# From Binary Hermitian Forms to parabolic cocycles of Euclidean Bianchi
groups

Cihan Karabulut<EMAIL_ADDRESS>William Paterson University, Wayne, New
Jersey 07470, USA

###### Abstract.

We study a family of functions defined in a very simple way as sums of powers
of binary Hermitian forms with coefficients in the ring of integers of an
Euclidean imaginary quadratic field $K$ with discriminant $d_{K}$. Using these
functions we construct a nontrivial cocycle belonging to the space of
parabolic cocycles on Euclidean Bianchi groups. We also show that the average
value of these functions is related to the special values of
$L(\chi_{d_{K}},s)$. Using the properties of these functions we give new and
computationally efficient formulas for computing some special values of
$L(\chi_{d_{K}},s)$.

###### Key words and phrases:

Binary Hermitian forms, special values, parabolic cohomology.

###### 2010 Mathematics Subject Classification:

Primary 11E39, 11M06, 11F75; Secondary 11J70

The author was partly supported by Assigned Release Time (ART) program for
research from William Paterson University

###### Contents

  1. 1 Introduction
  2. 2 Definition and Elementary Properties of $H_{k,\Delta}$
  3. 3 Continuity of $H_{k,\Delta}$
  4. 4 The average value of $H_{k,\Delta}$ and Cohen-Zagier type formulas
  5. 5 Cocycle property of $H_{k,\Delta}$

##  1\. Introduction

In [28], Zagier studies the following family of functions: Let $D>0$ be a non-
square integer satisfying $D\equiv 0,1$ (mod 4) and $k>0$ be an even integer.
Then, define $F_{k,D}:\mathbb{R}\rightarrow\mathbb{R}$ as follows,

$F_{k,D}(x):=\sum_{\begin{subarray}{c}\text{disc}(Q)=D\\\
a<0<Q(x)\end{subarray}}Q(x)^{k-1}$ (1.1)

where $Q(X)=aX^{2}+bX+c$ with $(a,b,c)\in\mathbb{Z}^{3}$ and
$\text{disc}(Q)=b^{2}-4ac$. When $D$ is a square discriminant, one has to add
a simple correction term consisting of the $k$-th Bernoulli polynomial. But
for the sake of brevity and simplicity, let us assume that $D$ is a non-square
discriminant.

Zagier shows that these functions have many surprising properties and are
intimately related to modular forms of integral weight and half integral
weight, certain values of Dirichlet $L$-functions, Diophantine approximation,
continued fractions, and Dedekind sums.

For example, Zagier finds that $F_{2,5}$ is constant with value
$F_{2,5}(x)=F_{2,D}(0)=2$ for any $x\in\mathbb{R}$ despite the fact that there
are infinitely many quadratic polynomials contributing to the sum when $x$ is
irrational. More generally, he proves the following theorem:

######  Theorem 1.1 (Zagier 1999).

Let $D$ be a positive non-square discriminant and $k$ a positive even integer.
Then

  1. (1)

$F_{2,D}(x)$ has a constant value $\alpha_{D}$ for all $x\in\mathbb{R}$,

  2. (2)

$F_{4,D}(x)$ has a constant value $\beta_{D}$ for all $x\in\mathbb{R}$,

  3. (3)

For $k\geq 6$, $F_{k,D}$ is no longer constant but is a linear combination,
with coefficients depending on $D$ and $k$, of a finite collection of
functions depending only on $k$.

To get an explicit formula for $\alpha_{D}$ and $\beta_{D}$, Zagier uses the
above theorem to write $\alpha_{D}$ as $F_{2,D}(0)$ and $\beta_{D}$ as
$F_{4,D}(0)$ which gives the following formulas:

$\alpha_{D}=\sum_{\begin{subarray}{c}b^{2}-4ac=D\\\
a<0<c\end{subarray}}c=\sum_{|b|<\sqrt{D}}\sigma_{1}\left(\frac{D-b^{2}}{4}\right),$
(1.2)

$\beta_{D}=\sum_{\begin{subarray}{c}b^{2}-4ac=D\\\
a<0<c\end{subarray}}c^{3}=\sum_{|b|<\sqrt{D}}\sigma_{3}\left(\frac{D-b^{2}}{4}\right)$
(1.3)

where $(a,b,c)\in\mathbb{Z}^{3}$. The sums involving the divisor functions
$\sigma_{1}$ and $\sigma_{3}$ have appeared in the literature before as the
values of Dirichlet $L$-function $L(\chi_{D},s)$ associated to the Kronecker
symbol $\chi_{D}:=\left(\frac{D}{\cdot}\right)$ at $s=-1$ and $s=-3$. In fact,
Cohen [3, 4] and Zagier [26] using the results of Siegel [22] showed that

$\displaystyle\alpha_{D}=$ $\displaystyle-5L(\chi_{D},-1),$ (1.4)
$\displaystyle\beta_{D}=$ $\displaystyle L(\chi_{D},-3).$ (1.5)

The formulas in (1.4) and (1.5), which we refer to as Cohen-Zagier type
formulas, are computationally very efficient and give a fast
$O(D^{1/2+\epsilon})$ method for computing the special values $s=-1$ and
$s=-3$. The usual formulas for computing special values in terms of finite
character sums are $O(D)$ and are not very practical for actual computations,
except for small conductors (see Remarks on page 194 in [5]).

In recent years, there has been a renewed interest in Zagier’s paper, with [1,
17, 18, 25] all further exploring the functions defined in (1.1). In this
paper, generalizing in a different direction, we introduce an analogue of
Zagier’s functions $F_{k,D}$ by using binary Hermitian forms and prove a
result similar to Theorem 1.1 for these functions. As an application, we get
new formulas, similar to the ones given in (1.4) and (1.5).

Let $K=\mathbb{Q}(\sqrt{-d})$ with $d=1,2,3,7,11$ be an Euclidean imaginary
quadratic field with discriminant $d_{K}$ and ${\mathcal{O}}_{d}$ be the ring
of integers of $K$. For a positive integer $\Delta>0$ which is not a norm of
an algebraic integer in ${\mathcal{O}}_{d}$ and an odd integer $k\geq 1$, we
define the function $H_{k,\Delta}:\mathbb{C}\to\mathbb{R}$ as

$H_{k,\Delta}(z):=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0<h(z)\end{subarray}}h(z)^{k}\\\ $ (1.6)

where $\displaystyle h(z)=a|z|^{2}+bz+\bar{b}\bar{z}+c$ with
$a,c\in\mathbb{Z}$ and $b\in{\mathcal{O}}_{d}$ is a binary Hermitian with
discriminant $-\Delta$. Our first result is the following theorem which is
analog of Theorem 1.1:

######  Theorem 1.2.

Let $\Delta$ be a positive integer which is not a norm of an algebraic integer
in ${\mathcal{O}}_{d}$ and $k\geq 1$ be an odd integer. Then

  1. (1)

$H_{1,\Delta}(z)$ has a constant value $\alpha_{1,\Delta}$ for all
$z\in\mathbb{C}$,

  2. (2)

$H_{3,\Delta}(z)$ has a constant value $\alpha_{3,\Delta}$ for all
$z\in\mathbb{C}$ when $d=1,3,7,$

  3. (3)

$H_{5,\Delta}(z)$ has a constant value $\alpha_{5,\Delta}$ for all
$z\in\mathbb{C}$ when $d=3$

where $\alpha_{k,\Delta}=H_{k,\Delta}(0)$.

The proof of Theorem 1.2 is presented in Section 2. The main idea in proving
Theorem 1.2 is the observation that any $z\in K$, since $K$ is Euclidean, can
be reduced to zero by a finite number of iterations of the transformations
$z\mapsto z+\lambda$ and $z\mapsto\frac{1}{z}$. The $H_{k,\Delta}$ is
invariant under the transformation $z\mapsto z+\lambda$ (see Proposition 2.2)
and transforms up to a certain polynomial $P_{k,\Delta}$ under the
transformation $z\mapsto\frac{1}{z}$ (Propositions 2.3). These polynomials
satisfy various functional equations, and we use them to deduce that they are
of a particular form when $k=1,3,5$. This allows us to conclude that for
suitable choices of $d$, $H_{k,\Delta}(z)=H_{k,\Delta}(0)$ for any $z\in K$
when $k=1,3,5$. The fact that $H_{1,\Delta}(z)=\alpha_{1,\Delta}$,
$H_{3,\Delta}(z)=\alpha_{3,\Delta}$ and $H_{5,\Delta}(z)=\alpha_{5,\Delta}$
for all $z\in\mathbb{C}$ is deduced using the continuity of $H_{k,\Delta}$
which is our next result.

######  Theorem 1.3.

Let $\Delta$ be a positive integer which is not a norm of an algebraic integer
in ${\mathcal{O}}_{d}$ and $k\geq 1$ be an odd integer. Then $H_{k,\Delta}$ is
continuous for all $z\in\mathbb{C}$.

We discuss the convergence and the continuity of $H_{k,\Delta}$ and present
the proof of Theorem 1.3 in Section 3. When $k\geq 3$, the proof of Theorem
1.3 (see Theorem 3.1) is straightforward since $h(z)=O(1/a_{h})$ for all $h$
occurring in Definition (1.6) and there are only $O(1)$ functions of the form
$h(z)=a|z|^{2}+bz+\bar{b}\bar{z}+c$ for each value of $a$. So, the sum
defining $H_{k,\Delta}$ converges at most like $\sum_{a>0}a^{-k}$. However,
this argument fails when $k=1$. In that case, we deduce the continuity of
$H_{k,\Delta}$ by showing that the binary Hermitian forms appearing in (1.6)
for a fixed $z\in\mathbb{C}$ decrease to zero exponentially. This is done by
describing the binary Hermitian forms appearing in (1.6) for a fixed
$z\in\mathbb{C}$ using the the nearest integer continued fraction algorithm of
$z$ over the Euclidean imaginary quadratic fields.

Since $H_{k,\Delta}$ is ${\mathcal{O}}_{d}$-invariant, it has a well defined
average value. In Section 4, we prove the following theorem which gives the
average value of $H_{k,\Delta}$ in terms of $L(\chi_{d_{K}},k)$ and
$\zeta_{\mathbb{Q}}(k)$:

######  Theorem 1.4.

Let $\Delta$ be a positive integer which is not a norm of an algebraic integer
in ${\mathcal{O}}_{d}$ and $k\geq 1$ be an odd integer. Then, the average
value of $H_{k,\Delta}$ is given by

$\langle
H_{k,\Delta}(z)\rangle_{av}=\frac{2\pi\Delta^{k+1}}{(k+1)\sqrt{|d_{K}|}}\theta(\Delta,k+1)\zeta_{\mathbb{Q}}(k+1)L(\chi_{d_{K}},k+2)^{-1}$
(1.7)

where $\theta(\Delta,k)$ is a finite Euler product whose factors depend on the
prime factors of $\Delta$ and $d_{K}$.

The proof (see Theorem 4.1) consists of showing that the integral of
$H_{k,\Delta}$ over the fundamental parallelogram $\mathcal{P}$ for the
lattice ${\mathcal{O}}_{d}$ is, up to a constant, equal to the zeta function
of binary Hermitian forms with coefficients in ${\mathcal{O}}_{d}$ and
discriminant $-\Delta$. These zeta functions were studied by Elstrodt,
Grunewald, and Mennicke [10, 11] while developing the theory of representation
numbers of binary Hermitian forms over ${\mathcal{O}}_{d}$ which parallels the
classical theory of representation of integers by binary quadratic forms over
$\mathbb{Z}$.

As an application of Theorem 1.2 and Theorem 1.4 we immediately get the
following formulas (see Corollary 4.2):

  1. (1)

Let $d=1,2,3,7,11$ and $d_{K}$ be the discriminant of
$K=\mathbb{Q}(\sqrt{d})$. Then,

$\displaystyle L(\chi_{d_{K}},3)=$
$\displaystyle\frac{\pi^{3}\Delta^{2}}{6\alpha_{2,\Delta}\sqrt{|d_{K}|}}\
\theta(\Delta,2),$ (1.8) $\displaystyle L(\chi_{d_{K}},-2)=$
$\displaystyle\frac{-|d_{K}|^{2}\Delta^{2}}{12\alpha_{2,\Delta}}\
\theta(\Delta,2).$ (1.9)

  2. (2)

Let $d=1,3,7$ and $d_{K}$ be the discriminant of $K=\mathbb{Q}(\sqrt{d})$.
Then,

$\displaystyle L(\chi_{d_{K}},5)=$
$\displaystyle\frac{\pi^{5}\Delta^{4}}{180\alpha_{4,\Delta}\sqrt{|d_{K}|}}\
\theta(\Delta,4),$ (1.10) $\displaystyle L(\chi_{d_{K}},-4)=$
$\displaystyle\frac{|d_{K}|^{4}\Delta^{4}}{120\alpha_{4,\Delta}}\
\theta(\Delta,4).$ (1.11)

  3. (3)

Let $d=3$ and $d_{K}$ be the discriminant of $K=\mathbb{Q}(\sqrt{d})$. Then,

$\displaystyle L(\chi_{d_{K}},7)=$
$\displaystyle\frac{\pi^{7}\Delta^{6}}{2835\alpha_{6,\Delta}\sqrt{3}}\
\theta(\Delta,6),$ (1.12) $\displaystyle L(\chi_{d_{K}},-6)=$
$\displaystyle\frac{-81\Delta^{6}}{28\alpha_{6,\Delta}}\ \theta(\Delta,6).$
(1.13)

where $\alpha_{1,\Delta}$,$\alpha_{1,\Delta}$ and $\alpha_{1,\Delta}$ are the
values of $H_{1,\Delta}(0)$, $H_{1,\Delta}(0)$ and $H_{1,\Delta}(0)$ given
explicitly by

$\displaystyle\alpha_{1,\Delta}:$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0<c\end{subarray}}c=\sum_{\begin{subarray}{c}N(b)<\Delta\end{subarray}}\sigma_{1}\left(\Delta-N(b)\right),$
(1.14) $\displaystyle\alpha_{3,\Delta}:$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0<c\end{subarray}}c^{3}=\sum_{\begin{subarray}{c}N(b)<\Delta\end{subarray}}\sigma_{3}\left(\Delta-N(b)\right),$
(1.15) $\displaystyle\alpha_{5,\Delta}:$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0<c\end{subarray}}c^{5}=\sum_{\begin{subarray}{c}N(b)<\Delta\end{subarray}}\sigma_{5}\left(\Delta-N(b)\right).$
(1.16)

The above formulas for special values can be seen as Cohen-Zagier type
formulas and as such they are computationally very efficient. In fact, using
our formulas to compute the special values at $s=-6,-4,-2,3,5,7$ requires only
the computation of $\theta(\Delta,k)$ and $\alpha_{k,\Delta}$ for $k=1,3,5$.
Furthermore, since the left hand side of our formulas is independent of
$\Delta$, we can choose the most optimal $\Delta$, i.e. usually the smallest
$\Delta$ that is not a norm, when computing $\theta(\Delta,k)$ and
$\alpha_{k,\Delta}$.

As a simple example, we use our formulas to compute the aforementioned special
values when $K=\mathbb{Q}(i)$ with ${\mathcal{O}}_{1}=\mathbb{Z}[i]$,
$d_{K}=-4$ and $\chi_{d_{K}}=\left(\frac{-4}{\cdot}\right)$. Letting
$\Delta=3$, a simple computation shows that $\theta(3,1)=\frac{5}{6}$,
$\theta(3,3)=\frac{425}{432}$, $\alpha_{1,3}=20$ and $\alpha_{3,3}=68$.
Finally, plugging all these into appropriate above formulas gives

$\displaystyle L(\chi_{-4},3)=$ $\displaystyle\frac{3\pi}{16}\
\zeta_{\mathbb{Q}}(2)=\frac{\pi^{3}}{32},$ $\displaystyle L(\chi_{-4},5)=$
$\displaystyle\frac{75\pi}{256}\ \zeta_{\mathbb{Q}}(4)=\frac{5\pi^{5}}{1536},$
$\displaystyle L(\chi_{-4},-2)=$ $\displaystyle\frac{-|-4|^{2}\cdot
3^{2}}{12\cdot 20}\cdot\frac{5}{6}=-\frac{1}{2},$ $\displaystyle
L(\chi_{-4},-4)=$ $\displaystyle\frac{32\cdot 3^{4}}{15\cdot
68}\cdot\frac{425}{432}=\frac{5}{2}.$

In the last section, we discuss the construction of a nontrivial parabolic
cocycle on the Euclidean Bianchi group
$\textbf{PSL}(2,\mathbb{{\mathcal{O}}}_{d})$ using $H_{k,\Delta}$. We carry
this out for each $d\in\\{1,2,3,7,11\\}$ by showing that the space of
parabolic cocycles on $\textbf{PSL}(2,\mathbb{{\mathcal{O}}}_{d})$ is
isomorphic to a certain space of polynomials $W_{k,k}$ which is defined using
the presentation of $\textbf{PSL}(2,\mathbb{{\mathcal{O}}}_{d})$ and the
cocycle condition. We then show that $P_{k,\Delta}$, which is the polynomial
that $H_{k,\Delta}$ transforms by under the map $z\mapsto\frac{1}{z}$, belongs
to $W_{k,k}$. Consequently, $P_{k,\Delta}$, which is defined using
$H_{k,\Delta}$, corresponds to a parabolic cocycle on
$\textbf{PSL}(2,\mathbb{{\mathcal{O}}}_{d})$. In view of the fact that
$P_{k,\Delta}$ is invariant under the transformation $z\mapsto uz$ where
$u\in{\mathcal{O}}_{d}$ is a unit, we study decomposition of $W_{k,k}$ as a
direct sum of the eigenspaces corresponding to the eigenvalues of the linear
operator associated to the linear map $z\mapsto uz$. This allows us to
precisely describe the subspace of $W_{k,k}$ containing $P_{k,\Delta}$. We
also compute the dimensions of $W_{k,k}$ and its subspace containing
$P_{k,\Delta}$ for some odd values of $k$ using a computer program that we
wrote in Sagemath [21]. In each case, based on the numerical evidence, we
conjecture dimension formulas for these spaces.

Finally, it is fair to say the cocycle property of $H_{k,\Delta}$ is perhaps
the most interesting and also the deepest property of $H_{k,\Delta}$ as it
connects $H_{k,\Delta}$ to cohomology of Euclidean Bianchi groups which are
central to the study of Bianchi modular forms. This connection to cohomology
is a promising one and is worth studying more. Accordingly, we end the section
with saying a few more words on this, and also describing some collaborative
work [13] and other related questions.

_Acknowledgements._ This work is continuation of my PhD thesis. I would like
to express my gratitude to my advisor, Gautam Chinta, for his support and
guidance throughout the completion of my thesis. I would also like to thank
Jorge Flórez, Tian An Wong and Don Zagier for helpful conversations. Finally,
I would like to thank the anonymous referee for a careful reading and numerous
suggestions which greatly improved the manuscript.

##  2\. Definition and Elementary Properties of $H_{k,\Delta}$

###  2.1. Preliminaries

In this section we establish some notation and conventions which are used
throughout the rest of this paper. We fix $d=1,2,3,7,11$ and let
$K=\mathbb{Q}(\sqrt{-d})$ be the corresponding Euclidean imaginary quadratic
number field with discriminant $d_{K}$. We let
${\mathcal{O}}_{d}={\mathcal{O}}_{K}$ be the ring of integers of $K$ and
choose $\\{1,\omega\\}$ with

$\displaystyle\omega:=\frac{d_{K}+\sqrt{d_{K}}}{2}$

as a fixed $\mathbb{Z}$-basis of ${\mathcal{O}}_{d}$.

Let $V_{n}(\mathbb{C})$ be the space of polynomials in $z$ of degree less than
or equal to $n\in\mathbb{Z}_{\geq 0}$ with coefficients in $\mathbb{C}$. For a
polynomial $P(z)$ in $V_{n}(\mathbb{C})$ and a matrix
$\gamma=\begin{pmatrix}a&b\\\ c&e\end{pmatrix}\in\textbf{SL}(2,\mathbb{C})$ we
have the right action

$(P|\gamma):=(cz+e)^{n}P\left(\frac{az+b}{cz+e}\right).$ (2.1)

Because $\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}$ acts trivially,
$V_{n}(\mathbb{C})$ is also $\textbf{PSL}(2,\mathbb{C})$-module. Now, consider
the $\textbf{PSL}(2,\mathbb{C})$-module
$V_{n,n}(\mathbb{C}):=V_{n}(\mathbb{C})\otimes_{\mathbb{C}}\mkern
1.5mu\overline{\mkern-1.5muV_{n}(\mathbb{C})\mkern-1.5mu}\mkern 1.5mu$ where
the overline indicates that the action of $\textbf{SL}(2,\mathbb{C})$ on the
second factor is twisted with complex conjugation; that is for $P(z,\bar{z})$
in $V_{n,n}(\mathbb{C})$ and a matrix $\gamma=\begin{pmatrix}a&b\\\
c&e\end{pmatrix}\in\textbf{PSL}(2,\mathbb{C})$ we have the right action

$(P|\gamma):=(cz+e)^{n}\mkern
1.5mu\overline{\mkern-1.5mu(cz+e)\mkern-1.5mu}\mkern
1.5mu^{n}P\left(\frac{az+b}{cz+e},\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{e}}\right).$
(2.2)

We also extend the action of $\textbf{PSL}(2,\mathbb{C})$ on
$V_{n,n}(\mathbb{C})$ by linearity to an action of the group ring
$\mathbb{Z}[\textbf{PSL}(2,\mathbb{C})]$. The groups
$\textbf{PSL}(2,\mathbb{{\mathcal{O}}}_{d})$ play an important role in our
work, so we let $\Gamma_{d}=\textbf{PSL}(2,\mathbb{{\mathcal{O}}}_{d})$.
Finally, to simplify the notation the letter $\mathbb{C}$ sometimes will be
omitted when referring to different $\textbf{PSL}(2,\mathbb{C})$-modules
throughout the paper.

###  2.2. Sums of Binary Hermitian Forms with coefficients in
${\mathcal{O}}_{d}$

Before we give the definition of $H_{k,\Delta}$ we recall some relevant facts
from the theory of binary Hermitian forms as discussed in [10, 11]. Let $A$ be
a $2\times 2$ matrix with entries in $\mathbb{C}$. $A$ is said to be a
Hermitian matrix if

$\displaystyle A=\bar{A}^{t}$

where $\bar{A}^{t}$ is obtained from $A$ by applying complex conjugation to
each of the entries and then taking the transpose or vice versa. Let $R$ be a
subring of $\mathbb{C}$ which is closed under conjugation, i.e. $R=\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu$. We denote by
$\mathcal{H}(R)$ the set of all $2\times 2$ Hermitian matrices with entries in
$R$. Trivially, $h\in\mathcal{H}(R)$ if and only if $h=\begin{pmatrix}a&b\\\
\bar{b}&c\end{pmatrix}$ with $a,c\in R\cap\mathbb{R}$ and $b\in R$. Every
$h\in\mathcal{H}(R)$ defines a binary Hermitian form with coefficients in $R$.
If $h=\begin{pmatrix}a&b\\\ \bar{b}&c\end{pmatrix}$ then the associated binary
Hermitian form is the map $h:\mathbb{C\times C}\rightarrow\mathbb{R}$ defined
by,

$h(z,w)=\left(\begin{matrix}z&w\\\
\end{matrix}\right)\left(\begin{matrix}a&b\\\
\bar{b}&c\end{matrix}\right)\left(\begin{matrix}\bar{z}\\\ \mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\\\
\end{matrix}\right)=a|z|^{2}+bz\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu+\bar{b}\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5muw+c|w|^{2}.$

We shall often call an element $h\in\mathcal{H}(R)$ a binary Hermitian form
with coefficients in $R$, and we shall use the notation
$a_{h},b_{h},\bar{b}_{h},c_{h}$ to refer to the individual matrix entries of
$h$ when the need arises. The discriminant $\Delta(h)$ of $h\in\mathcal{H}(R)$
is defined as

$\displaystyle\Delta(h)=\text{det}(h),$

and $\mathcal{H}(R,\Delta)$ denotes the set of all $h\in\mathcal{H}(R)$ with
$\Delta(h)=\Delta$.

Similar to the action of the $\textbf{GL}(2,\mathbb{Z})$ on binary quadratic
forms, we have the action of $\textbf{GL}(2,R)$ on $\mathcal{H}(R)$ given by
the formula

$\sigma(h)=\mkern 1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern
1.5mu^{t}h\sigma$ (2.3)

where $\sigma\in\textbf{GL}(2,R)$ and $h\in\mathcal{H}(R)$.

######  Remark 2.1.

The action defined in (2.3) differs slightly from that of [10, 11], but it is
more convenient for our purposes.

For our purposes we let $R={\mathcal{O}}_{d}$ and define $H_{k,\Delta}$ as
follows.

######  Definition 1.

Let $k\geq 1$ be an odd integer and $\Delta$ be a positive integer that is not
a norm of an algebraic integer in ${\mathcal{O}}_{d}$. Then, we define the
function $H_{k,\Delta}:\mathbb{C}\rightarrow\mathbb{R}$ as

$\displaystyle H_{k,\Delta}(z):$
$\displaystyle=\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathbb{{\mathcal{O}}}_{d},-\Delta)\\\
a_{h}<0\end{subarray}}\max\left(0,h(z,1)^{k}\right)$ (2.4)
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right)$
(2.5)

where $a,c\,\in\,\mathbb{Z},\,b\,\in\,{\mathcal{O}}_{d}$, and
$N(b)=|b|^{2}=b\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu$
is the usual norm on $K$.

We assume that $k\geq 1$ is an odd integer and $\Delta$ is a positive integer
that is not a norm of an algebraic integer in ${\mathcal{O}}_{d}$ for the rest
of the paper unless otherwise is stated.

The convergence and the continuity of $H_{k,\Delta}$ is discussed in section
3. In particular, we show that $H_{k,\Delta}$ converges and is a continuous
function for any $k\geq 1$, $\Delta>0$ as in the definition of $H_{k,\Delta}$.
Let us assume for now that $H_{k,\Delta}$ converges and is a continuous
function for all $z\in\mathbb{C}$.

Note that the set of all $h(z,1)$ appearing in equation (2.4) is a subset of
$V_{1,1}({\mathcal{O}}_{d})$ and an elementary computation shows that the
action of $\gamma=\begin{pmatrix}p&q\\\ r&s\end{pmatrix}\in\Gamma_{d}$ on
$h(z,1)$ as an element of $V_{1,1}({\mathcal{O}}_{d})$ corresponds to the
action of $\bar{\gamma}\in\Gamma_{d}$ on $h$ as an element of
$\mathcal{H}(\mathcal{O}_{d},\Delta)$. That is, if $P(z,\bar{z})=h(z,1)$ then

$\displaystyle P|\gamma=\left(\begin{matrix}z&1\\\ \end{matrix}\right)\mkern
1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern
1.5mu(h)\left(\begin{matrix}\bar{z}\\\ 1\end{matrix}\right).$

Using this observation we can extend the action of
$\textbf{PSL}(2,\mathbb{C})$ to $H_{k,\Delta}$ by defining

$\displaystyle H_{k,\Delta}|\gamma$
$\displaystyle:=\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathbb{{\mathcal{O}}}_{d},-\Delta)\\\
a_{h}<0\end{subarray}}\max\left(0,\left(\left(\begin{matrix}z&1\\\
\end{matrix}\right)\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern
1.5mu(h)\left(\begin{matrix}\bar{z}\\\ 1\end{matrix}\right)\right)^{k}\right)$
$\displaystyle=|rz+s|^{2k}H_{k,\Delta}\left(\frac{pz+q}{rz+s}\right)$

Let us now discuss some properties of $H_{k,\Delta}$ and its behavior under
the action of certain elements of $\Gamma_{d}$.

######  Proposition 2.2.

  * (1)

$H_{k,\Delta}(\bar{z})=H_{k,\Delta}(z)$ for any $z\in\mathbb{C}$.

  * (2)

$H_{k,\Delta}(uz)=H_{k,\Delta}(z)$ for any unit $u\in{\mathcal{O}}_{d}$.

  * (3)

$H_{k,\Delta}(z+\lambda)=H_{k,\Delta}(z)$ for any
$\lambda\in{\mathcal{O}}_{d}$.

###### Proof.

We have

$\displaystyle H_{k,\Delta}(\bar{z})$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0\end{subarray}}\max\left(0,\left(a|\bar{z}|^{2}+b\bar{z}+\bar{b}z+c\right)^{k}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right)$

since if $b\in{\mathcal{O}}_{d}$ is a solution to $N(b)-ac=\Delta$ then so is
$\bar{b}\in{\mathcal{O}}_{d}$. The proof of $H_{k,\Delta}(uz)=H_{k,\Delta}(z)$
is similar since if $u\in{\mathcal{O}}_{d}$ is a unit then

$a|uz|^{2}+buz+\bar{b}\mkern 1.5mu\overline{\mkern-1.5muuz\mkern-1.5mu}\mkern
1.5mu+c=a|z|^{2}+buz+\bar{b}\mkern
1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu\bar{z}+c,$

and $b\in{\mathcal{O}}_{d}$ can be replaced by $bu\in{\mathcal{O}}_{d}$ or
$\bar{b}\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern
1.5mu\in{\mathcal{O}}_{d}$ in the equation $N(b)-ac=\Delta$.

As for the proof of the last statement, notice that
$H_{k,\Delta}(z+\lambda)=H_{k,\Delta}|A$ where $A=\begin{pmatrix}1&\lambda\\\
0&1\end{pmatrix}\in\Gamma_{d}.$ Since

$\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu(h)=A^{t}h\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu=\begin{pmatrix}a&a\mkern
1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu+b\\\
a\lambda+\bar{b}&a\lambda\mkern
1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern
1.5mu+b\lambda+\bar{b}\mkern
1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu+c\end{pmatrix}$

is just another binary Hermitian form with discriminant $\Delta$ and negative
first entry, we see that $H_{k,\Delta}$ is invariant under the action of $A$.
∎

As mentioned in the introduction, we use the Euclidean algorithm, which
consists of finite iterations of the maps $z\mapsto z+\lambda$ and
$z\mapsto\frac{1}{z}$, to prove that $H_{k,\Delta}$ is constant in some
special cases. We already saw that $H_{k,\Delta}$ is invariant under the map
$z\mapsto z+\lambda$, but we will see next that $H_{k,\Delta}$ is not
invariant under map $z\mapsto\frac{1}{z}$ and transforms by a polynomial which
is defined as:

######  Definition 2.

We define the polynomials $P_{k,\Delta}$ in $z$ and $\bar{z}$ as

$\displaystyle P_{k,\Delta}(z,\bar{z}):$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\left(az\bar{z}+bz+\bar{b}\bar{z}+c\right)^{k}$ (2.6)

where $b\in{\mathcal{O}}_{d}$ and $a,c\in\mathbb{Z}.$ Notice that
$P_{k,\Delta}\in V_{k,k}({\mathcal{O}}_{d})$.

######  Proposition 2.3.

Let $S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$ and ${\bf
1}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$. Then,

$\displaystyle H_{k,\Delta}|(S-{\bf 1})=P_{k,\Delta}$

where $|(S-{\bf 1})$ indicates the action of the element $S-{\bf 1}$ in the
group ring $\mathbb{Z}[\textbf{PSL}(2,\mathbb{C})]$ on $H_{k,\Delta}$.

###### Proof.

We first prove that $H_{k,\Delta}|(S-{\bf 1})=P_{k,\Delta}$ for $z\in K$ and
use the continuity of $H_{k,\Delta}$ to conclude that it holds for all
$z\in\mathbb{C}$. Notice that when $z\in K$, the number of terms appearing in
the sum for $H_{k,\Delta}(z)$ is finite. To see this, suppose $\displaystyle
h(z,1)=a|z|^{2}+bz+\bar{b}\bar{z}+c\in\mathcal{H}(\mathbb{{\mathcal{O}}}_{d},-\Delta)$
occurs in the sum for $H_{k,\Delta}(z)$ for a fixed $\displaystyle
z=\frac{p}{q}+\frac{r}{s}\sqrt{-d}$ with $p,q,r,s\in\mathbb{Z}$. Then using
the identity $-\Delta(h)=N(\bar{b}_{h}+a_{h}z)-a_{h}h(z,1)$ one gets,

$\displaystyle\Delta
q^{2}s^{2}=N(qs\bar{b}+aw)+|a|\left|a|w|^{2}+qsbw+qs\bar{b}\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu+cq^{2}s^{2}\right|$

where $w=ps+qr\sqrt{-d}$. Since $\left|a|w|^{2}+qsbw+qs\bar{b}\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu+cq^{2}s^{2}\right|\geq
1$, the last equation implies that $|a|\leq\Delta q^{2}s^{2}$ which shows that
$a$ is bounded, and this in turn bounds $c$ and $N(b)$ as well. Thus only
finitely many $h(z,1)$ appear in the sum for $H_{k,\Delta}(z)$ when $z\in K$.
On the other hand, the action of $S-{\bf 1}$ on $H_{k,\Delta}$ is given by

$H_{k,\Delta}|(S-{\bf 1})=H_{k,\Delta}|S-H_{k,\Delta},$

which can be computed directly as

$\displaystyle=|z|^{2k}H_{k,\Delta}\left(\frac{-1}{z}\right)-H_{k,\Delta}(z)$
$\displaystyle=|z|^{2k}H_{k,\Delta}\left(\frac{1}{z}\right)-H_{k,\Delta}(z)\quad\text{(since
$H_{k,\Delta}(-z)=H_{k,\Delta}(z))$}$
$\displaystyle=|z|^{2k}\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0\end{subarray}}\max\left(0,\left(\frac{a}{|z|^{2}}+\frac{b}{z}+\frac{\bar{b}}{\bar{z}}+c\right)^{k}\right)-\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0\end{subarray}}\max\left(0,\left(a+b\bar{z}+\bar{b}z+c|z|^{2}\right)^{k}\right)-\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right)-\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right).$

Notice that the summands with $a$ and $c$ both negative in the two previous
sums cancel. Moreover $a\neq 0$ and $c\neq 0$ because $\Delta$ is not a norm
in ${\mathcal{O}}_{d}$. Therefore, we have

$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right)-\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0<c\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right).$

Now, applying the fact that $\max(0,X)=-\min(0,-X)$ for any $X\in\mathbb{R}$
gives

$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right)+\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0<c\end{subarray}}\min\left(0,\left(-a|z|^{2}-bz-\bar{b}\bar{z}-c\right)^{k}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\max\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right)+\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\min\left(0,\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}\right).$

Finally, because $\max(0,X)+\min(0,X)=X$ for any $X\in\mathbb{R}$ we get

$\displaystyle H_{k,\Delta}|(S-{\bf 1})$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{k}.$

∎

Let us now define the following numbers which are the values of $H_{k,\Delta}$
at $z=0$:

$\displaystyle\alpha_{k,\Delta}:$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
a<0<c\end{subarray}}c^{k}$ (2.7)
$\displaystyle=\sum_{\begin{subarray}{c}N(b)<\Delta\end{subarray}}\sigma_{k}\left(\Delta-N(b)\right).$
(2.8)

######  Corollary 2.4.

$H_{1,\Delta}|(S-{\bf
1})=P_{1,\Delta}(z,\bar{z})=\alpha_{1,\Delta}(z\bar{z}-1).$

###### Proof.

When $k=1$, by Proposition 2.3, we get

$H_{1,\Delta}|(S-{\bf 1})=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}az\bar{z}+bz+\bar{b}\bar{z}+c.$

The coefficient of $z$ and $\bar{z}$ is zero because if $b$ and $\bar{b}$ are
solutions to $N(b)-ac=\Delta$ then so are $-b$ and $-\bar{b}$. Finally, it is
easy to see the coefficient of $z\bar{z}$ term is $\alpha_{1,\Delta}$ and the
coefficient of the constant term is $-\alpha_{1,\Delta}$ since $c<0$. In other
words, we have

$H_{1,\Delta}|(S-{\bf 1})=\alpha_{1,\Delta}(z\bar{z}-1).$

∎

We now have everything we need to show that $H_{1,\Delta}$ is constant.

######  Theorem 2.5.

Suppose that $H_{1,\Delta}$ is a continuous function. Then,
$H_{1,\Delta}(z)=\alpha_{1,\Delta}$ for all $z\in\mathbb{C}$.

###### Proof.

Let $\Theta_{1,\Delta}(z):=H_{1,\Delta}(z)-\alpha_{1,\Delta}$. Then,

$\Theta_{1,\Delta}(z+\lambda)=\Theta_{1,\Delta}(z)\quad\text{and}\quad|z|^{2}\Theta_{1,\Delta}\left(\frac{1}{z}\right)=\Theta_{1,\Delta}(z),$

where $\lambda\in{\mathcal{O}}_{d}$ and $z\in K$.

Since ${\mathcal{O}}_{d}$ has an Euclidean algorithm, any $z\in K$ can be
reduced to zero by a finite number of iterations of the transformations
$z\mapsto z+\lambda$ and $\displaystyle z\mapsto\frac{1}{z}$. This shows that

$\Theta_{1,\Delta}(z)=\Theta_{1,\Delta}(0)=0$

for all $z\in K$ which implies that $H_{1,\Delta}(z)=\alpha_{1,\Delta}$ for
all $z\in K$. Assuming the continuity, we conclude that
$H_{1,\Delta}(z)=\alpha_{1,\Delta}$ for all $z\in\mathbb{C}$. ∎

The key ingredient that makes the proof of Theorem 2.5 work is the fact that
$P_{1,\Delta}(z,\bar{z})=\alpha_{1,\Delta}(z\bar{z}-1)$. If we want to use the
same argument to show that $H_{k,\Delta}$ is also constant when $k=3,5$ for
the relevant values of $d$ then we need to establish that $P_{3,\Delta}$ and
$P_{5,\Delta}$ have the same form as $P_{1,\Delta}$. In the following
proposition we list some identities that the polynomials $P_{k,\Delta}$
satisfy. We use some of these identities in Lemma 2.7 below to show that
$P_{3,\Delta}$ and $P_{5,\Delta}$ have the desired form. Some of these
identities, especially the first two, are quite clear and can be surmised
easily given the definition of $P_{k,\Delta}$. However, the last two may seem
strange, and may look as if they are being pulled out of a hat. But, that is
not the case, and they come from combining some of the relations that define
the vector space of polynomials containing $P_{k,\Delta}$. We have chosen
them, granted in a seemingly ad-hoc manner, purely for the purpose of deducing
that $P_{3,\Delta}$ and $P_{5,\Delta}$ have the desired form. We discuss these
vector spaces for each $d$ in the last section, and show that they can be
identified with the space of parabolic 1-cocycles for $\Gamma_{d}$ which is
related to Bianchi cusp forms on $\Gamma_{d}$ via generalized Eichler-Shimura
isomorphism. We also compute their dimensions for small degrees using a
computer program written by the author in SageMath [21], and in each case also
conjecture a formula for the dimension of these spaces. Indeed, the computer
calculations show that the vector space containing $P_{k,\Delta}$ is one
dimensional for all $d$ when $k=1$, for $d=1,3,7$ when $k=3$, and for $d=3$
when $k=5$.

######  Proposition 2.6.

Let $\varepsilon=\left(\begin{smallmatrix}-1&0\\\
0&1\end{smallmatrix}\right),\,T=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right),\,T_{\omega}=\left(\begin{smallmatrix}1&\omega\\\
0&1\end{smallmatrix}\right)$ where $\omega=\frac{1+\sqrt{-7}}{2}$. Then,

  * (1)

$P_{k,\Delta}(\bar{z},z)=P_{k,\Delta}(z,\bar{z})$.

  * (2)

$P_{k,\Delta}(uz,\bar{u}\bar{z})=P_{k,\Delta}(z,\bar{z})$ for any unit
$u\in{\mathcal{O}}_{d}$.

  * (3)

$P_{k,\Delta}|({\bf 1}+S)=0$.

  * (4)

$P_{k,\Delta}|({\bf 1}+TS\varepsilon-T)=0$.

  * (5)

Additionally, when $P_{k,\Delta}\in V_{k,k}({\mathcal{O}}_{7})$ we have

$P_{k,\Delta}|({\bf
1}-T_{\omega}-ST^{-1}T_{\omega}S-TT_{\omega}^{-1}ST_{\omega})=0.$

###### Proof.

Part (1) and (2) are clear from the equation (2.6) once we notice that if
$b\in{\mathcal{O}}_{d}$ is a solution to $N(b)-ac=\Delta$ then so are
$\bar{b}\in{\mathcal{O}}_{d}$ and $bu\in{\mathcal{O}}_{d}$.

For part (3) we have

$\displaystyle P_{k,\Delta}|(S+{\bf 1})=H_{k,\Delta}|(S-{\bf 1})(S+{\bf
1})=H_{k,\Delta}|(S^{2}-S+S+{\bf 1})=H_{k,\Delta}|(S^{2}-{\bf 1})=0$

since $S^{2}={\bf 1}$.

As for part (4) we make use of the identities

$\displaystyle\varepsilon S=S\varepsilon,\ \ T\varepsilon=\varepsilon T^{-1},\
\ TSTS=ST^{-1},\ \
H_{k,\Delta}|\varepsilon=H_{k,\Delta}|T=H_{k,\Delta}|T^{-1}=H_{k,\Delta},$

and calculate

$\displaystyle P_{k,\Delta}|({\bf 1}+TS\varepsilon-T)=$ $\displaystyle
H_{k,\Delta}|(S-{\bf 1})({\bf 1}+TS\varepsilon-T)$ $\displaystyle=$
$\displaystyle H_{k,\Delta}|(S-{\bf 1}+STS\varepsilon-TS\varepsilon-ST+T)$
$\displaystyle=$ $\displaystyle H_{k,\Delta}|(S+TSTS\varepsilon-\varepsilon
T^{-1}S-ST)$ $\displaystyle=$ $\displaystyle H_{k,\Delta}(ST^{-1}\varepsilon-
ST)\quad$ $\displaystyle=$ $\displaystyle 0.$

Finally for the proof of part (5) calculating $(S-{\bf 1})|({\bf
1}-T_{\omega}-ST^{-1}T_{\omega}S-TT_{\omega}^{-1}ST_{\omega})$ gives

$\displaystyle S-{\bf
1}-ST_{\omega}+T_{\omega}-T^{-1}T_{\omega}S+ST^{-1}T_{\omega}S-STT_{\omega}^{-1}ST_{\omega}+TT_{\omega}^{-1}ST_{\omega},$

and the result follows after applying the following identities

$\displaystyle
H_{k,\Delta}|T_{\omega}=H_{k,\Delta}|T_{\omega}^{-1}=H_{k,\Delta}|T^{-1}T_{\omega}=H_{k,\Delta},\quad
STT_{\omega}^{-1}ST_{\omega}=T_{\omega}^{-1}ST^{-1}T_{\omega}S.$

∎

######  Lemma 2.7.

  1. (1)

$P_{3,\Delta}(z,\bar{z})=\alpha_{3,\Delta}(z^{3}\bar{z}^{3}-1)$ when
$d=1,3,7$.

  2. (2)

$P_{5,\Delta}(z,\bar{z})=\alpha_{5,\Delta}(z^{5}\bar{z}^{5}-1)$ when $d=3$.

###### Proof.

By Definition 2 we have

$P_{3,\Delta}(z,\bar{z})=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\left(a|z|^{2}+bz+\bar{b}\bar{z}+c\right)^{3}$

where $b\in{\mathcal{O}}_{d}$ and $a,c\in\mathbb{Z}.$ Using the multinomial
theorem we can write $P_{3,\Delta}(z)$ in the form

$\displaystyle P_{3,\Delta}(z,\bar{z})$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\left(\sum_{k_{1}+k_{2}+k_{3}+k_{4}=3}\binom{3}{k_{1},k_{2},k_{3},k_{4}}a^{k_{1}}b^{k_{2}}\bar{b}^{k_{3}}c^{k_{4}}|z|^{2k_{1}}z^{k_{2}}\bar{z}^{k_{3}}\right)$
$\displaystyle=\sum_{k_{1}+k_{2}+k_{3}+k_{4}=3}\binom{3}{k_{1},k_{2},k_{3},k_{4}}\left(\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}a^{k_{1}}b^{k_{2}}\bar{b}^{k_{3}}c^{k_{4}}\right)|z|^{2k_{1}}z^{k_{2}}\bar{z}^{k_{3}}.$

We now use Proposition 2.6 to deduce restrictions on the coefficients of
$P_{3,\Delta}$. Notice that $P_{3,\Delta}$ is not identically zero. The units
in ${\mathcal{O}}_{1}$ are $-1$ and $i$. So, by Proposition 2.6 we have

$\displaystyle P_{3,\Delta}(-z,-\bar{z})=P_{3,\Delta}(z,\bar{z}),\quad
P_{3,\Delta}(iz,\bar{i}\bar{z})=P_{3,\Delta}(z,\bar{z}).$

$P_{3,\Delta}(-z,-\bar{z})=P_{3,\Delta}(z,\bar{z})$ implies that
$2|(k_{2}+k_{3})$ which, in turn, implies $k_{2}+k_{3}=0,2$, because
$k_{2}+k_{3}\leq 3$. On the other hand,
$P_{3,\Delta}(iz,\bar{i}\bar{z})=P_{3,\Delta}(z,\bar{z})$ implies that either
$k_{3}$ is odd and $k_{2}+k_{3}\equiv 2$ (mod 4) or $k_{3}$ is even and
$k_{2}+k_{3}\equiv 0$ (mod 4). Since $k_{2}+k_{3}=0,2$, in the first case we
get that $k_{2}=k_{3}=1$ and in the second case we get that $k_{2}=k_{3}=0.$

When $d=3$, $-1,\zeta=e^{\frac{2\pi i}{3}}\in{\mathcal{O}}_{3}$ are units, and
again by part (2) of Proposition 2.6 we have that

$\displaystyle P_{3,\Delta}(-z,-\bar{z})=P_{3,\Delta}(z,\bar{z}),\quad
P_{3,\Delta}(\zeta z,\bar{\zeta}\bar{z})=P_{3,\Delta}(z,\bar{z}).$

Again, $P_{3,\Delta}(-z,-\bar{z})=P_{3,\Delta}(z,\bar{z})$ implies that
$k_{2}+k_{3}=0,2$. On the other hand, $P_{3,\Delta}(\zeta
z,\zeta\bar{z})=P_{3,\Delta}(z,\bar{z})$ implies that $k_{2}+2k_{3}\equiv 0$
(mod 3). Since $k_{2}+k_{3}=0,2$, we get that $k_{2}=k_{3}=1$ or
$k_{2}=k_{3}=0.$ Therefore, in both cases we have

$\displaystyle P_{3,\Delta}(z,\bar{z})$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\left(\sum_{k_{1}+k_{4}=3}\binom{3}{k_{1},0,0,k_{4}}a^{k_{1}}c^{k_{4}}|z|^{2k_{1}}\right)+$
$\displaystyle\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\left(\sum_{k_{1}+k_{4}=1}\binom{3}{k_{1},1,1,k_{4}}a^{k_{1}}|b|^{2}c^{k_{4}}|z|^{2k_{1}+2}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\left(a^{3}|z|^{6}+\left(6a|b|^{2}+3a^{2}c\right)|z|^{4}+\left(6c|b|^{2}+3ac^{2}\right)|z|^{2}+c^{3}\right)$
$\displaystyle=\alpha_{3,\Delta}|z|^{6}+\alpha_{3,\Delta}^{\prime}|z|^{4}+\alpha_{3,\Delta}^{\prime\prime}|z|^{2}-\alpha_{3,\Delta}$

for some
$\alpha_{3,\Delta}^{\prime},\alpha_{3,\Delta}^{\prime\prime}\in\mathbb{Z}$
depending on $\Delta$.

Because $P_{3,\Delta}|(S+{\bf 1})=0$, we see that
$\alpha_{3,\Delta}^{\prime}=-\alpha_{3,\Delta}^{\prime\prime}$ which gives

$P_{3,\Delta}(z,\bar{z})=\alpha_{3,\Delta}|z|^{6}+\alpha_{3,\Delta}^{\prime}|z|^{4}-\alpha_{3,\Delta}^{\prime}|z|^{2}-\alpha_{3,\Delta}.$

By part (4) of Proposition 2.6 we have

$P_{3,\Delta}(z+1,\bar{z}+1)-P_{3,\Delta}(z,\bar{z})=|z|^{6}P_{3,\Delta}\left(\frac{z+1}{z},\frac{\bar{z}+1}{\bar{z}}\right).$
(2.9)

A direct calculation shows that $|z|^{6}-1$ satisfies the equation (2.9) but
that $|z|^{4}-|z|^{2}$ does not. Thus, the middle terms must vanish (i.e.
$\alpha_{3,\Delta}^{\prime}=0$), and we get

$P_{3,\Delta}(z)=\alpha_{3,\Delta}|z|^{6}-\alpha_{3,\Delta}=\alpha_{3,\Delta}(|z|^{6}-1)$

which proves the first assertion for $d=1,3$.

It remains to prove the first assertion when $d=7$. Since
$P_{3,\Delta}(-z,-\bar{z})=P_{3,\Delta}(z,\bar{z})$, we know that
$k_{2}+k_{3}=0,2$ which implies that $(k_{2},k_{3})=(0,0),(0,2),(2,0),(1,1)$.
Using these values for $k_{2}$ and $k_{3}$ in the equation (2.2) gives

$P_{3,\Delta}(z,\bar{z})=\alpha_{3,\Delta}|z|^{6}+\delta_{1}|z|^{4}+\delta_{2}|z|^{2}+\delta_{3}|z|^{2}\bar{z}^{2}+\mkern
1.5mu\overline{\mkern-1.5mu\delta_{3}\mkern-1.5mu}\mkern
1.5mu|z|^{2}z^{2}+\delta_{4}z^{2}+\mkern
1.5mu\overline{\mkern-1.5mu\delta_{4}\mkern-1.5mu}\mkern
1.5mu\bar{z}^{2}+\alpha_{3,\Delta}$

where $\delta_{1},\delta_{2}\in\mathbb{Z}$ and
$\delta_{3},\delta_{4}\in{\mathcal{O}}_{7}.$ Because $P_{3,\Delta}|(S+{\bf
1})=0$, we see that $\delta_{1}=-\delta_{2},\ \delta_{3}=-\delta_{4}$, and
thus

$P_{3,\Delta}(z,\bar{z})=\alpha_{3,\Delta}(|z|^{6}-1)+\delta_{1}(|z|^{4}-|z|^{2})+\delta_{3}(|z|^{2}\bar{z}^{2}-z^{2})+\mkern
1.5mu\overline{\mkern-1.5mu\delta_{3}\mkern-1.5mu}\mkern
1.5mu(|z|^{2}z^{2}-\bar{z}^{2}).$

Now, let

$F(z):=P_{3,\Delta}(z+1,\bar{z}+1)-P_{3,\Delta}(z,\bar{z})-|z|^{6}P_{3,\Delta}\left(\frac{z+1}{z},\frac{\bar{z}+1}{\bar{z}}\right)$

and by Proposition 2.6 we know that $F(z)=0$ for any $z\in\mathbb{C}$. Since
$F(2)=-120\delta_{1}-240\delta_{3}$ we get that $\delta_{1}=-2\delta_{3}$,
which also implies that $\delta_{3}=\mkern
1.5mu\overline{\mkern-1.5mu\delta_{3}\mkern-1.5mu}\mkern 1.5mu$. So,

$P_{3,\Delta}(z,\bar{z})=\alpha_{3,\Delta}(|z|^{6}-1)-2\delta_{3}(|z|^{4}-|z|^{2})+\delta_{3}(|z|^{2}\bar{z}^{2}+|z|^{2}z^{2}-z^{2}-\bar{z}^{2}).$

Now, we let

$\displaystyle G(z):=$ $\displaystyle
P_{3,\Delta}(z,\bar{z})-P_{3,\Delta}(z+\omega,\bar{z}+\bar{\omega})-|\bar{\omega}z+1|^{6}P_{3,\Delta}\left(\frac{z}{\bar{\omega}z+1},\frac{\bar{z}}{\omega\bar{z}+1}\right)$
$\displaystyle-$
$\displaystyle|z+\omega|^{6}P_{3,\Delta}\left(\frac{\bar{\omega}z+1}{z+\omega},\frac{\omega\bar{z}+1}{\bar{z}+\bar{\omega}}\right),$

and again by Proposition 2.6 part (5) we know that $G(z)=0$ for any
$z\in\mathbb{C}$. Finally, since $G(2)=294\delta_{3}$ we see that
$\delta_{3}=0$ and thus

$P_{3,\Delta}(z,\bar{z})=\alpha_{3,\Delta}(|z|^{6}-1)$

which completes the proof of the first assertion.

As for the second claim, we know that

$P_{5,\Delta}(z,\bar{z})=\sum_{k_{1}+k_{2}+k_{3}+k_{4}=5}\binom{5}{k_{1},k_{2},k_{3},k_{4}}\left(\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}a^{k_{1}}b^{k_{2}}\bar{b}^{k_{3}}c^{k_{4}}\right)|z|^{2k_{1}}z^{k_{2}}\bar{z}^{k_{3}}$

where $2|(k_{2}+k_{3})$ and $k_{2}+2k_{3}\equiv 0$ (mod 3). Thus,

$P_{5,\Delta}(z,\bar{z})=\alpha_{5,\Delta}|z|^{10}+\delta_{1}|z|^{8}+\delta_{2}|z|^{6}+\delta_{3}|z|^{4}+\delta_{4}|z|^{2}-\alpha_{5,\Delta}$

where $\delta_{1},\delta_{2},\delta_{3},\delta_{4}\in\mathbb{Z}.$ Since
$P_{5,\Delta}|(S+{\bf 1})=0$, we have that $\delta_{1}=-\delta_{4}$ and
$\delta_{2}=-\delta_{3}$. So,

$P_{5,\Delta}(z,\bar{z})=\alpha_{5,\Delta}(|z|^{10}-1)+\delta_{1}(|z|^{8}-|z|^{2})+\delta_{2}(|z|^{6}-|z|^{4}).$

Finally, we again let

$F(z):=P_{5,\Delta}(z+1,\bar{z}+1)-P_{5,\Delta}(z,\bar{z})-|z|^{10}P_{5,\Delta}\left(\frac{z+1}{z},\frac{\bar{z}+1}{\bar{z}}\right)$

and by Proposition 2.6 we know that $F(z)=0$ for any $z\in\mathbb{C}$.
Evaluating $F$ at $z=2$ and $z=2i$ gives the system of equations

$\displaystyle-17640\delta_{1}-5880\delta_{2}$ $\displaystyle=0$
$\displaystyle-852\delta_{1}-348\delta_{2}$ $\displaystyle=0$

which has $\delta_{1}=\delta_{2}=0$ as solutions. Thus,

$P_{5,\Delta}(z,\bar{z})=\alpha_{5,\Delta}(|z|^{10}-1)$

which concludes the proof. ∎

######  Theorem 2.8.

Suppose that $H_{3,\Delta}$ and $H_{5,\Delta}$ are continuous functions. Then,

  1. (1)

$H_{3,\Delta}(z)$ has a constant value $\alpha_{3,\Delta}$ for all
$z\in\mathbb{C}$ when $d=1,3,7$.

  2. (2)

$H_{5,\Delta}(z)$ has a constant value $\alpha_{5,\Delta}$ for all
$z\in\mathbb{C}$ when $d=3$.

###### Proof.

Let $\Theta_{3,\Delta}(z):=\Theta_{3,\Delta}(z)-\alpha_{3,\Delta}$. Then, the
same argument as in the proof of Theorem 2.5 shows that
$\Theta_{3,\Delta}(z)=\Theta_{3,\Delta}(0)=0$ for all $z\in K$ and thus
$H_{3,\Delta}(z)=\alpha_{3,\Delta}$ for all $z\in K$. Since $H_{3,\Delta}(z)$
is continuous, we conclude that $H_{3,\Delta}(z)=\alpha_{3,\Delta}$ for all
$z\in\mathbb{C}$. The proof of the second claim is exactly the same.

∎

##  3\. Continuity of $H_{k,\Delta}$

Proving that $H_{k,\Delta}$ is a continuous function when $k\geq 3$ is
straightforward since in that case the sum defining $H_{k,\Delta}$ converges
absolutely and uniformly as we show below. The argument we use to show the
continuity for $k\geq 3$ does not apply for $k=1$, since the bound we use to
show the convergence and uniform continuity is not sharp enough. Since the
summands that appear in the sum defining $H_{1,\Delta}$ depend on $z$, to show
the continuity of $H_{1,\Delta}$ a more explicit description of these summands
in terms of $z$ is needed. The same issue also arises when trying to prove
that $F_{2,D}$ is continuous. Zagier gives an argument for the convergence of
$F_{2,D}$ and mentions that the continuity of $F_{2,D}$ can be proved in an
elementary way, but he does not provide an argument. However, based on
computer experiments he observes that for a given $x$ there is in fact a
strong relation between the summands that appear in the sum for $F_{2,D}(x)$
and the continued fraction expansion of $x$. More specifically, he observes
that the summands which appear in the sum defining $F_{2,D}(x)$ for a given
$x$ belong to a union of two lists whose elements tend to zero exponentially
quickly, and each summand in each list is obtained from the previous one by
applying an element of $\textbf{SL}(2,\mathbb{Z})$ coming from the continued
fraction expansion of $x$. This observation was shown to be true by Bengoechea
in [1], and as a result of this she obtains a direct proof of the continuity
of $F_{2,D}$ by showing that $F_{2,D}$ converges exponentially quickly. To
prove the continuity of $H_{1,\Delta}$ we use Bengoechea’s methods but the
difficulty with that approach is finding a continued fraction algorithm for
complex numbers that is similar to the classical continued fraction (i.e.,
nearest integer continued fraction) expansion of real numbers which Bengoechea
uses in her paper. Such an algorithm was developed by Hurwitz [16] and has
been studied recently by a number of different people (see [2, 7, 8, 9, 15],
and other references cited therein).

###  3.1. Continuity of $H_{k,\Delta}$ for $k\geq 3$

######  Theorem 3.1.

For $k\geq 3$, $H_{k,\Delta}$ is a continuous function.

###### Proof.

We can write $H_{k,\Delta}$ as:

$H_{k,\Delta}(z)=\sum_{a<0}\left(\sum_{N(b)-ac=\Delta}\max\left(0,(a|z|^{2}+bz+\bar{b}\bar{z}+c)^{k}\right)\right).$

The identity $\Delta=N(\bar{b}+az)+|a|(a|z|^{2}+bz+\bar{b}\bar{z}+c),$ and the
inequalities $a<0<a|z|^{2}+bz+\bar{b}\bar{z}+c$ together imply that
$N(\bar{b}+az)<\sqrt{\Delta}$. So for each $z\in\mathbb{C}$, $\bar{b}$ is an
algebraic integer belonging to a disk of radius $\sqrt{\Delta}$ centered at
$-2az$. On the other hand the number of algebraic integers in a disk of fixed
radius is finite. Thus, for each $a$ the number of $b$’s satisfying the
conditions of the sum is finite and independent of $z$ which shows that for
each $a$ the number functions appearing in the inner sum is finite and
independent of $z$. Also, for each $a$ the functions appearing in the inner
sum are bounded by $\frac{\Delta}{|a|}$. So, for each $a$ and all $z$ the
inner sum is the sum of finitely many continuous functions
$h(z,1)^{k}=(a|z|^{2}+bz+\bar{b}\bar{z}+c)^{k}$ that are bounded by
$\left(\frac{\Delta}{|a|}\right)^{k}$. Hence by the Weierstrass M-test, the
sum in the definition of $H_{k,\Delta}$ converges absolutely and uniformly and
is a continuous function for $k\geq 3$.

∎

###  3.2. Nearest Integer Continued Fraction Algorithm for the Euclidean
Imaginary Quadratic Fields.

Let $z$ be a complex number. We denote by $\lfloor z\rceil$ the integer in
${\mathcal{O}}_{d}$ that is nearest to $z$ with respect to Euclidean distance
in complex plane, rounding down, in both the real and the imaginary components
to break ties. The nearest integer continued fraction algorithm for the
Euclidean imaginary quadratic fields proceeds by steps of the form

$z_{0}=z,\quad\alpha_{n}=\lfloor z_{n}\rceil,\quad
z_{n+1}=\frac{1}{z_{n}-\alpha_{n}}\quad$ (3.1)

where $\alpha_{n}=a_{n}+b_{n}\omega\in{\mathcal{O}}_{d}$ and $n\geq 0$. If
$z\in K$ the algorithm terminates when, as must eventually occur, $z_{n}=0$.
If initially $z\notin K$ then the algorithm continues indefinitely. Once the
partial quotient $\alpha_{n}$’s have been computed, the partial convergents
are computed using the usual formula from the real case:

$p_{-2}=0,\quad p_{-1}=1,\quad p_{n}=\alpha_{n}p_{n-1}+p_{n-2}$ (3.2)
$q_{-2}=1,\quad q_{-1}=0,\quad q_{n}=\alpha_{n}q_{n-1}+q_{n-2}.$ (3.3)

These numbers satisfy the equation

$p_{n+1}q_{n}-p_{n}q_{n+1}=(-1)^{n},$

and $\displaystyle\lim_{n\to\infty}\frac{p_{n}}{q_{n}}=z$.

Similar to the real case, let us also define the numbers $\delta_{n}$
associated to continued fraction expansion of $z$. The numbers $\delta_{n}$
($n\geq-1$) are defined inductively by

$\delta_{-1}=z,\quad\delta_{0}=1,\quad\delta_{n+1}=\delta_{n-1}-\alpha_{n}\delta_{n},$
(3.4)

and satisfy

$\frac{\delta_{n}}{\delta_{n-1}}=\frac{1}{z_{n}}=z_{n-1}-\lfloor
z_{n-1}\rceil.$ (3.5)

If $z\in K$, then $z=\displaystyle\frac{p_{n}}{q_{n}}$ for some $n$ and the
recurrence stops with $\delta_{n+1}=0$. If $z\notin K$, then $\delta_{n}$’s
decay to zero exponentially quickly in $n$.

Finally, let us define the set of matrices associated to the continued
fraction of $z$

$\displaystyle\hat{\Gamma}_{d}(z):=\left\\{\gamma_{n}\in\hat{\Gamma}_{d}\,:\,\gamma_{n}=\left(\begin{array}[]{cc}q_{n-2}&-p_{n-2}\\\
-q_{n-1}&p_{n-1}\end{array}\right)\right\\}$

where $\hat{\Gamma}_{d}=\textbf{PGL}(2,{\mathcal{O}}_{d})$.

###  3.3. Continuity of $H_{1,\Delta}$

For a fixed $z\in\mathbb{C}$ denote the set of functions appearing in the sum
for $H_{1,\Delta}(z)$ by

$\Omega_{\Delta}(z)=\left\\{h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)\ \big{|}\
a_{h}<0<h(z,1)\right\\}.$

Let $\mathcal{A}$ be an equivalence class in
$\mathcal{H}(\mathcal{O}_{d},-\Delta)/\hat{\Gamma}_{d}$, and define

$\displaystyle\mathcal{A}(z)$ $\displaystyle:=\\{h\in\mathcal{A}\ \big{|}\
a_{h}<0<h(z,1)\\},$ (3.6) $\displaystyle\hat{\mathcal{A}}$
$\displaystyle:=\\{h\in\mathcal{A}\ \big{|}\ h(0,1)<0<a_{h}\\}.$ (3.7)

Then, the sum defining $H_{1,\Delta}$ can be written as

$H_{1,\Delta}(z)=\sum_{\mathcal{A}\in\mathcal{H}(\mathcal{O}_{d},-\Delta)/\hat{\Gamma}_{d}}\left(\sum_{h\in\mathcal{A}(z)}h(z,1)\right).$
(3.8)

The next proposition gives a description of $\mathcal{A}(z)$, which is enough
to prove the continuity of $H_{1,\Delta}$.

######  Proposition 3.2.

Let $z\in\mathbb{C}$ be fixed, $\mathcal{A}$ be an equivalence class in
$\mathcal{H}(\mathcal{O}_{d},-\Delta)$ and

$\displaystyle\mathcal{B}(z)=\left\\{(f,\gamma)\in\hat{\mathcal{A}}\times\hat{\Gamma}_{d}(z)\
|\ f(\gamma(\infty),1)<0<f(\gamma(z),1)\right\\},$

where for $\gamma=\begin{pmatrix}r&s\\\ t&v\end{pmatrix}$ we let

$\displaystyle\gamma(z):=\frac{rz+s}{tz+v}\quad\text{and}\quad\gamma(\infty):=\lim_{z\to\infty}\gamma{(z)}=\frac{r}{t}.$

Then, the map $\Phi:\mathcal{B}(z)\rightarrow\mathcal{A}(z)$ defined by
$\Phi(f,\gamma)=\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern
1.5mu(f)$ is a surjection.

###### Proof.

First we check that the map is well defined. Let

$f=\begin{pmatrix}a&b\\\ \bar{b}&c\end{pmatrix}\in\hat{\mathcal{A}}{\ \
\text{and}\ \ }\gamma=\begin{pmatrix}r&s\\\
t&v\end{pmatrix}\in\hat{\Gamma}_{d}(z).$

Then

$\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern
1.5mu(f)=\gamma^{t}f\mkern
1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern
1.5mu=\begin{pmatrix}r&t\\\ s&v\end{pmatrix}\begin{pmatrix}a&b\\\
\bar{b}&c\end{pmatrix}\begin{pmatrix}\bar{r}&\bar{s}\\\
\bar{t}&\bar{v}\end{pmatrix},$

and we see that

$a_{\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern
1.5mu(f)}=a|r|^{2}+br\bar{t}+\bar{b}\bar{r}t+c|t|^{2},$

which is equal to

$|t|^{2}f\left(\frac{r}{t},1\right)=|t|^{2}f(\gamma(\infty),1).$

But by assumption $f(\gamma(\infty),1)<0$, thus $a_{\mkern
1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu(f)}<0.$ Also,
$f(\gamma(z),1)>0$ implies that $\mkern
1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu(f)(z,1)>0$ since
$\displaystyle\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern
1.5mu(f)(z,1)=|tz+v|^{2}f(\gamma(z),1)$. So, $\Phi$ is well defined.

To show that $\Phi$ is surjective; let $h\in\mathcal{A}(z)$ and consider the
identity

$-\Delta(h)=N(\bar{b}_{h}+a_{h}Z)+|a_{h}|h(Z,1)$

which holds for any $Z\in\mathbb{C}$. Using this identity we see that the set
of zeros of $h(Z,1)$ satisfy the equation

$\Delta=N(\bar{b}_{h}+a_{h}Z)=a_{h}^{2}\left|Z+\frac{\bar{b}_{h}}{a_{h}}\right|^{2},$
(3.9)

which describes a circle in the complex plane centered at
$-\frac{\bar{b}_{h}}{a_{h}}$ with radius $\frac{\sqrt{\Delta}}{|a_{h}|}.$
Let $B$ be the open disk centered at $-\frac{\bar{b}_{h}}{a_{h}}$ with radius
$\frac{\sqrt{\Delta}}{|a_{h}|}$. From equation (3.9) it is clear that
$N(\bar{b}_{h}+a_{h}Z)<\Delta$ if $Z\in B$ and $N(\bar{b}_{h}+a_{h}Z)>\Delta$
if $Z\notin B$. Hence $h(Z,1)>0$ if $Z\in B$ and $h(Z,1)<0$ if $Z\notin B$ and
in particular $z\in B$, since $h\in\mathcal{A}(z)$ implies that $h(z,1)>0$.
Let $p_{n}$ and $q_{n}$ be the partial convergents of $z$ and because
$\displaystyle\frac{p_{n}}{q_{n}}\rightarrow z$ as $n\rightarrow\infty$, we
can find a positive integer $n$ large enough so that
$\displaystyle\frac{p_{n-1}}{q_{n-1}}\notin B$ but
$\displaystyle\frac{p_{n}}{q_{n}},\frac{p_{n+1}}{q_{n+1}},\frac{p_{n+2}}{q_{n+2}},\ldots\in
B$. If no such $n$ exists, we set $n=0$. In both cases we have

$h\left(\frac{p_{n-1}}{q_{n-1}},1\right)<0<h\left(\frac{p_{n}}{q_{n}},1\right)$

where we observe that $h(\infty,1)<0$ when $n=0$, since $a_{h}<0$. Let

$\gamma=\gamma_{n+1}=\left(\begin{array}[]{cc}q_{n-1}&-p_{n-1}\\\
-q_{n}&p_{n}\end{array}\right)\in\Gamma_{d}(z)\implies\gamma^{-1}=\left(\begin{array}[]{cc}p_{n}&p_{n-1}\\\
q_{n}&q_{n-1}\end{array}\right),$

and

$f=\overline{\gamma^{-1}}(h)=\begin{pmatrix}p_{n}&q_{n}\\\
p_{n-1}&q_{n-1}\end{pmatrix}\begin{pmatrix}a&b\\\
\bar{b}&c\end{pmatrix}\begin{pmatrix}\bar{p}_{n}&\bar{p}_{n-1}\\\
\bar{q}_{n}&\bar{q}_{n-1}\end{pmatrix}.$

Then, $\displaystyle
f(0,1)=|q_{n-1}|^{2}h\left(\frac{p_{n-1}}{q_{n-1}},1\right)<0$ and
$\displaystyle a_{f}=|q_{n}|^{2}h\left(\frac{p_{n}}{q_{n}},1\right)>0$. Thus
$f\in\hat{\mathcal{A}}$. Moreover $\mkern
1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu(f)=h$ which
completes the proof of surjectivity. ∎

######  Theorem 3.3.

$H_{1,\Delta}$ is a continuous function for all $z\in\mathbb{C}$.

###### Proof.

From the equation (3.8) we see that $H_{1,\Delta}(z)$ is the sum of the sums
$\displaystyle\sum_{h\in\mathcal{A}(z)}h(z,1)$ over all the
$\hat{\Gamma}_{d}$-equivalence classes in
$\mathcal{H}(\mathcal{O}_{d},-\Delta)$. By Proposition 3.2, we have that

$\sum_{h\in\mathcal{A}(z)}h(z,1)\leq\sum_{f\in\mathcal{A}(0)}\sum_{\begin{subarray}{c}\gamma\in\hat{\Gamma}_{d}(z)\\\
f(\gamma(\infty),1)<0\\\ f(\gamma(z),1)>0\end{subarray}}\mkern
1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu(f)(z,1).$ (3.10)

There are finitely many $f$ in $\mathcal{A}(0)$. So, the outer sum on the
right side of inequality (3.10) is finite. Furthermore, the summands in the
inner sum are of the form

$\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern
1.5mu_{n}f(z,1)=a_{f}|\bar{\delta}_{n-1}|^{2}+b_{f}\delta_{n-1}\bar{\delta}_{n}+\bar{b}_{f}\bar{\delta}_{n-1}\delta_{n}+c_{f}|\delta_{n}|^{2}$

for some $n\geq 1$. The sequence $\delta_{n}=|p_{n-1}-q_{n-1}z|$ stops if
$z\in K$ and decreases to 0 exponentially quickly in $n$ if $z\notin K$. So,
the summands of the double sum on the right side of the equation (3.10)
converge to $0$ exponentially quickly in $n$, which implies that
$H_{1,\Delta}(z)$ converges exponentially quickly in $n$ for any
$z\in\mathbb{C}$. Thus $H_{1,\Delta}(z)$ is a continuous function for all
$z\in\mathbb{C}$. ∎

##  4\. The average value of $H_{k,\Delta}$ and Cohen-Zagier type formulas

As we saw in Proposition 2.2 part (3), the function $H_{k,\Delta}$ is
${\mathcal{O}}_{d}$-invariant, and so it has a well-defined average value
which we denote by $\langle H_{k,\Delta}(z)\rangle_{av}$. We let $\mathcal{P}$
denote the fundamental parallelogram for the lattice ${\mathcal{O}}_{d}$ and
$|\mathcal{P}|$ denote the Euclidean area enclosed by $\mathcal{P}$. Since
$\left\\{1,\omega=\frac{d_{K}+\sqrt{d_{K}}}{2}\right\\}$ is a
$\mathbb{Z}$-basis of ${\mathcal{O}}_{d}$, we see that
$|\mathcal{P}|=\frac{\sqrt{|d_{K}|}}{2}$. We also define the following
subgroup of $\textbf{PSL}(2,{\mathcal{O}}_{d})$

$\tilde{\Gamma}_{d}:=\left\\{\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix}\
\bigg{|}\ \lambda\in{\mathcal{O}}_{d}\right\\}.$

Notice that the set of binary Hermitian forms corresponding to Hermitian
matrices in the orbit of $h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)$ under the
action of $\tilde{\Gamma}_{d}$ is given by

$\tilde{\Gamma}_{d}(h)=\left\\{h(z+\lambda,1)\ |\
\lambda\in{\mathcal{O}}_{d}\right\\}.$

Finally, we define the following zeta function. For $\Delta,n\in\mathbb{Z}$,
we set

$r(\Delta,n):=\\#\\{\beta\in{\mathcal{O}}_{d}/n{\mathcal{O}}_{d}\ |\
\beta\bar{\beta}+\Delta\equiv 0\ (\text{mod}\ n)\\}$

where we denote by $\\#S$ the cardinality of a set $S$ and define

$Z(\Delta,s):=\sum_{n=1}^{\infty}\frac{r(\Delta,n)}{n^{s+1}}.$

This zeta function is studied by Elstrodt, Grunewald and Mennicke [10, 11] in
connection with representations numbers of binary Hermitian forms with
coefficients in ${\mathcal{O}}_{d}$. They show that

$Z(\Delta,s)=\begin{cases}\zeta_{K}(s)L(\chi_{d_{K}},s+1)^{-1}&\text{if
}\Delta=0\\\
\theta(\Delta,s)\zeta_{\mathbb{Q}}(s)L(\chi_{d_{K}},s+1)^{-1}&\text{if
}\Delta\neq 0\\\ \end{cases}$ (4.1)

where $\zeta_{K}(s)$ denotes the usual zeta function of $K$ and
$\theta(\Delta,s)$ is a finite Euler product given by

$\theta(\Delta,s)=\prod_{p|d_{k}\Delta}R_{p}(\Delta,p^{-1-s})$ (4.2)

with

$R_{p}(\Delta,X)=\begin{cases}\frac{1-\left(\left(\frac{d_{K}}{p}\right)(pX)\right)^{t+1}}{1-\left(\frac{d_{K}}{p}\right)pX}&\text{for
}p\nmid d_{K},\ p^{t}\mathrel{\|}\Delta,\\\
1+\left(\frac{-|D_{0}|^{t}\Delta_{0}}{p}\right)(pX)^{t+1}&\text{for }p\mid
d_{K},\ p\neq 2,\ p^{t}\mathrel{\|}\Delta,\\\
1+\left(\frac{8}{\Delta_{0}D^{t}_{2}}\right)(2X)^{t+3}&\text{for }p=2,\ 4\mid
d_{K},\ D_{1}\equiv 2(8),\ 2^{t}\mathrel{\|}\Delta,\\\
1-\left(\frac{-8}{\Delta_{0}D^{t}_{2}}\right)(2X)^{t+3}&\text{for }p=2,\ 4\mid
d_{K},\ D_{1}\equiv 6(8),\ 2^{t}\mathrel{\|}\Delta,\\\
1-\left(\frac{-4}{\Delta_{0}D^{t}_{2}}\right)(2X)^{t+2}&\text{for }p=2,\ 4\mid
d_{K},\ D_{1}\equiv 3\ \text{or}\ 7(8),\ 2^{t}\mathrel{\|}\Delta,\\\
\end{cases}$

where $D_{0}:=d_{K}/p$, $\Delta_{0}:=p^{-t}\Delta$, and where for $d_{K}\equiv
0\pmod{4}$

$D_{1}:=\frac{d_{K}}{4},\quad\quad
D_{2}:=\begin{cases}-\frac{D_{1}}{2}&\text{if }D_{1}\equiv 2\pmod{4},\\\
\frac{1-D_{1}}{2}&\text{if }D_{1}\equiv 3\pmod{4}.\end{cases}$

######  Theorem 4.1.

Let $k$ and $\Delta$ be as in Definition 1. Then, the average value of
$H_{k,\Delta}$ is given by

$\langle
H_{k,\Delta}(z)\rangle_{av}=\frac{2\pi\Delta^{k+1}}{(k+1)\sqrt{|d_{K}|}}\,Z(-\Delta,k+1).$
(4.3)

###### Proof.

We have

$\langle
H_{k,\Delta}(z)\rangle_{av}=\frac{1}{|\mathcal{P}|}\int_{\mathcal{P}}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)\\\
a_{h}<0\end{subarray}}\max(0,h(z,1)^{k})\,dx\,dy$

Since $H_{k,\Delta}$ is continuous, it converges uniformly on $\mathcal{P}$.
Thus, we can interchange the integral and the sum to get

$\displaystyle\langle H_{k,\Delta}(z)\rangle_{av}$
$\displaystyle=\frac{1}{|\mathcal{P}|}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)\\\
a_{h}<0\end{subarray}}\int_{\mathcal{P}}\max(0,h(z,1)^{k})\,dx\,dy$
$\displaystyle=\frac{1}{|\mathcal{P}|}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)/\tilde{\Gamma}_{d}\\\
a_{h}<0\end{subarray}}\left(\sum_{\lambda\in{\mathcal{O}}_{d}}\int_{\mathcal{P}}\max(0,h(z+\lambda,1)^{k})\right)\,dx\,dy$
$\displaystyle=\frac{1}{|\mathcal{P}|}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)/\tilde{\Gamma}_{d}\\\
a_{h}<0\end{subarray}}\int_{\mathbb{C}}\max(0,h(z,1)^{k})\,dx\,dy$
$\displaystyle=\frac{1}{|\mathcal{P}|}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)/\tilde{\Gamma}_{d}\\\
a_{h}<0\end{subarray}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\max(0,h(z,1)^{k})\,dx\,dy.$

Making the substitution $z=\frac{-\bar{b}_{h}+w\sqrt{\Delta}}{a_{h}}$ where
$w=u+iv\in\mathbb{C}$ produces

$\displaystyle\langle H_{k,\Delta}(z)\rangle_{av}$
$\displaystyle=\frac{1}{|\mathcal{P}|}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)/\tilde{\Gamma}_{d}\\\
a_{h}<0\end{subarray}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\max\left(0,\left(\frac{(w\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu-1)\Delta}{a_{h}}\right)^{k}\right)\frac{\Delta}{a_{h}^{2}}\,du\,dv$
$\displaystyle=\frac{\Delta^{k+1}}{|\mathcal{P}|}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)/\tilde{\Gamma}_{d}\\\
a_{h}<0\end{subarray}}\frac{1}{|a_{h}|^{k+2}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\max\left(0,(1-w\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu)^{k}\right)\,du\,dv$
$\displaystyle=\frac{\Delta^{k+1}}{|\mathcal{P}|}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)/\tilde{\Gamma}_{d}\\\
a_{h}<0\end{subarray}}\frac{1}{|a_{h}|^{k+2}}\iint\limits_{R}(1-u^{2}-v^{2})^{k}\,du\,dv$

where $R$ is the unit circle centered at the origin. The double integral can
be evaluated using polar coordinates to get

$\iint\limits_{R}(1-u^{2}-v^{2})^{k}\,du\,dv=\int_{0}^{2\pi}\int_{0}^{1}(1-r^{2})^{k}r\,dr\,d\theta=\frac{\pi}{k+1}.$

So, we obtain

$\displaystyle\langle H_{k,\Delta}(z)\rangle_{av}$
$\displaystyle=\frac{\Delta^{k+1}}{|\mathcal{P}|}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)/\tilde{\Gamma}_{d}\\\
a_{h}<0\end{subarray}}\frac{1}{|a_{h}|^{k+2}}\frac{\pi}{k+1}$
$\displaystyle=\frac{2\pi\Delta^{k+1}}{(k+1)\sqrt{|d_{K}|}}\sum_{\begin{subarray}{c}h\in\mathcal{H}(\mathcal{O}_{d},-\Delta)/\tilde{\Gamma}_{d}\\\
a_{h}<0\end{subarray}}\frac{1}{|a_{h}|^{k+2}}$

Notice that, the action of $\tilde{\Gamma}_{d}$ on $h=\begin{pmatrix}a&b\\\
\bar{b}&c\end{pmatrix}$ corresponds to simply shifting $b$ and $\bar{b}$ by
multiples of $a$ in ${\mathcal{O}}_{d}$ since

$\gamma(h)=\begin{pmatrix}a&a\lambda+b\\\ a\mkern
1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern
1.5mu+\bar{b}&a\lambda\mkern
1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu+b\mkern
1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern
1.5mu+\bar{b}\lambda+c\end{pmatrix}$

where $\gamma=\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix}.$ Since $a_{h}$ is
fixed under the action of $\tilde{\Gamma}_{d}$, the number of equivalence
classes in $\mathcal{H}(\mathcal{O}_{d},-\Delta)/\tilde{\Gamma}_{d}$ with
$a_{h}=a$ is precisely

$\\#\\{b\in{\mathcal{O}}_{d}/a{\mathcal{O}}_{d}\ |\ b\bar{b}\equiv\Delta\
(\text{mod}\ a)\\}=r(-\Delta,a).$

Thus, we obtain

$\displaystyle\langle H_{k,\Delta}(z)\rangle_{av}$
$\displaystyle=\frac{2\pi\Delta^{k+1}}{(k+1)\sqrt{|d_{K}|}}\sum_{a=1}^{\infty}\frac{r(-\Delta,a)}{a^{k+2}}.$

∎

As an application of Theorem 4.1 we get the following explicit formulas for
the special values.

######  Corollary 4.2.

  1. (1)

Let $d=1,2,3,7,11$ and $d_{K}$ be the discriminant of
$K=\mathbb{Q}(\sqrt{-d})$. Then,

$\displaystyle L(\chi_{d_{K}},3)=$
$\displaystyle\frac{\pi^{3}\Delta^{2}}{6\alpha_{2,\Delta}\sqrt{|d_{K}|}}\
\theta(\Delta,2),$ (4.4) $\displaystyle L(\chi_{d_{K}},-2)=$
$\displaystyle\frac{-|d_{K}|^{2}\Delta^{2}}{12\alpha_{2,\Delta}}\
\theta(\Delta,2).$ (4.5)

  2. (2)

Let $d=1,3,7$ and $d_{K}$ be the discriminant of $K=\mathbb{Q}(\sqrt{-d})$.
Then,

$\displaystyle L(\chi_{d_{K}},5)=$
$\displaystyle\frac{\pi^{5}\Delta^{4}}{180\alpha_{4,\Delta}\sqrt{|d_{K}|}}\
\theta(\Delta,4),$ (4.6) $\displaystyle L(\chi_{d_{K}},-4)=$
$\displaystyle\frac{|d_{K}|^{4}\Delta^{4}}{120\alpha_{4,\Delta}}\
\theta(\Delta,4).$ (4.7)

  3. (3)

Let $d=3$ and $d_{K}$ be the discriminant of $K=\mathbb{Q}(\sqrt{-d})$. Then,

$\displaystyle L(\chi_{d_{K}},7)=$
$\displaystyle\frac{\pi^{7}\Delta^{6}}{2835\alpha_{6,\Delta}\sqrt{3}}\
\theta(\Delta,6),$ (4.8) $\displaystyle L(\chi_{d_{K}},-6)=$
$\displaystyle\frac{-81\Delta^{6}}{28\alpha_{6,\Delta}}\ \theta(\Delta,6).$
(4.9)

###### Proof.

Since $H_{k,\Delta}$ is constant for $k=1$ and since
$\zeta_{\mathbb{Q}}(2)=\frac{\pi^{2}}{6}$, we immediately get the formula in
the equation (4.4) using Theorem 4.1 and the equation (4.1). To get the
formula in the equation (4.5) we use the functional equation

$L(\chi_{d_{K}},1-s)=2|d_{K}|^{s-1/2}(2\pi)^{-s}\Gamma(s)\sin\left(\frac{s\pi}{2}\right)L(\chi_{d_{K}},s).$

The other formulas are obtained similarly using
$\zeta_{\mathbb{Q}}(4)=\frac{\pi^{4}}{90}$ and
$\zeta_{\mathbb{Q}}(6)=\frac{\pi^{6}}{945}$.

∎

##  5\. Cocycle property of $H_{k,\Delta}$

Another important property of $H_{k,\Delta}$ is that it can be used to
construct a nontrivial cocycle belonging to cohomology group
$H^{1}(\Gamma_{d},V_{k,k}(\mathbb{C}))$. The groups
$\Gamma=\textbf{PSL}(2,\mathbb{{\mathcal{O}}})$ where ${\mathcal{O}}$ is the
ring of integers of any imaginary quadratic field are called Bianchi groups
and their cohomology groups are fundamental to the study of Bianchi modular
forms (see [24]).

Let us now define $H^{1}(\Gamma_{d},V_{n,n})$. A map $f:\Gamma_{d}\to V_{n,n}$
is called a $1$-cocycle if it satisfies

$f(\gamma_{1}\gamma_{2})=f(\gamma_{1})|\gamma_{2}+f(\gamma_{2})$ (5.1)

for all $\gamma_{1},\gamma_{2}\in\Gamma_{d}$ and where $|\gamma_{2}$ denotes
the action of $\gamma_{2}$ on $f(\gamma_{1})$ as defined in equation (2.2).
For a fixed $P\in V_{n,n}$ the map

$\gamma\mapsto P|\gamma-P$ (5.2)

is a 1-cocycle and is called a 1-coboundary corresponding to $P$. The set of
1-cocycles denoted by $C(\Gamma_{d},V_{n,n})$ and the set of 1-coboundaries
denoted by $B(\Gamma_{d},V_{n,n})$ are $\mathbb{C}$-vector spaces. We have
that $B(\Gamma_{d},V_{n,n})\subset C(\Gamma_{d},V_{n,n})$, and the (first)
cohomology group $H^{1}(\Gamma_{d},V_{n,n})$ of $\Gamma_{d}$ with coefficients
in $V_{n,n}$ is defined to be

$H^{1}(\Gamma_{d},V_{n,n})=\frac{C(\Gamma_{d},V_{n,n})}{B(\Gamma_{d},V_{n,n})}.$
(5.3)

In general, if $G$ is a finitely presented group and $M$ is an $RG$-module for
a commutative ring $R$ then $H^{1}(G,M)$ can be computed using the
presentation of the group. An illustration of how this is carried out is given
in [23] through an example (see also [14]). Since Bianchi groups are finitely
presented this method is used in [23, 14] to study the structure of
$H^{1}(\Gamma,V_{n,n})$. Using the same method we will define a certain space
of cocycles in $H^{1}(\Gamma_{d},V_{k,k})$ that contains the cocycle
constructed using $H_{k,\Delta}$.

Recall that our function $H_{k,\Delta}$ is defined when ${\mathcal{O}}_{d}$ is
the ring of integers of an Euclidean imaginary quadratic field, i.e when
$d=1,2,3,7,11$. So, we fix $\omega=i$ in ${\mathcal{O}}_{1}$,
$\omega=i\sqrt{2}$ in ${\mathcal{O}}_{2}$, $\omega=\frac{-1+i\sqrt{3}}{2}$ in
${\mathcal{O}}_{3}$, $\omega=\frac{1+i\sqrt{7}}{2}$ in ${\mathcal{O}}_{7}$,
$\omega=\frac{1+i\sqrt{11}}{2}$ in ${\mathcal{O}}_{11}$. We also let

$\displaystyle S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},\hskip
7.22743ptT=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\hskip
7.22743ptT_{\omega}=\begin{pmatrix}1&\omega\\\ 0&1\end{pmatrix}.$

###  5.1.  The case $d=1$

The following presentation of $\Gamma_{1}$ is given in [12]

$\displaystyle\Gamma_{1}=\langle\,S,T,T_{\omega},L\,\big{|}\,S^{2}=L^{2}=(SL)^{2}=(TL)^{2}=(T_{\omega}L)^{2}=(ST)^{3}$
$\displaystyle=(T_{\omega}SL)^{3}=[T,T_{\omega}]={\bf 1}\,\rangle$

where $L=\begin{pmatrix}i&0\\\ 0&-i\end{pmatrix}$, and $[T,T_{\omega}]$ is the
commutator of $T$ and $T_{\omega}$.

We are interested in the cocycles belonging to the subspace
$C_{p}(\Gamma_{1},V_{k,k})$ of $C(\Gamma_{1},V_{k,k})$ defined as

$\displaystyle C_{p}(\Gamma_{1},V_{k,k})=\\{f\in
C(\Gamma_{1},V_{k,k})\,|\,f(T)=f(T_{\omega})=f(L)=0\\}.$

The analogous space of cocycles in the classical case of
$\textbf{PSL}(2,\mathbb{Z})$ is called the space of parabolic 1-cocycles and
hence the subscript $p$ in the notation $C_{p}(\Gamma_{1},V_{k,k}).$ The next
proposition shows that $C_{p}(\Gamma_{1},V_{k,k})$ can be identified with the
following subspace $W_{k,k}$ of $V_{k,k}$:

$\displaystyle W_{k,k}$ $\displaystyle:=\\{P\in V_{k,k}\,:\,P|({\bf
1}+S)=P|({\bf 1}-L)=P|({\bf 1}+U+U^{2})=P|({\bf 1}+E+E^{2})=0\\}$
$\displaystyle=ker({\bf 1}+S)\cap ker({\bf 1}-L)\cap ker({\bf 1}+U+U^{2})\cap
ker({\bf 1}+E+E^{2})$

where $U=TS$ and $E=T_{\omega}SL$.

######  Proposition 5.1.

The map sending $f\in C_{p}(\Gamma_{1},V_{k,k})$ to $P_{S}\in V_{k,k}$ is a
$\mathbb{C}$-vector space isomorphism from $C_{p}(\Gamma_{1},V_{k,k})$ to
$W_{k,k}$.

###### Proof.

The proof is similar to that of the classical case which can be found in [6,
Lemma 11.8.9]). Let $f\in C_{p}(\Gamma_{1},V_{k,k})$ such that
$f(\gamma)=P_{\gamma}\in V_{k,k}$ and assume that $f\mapsto f(S)=P_{S}$. Since
$S^{2}={\bf 1}$, the cocycle property of $f$ gives

$\displaystyle 0=f(S^{2})=f(S)|S+f(S)\implies P_{S}|S+P_{S}=P_{S}|({\bf
1}+S)=0.$

Again using the cocycle property of $f$ and the fact that $f(L)=0$ leads to

$\displaystyle f(SL)$ $\displaystyle=f(S)|L+f(L)=P_{S}|L$ $\displaystyle
f(LS)$ $\displaystyle=f(L)|S+f(S)=P_{S}$

But $SL=LS$ and hence $P_{S}|({\bf 1}-L)=0$.

The relation $(ST)^{3}={\bf 1}$ gives

$\displaystyle f(ST)|(ST)^{2}+f(ST)|(ST)+f(ST)=0,$

and since $f(ST)=f(S)|T+f(T)=P_{S}|T$ we obtain

$\displaystyle(P_{S}|T)|(ST)^{2}+(P_{S}|T)|ST+P_{S}|T$ $\displaystyle=0$
$\displaystyle P_{S}|TSTST+P_{S}|TST+P_{S}|T$ $\displaystyle=0$
$\displaystyle(P_{S}|U^{2}+P_{S}|U+P_{S})|T$ $\displaystyle=0.$

This shows that $P_{S}|({\bf 1}+U+U^{2})\in ker(T)$ implying that $P_{S}|({\bf
1}+U+U^{2})$ is the zero polynomial.

Similarly, $P_{S}|({\bf 1}+E+E^{2})=0$ follows from the fact that
$f(T_{\omega}SL)=P_{S}|L$ and $P_{S}|L=P_{S}$ which proves that $P_{S}\in
W_{k,k}$.

If $P_{S}=0$, then $f$ vanishes on all the generators of $\Gamma_{1}$ and the
cocycle property of $f$ implies that $f(\gamma)=P_{\gamma}=0$ for all
$\gamma\in\Gamma_{1}$. Thus the map $f\mapsto f(S)=P_{S}$ is injective.

Finally, we need to show that the map $f\mapsto f(S)=P_{S}$ is surjective. Let
$P\in W_{k,k}$ and set $f(S)=P|S$ for $f\in C_{p}(\Gamma_{1},V_{k,k})$. Then,
it is easily seen that $f$ agrees with the defining relations of $\Gamma_{1}$,
i.e., $f(S^{2})=0$ and the same holds for the other defining relations. The
surjectivity then follows from the fact that

$\displaystyle
S^{2}=L^{2}=(SL)^{2}=(TL)^{2}=(T_{\omega}L)^{2}=(ST)^{3}=(T_{\omega}SL)^{3}=[T,T_{\omega}]={\bf
1}\ $

generates all relations for $\Gamma_{1}$. ∎

Let us now discuss how the function $H_{k,\Delta}$ can be used to obtain a
nontrivial element of $C_{p}(\Gamma_{1},V_{k,k})$. Recall in Proposition 2.3
we showed that

$\displaystyle H_{k,\Delta}|({\bf
1}-S)=P_{k,\Delta}(z,\bar{z})=\sum_{\begin{subarray}{c}N(b)-ac=\Delta\\\
c<0<a\end{subarray}}\left(az\bar{z}+bz+\bar{b}\bar{z}+c\right)^{k}$

where $b\in{\mathcal{O}}_{1}$ and $a,c\in\mathbb{Z}.$ In what follows we will
show, that the polynomial $P_{k,\Delta}$ belongs to $W_{k,k}$ so that the map
$f:\Gamma_{1}\to V_{k,k}$ defined by

$\displaystyle\gamma\mapsto(H_{k,\Delta}|({\bf 1}-S))|\gamma$

is a cocycle belonging to $C_{p}(\Gamma_{1},V_{k,k})$. Indeed, we will
precisely describe the subspace of $W_{k,k}$ that $P_{k,\Delta}$ belongs to.
We will also carry out the same kind of analysis for the other values of $d$
as well, although the description of $W_{k,k}$ and its subspace vary depending
on the value of $d$.

The element $\varepsilon_{1}=\begin{pmatrix}i&0\\\
0&1\end{pmatrix}\in\textbf{PGL}(2,{\mathcal{O}}_{1})$ acts on $V_{k,k}$ by
$P|\varepsilon_{1}=P(iz,\overline{i}\bar{z})$ and splits $V_{k,k}$ into a
direct sum of the spaces defined as follows: The linear operator induced by
the action of $\varepsilon_{1}$ on $V_{k,k}$ has four eigenvalues, namely $\pm
1,\pm i$ and four eigenspaces corresponding to these eigenvalues. We let
$V_{k,k}^{1},V_{k,k}^{-1},V_{k,k}^{i},V_{k,k}^{-i}$ be the eigenspaces
corresponding to the eigenvalues $\pm 1,\pm i$ respectively. Then

$\displaystyle V_{k,k}^{1}=$ $\displaystyle\\{P\in
V_{k,k}\,|\,P(iz,\overline{i}\bar{z})=P(z,\bar{z})\\}$ $\displaystyle
V_{k,k}^{-1}=$ $\displaystyle\\{P\in
V_{k,k}\,|\,P(iz,\overline{i}\bar{z})=-P(z,\bar{z})\\}$ $\displaystyle
V_{k,k}^{i}=$ $\displaystyle\\{P\in
V_{k,k}\,|\,P(iz,\overline{i}\bar{z})=iP(z,\bar{z})\\}$ $\displaystyle
V_{k,k}^{-i}=$ $\displaystyle\\{P\in
V_{k,k}\,|\,P(iz,\overline{i}\bar{z})=-iP(z,\bar{z})\\}$

and therefore $V_{k,k}=V_{k,k}^{1}\oplus V_{k,k}^{-1}\oplus V_{k,k}^{i}\oplus
V_{k,k}^{-i}$.

The next proposition shows that $W_{k,k}$ is stable under the action of
$\varepsilon_{1}$.

######  Proposition 5.2.

$W_{k,k}|\varepsilon_{1}=W_{k,k}$ and hence

$\displaystyle W_{k,k}=W_{k,k}^{1}\oplus W_{k,k}^{-1}\oplus W_{k,k}^{i}\oplus
W_{k,k}^{-i}$

where $W_{k,k}^{e}=W_{k,k}\cap V_{k,k}^{e}$ for $e=\pm 1,\pm i.$

###### Proof.

Let $P\in W_{k,k}$. From $LS\varepsilon_{1}=\varepsilon_{1}S$ we get that

$\displaystyle(P|\varepsilon_{1})|({\bf 1}+S)$
$\displaystyle=P|\varepsilon_{1}+P|LS\varepsilon_{1}$
$\displaystyle=(P+P|LS)|\varepsilon_{1}.$

But, $P|L=P$ which gives

$\displaystyle(P|\varepsilon_{1})|({\bf
1}+S)=(P+P|S)|\varepsilon_{1}=0\implies P|\varepsilon_{1}\in ker({\bf 1}+S).$

Because $\varepsilon_{1}L=L\varepsilon_{1}$ we immediately get that
$P|\varepsilon_{1}\in ker({\bf 1}-L)$. From
$\varepsilon_{1}U\varepsilon_{1}^{3}=SE^{2}S$,
$\varepsilon_{1}U^{2}\varepsilon_{1}^{3}=SES$ and $P|S=-P$ we get that

$\displaystyle(P|\varepsilon_{1})|({\bf 1}+U+U^{2})$
$\displaystyle=P|\varepsilon_{1}+P|\varepsilon_{1}U+P|\varepsilon U^{2}$
$\displaystyle=P|\varepsilon_{1}+P|SE^{2}S\varepsilon_{1}+P|SES\varepsilon_{1}$
$\displaystyle=(P+P|SE^{2}S+P|SES)|\varepsilon_{1}$
$\displaystyle=(-P|S-P|E^{2}S-P|ES)|\varepsilon_{1}$
$\displaystyle=(-P|-P|E^{2}-P|E)|S\varepsilon_{1}$ $\displaystyle=0$

and hence $P|\varepsilon_{1}\in ker({\bf 1}+U+U^{2})$.

Finally, because $\varepsilon_{1}E\varepsilon_{1}^{3}=SU^{2}S$ and
$\varepsilon_{1}E^{2}\varepsilon_{1}^{3}=SUS$ the preceding argument shows
that $P|\varepsilon_{1}\in ker({\bf 1}+E+E^{2})$ which completes the proof of
$W_{k,k}|\varepsilon_{1}=W_{k,k}$ so that

$\displaystyle W_{k,k}=W_{k,k}^{1}\oplus W_{k,k}^{-1}\oplus W_{k,k}^{i}\oplus
W_{k,k}^{-i}.$

∎

######  Proposition 5.3.

The polynomial $P_{k,\Delta}$ belongs to the subspace $W_{k,k}^{1}$ and
therefore the map $f:\Gamma_{1}\to V_{k,k}$ defined by

$\displaystyle\gamma\mapsto(H_{k,\Delta}|({\bf 1}-S))|\gamma$

is a cocycle belonging to $C_{p}(\Gamma_{1},V_{k,k})$.

###### Proof.

Since $S^{2}={\bf 1}$ we immediately get

$\displaystyle P_{k,\Delta}|({\bf 1}+S)=(H_{k,\Delta}|({\bf 1}-S)|({\bf
1}+S)=0\implies P_{k,\Delta}\in ker({\bf 1}+S).$

Recall from Proposition 2.2 that $H_{k,\Delta}|L=H_{k,\Delta}$ and since
$SL=LS$ it follows that $P_{k,\Delta}\in ker({\bf 1}-L)$. Also from the same
proposition we have

$\displaystyle H_{k,\Delta}|T=H_{k,\Delta}\quad\text{and}\quad
H_{k,\Delta}|T_{\omega_{1}}=H_{k,\Delta}.$

Since $TS=U$ and $H_{k,\Delta}|T=H_{k,\Delta}$ notice that

$\displaystyle
P_{k,\Delta}=H_{k,\Delta}-H_{k,\Delta}|S=H_{k,\Delta}-(H_{k,\Delta}|T)|S=H_{k,\Delta}|({\bf
1}-U).$

But $U^{3}={\bf 1}$ and so

$\displaystyle(H_{k,\Delta}|({\bf 1}-U))|({\bf 1}+U+U^{2})=H_{k,\Delta}|({\bf
1}-U+U-U^{2}+U^{2}-U^{3})=0,$

which shows that $P_{k,\Delta}\in ker({\bf 1}+U+U^{2})$. Again, using
$H_{k,\Delta}|T_{\omega}=H_{k,\Delta}$, $H_{k,\Delta}|L=H_{k,\Delta}$ and
$LS=SL$ gives

$\displaystyle
P_{k,\Delta}=H_{k,\Delta}-H_{k,\Delta}|S=H_{k,\Delta}-(H_{k,\Delta}|T_{\omega}L)|S=H_{k,\Delta}|({\bf
1}-E).$

Since $E^{3}={\bf 1}$ the preceding argument shows that $P_{k,\Delta}\in
ker({\bf 1}+E+E^{2})$ implying that $P_{k,\Delta}\in W_{k,k}$. Finally, by
part (2) of Proposition 2.6 we have

$\displaystyle P_{k,\Delta}|\varepsilon_{1}=P_{k,\Delta},$

which concludes the proof that $P_{k,\Delta}\in W_{k,k}^{1}$. ∎

We wrote a computer program and made use of SageMath [21] to compute the
dimensions of $W_{k,k}$ and $W_{k,k}^{e}$ for $1\leq k\leq 31$. For those
values of $k$ we found that
$\textbf{dim}(W_{k,k}^{i})=\textbf{dim}(W_{k,k}^{-i})=0$. The data about the
dimensions of $W_{k,k}$, $W_{k,k}^{1}$, and $W_{k,k}^{-1}$ for $1\leq k\leq
31$ is as follows:

$k$ |  | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\textbf{dim}(W_{k,k}^{1})$ |  | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 7 | 7 | 8 | 8
$\textbf{dim}(W_{k,k}^{-1})$ |  | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 3
$\textbf{dim}(W_{k,k})$ |  | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 7 | 7 | 8 | 9 | 9 | 10 | 11

Since $\textbf{dim}(W_{k,k}^{i})=\textbf{dim}(W_{k,k}^{-i})=0$ for $1\leq
k\leq 31$, according to Proposition 5.2 we must have

$\displaystyle\textbf{dim}(W_{k,k}^{1})=\textbf{dim}(W_{k,k}^{1})+\textbf{dim}(W_{k,k}^{-1}).$

The data in each row of the above table was computed independent of this fact,
and thankfully our data agrees with it. Based on this numerical evidence we
are led to the following conjecture.

######  Conjecture 5.4.

For odd $k\geq 1$ we have that

$\displaystyle\textbf{dim}(W_{k,k}^{1})$
$\displaystyle=\left\lfloor\frac{k-1}{4}\right\rfloor+1,$
$\displaystyle\textbf{dim}(W_{k,k})$
$\displaystyle=\left\lfloor\frac{k-1}{3}\right\rfloor+1.$

###  5.2.  The case $d=3$

The following presentation of $\Gamma_{3}$ is given in [12]

$\displaystyle\Gamma_{3}=\langle\,S,T,T_{\omega},L\,\big{|}\,S^{2}=L^{3}=(SL)^{2}=(ST)^{3}=(T_{\omega}SL)^{3}=[T,T_{\omega}]={\bf
1},$ $\displaystyle
L^{-1}T_{\omega}L=T,\,\,L^{-1}TL=T^{-1}T_{\omega}^{-1}\,\rangle$

where $L=\begin{pmatrix}\omega^{2}&0\\\ 0&\omega\end{pmatrix}.$ The space of
cocycles vanishing on parabolic elements of $\Gamma_{3}$ is defined to be

$\displaystyle C_{p}(\Gamma_{3},V_{k,k})=\\{f\in
C(\Gamma_{3},V_{k,k})\,|\,f(T)=f(T_{\omega})=f(L)=0\\}$

which again can be identified with a subspace of $V_{k,k}$.

######  Proposition 5.5.

The map sending $f\in C_{p}(\Gamma_{3},V_{k,k})$ to $P_{S}\in V_{k,k}$ is a
$\mathbb{C}$-vector space isomorphism from $C_{p}(\Gamma_{3},V_{k,k})$ to
$W_{k,k}$ where

$\displaystyle W_{k,k}:=ker({\bf 1}+S)\cap ker({\bf 1}-L)\cap ker({\bf
1}+U+U^{2})\cap ker({\bf 1}+E+E^{2}),$

with $U=TS$ and $E=T_{\omega}SL$.

###### Proof.

Let $f\in C_{p}(\Gamma_{3},V_{k,k})$ such that $f(\gamma)=P_{\gamma}\in
V_{k,k}$ and assume that $f\mapsto f(S)=P_{S}$. The proof that

$\displaystyle P_{S}\in ker({\bf 1}+S),\hskip 7.22743ptP_{S}\in ker({\bf
1}+U+U^{2}),\hskip 7.22743ptP_{S}\in ker({\bf 1}+E+E^{2})$

proceeds as in the proof of Proposition 5.1. We also have

$\displaystyle(SL)^{2}={\bf 1}\implies P_{S}|LSL+P_{S}|L=0.$

But $SL=L^{-1}S$ and so

$\displaystyle P_{S}|LSL+P_{S}|L=0\implies P_{S}|S+P_{S}|L=0\implies
P_{S}|L=P_{S}$

which shows that $P_{S}\in ker({\bf 1}-L).$

Finally, we do not get any new equations from the remaining relations and the
rest of the proof proceeds as in the proof of Proposition 5.1. ∎

Let $\varepsilon_{3}=\begin{pmatrix}\zeta&0\\\
0&1\end{pmatrix}\in\textbf{PGL}(2,{\mathcal{O}}_{3})$ where
$\zeta=e^{\frac{\pi i}{3}}$. Then $\varepsilon_{3}$ acts on $V_{k,k}$ by
$P|\varepsilon_{3}=P(\varepsilon_{3}z,\overline{\varepsilon_{3}}\bar{z})$. The
linear operator induced by the action of $\varepsilon_{3}$ on $V_{k,k}$ has
six eigenvalues and these eigenvalues are given by the sixth roots of unity
$\zeta^{j}$ for $1\leq j\leq 6$. We let $V_{k,k}^{\zeta^{j}}$ be the
eigenspace corresponding to $\zeta^{j}$. Then

$\displaystyle V_{k,k}^{\zeta^{j}}=\\{P\in V_{k,k}\,|\,P(\zeta
z,\overline{\zeta}\bar{z})=\zeta^{j}P(z,\bar{z})\\}$

and $\displaystyle V_{k,k}=\bigoplus_{1\leq j\leq 6}V_{k,k}^{\zeta^{j}}$. The
next proposition is the analog of the Proposition 5.2 for case of $d=3$.

######  Proposition 5.6.

$W_{k,k}|\varepsilon_{3}=W_{k,k}$ and hence

$\displaystyle W_{k,k}=\bigoplus_{1\leq j\leq 6}W_{k,k}^{\zeta^{j}}$

where $W_{k,k}^{\zeta^{j}}=W_{k,k}\cap V_{k,k}^{\zeta^{j}}$ for $1\leq j\leq
6$.

###### Proof.

The proof is identical to that of Proposition 5.2 after observing the
following identities

$\displaystyle LS\varepsilon_{3}=\varepsilon_{3}S,\hskip
14.45377ptL\varepsilon_{3}=\varepsilon_{3}L,\hskip
14.45377pt\varepsilon_{3}U\varepsilon_{3}^{5}=SE^{2}S,$
$\displaystyle\varepsilon_{3}U^{2}\varepsilon_{3}^{5}=SES,\hskip
14.45377pt\varepsilon_{e}E\varepsilon_{3}^{5}=SU^{2}S,\hskip
14.45377pt\varepsilon_{3}E^{2}\varepsilon_{3}^{5}=SUS.$

∎

######  Proposition 5.7.

The polynomial $P_{k,\Delta}$ belongs to the subspace $W_{k,k}^{1}$ and
therefore the map $f:\Gamma_{3}\to V_{k,k}$ defined by

$\displaystyle\gamma\mapsto(H_{k,\Delta}|({\bf 1}-S))|\gamma$

is a cocycle belonging to $C_{p}(\Gamma_{3},V_{k,k})$.

###### Proof.

The proof is identical to the proof of Proposition 5.3. ∎

The computer calculations we performed using our program for $1\leq k\leq 31$
showed that $\textbf{dim}(W_{k,k})=\textbf{dim}(W_{k,k}^{1})$ and
$\textbf{dim}(W_{k,k}^{j})=0$ for $2\leq j\leq 6$. Below is a table that gives
the dimension of $W_{k,k}^{1}$ for $1\leq k\leq 31$

$k$ |  | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\textbf{dim}(W_{k,k}^{1})$ |  | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 6

Based on this numerical data we are led to the following conjecture.

######  Conjecture 5.8.

For odd $k\geq 1$ we have that
$\displaystyle\textbf{dim}(W_{k,k}^{1})=\left\lfloor\frac{k}{6}\right\rfloor+1.$

###  5.3.  The case $d=2$

The following presentation of $\Gamma_{2}$ is given in [12]

$\displaystyle\Gamma_{2}=\langle\,S,T,T_{\omega}\,\big{|}\,S^{2}=(ST)^{3}=(T_{\omega}^{-1}ST_{\omega}S)^{2}=[T,T_{\omega}]={\bf
1}\rangle.$

The space of cocycles vanishing on parabolic elements of $\Gamma_{2}$ is
defined to be

$\displaystyle C_{p}(\Gamma_{2},V_{k,k})=\\{f\in
C(\Gamma_{2},V_{k,k})\,|\,f(T)=f(T_{\omega})=0\\}$

which again can be identified with a subspace of $V_{k,k}$.

######  Proposition 5.9.

The map sending $f\in C_{p}(\Gamma_{2},V_{k,k})$ to $P_{S}\in V_{k,k}$ is a
$\mathbb{C}$-vector space isomorphism from $C_{p}(\Gamma_{2},V_{k,k})$ to
$W_{k,k}$ where

$\displaystyle W_{k,k}:=ker({\bf 1}+S)\cap ker({\bf 1}+U+U^{2})\cap ker({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S),$

with $U=TS$.

###### Proof.

Let $f\in C_{p}(\Gamma_{2},V_{k,k})$ such that $f(\gamma)=P_{\gamma}\in
V_{k,k}$ and assume that $f\mapsto f(S)=P_{S}$. The proof that

$\displaystyle P_{S}\in ker({\bf 1}+S),\hskip 7.22743ptP_{S}\in ker({\bf
1}+U+U^{2})$

proceeds as in the proof of Proposition 5.1. As for the proof of $P_{S}\in
ker({\bf 1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S)$ we observe
that

$\displaystyle
0=f((T_{\omega}^{-1}ST_{\omega}S)^{2})=P_{S}|T_{\omega}ST_{\omega}^{-1}ST_{\omega}S+P_{S}|T_{\omega}^{-1}ST_{\omega}S+P_{S}|T_{\omega}S+P_{S}$

and since $T_{\omega}ST_{\omega}^{-1}ST_{\omega}S=ST_{\omega}$ the result
follows. Finally, we do not get any new equations from the remaining relations
and the rest of the proof proceeds as in the proof of Proposition 5.1. ∎

Let $\varepsilon=\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\in\textbf{PGL}(2,{\mathcal{O}}_{K})$. Then $\varepsilon$ acts
on $V_{k,k}$ by $P|\varepsilon=P(-z,-\bar{z})$ and splits $V_{k,k}$ up into
the direct sum of the spaces $V_{k,k}^{1}$ and $V_{k,k}^{-1}$ where

$\displaystyle V_{k,k}^{1}=\\{P\in V_{k,k}\,|\,P(-z,-\bar{z})=P(z,\bar{z})\\}$
$\displaystyle V_{k,k}^{-1}=\\{P\in
V_{k,k}\,|\,P(-z,-\bar{z})=-P(z,\bar{z})\\}$

and $\displaystyle V_{k,k}=V_{k,k}^{1}\oplus V_{k,k}^{-1}$.

######  Proposition 5.10.

$W_{k,k}|\varepsilon=W_{k,k}$ and hence

$\displaystyle W_{k,k}=W_{k,k}^{1}\oplus W_{k,k}^{-1}$

where $W_{k,k}^{\pm 1}=W_{k,k}\cap V_{k,k}^{\pm 1}$.

###### Proof.

Let $P\in W_{k,k}$. Using the identities

$\displaystyle\varepsilon S=S\varepsilon,\quad\varepsilon
U=SU^{2}S\varepsilon$

we immediately see that $P|\varepsilon$ belongs to $ker({\bf 1}+S)$ and
$ker({\bf 1}+U+U^{2})$.

It remains to show that $P|\varepsilon\in ker({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S)$. Notice that
$\varepsilon T_{\omega}=T_{\omega}^{-1}\varepsilon$ and so

$\displaystyle(P|\varepsilon)|({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S)=(P|({\bf
1}+ST_{\omega}^{-1}+T_{\omega}^{-1}S+T_{\omega}ST_{\omega}^{-1}S))|\varepsilon.$

Now observe that

$\displaystyle 0=$ $\displaystyle P|({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S)$ $\displaystyle=$
$\displaystyle(P|({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S))|T_{\omega}^{-1}S$
$\displaystyle=$ $\displaystyle P|(T_{\omega}^{-1}S+{\bf
1}+T_{\omega}ST_{\omega}^{-1}S+T_{\omega}^{-1}ST_{\omega}ST_{\omega}^{-1}S),$

but $T_{\omega}^{-1}ST_{\omega}ST_{\omega}^{-1}S=ST_{\omega}^{-1}$ which shows
that

$\displaystyle P|({\bf
1}+ST_{\omega}^{-1}+T_{\omega}^{-1}S+T_{\omega}ST_{\omega}^{-1}S)=0.$

It follows that $P|\varepsilon\in ker({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S).$ ∎

######  Proposition 5.11.

The polynomial $P_{k,\Delta}$ belongs to the subspace $W_{k,k}^{1}$ and
therefore the map $f:\Gamma_{2}\to V_{k,k}$ defined by

$\displaystyle\gamma\mapsto(H_{k,\Delta}|({\bf 1}-S))|\gamma$

is a cocycle belonging to $C_{p}(\Gamma_{2},V_{k,k})$.

###### Proof.

The proof that $P_{k,\Delta}\in ker({\bf 1}+S)$ and $P_{k,\Delta}\in ker({\bf
1}+U+U^{2})$ is same as the one given in the proof of Proposition 5.3. Let us
now calculate $P_{k,\Delta}|({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S)$ which is equal to

$\displaystyle=$ $\displaystyle(H_{k,\Delta}|({\bf 1}-S))|({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S)$ $\displaystyle=$
$\displaystyle H_{k,\Delta}|({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S-S-
T_{\omega}-ST_{\omega}S-ST_{\omega}^{-1}ST_{\omega}S)$ $\displaystyle=$
$\displaystyle H_{k,\Delta}|({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S-S-
T_{\omega}-ST_{\omega}S-T_{\omega}^{-1}ST_{\omega}).$

Since $H_{k,\Delta}|T_{\omega}=H_{k,\Delta}$ and
$H_{k,\Delta}|T_{\omega}^{-1}=H_{k,\Delta}$ we see that

$\displaystyle H_{k,\Delta}|({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S-S-
T_{\omega}-ST_{\omega}S-T_{\omega}^{-1}ST_{\omega})=0$

which shows that $P_{k,\Delta}\in ker({\bf
1}+ST_{\omega}+T_{\omega}S+T_{\omega}^{-1}ST_{\omega}S)$. Finally, by part (2)
of Proposition 2.6 we have

$\displaystyle P_{k,\Delta}|\varepsilon=P_{k,\Delta}$

which concludes the proof that $P_{k,\Delta}\in W_{k,k}^{1}$. ∎

Our data again showed that for $1\leq k\leq 31$
$\textbf{dim}(W_{k,k})=\textbf{dim}(W_{k,k}^{1})$ and
$\textbf{dim}(W_{k,k}^{-1})=0$. The following table gives the dimension of
$W_{k,k}^{1}$ for $1\leq k\leq 31$.

$k$ |  | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\textbf{dim}(W_{k,k}^{1})$ |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16

Based on this numerical data we are led to the following conjecture.

######  Conjecture 5.12.

For odd $k\geq 1$ we have that
$\displaystyle\textbf{dim}(W_{k,k}^{1})=\frac{k+1}{2}.$

###  5.4.  The case $d=7$

The following presentation for $\Gamma_{7}$ is given in [12]

$\displaystyle\Gamma_{7}=\langle\,S,T,T_{\omega}\,\big{|}\,S^{2}=(ST)^{3}=(T_{\omega}^{-1}ST_{\omega}ST)^{2}=[T,T_{\omega}]={\bf
1}\rangle.$

The presentation for $\Gamma_{7}$ is identical to that of $\Gamma_{2}$ except
for a minor difference where the element $T$ appears in the relation
$(T_{\omega}^{-1}ST_{\omega}ST)^{2}$. Consequently, we can not use any of the
arguments from the $d=2$ case. Moreover, this minor difference leads to a
slightly more complicated description of the kernel of the linear map
associated to the relation $(T_{\omega}^{-1}ST_{\omega}ST)^{2}$. Nonetheless,
we can still define the space of parabolic cocycles on $\Gamma_{7}$ which is

$\displaystyle C_{p}(\Gamma_{7},V_{k,k})=\\{f\in
C(\Gamma_{7},V_{k,k})\,|\,f(T)=f(T_{\omega})=0\\}$

and show that it can be identified it with a subspace of $V_{k,k}$ as follows.

######  Proposition 5.13.

The map sending $f\in C_{p}(\Gamma_{7},V_{k,k})$ to $P_{S}\in V_{k,k}$ is a
$\mathbb{C}$-vector space isomorphism from $C_{p}(\Gamma_{7},V_{k,k})$ to
$W_{k,k}$ where

$\displaystyle W_{k,k}:=ker({\bf 1}+S)\cap ker({\bf 1}+U+U^{2})\cap
ker(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega}),$

with $U=TS$.

###### Proof.

Let $f\in C_{p}(\Gamma_{7},V_{k,k})$ such that $f(\gamma)=P_{\gamma}\in
V_{k,k}$ and assume that $f\mapsto f(S)=P_{S}$. The proof that

$\displaystyle P_{S}\in ker({\bf 1}+S),\hskip 7.22743ptP_{S}\in ker({\bf
1}+U+U^{2})$

proceeds as in the proof of Proposition 5.1. As for the proof of $P_{S}\in
ker(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$ we observe that

$\displaystyle
0=f((T_{\omega}^{-1}ST_{\omega}ST)^{2})=P_{S}|T_{\omega}STT_{\omega}^{-1}ST_{\omega}ST+P_{S}|TT_{\omega}^{-1}ST_{\omega}ST+P_{S}|T_{\omega}ST+P_{S}|T$

and since

$\displaystyle T_{\omega}STT_{\omega}^{-1}ST_{\omega}ST=ST_{\omega},\quad
TT_{\omega}^{-1}ST_{\omega}ST=ST_{\omega}^{-1}ST_{\omega}$

the result follows. Finally, we do not get any new equations from the
remaining relations and the rest of the proof proceeds as in the proof of
Proposition 5.1. ∎

$W_{k,k}$ is stable under the action of $\varepsilon=\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}$ and splits as in the case of $d=2$.

######  Proposition 5.14.

$W_{k,k}|\varepsilon=W_{k,k}$ and hence

$\displaystyle W_{k,k}=W_{k,k}^{1}\oplus W_{k,k}^{-1}$

where $W_{k,k}^{\pm 1}=W_{k,k}\cap V_{k,k}^{\pm 1}$.

###### Proof.

Let $P\in W_{k,k}$. The proof that $P|\varepsilon$ belongs to $ker({\bf 1}+S)$
and $ker({\bf 1}+U+U^{2})$ is same as the case of $d=2$ since the identities

$\displaystyle\varepsilon S=S\varepsilon,\quad\varepsilon
U=SU^{2}S\varepsilon$

still hold.

It remains to show that $P|\varepsilon\in
ker(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$. Notice that

$\displaystyle\varepsilon T=T^{-1}\varepsilon,\quad\varepsilon
T_{\omega}=T_{\omega}^{-1}\varepsilon,$

and so

$\displaystyle(P|\varepsilon)|(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})=(P|(T^{-1}+ST_{\omega}^{-1}+T_{\omega}^{-1}ST^{-1}+ST_{\omega}ST_{\omega}^{-1}))|\varepsilon.$

Now, since

$\displaystyle S^{2}={\bf 1},\quad
T_{\omega}^{-1}T^{-1}=T^{-1}T_{\omega}^{-1},\quad\text{and}\quad P|S=-P,$

we have

$\displaystyle 0=$
$\displaystyle(-P)|(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$
$\displaystyle=$
$\displaystyle((-P)|(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega}))|T_{\omega}^{-1}T^{-1}$
$\displaystyle=$
$\displaystyle(-P)|(T_{\omega}^{-1}+ST^{-1}+T_{\omega}ST_{\omega}^{-1}+ST_{\omega}^{-1}ST^{-1})$
$\displaystyle=$
$\displaystyle(P|S)|(T_{\omega}^{-1}+ST^{-1}+T_{\omega}ST_{\omega}^{-1}+ST_{\omega}^{-1}ST^{-1})$

which shows that

$\displaystyle
P|(T^{-1}+ST_{\omega}^{-1}+T_{\omega}^{-1}ST^{-1}+ST_{\omega}ST_{\omega}^{-1})=0.$

It follows that $P|\varepsilon\in
ker(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega}).$ ∎

######  Proposition 5.15.

The polynomial $P_{k,\Delta}$ belongs to the subspace $W_{k,k}^{1}$ and
therefore the map $f:\Gamma_{7}\to V_{k,k}$ defined by

$\displaystyle\gamma\mapsto(H_{k,\Delta}|({\bf 1}-S))|\gamma$

is a cocycle belonging to $C_{p}(\Gamma_{7},V_{k,k})$.

###### Proof.

The proof that $P_{k,\Delta}\in ker({\bf 1}+S)$ and $P_{k,\Delta}\in ker({\bf
1}+U+U^{2})$ is same as the one given in the proof of Proposition 5.3. Let us
now calculate
$P_{k,\Delta}|(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$ which
is equal to

$\displaystyle=$ $\displaystyle(H_{k,\Delta}|({\bf
1}-S))|(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$
$\displaystyle=$ $\displaystyle
H_{k,\Delta}|(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega}-ST-
T_{\omega}-ST_{\omega}ST-T_{\omega}^{-1}ST_{\omega})$ $\displaystyle=$
$\displaystyle
H_{k,\Delta}|(T+ST_{\omega}+T_{\omega}ST+TT_{\omega}^{-1}ST_{\omega}ST-ST-
T_{\omega}-ST_{\omega}ST-T_{\omega}^{-1}ST_{\omega}).$

Since $H_{k,\Delta}|T_{\omega}=H_{k,\Delta}$ and
$H_{k,\Delta}|T_{\omega}^{-1}=H_{k,\Delta}$ we see that

$\displaystyle
H_{k,\Delta}|(T+ST_{\omega}+T_{\omega}ST+TT_{\omega}^{-1}ST_{\omega}ST-ST-
T_{\omega}-ST_{\omega}ST-T_{\omega}^{-1}ST_{\omega})=0$

which shows that $P_{k,\Delta}\in
ker(T+ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$. Finally, by part
(2) of Proposition 2.6 we have

$\displaystyle P_{k,\Delta}|\varepsilon=P_{k,\Delta}$

which concludes the proof that $P_{k,\Delta}\in W_{k,k}^{1}$. ∎

Unlike the previous cases, we calculated the dimensions of $W_{k,k}^{1}$ for
$1\leq k\leq 27$ due to lack of computing power at our disposal. In fact, it
took over a week on my laptop for the computer program that I wrote to compute
the dimension of $W_{k,k}^{1}$ for $k=27$. Regardless, here is the table
containing our data for the case of $d=7$.

$k$ |  | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\textbf{dim}(W_{k,k}^{1})$ |  | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 7 | 7 | 8 | 9 | 9

We also observed the same behavior as the previous two cases with respect to
the dimensions of $\textbf{dim}(W_{k,k})$ and $\textbf{dim}(W_{k,k}^{-1})$
which is $\textbf{dim}(W_{k,k})=\textbf{dim}(W_{k,k}^{1})$ and
$\textbf{dim}(W_{k,k}^{-1})=0$ for $1\leq k\leq 19$. We again record the
following conjecture based on the computational evidence.

######  Conjecture 5.16.

For odd $k\geq 1$ we have that
$\displaystyle\textbf{dim}(W_{k,k}^{1})=\left\lfloor\frac{k-1}{3}\right\rfloor+1.$

###  5.5.  The case $d=11$

We use the following presentation of $\Gamma_{11}$ given in [12]

$\displaystyle\Gamma_{11}=\langle\,S,T,T_{\omega}\,\big{|}\,S^{2}=(ST)^{3}=(T_{\omega}^{-1}ST_{\omega}ST)^{3}=[T,T_{\omega}]={\bf
1}\rangle$

As usual, the space of parabolic cocycles on $\Gamma_{11}$ is defined to be

$\displaystyle C_{p}(\Gamma_{11},V_{k,k})=\\{f\in
C(\Gamma_{11},V_{k,k})\,|\,f(T)=f(T_{\omega})=0\\},$

which again can be identified it with a subspace of $V_{k,k}$ as follows.

######  Proposition 5.17.

The map sending $f\in C_{p}(\Gamma_{11},V_{k,k})$ to $P_{S}\in V_{k,k}$ is a
$\mathbb{C}$-vector space isomorphism from $C_{p}(\Gamma_{11},V_{k,k})$ to
$W_{k,k}$ where

$\displaystyle W_{k,k}:=ker({\bf 1}+S)\cap ker({\bf 1}+U+U^{2})\cap
ker(T+ST_{\omega}+TE+ST_{\omega}E^{-1}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega}),$

with $U=TS$ and $E=T_{\omega}^{-1}ST_{\omega}ST$.

###### Proof.

Let $f\in C_{p}(\Gamma_{11},V_{k,k})$ such that $f(\gamma)=P_{\gamma}\in
V_{k,k}$ and assume that $f\mapsto f(S)=P_{S}$. Again, the fact that

$\displaystyle P_{S}\in ker({\bf 1}+S),\hskip 7.22743ptP_{S}\in ker({\bf
1}+U+U^{2})$

follows as before. For the proof of $P_{S}\in
ker(T+ST_{\omega}+TE+ST_{\omega}E^{-1}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$,
we observe that

$\displaystyle
f(T_{\omega}^{-1}ST_{\omega}ST)=f(E)=P_{S}|T_{\omega}ST+P_{S}|T.$

The relation $(T_{\omega}^{-1}ST_{\omega}ST)^{3}=E^{3}={\bf 1}$ gives

$\displaystyle 0$ $\displaystyle=f((T_{\omega}^{-1}ST_{\omega}ST)^{3})$
$\displaystyle=f(E)|E^{2}+f(E)|E+F(E)$
$\displaystyle=(P_{S}|T_{\omega}ST+P_{S}|T)|E^{-1}+(P_{S}|T_{\omega}ST+P_{S}|T)|E+P_{S}|T_{\omega}ST+P_{S}|T\quad(E^{2}=E^{-1}),$

and since

$\displaystyle T_{\omega}STE^{-1}=ST_{\omega},\quad
TE^{-1}=ST_{\omega}^{-1}ST_{\omega}$

the result follows. Finally, we do not get any new equations from the
remaining relations and the rest of the proof proceeds as in the proof of
Proposition 5.1. ∎

$W_{k,k}$ is stable under the action of $\varepsilon=\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}$ and splits as in the case of $d=2,7$.

######  Proposition 5.18.

$W_{k,k}|\varepsilon=W_{k,k}$ and hence

$\displaystyle W_{k,k}=W_{k,k}^{1}\oplus W_{k,k}^{-1}$

where $W_{k,k}^{\pm 1}=W_{k,k}\cap V_{k,k}^{\pm 1}$.

###### Proof.

Let $P\in W_{k,k}$. We just need to show that $P|\varepsilon$ belongs to
$ker(T+ST_{\omega}+TE+ST_{\omega}E^{-1}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$
since the proof of the fact that $P|\varepsilon$ belongs to $ker({\bf 1}+S)$
and $ker({\bf 1}+U+U^{2})$ is same as before because the identities

$\displaystyle\varepsilon S=S\varepsilon,\quad\varepsilon
U=SU^{2}S\varepsilon$

still hold.

So, using

$\displaystyle\varepsilon T=T^{-1}\varepsilon,\quad\varepsilon
T_{\omega}=T_{\omega}^{-1}\varepsilon,$

along with $E=T_{\omega}^{-1}ST_{\omega}ST$ and
$E^{-1}=T^{-1}ST_{\omega}^{-1}ST_{\omega}$, to calculate
$(P|\varepsilon)|(T+ST_{\omega}+TE+ST_{\omega}E^{-1}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$
gives

$\displaystyle=(P|\varepsilon)|(T+ST_{\omega}+TT_{\omega}^{-1}ST_{\omega}ST+ST_{\omega}T^{-1}ST_{\omega}^{-1}ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$
$\displaystyle=(P|(T^{-1}+ST_{\omega}^{-1}+T^{-1}T_{\omega}ST_{\omega}^{-1}ST^{-1}+ST_{\omega}^{-1}TST_{\omega}ST_{\omega}^{-1}+T_{\omega}^{-1}ST^{-1}+ST_{\omega}ST_{\omega}^{-1}))|\varepsilon$

Now, since

$\displaystyle S^{2}={\bf 1},\quad
T_{\omega}^{-1}T^{-1}=T^{-1}T_{\omega}^{-1},\quad\text{and}\quad P|S=-P,$

we have

$\displaystyle 0=$
$\displaystyle(-P)|(T+ST_{\omega}+TT_{\omega}^{-1}ST_{\omega}ST+ST_{\omega}T^{-1}ST_{\omega}^{-1}ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$
$\displaystyle=$
$\displaystyle(-P)|(T+ST_{\omega}+TT_{\omega}^{-1}ST_{\omega}ST+ST_{\omega}T^{-1}ST_{\omega}^{-1}ST_{\omega}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})|T_{\omega}^{-1}T^{-1}$
$\displaystyle=$
$\displaystyle(-P)|(T_{\omega}^{-1}+ST^{-1}+TT_{\omega}^{-1}ST_{\omega}ST_{\omega}^{-1}+ST_{\omega}T^{-1}ST_{\omega}^{-1}ST^{-1}+T_{\omega}ST_{\omega}^{-1}+ST_{\omega}^{-1}ST^{-1})$
$\displaystyle=$
$\displaystyle(P|S)|(T_{\omega}^{-1}+ST^{-1}+TT_{\omega}^{-1}ST_{\omega}ST_{\omega}^{-1}+ST_{\omega}T^{-1}ST_{\omega}^{-1}ST^{-1}+T_{\omega}ST_{\omega}^{-1}+ST_{\omega}^{-1}ST^{-1})$
$\displaystyle=$ $\displaystyle
P|(ST_{\omega}^{-1}+T^{-1}+STT_{\omega}^{-1}ST_{\omega}ST_{\omega}^{-1}+T_{\omega}T^{-1}ST_{\omega}^{-1}ST^{-1}+ST_{\omega}ST_{\omega}^{-1}+T_{\omega}^{-1}ST^{-1}).$

But,

$\displaystyle STT_{\omega}^{-1}ST_{\omega}ST_{\omega}^{-1}$
$\displaystyle=ST_{\omega}^{-1}TST_{\omega}ST_{\omega}^{-1}$ $\displaystyle
T_{\omega}T^{-1}ST_{\omega}^{-1}ST^{-1}$
$\displaystyle=T^{-1}T_{\omega}ST_{\omega}^{-1}ST^{-1}$

which shows that

$\displaystyle
P|(T^{-1}+ST_{\omega}^{-1}+T^{-1}T_{\omega}ST_{\omega}^{-1}ST^{-1}+ST_{\omega}^{-1}TST_{\omega}ST_{\omega}^{-1}+T_{\omega}^{-1}ST^{-1}+ST_{\omega}ST_{\omega}^{-1})=0.$

It follows that $P|\varepsilon\in
ker(T+ST_{\omega}+TE+ST_{\omega}E^{-1}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega}).$
∎

######  Proposition 5.19.

The polynomial $P_{k,\Delta}$ belongs to the subspace $W_{k,k}^{1}$ and
therefore the map $f:\Gamma_{11}\to V_{k,k}$ defined by

$\displaystyle\gamma\mapsto(H_{k,\Delta}|({\bf 1}-S))|\gamma$

is a cocycle belonging to $C_{p}(\Gamma_{11},V_{k,k})$.

###### Proof.

The proof that $P_{k,\Delta}\in ker({\bf 1}+S)$ and $P_{k,\Delta}\in ker({\bf
1}+U+U^{2})$ is same as before. To show that $P_{k,\Delta}\in
ker(T+ST_{\omega}+TE+ST_{\omega}E^{-1}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$
we calculate

$\displaystyle(H_{k,\Delta}|({\bf
1}-S))|(T+ST_{\omega}+TE+ST_{\omega}E^{-1}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega}).$

Again, using the identities $H_{k,\Delta}|T_{\omega}=H_{k,\Delta}$ and
$H_{k,\Delta}|T_{\omega}^{-1}=H_{k,\Delta}$ together with
$E=T_{\omega}^{-1}ST_{\omega}ST$ and
$E^{-1}=T^{-1}ST_{\omega}^{-1}ST_{\omega}$ we end up with

$\displaystyle(H_{k,\Delta}|({\bf
1}-S))|(T+ST_{\omega}+TE+ST_{\omega}E^{-1}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})=H_{k,\Delta}|(ST_{\omega}E^{-1}-STE).$

But

$\displaystyle ST_{\omega}E^{-1}=T_{\omega}STE,$

and so

$\displaystyle
H_{k,\Delta}|(ST_{\omega}E^{-1}-STE)=H_{k,\Delta}|(T_{\omega}STE-STE)=0$

which shows that $P_{k,\Delta}\in
ker(T+ST_{\omega}+TE+ST_{\omega}E^{-1}+T_{\omega}ST+ST_{\omega}^{-1}ST_{\omega})$.
Finally, by part (2) of Proposition 2.6 we have

$\displaystyle P_{k,\Delta}|\varepsilon=P_{k,\Delta}$

which concludes the proof that $P_{k,\Delta}\in W_{k,k}^{1}$. ∎

Again, the lack of computational resources and the complicated description of
the third kernel limited our dimension calculations to the range $1\leq k\leq
21$. In this range the data again showed that
$\textbf{dim}(W_{k,k})=\textbf{dim}(W_{k,k}^{1})$ and
$\textbf{dim}(W_{k,k}^{-1})=0$. Here is the table containing our data for
$d=11$ case.

$k$ |  | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21
---|---|---|---|---|---|---|---|---|---|---|---|---
$\textbf{dim}(W_{k,k}^{1})$ |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11

Based on this numerical data we are led to the following conjecture.

######  Conjecture 5.20.

For odd $k\geq 1$ we have that
$\displaystyle\textbf{dim}(W_{k,k}^{1})=\frac{k+1}{2}.$

###  5.6. Some concluding remarks and future work

We close this section by elaborating on the importance of the conjectural
formulas we have stated and how they relate to Bianchi cusp forms which are
the analogs of cusp forms for imaginary quadratic fields. We also list some
questions (some of which are being considered in [13]) that may have occurred
to the reader but are left unanswered.

We will not define most of the objects we refer to in this subsection and be
intentionally vague about them due to the fact giving precise definitions
require a lot of sophisticated mathematical machinery. We also do not want to
stray too much from the main theme of this paper which is to study the
function $H_{k,\Delta}$. However, some good references that discuss some of
the things we refer to are [19, 20, 24].

We begin with the generalized Eichler-Shimura Isomorphism which states that

$\displaystyle H^{1}_{\text{cusp}}(\Gamma,V_{n,n})\simeq S_{n+2}(\Gamma)$

where $\Gamma$ is a Bianchi group, $S_{n}(\Gamma)$ is the vector space of
Bianchi cusp forms of weight $n$ on $\Gamma$, and $H^{1}_{\text{cusp}}$ refers
to the cuspidal cohomology of $\Gamma$. By translating the sheaf cohomology
into group cohomology one can show that

$\displaystyle H^{1}_{\text{cusp}}(\Gamma,V_{n,n})\simeq
H^{1}_{\text{par}}(\Gamma,V_{n,n})$

where $H^{1}_{\text{par}}$ is the parabolic cohomology defined as the quotient
of parabolic cocycles on $\Gamma$ by parabolic coboundaries on $\Gamma$. We
already saw that when $\Gamma=\Gamma_{d}$ the space of parabolic cocycles can
be identified with $W_{k,k}$. Also, one can easily show that the space of
parabolic coboundaries can be identified with constant polynomials implying
that it is of dimension 1. Consequently,

$\displaystyle\textbf{dim}(S_{k}(\Gamma_{d}))=\textbf{dim}(W_{k-2,k-2})-1.$

So, if the conjectural formulas we have stated are true then, at least in the
case when $k$ is odd, one immediately obtains explicit formulas, depending
only on $k$, for the dimension of $S_{k}(\Gamma_{d})$. As far as we can tell,
no explicit dimension formulas for $S_{k}(\Gamma)$ are known. However, Finis
et al.[14] have carried out extensive computations on the dimension of various
cohomology groups associated to Bianchi groups which provides a way of
checking the validity of our data. We are encouraged by the fact that our data
does agree with theirs in the overlapping cases. Indeed, we checked our data
for the $\textbf{dim}H^{1}_{\text{cusp}}(\Gamma_{d},V_{k,k})$ against the
data111The table gives the dimension of
$\textbf{dim}H^{1}(\Gamma_{d},V_{k,k})$, but the
$\textbf{dim}H^{1}_{\text{cusp}}(\Gamma_{d},V_{k,k})$ can be easily obtained
using the formula
$\textbf{dim}H^{1}(\Gamma_{d},V_{k,k})-\textbf{dim}H^{1}_{\text{cusp}}(\Gamma_{d},V_{k,k})=\nu_{K,k}h_{K}$
which is given on the same page. provided in Table 1 on pg. 53 in [14], and
both sets of data are in agreement for $k$ odd and in the range $1\leq k\leq
15$.

Finally, we discuss some future work and questions:

  * 1)

The foremost question is obviously whether the formulas we have stated are
true or not. Currently, we are working to answer this question in [13]. We are
also trying to find and prove dimension formulas when $k$ is even by looking
at the experimental data. Another aspect of our work in [13] is to define the
analog of the period map from the classical case of
$\textbf{PSL}(2,\mathbb{Z})$ for the Bianchi cusp forms.

  * 2)

There is an action of Hecke operators on $H^{1}_{\text{cusp}}(\Gamma,V_{n,n})$
and the isomorphism

$\displaystyle H^{1}_{\text{cusp}}(\Gamma,V_{n,n})\simeq S_{n+2}(\Gamma)$

is a Hecke module isomorphism. It will be interesting to transfer the action
of Hecke operators to the space $W_{k,k}$ and explore its consequences.
Indeed, Zagier [27] has done this for the space of period polynomials which is
defined similarly as $W_{k,k}$ and has obtained interesting results for the
traces of Hecke operators. We plan to carry this out in [13] or in a
subsequent paper.

  * 3)

Reader may be wondering if there are any other values of $k$, other than the
ones we have found, for which $H_{k,\Delta}$ is constant. The anonymous
referee also asked about this. The answer is most likely no since it is shown
in [14] that the dimension of $H^{1}_{\text{cusp}}(\Gamma,V_{n,n})$ grows at
least linearly in $n$ which means that the dimension of the space of parabolic
cocycles containing $P_{k,\Delta}$ grows at least linearly in $n$ as well, and
$H_{k,\Delta}$ is constant precisely when the subspace containing
$P_{k,\Delta}$ in the direct sum decomposition of the space of parabolic
cocycles is one dimensional. This, along with computational evidence we have
gathered, strongly suggest that there are no other $k$ values, other than the
ones we have found, for which $H_{k,\Delta}$ is constant.

  * 4)

Finally, the reader may also be wondering whether the kernels of the linear
maps defining $W_{k,k}$ can be combined into a single or fewer kernels. We
have not pursued this, but it is likely to be true. Indeed, this is true for
the space of parabolic cocycles on $\textbf{PSL}(2,\mathbb{Z})$ defined as the
intersection of $ker({\bf 1}+S)$ and $ker({\bf 1}+U+U^{2})$ where $S$ and $U$
are the generators of $\textbf{PSL}(2,\mathbb{Z})$. It can be easily checked,
which is left as an exercise in [28], that

$\displaystyle f\in ker({\bf 1}+S)\cap ker({\bf 1}+U+U^{2})\iff f\in ker({\bf
1}-US-U^{2}S)$

where $f$ is a parabolic cocycle on $\textbf{PSL}(2,\mathbb{Z})$.

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