we have the following relation:

$\Psi(0,0)(\bar{A})=-\Psi(0,0)(A).$

The bar means that we take the complex conjugates of the entries of the matrix
$A$. More generally, Ito proves that

$\Psi(u,v)(\bar{A})=\frac{-1}{N^{2}}\sum_{s,t\in\frac{1}{N}\mathcal{O}/\mathcal{O}}\phi(s\bar{v}-t\bar{u})\Psi(s,t)(A);$

where $\phi(z)\mathrel{\mathop{\ordinarycolon}}=\exp(2\pi i(z-\bar{z})/D)$
with $D$ the discriminant of $K$. Observe that when $(s,t)=(u,v)$ or
$(s,t)=(0,0)$, we have $\phi(s\bar{v}-t\bar{u})=1$. Using this, we write this
summation more suggestively:

$\Psi(u,v)(\bar{A})=\frac{-1}{N^{2}}\left[\Psi(0,0)(A)+\Psi(u,v)(A)+\left(\sum_{\begin{subarray}{c}s,t\in\frac{1}{N}\mathcal{O}/\mathcal{O}\\\
(s,t)\neq(u,v)\\\
(s,t)\neq(0,0)\end{subarray}}\phi(s\bar{v}-t\bar{u})\Psi(s,t)(A)\right)\right].$

We now describe the action of complex conjugation $\sigma$ on the Sczech
cocycles as given by

$\sigma(\Psi(u,v))(A)\mathrel{\mathop{\ordinarycolon}}=\Psi(u,v)(\bar{A}).$

We see that $\sigma(\Psi(u,v))$ is expressed as a summation over all the
Sczech cocycles. Consider the operator $\sigma$ as a linear operator on the
formal space $\mathbb{C}\left[\Psi_{N}\right]$ for which the Sczech cocycles
are taken as a basis.

The pair $(0,0)$ $\in$ $\left(\frac{1}{N}\mathcal{O}/\mathcal{O}\right)^{2}$
never corresponds to a cusp of $\Gamma_{1}(N)$, so let us eliminate the term
$\Psi(0,0)$ from the summation. Using Ito’s summation formula for the case
$(u,v)=(0,0)$, we get

$\Psi(0,0)(\bar{A})=\frac{-1}{N^{2}}\left[\Psi(0,0)(A)+\left(\sum_{\begin{subarray}{c}s,t\in\frac{1}{N}\mathcal{O}/\mathcal{O}\\\
(s,t)\neq(0,0)\end{subarray}}\Psi(s,t)(A)\right)\right].$

Now, we know the relation $\Psi(0,0)(\bar{A})=-\Psi(0,0)(A)$, we get

$\Psi(0,0)(A)=\frac{1}{N^{2}-1}\sum_{\begin{subarray}{c}s,t\in\frac{1}{N}\mathcal{O}/\mathcal{O}\\\
(s,t)\neq(0,0)\end{subarray}}\Psi(s,t)(A).$

Now for $(u,v)\neq(0,0)$, we have

$\Psi(u,v)(\bar{A})=\frac{-1}{N^{2}}\left[\Psi(0,0)(A)+\left(\sum_{\begin{subarray}{c}s,t\in\frac{1}{N}\mathcal{O}/\mathcal{O}\\\
(s,t)\neq(0,0)\end{subarray}}\phi(s\bar{v}-t\bar{u})\Psi(s,t)(A)\right)\right].$

Substituting $\Psi(0,0)(A)$, we get the following expression

$\displaystyle\Psi(u,v)(\bar{A})=\frac{-1}{\left(N^{2}\right)\left(N^{2}-1\right)}\sum_{\begin{subarray}{c}s,t\in\frac{1}{N}\mathcal{O}/\mathcal{O}\\\
(s,t)\neq(0,0)\end{subarray}}\Psi(s,t)(A)$
$\displaystyle+\frac{-1}{N^{2}}\sum_{\begin{subarray}{c}s,t\in\frac{1}{N}\mathcal{O}/\mathcal{O}\\\
(s,t)\neq(0,0)\end{subarray}}\phi(s\bar{v}-t\bar{u})\Psi(s,t)(A).$

Since $\Psi(-u,-v)=\Psi(u,v)$, we define $\mathbb{C}\left[\Psi_{N}^{*}\right]$
to be the formal vector space generated by the basis

$\left\\{\Psi(u,v)\mid(u,v)\in\left(\frac{1}{N}\mathcal{O}/\mathcal{O}\right)^{2}/\pm
1,(u,v)\neq(0,0)\quad\text{and}\ u\text{ modulo}\ gcd(v,N)\right\\}.$

Having eliminated $\Psi(0,0)$ in the above equation, we now regard $\sigma$ as
a linear operator on the formal space $\mathbb{C}\left[\Psi_{N}^{*}\right]$.

Recall that $\Psi(-u,-v)=\Psi(u,v)$ and
$\phi(-u\bar{v}+v\bar{u})=\phi(u\bar{v}-v\bar{u})=1$. We see that the
coefficient of the summand $\Psi(u,v)(A)$ on the right-hand side of the
equality is

$\frac{-1}{\left(N^{2}\right)\left(N^{2}-1\right)}+\frac{-1}{N^{2}}=\frac{-2}{N^{2}-1}.$

This implies that the trace of $\sigma$ on
$\mathbb{C}\left[\Psi_{N}^{*}\right]$ is
$\left(\\#\mathcal{C}_{N}\right).\frac{-2}{N^{2}-1}.$ For $N=p$, we have
$\\#\mathcal{C}_{p}=p^{2}-1$. The trace of $\sigma$ on
$\mathbb{C}\left[\Psi_{p}^{*}\right]$ is
$\left(p^{2}-1\right).\frac{-2}{p^{2}-1}=-2$. For $N=p^{n}$ with $n>1$, we
have
$\\#\mathcal{C}_{p^{n}}=|\bar{P}_{+}\backslash\mathrm{SL}_{2}(\mathbf{\mathcal{O}}/p^{n}\mathbf{\mathcal{O}})/\bar{P}|$.
The trace of $\sigma$ on $\mathbb{C}\left[\Psi_{p}^{*}\right]$ is
$\left(\\#\mathcal{C}_{p^{n}}\right).\frac{-2}{p^{2}-1}$.

∎

######  Proposition 16.

Let $p$ be a rational prime that is unramified in $K$ and let
$\Gamma_{1}\left(p^{n}\right)$ denote the subgroup of level $(p)^{n}$ of a
Bianchi group $\mathrm{SL}_{2}(\mathcal{O})$. Then

  1. (1)

$\operatorname{dim}\mathrm{H}_{\text{cusp
}}^{1}\left(\Gamma_{1}\left(p^{n}\right),E_{k,k}\right)\gg k$

as $k$ increases and $n$ is fixed.

  2. (2)

Assume further that the class number of $K$ is one. We have the following
asymptotic bound.

$\operatorname{dim}\mathrm{H}_{\text{cusp
}}^{1}\left(\Gamma_{1}\left(p^{n}\right),\mathbb{C}\right)\gg p^{3n}$

as $n$ increases.

###### Proof.

From [[12], proposition $2.2$] , recall that

$\operatorname{dim}\mathrm{H}_{cusp}^{1}\left(\Gamma,E_{k,k}\right)\geq\frac{1}{2}\left(L(\sigma,\Gamma,k)+\operatorname{tr}\left(\sigma_{\text{Eis
}}^{1},\Gamma,k\right)-\operatorname{tr}\left(\sigma_{\text{Eis
}}^{2},\Gamma,k\right)\right).$

When $\Gamma$ is fixed, by Proposition 10, the dimension of the Eisenstein
part of the cohomology is the same for every weight $k>0$. Hence, the
asymptotic for (1) is given by corollary 2.1. The claim in (2) follows
directly from Theorem 3 and Theorem 4, together with the Lefschetz number
formula provided in corollary 2.1. ∎

In a recent paper [14], Weibo Fu calculated the bounds (both upper and lower)
of cuspidal cohomology without utilizing the Eisenstein part. However, our
calculation (similar to Sengün-Türkelli) shows that the Eisenstein part of the
cohomology can also be used to make conclusions of the cuspidal part. Note
that in loc. cit also, the author used both (completed) homology and
cohomology groups and an interesting result of Marshall[22] comparing the
dimensions of the spaces over $\mathbb{Q}_{p}$ and $\mathbb{C}$.

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