# Quasimodular forms and modular differential equations which are not apparent
at cusps: I

Chang-Shou Lin and Yifan Yang  Center for Advanced Study in Theoretical
Sciences (CASTS), National Taiwan University, Taipei, Taiwan 10617.
<EMAIL_ADDRESS>Department of Mathematics, National Taiwan University
and National Center for Theoretical Sciences, Taipei, Taiwan 10617.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we explore a two-way connection between quasimodular forms of
depth $1$ and a class of second-order modular differential equations with
regular singularities on the upper half-plane and the cusps. Here we consider
the cases $\Gamma=\Gamma_{0}^{+}(N)$ generated by $\Gamma_{0}(N)$ and the
Atkin-Lehner involutions for $N=1,2,3$
($\Gamma_{0}^{+}(1)=\mathrm{SL}(2,\mathbb{Z})$). Firstly, we note that a
quasimodular form of depth $1$, after divided by some modular form with the
same weight, is a solution of a modular differential equation. Our main
results are the converse of the above statement for the groups
$\Gamma_{0}^{+}(N)$, $N=1,2,3$.

###### 2010 Mathematics Subject Classification:

Primary 11F11; secondary 11F25, 11F37, 34M03, 34M35

##  1\. Introduction

Let $\Gamma$ be a discrete subgroup of $\mathrm{SL}(2,\mathbb{R})$
commensurable with $\mathrm{SL}(2,\mathbb{Z})$. The aim of our study is to
explore a two-way connection between quasimodular forms with depth $\ell$ on
$\Gamma$ and modular differential equations of order $\ell+1$. In this paper,
we shall consider the case $\ell=1$.

The notion of quasimodular forms was first introduced by Kaneko and Zagier
[9]. For example, if $\Gamma=\mathrm{SL}(2,\mathbb{Z})$, a function $f(z)$ on
the upper half-plane $\mathbb{H}$ is called a quasimodular form of weight $k$
and depth $\leq 1$ if

(1.1) $f(z)=f_{1}(z)E_{2}(z)+f_{0}(z),\qquad z\in\mathbb{H},$

for some modular forms $f_{0}(z)$ and $f_{1}(z)$ of weight $k$ and $k-2$,
respectively, on $\mathrm{SL}(2,\mathbb{Z})$, where $E_{2}(z)$ is the
classical Eisenstein series of weight $2$ on $\mathrm{SL}(2,\mathbb{Z})$ with
$E_{2}(\infty)=1$.

For $f(z)$ of (1.1), Pellarin [10] introduced the Wronskian

(1.2) $W_{f}(z)=\det\begin{pmatrix}zf(z)+\frac{6}{\pi i}f_{1}(z)&f(z)\\\
\left(zf(z)+\frac{6}{\pi i}f_{1}(z)\right)^{(1)}&f^{(1)}(z)\end{pmatrix},$

where $g^{(k)}$ is the $k$-th derivative of a function $g$ with respect to
$z$, and a straightforward computation finds that $W_{f}(z)$ is a modular form
of weight $2k$ (see [10] or Section 2 for more details). Let

$g_{1}(z)=\frac{zf(z)+6f_{1}(z)/\pi i}{\sqrt{W_{f}(z)}}\quad\text{and}\quad
g_{2}(z)=\frac{f(z)}{\sqrt{W_{f}(z)}}.$

Then we have $\det\left(\begin{smallmatrix}g_{1}&g_{2}\\\
g_{1}^{\prime}&g_{2}^{\prime}\end{smallmatrix}\right)=1$, and hence
$\det\left(\begin{smallmatrix}g_{1}&g_{2}\\\
g_{1}^{\prime\prime}&g_{2}^{\prime\prime}\end{smallmatrix}\right)=0$, by a
further differentiation. Thus, both $g_{1}$ and $g_{2}$ are solutions of

(1.3) $y^{\prime\prime}=-4\pi^{2}Q(z)y,\qquad z\in\mathbb{H},$

where

(1.4)
$Q(z)=-\frac{1}{4\pi^{2}}\frac{g_{1}^{\prime\prime}}{g_{1}}=-\frac{1}{4\pi^{2}}\frac{g_{2}^{\prime\prime}}{g_{2}}.$

Although generally, $g_{i}$, $i=1,2$, are not single-valued functions, it is
easy to see that $Q(z)$ is a single-valued meromorphic function on
$\mathbb{H}$.

Equation (1.3) is a second-order ODE in a complex variable. Such a
differential equation is called Fuchsian if the order of pole of $Q(z)$ is
$\leq 2$. Let $z_{0}$ be a singular point and assume that
$Q(z)=A(z-z_{0})^{-2}+O\left((z-z_{0})^{-1}\right)$. Then the indicial
equation at $z_{0}$ is

$\rho(\rho-1)=-4\pi^{2}A,$

and its roots are $-\alpha$ and $\alpha+1$ for some $\alpha\in\mathbb{C}$.
These two roots are called the _local exponents_ of (1.3) (or of $Q(z)$). If
$\alpha\in\frac{1}{2}\mathbb{Z}$, then equation (1.3) might have a solution
with a logarithmic singularity near $z_{0}$. The (regular) singular point
$z_{0}$ is called _apparent_ if (a) $\alpha\in\frac{1}{2}\mathbb{Z}$ and (b)
any solution of (1.3) has no logarithmic singularity at $z_{0}$. If (a) holds,
but (b) does not, we will simply say $z_{0}$ is _not apparent_. In this paper,
we always assume that the local exponents of any singularity
$z_{0}\in\mathbb{H}$ are integers or half-integers, which will be denoted by
$1/2\pm\kappa_{z_{0}}$, $\kappa_{z_{0}}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$. It
is easy to see that (i) if $\kappa_{z_{0}}=0$, then $z_{0}$ is always not
apparent, and (ii) if $\kappa_{z_{0}}=1/2$, then $z_{0}$ is apparent if and
only if $Q(z)$ is holomorphic at $z_{0}$.

We start with the following observation.

######  Theorem 1.1.

Let $Q(z)$ be defined (1.4). Then $Q(z)$ is a meromorphic modular form of
weight $4$ on $\mathrm{SL}(2,\mathbb{Z})$. Moreover, the following hold.

  1. (i)

The ODE (1.3) is Fuchsian on $\mathbb{H}\cup\\{\text{cusps}\\}$.

  2. (ii)

Any pole of $Q$ on $\mathbb{H}$ is an apparent singularity for (1.3).

  3. (iii)

The cusp $\infty$ is not apparent.

  4. (iv)

Let $1/2\pm\kappa_{i}$ and $1/2\pm\kappa_{\rho}$,
$\kappa_{i},\kappa_{\rho}\in\frac{1}{2}\mathbb{N}$, be the local exponents of
(1.3) at $i=\sqrt{-1}$ and $\rho=(1+\sqrt{-3})/2$, respectively. Then
$(2\kappa_{i},2)=(2\kappa_{\rho},3)=1$.

This theorem is a special case of Theorem 3.3 and Proposition 3.6, proved in
Section 3.

The unexpected result of Theorem 1.1 is the modularity of $Q$. In view of
Theorem 1.1, a natural question is whether the converse of Theorem 1.1 holds
or not. This is the main question studied in this paper, not only for the
group $\mathrm{SL}(2,\mathbb{Z})$. Our methods for proving the main results
depend on the structure of $\Gamma$ and its modular forms. Thus, in this
paper, we restrict our attention to $\Gamma=\Gamma_{0}^{+}(N)$, $N=1,2,3$,
where $\Gamma_{0}^{+}(N)$ is the group generated by $\Gamma_{0}(N)$ and all
the Atkin-Lehner involutions. (Note that
$\Gamma_{0}^{+}(1)=\mathrm{SL}(2,\mathbb{Z})$. In the case $N=2,3$,
$\Gamma_{0}^{+}(N)$ is simply the group generated by $\Gamma_{0}(N)$ and
$\frac{1}{\sqrt{N}}\left(\begin{smallmatrix}0&-1\\\
N&0\end{smallmatrix}\right)$.) For such a group $\Gamma$, there are only one
cusp $\infty$ and two elliptic points $\rho_{i}$, $i=1,2$, where the order of
$\rho_{1}$ is $2$, and the order of $\rho_{2}\in\\{3,4,6\\}$, depending on
$N$. Throughout the paper, we refer to

(1.5) $y^{\prime\prime}(z)=-4\pi^{2}Q(z)y(z)$

as a modular ordinary differential equation (MODE) if $Q(z)$ is a nonzero
_meromorphic modular form of weight $4$_ on $\Gamma$. Motivated by Theorem
1.1, we consider equations (1.5) satisfying the following condition
($\text{H}_{\Gamma}$).

($\text{H}_{\Gamma}$) (i) The ODE (1.5) is Fuchsian on $\mathbb{H}$ and all
cusps. (ii) The local exponents of (1.5) at any singular point $z_{0}$ on
$\mathbb{H}$ are $1/2\pm\kappa_{z_{0}}$,
$\kappa_{z_{0}}\in\frac{1}{2}\mathbb{N}$. Moreover, $z_{0}$ is apparent for
(1.5). (iii) Let $e_{\rho_{i}}$, $i=1,2$, be the order of the elliptic point
$\rho_{i}$ and $1/2\pm\kappa_{\rho_{i}}$,
$\kappa_{\rho_{i}}\in\frac{1}{2}\mathbb{N}$, be the local exponents of (1.5)
at $\rho_{i}$. Then $(2\kappa_{\rho_{i}},e_{i})=1$ for $i=1,2$.

In Section 3, we will briefly explain the basic notions related to ODE in a
complex variable. (We refer to [5] for an introduction to elementary theory of
complex ODE.) Here we shall give some explanation pertaining to our condition
($\text{H}_{\Gamma}$). Firstly, suppose that the cusp $\infty$ has width $N$.
Let $q_{N}=e^{2\pi iz/N}$. Then (1.5) can be written as

(1.6) $\left(q_{N}\frac{d}{dq_{N}}\right)^{2}y=N^{2}Qy$

since $d/dz=2\pi iN^{-1}q_{N}d/dq_{N}$. For other cusps $s$, we choose
$\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathrm{SL}(2,\mathbb{Z})$ such that
$\gamma\infty=s$. Set

(1.7) $\hat{y}(z)=\left(y\big{|}_{-1}\gamma\right)(z):=(cz+d)y(\gamma z).$

Then $\hat{y}$ satisfies

(1.8)
$\hat{y}^{\prime\prime}=-4\pi^{2}\hat{Q}(z)\hat{y},\qquad\hat{Q}(z)=\left(Q\big{|}_{4}\gamma\right)(z).$

By (1.6) and (1.8), we see that (1.5) is Fuchsian at a cusp $s$ if and only if
$Q$ is _holomorphic at_ $s$, and the local expondents at $s$ are
$\pm\kappa_{s}$, where

(1.9) $\kappa_{s}=N\sqrt{Q(s)}.$

Secondly, suppose that $z_{0}\in\mathbb{H}$ is a singular point and (1.5) is a
modular ODE. Then the local exponents at $\gamma z_{0}$, $\gamma\in\Gamma$,
are the same as those at $z_{0}$. Also, if $z_{0}$ is apparent, then so is
$\gamma z_{0}$. See [4] for a proof.

For a cusp $s$, we can define a similar notion of apparentness (or non-
apparentness) when $\kappa_{s}$ in (1.9) is in $\frac{1}{2}\mathbb{Z}_{\geq
0}$. Suppose that at a cusp $s$ we have
$\kappa_{s}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$. Then (1.5) always has a unique
solution $y_{+}(z)$ of the form

(1.10) $y_{+}(z)=q_{N}^{\kappa_{s}}\left(1+\sum_{j\geq
1}c_{j}q_{N}^{j}\right)$

An elementary theorem in theory of complex ODE says that if $s$ is not
apparent and $y(z)$ is a solution of (1.5) independent of $y_{+}(z)$, then
there exists $e\neq 0\in\mathbb{C}$ and $m(z)$ such that

(1.11) $y(z)=ezy_{+}(z)+m(z),\qquad
m(z)=q_{N}^{-\kappa_{s}}\sum_{j=0}^{\infty}\hat{c}_{j}q_{N}^{j},\quad\hat{c}_{0}\neq
0.$

We now introduce a representation of $\Gamma$ associated to a modular ODE as
follows. It is an elementary fact that any (local) solution $y(z)$ of (1.5)
can be defined on the whole $\mathbb{H}$ through analytic continuation. Fix a
point $z_{0}\in\mathbb{H}$ that is not a singular point of $Q(z)$. Let $U$ be
a simply connected (small) open set in $\mathbb{H}$ containing $z_{0}$, but
not any singularities of $Q(z)$. For $\gamma\in\Gamma$, choose a path $\sigma$
from $z_{0}$ to $\gamma z_{0}$ and consider the analytic continuation of
$y(z)$, $z\in U$, along the path. Then $y(\gamma z)$ is well-defined in $U$.
We define $y\big{|}_{-1}\gamma$ by

$\left(y\big{|}_{-1}\gamma\right)(z):=(cz+d)y(\gamma z),\qquad z\in U.$

Then $y\big{|}_{-1}\gamma$ is also a solution of (1.5). This can be proved by
a direct computation or by using Bol’s identity in literature [1]. By choosing
a fundamental system $(y_{1}(z),y_{2}(z))$ of solutions, it gives a matrix
$\hat{\gamma}\in\mathrm{SL}(2,\mathbb{C})$ such that

$\begin{pmatrix}y_{1}\big{|}_{-1}\gamma\\\
y_{2}\big{|}_{-1}\gamma\end{pmatrix}(z)=\hat{\gamma}\begin{pmatrix}y_{1}(z)\\\
y_{2}(z)\end{pmatrix}.$

(The reason why $\hat{\gamma}$ has determinant $1$ is due to the fact that
$y_{1}|_{-1}\gamma$ and $y_{2}|_{-1}\gamma$ has the same Wronskian as $y_{1}$
and $y_{2}$.) Of course, this matrix $\hat{\gamma}$ depends on the choice of
the path $\sigma$. However, under the condition ($\text{H}_{\Gamma}$), all
local monodromy matrices are $\pm I$. Hence, different choices of $\sigma$
will only possibly change $\hat{\gamma}$ to $-\hat{\gamma}$. From this, we see
that there is a well-defined homomorphism $\rho$ from $\Gamma$ to
$\mathrm{PSL}(2,\mathbb{C})$ such that

(1.12) $\begin{pmatrix}y_{1}\big{|}_{-1}\gamma\\\
y_{2}\big{|}_{-1}\gamma\end{pmatrix}(z)=\pm\rho(\gamma)\begin{pmatrix}y_{1}(z)\\\
y_{2}(z)\end{pmatrix}$

for all $\gamma\in\Gamma$, where $y_{j}(\gamma z)$, $j=1,2$, are always
understood to take the same path for the analytic continuation. This
homomorphism $\rho$ will be called the _Bol representation_ in this paper. For
the group $\Gamma$ under consideration in this paper, the Bol representation
can be lifted to a homomorphism $\hat{\rho}$ from $\Gamma$ to
$\mathrm{GL}(2,\mathbb{C})$. To achieve this, we let $F(z)^{2}$ be a modular
form of weight $2(\ell+1)$ with some character for some integer $\ell$ such
that $\hat{y}(z)=F(z)y(z)$ is well-defined on $\mathbb{H}$ for any solution
$y(z)$ of (1.5). Note that since $\hat{y}(z)$ is single-valued, for any
$\gamma\in\Gamma$, there exists
$\hat{\rho}(\gamma)\in\mathrm{GL}(2,\mathbb{C})$ such that

$\begin{pmatrix}\hat{y}_{1}\big{|}_{\ell}\gamma\\\
y_{2}\big{|}_{\ell}\gamma\end{pmatrix}=\hat{\rho}(\gamma)\begin{pmatrix}\hat{y}_{1}\\\
\hat{y}_{2}\end{pmatrix}.$

Obviously, $\hat{\rho}$ is a lifting of $\rho$. We remark that there are many
possible choices for $F(z)$. In Sections 4–6, we will choose $F(z)$ such that
$\hat{y}(z)$ is holomorphic on $\mathbb{H}$ and at cusps. We emphasize that
the integer $\ell$ will give rise to some important information in this series
of papers. For example, we have $\det\hat{\rho}(\gamma)=1$ for all
$\gamma\in\Gamma$ if and only if $\ell$ is odd.

From now on, we assume that $\kappa_{\infty}\in\frac{1}{2}\mathbb{Z}_{\geq
0}$, where $\pm\kappa_{\infty}$ are the local exponents of (1.5) at the cusp
$\infty$, and $\\{z_{1},\ldots,z_{m}\\}$ is the set of poles of $Q(z)$ such
that $z_{i}$ is not an elliptic point and $z_{i}$ is not $\Gamma$-equivalent
to $z_{j}$ for $i\neq j$. Let $1/2\pm\kappa_{j}$ be the local exponents of
(1.5) at $z_{j}$.

When $\Gamma=\mathrm{SL}(2,\mathbb{Z})$, the integer $\ell$ in the lift of the
Bol representation is defined by

(1.13)
$\ell=-1+12\kappa_{\infty}+4\left(\kappa_{\rho}-\frac{1}{2}\right)+6\left(\kappa_{i}-\frac{1}{2}\right)+12\sum_{j=1}^{m}\left(\kappa_{j}-\frac{1}{2}\right),$

where $i=\sqrt{-1}$ and $\rho=(1+\sqrt{-3})/2$ are the elliptic points and
$1/2\pm\kappa_{\rho}$ and $1/2\pm\kappa_{i}$ are the local exponents of (1.5)
at $\rho$ and $i$, respectively. Since all the $\kappa$’s are in
$\frac{1}{2}\mathbb{Z}$, $\ell$ is always an integer. Furthermore, _the
integer $\ell$ is odd if and only if $\kappa_{i}\in\frac{1}{2}+\mathbb{Z}$_.
Thus, ($\text{H}_{\Gamma}$) implies that $\ell$ is odd.

Let $F(z)$ be defined by (4.6). Then $F(z)^{2}$ is a modular form of weight
$2(\ell+1)$ such that for any solution $y(z)$ of (1.5), the function
$\hat{y}(z)=F(z)y(z)$ is a single-valued holomorphic function on $\mathbb{H}$.
We use the standard notations $T=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)$, $S=\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right)$, and $R=TS=\left(\begin{smallmatrix}1&-1\\\
1&0\end{smallmatrix}\right)$. They satisfy $S^{2}=R^{3}=-I$. Our first main
result asserts that the converse of Theorem 1.1 holds true.

######  Theorem 1.2.

Let $\Gamma=\mathrm{SL}(2,\mathbb{Z})$. Suppose that $Q(z)$, not identically
$0$, satisfies ($\text{H}_{\Gamma}$) with
$\kappa_{\infty}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$. Then the following
statements hold true.

  1. (i)

The cusp $\infty$ is not apparent.

  2. (ii)

Let $y_{+}(z)$ be the solution of (1.5) of the form (1.10) and set
$y_{1}(z)=\left(\hat{y}_{+}|_{\ell}S\right)(z)/F(z)$ and $y_{2}(z)=y_{+}(z)$.
Then $y_{1}(z)$ and $y_{2}(z)$ form a fundamental system of solutions of
(1.5). Moreover, we have

$\hat{y}_{1}(z)=z\hat{y}_{2}(z)+\hat{m}_{1}(z)$

for some modular form $\hat{m}_{1}(z)$ of weight $\ell-1$ on
$\mathrm{SL}(2,\mathbb{Z})$.

  3. (iii)

Using $y_{1}(z)$ and $y_{2}(z)$ as the basis, Bol’s representation satisfies

$\hat{\rho}(\gamma)=\gamma$

for all $\gamma\in\mathrm{SL}(2,\mathbb{Z})$. In particular, the ratio
$h(z)=y_{1}(z)/y_{2}(z)$ is equivariant.

  4. (iv)

Write $\hat{y}_{+}(z)=\frac{\pi i}{6}\hat{m}_{1}(z)E_{2}(z)+\hat{m}_{2}(z)$.
Then $\hat{m}_{2}(z)$ is a modular form of weight $\ell+1$ with respect to
$\mathrm{SL}(2,\mathbb{Z})$. In other words, $\hat{y}_{+}(z)$ is a
quasimodular form of weight $\ell+1$ and depth $1$ on
$\mathrm{SL}(2,\mathbb{Z})$.

######  Remark 1.3.

Since (1.5) is assumed to be apparent at any pole of $Q$ on $\mathbb{H}$, the
ratio $h(z)$ of any two linearly independent solution of (1.5) is a
meromorphic (single-valued) function on $\mathbb{H}$. We say a meromorphic
function $h(z)$ on $\mathbb{H}$ is _equivariant_ with respect to
$\mathrm{SL}(2,\mathbb{Z})$ if $h(z)$ satisfies

$h(\gamma z)=\gamma\cdot h(z):=\frac{ah(z)+b}{ch(z)+d}$

for all $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathrm{SL}(2,\mathbb{Z})$ (see [13]).

We note that $y_{1}(z)$ is a second solution of (1.5), so $y_{1}(z)$ has an
expression of the form (1.11) for some $e\neq 0$. Then Part (ii) says that
$e=1$. It is surprising to see that if the second solution is suitably chosen,
then $m(z)$ of (1.11) is a meromorphic modular form of weight $-2$.

For the case $\Gamma=\Gamma_{0}^{+}(N)$, $N=2,3$, we define

$T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\quad
S=\frac{1}{\sqrt{N}}\begin{pmatrix}0&-1\\\ N&0\end{pmatrix},\quad
R=TS=\frac{1}{\sqrt{N}}\begin{pmatrix}N&-1\\\ N&0\end{pmatrix}$

such that

$S^{2}=R^{2N}=-I.$

The group $\Gamma_{0}^{+}(N)$, $N=2,3$, has one cusp $\infty$ with width $1$
and two elliptic point $\rho_{1}$ of order $2$ and $\rho_{2}$ of order $2N$,
which are the fixed points of $S$ and $R$, respectively, i.e.,

(1.14)
$\rho_{1}=\frac{i}{\sqrt{N}},\qquad\rho_{2}=\begin{cases}(1+i)/2,&\text{if
}N=2,\\\ (3+\sqrt{-3})/6,&\text{if }N=3.\end{cases}$

The integer $\ell$ in the lifting in the case $\Gamma_{0}^{+}(2)$ is

$\ell=-1+8\kappa_{\infty}+4\left(\kappa_{\rho_{1}}-\frac{1}{2}\right)+2\left(\kappa_{\rho_{2}}-\frac{1}{2}\right)+8\sum_{j=1}^{m}\left(\kappa_{j}-\frac{1}{2}\right).$

Note that ($\text{H}_{\Gamma}$) implies that both $\kappa_{\rho_{1}}-1/2$ and
$\kappa_{\rho_{2}}-1/2$ are nonnegative integers. Hence $\ell$ is an odd
integer. Let $F(z)$ be defined by (5.4) and for a solution $y(z)$ of (1.5),
set $\hat{y}(z)=F(z)y(z)$. The main theorem for $\Gamma_{0}^{+}(2)$ is as
follows.

######  Theorem 1.4.

Let $\Gamma=\Gamma_{0}^{+}(2)$. Suppose that $Q(z)$ satisfies
($\text{H}_{\Gamma}$) with $\kappa_{\infty}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$.
Then the following statements hold.

  1. (i)

The cusp $\infty$ is not apparent for (1.5).

  2. (ii)

Let $y_{1}(z)=\left(\hat{y}_{+}|_{\ell}S\right)(z)/F(z)$ and
$y_{2}(z)=y_{+}(z)$, where $y_{+}(z)$ is the unique solution of (1.5) of the
form (1.10). Then $\\{y_{1}(z),y_{2}(z)\\}$ is a fundamental solution of
(1.5). Moreover, we have

(1.15)
$\hat{y}_{1}(z)=\left(\frac{2}{\ell}\right)\sqrt{2}z\hat{y}_{2}(z)+\hat{m}_{1}(z)$

for some modular form $\hat{m}_{1}(z)$ in
$\mathscr{M}_{\ell-1}(\Gamma_{0}^{+}(2),\left(\frac{2}{\ell}\right))$, where
$\left(\frac{2}{\ell}\right)$ is the Legendre symbol whose values are given by

$\left(\frac{2}{\ell}\right)=\begin{cases}1,&\text{if }\ell\equiv 1,7\
\mathrm{mod}\ 8,\\\ -1,&\text{if }\ell\equiv 3,5\ \mathrm{mod}\ 8.\end{cases}$

  3. (iii)

The ratio $h(z)=\left(\frac{2}{\ell}\right)y_{1}(z)/\sqrt{2}y_{2}(z)$ is
equivariant with respect to $\Gamma_{0}^{+}(2)$. That is, for all
$\gamma\in\Gamma_{0}^{+}(2)$, we have $h(\gamma z)=\gamma\cdot h(z)$.

  4. (iv)

Let $M_{2}^{\ast}(z)=(2E_{2}(2z)+E_{2}(z))/3$ and write $\hat{y}_{+}(z)$ as

$\hat{y}_{+}(z)=\left(\frac{2}{\ell}\right)\frac{\pi
i}{4\sqrt{2}}\hat{m}_{1}(z)M_{2}^{\ast}(z)+\hat{m}_{2}(z).$

Then $\hat{m}_{2}(z)$ is a modular form in
$\mathscr{M}_{\ell+1}(\Gamma_{0}^{+}(2),\left(\frac{2}{\ell}\right))$.
Consequently, $\hat{y}_{+}(z)$ is a quasimodular form in
$\widetilde{\mathscr{M}}_{\ell+1}^{\leq
1}(\Gamma_{0}^{+}(2),\left(\frac{2}{\ell}\right))$.

Note that $M_{2}^{\ast}(z)$ is a quasimodular form of weight $2$ and depth $1$
on $\Gamma_{0}^{+}(2)$. Also, $\mathscr{M}_{k}(\Gamma_{0}^{+}(2),\pm 1)$
denotes the space of modular forms $f(z)$ of weight $k$ on $\Gamma_{0}(2)$
such that $\left(f|_{k}S\right)(z)=\pm f(z)$. (The reason for the choice of
the notation $\mathscr{M}_{k}(\Gamma_{0}^{+}(2),\pm)$ instead of the more
common $\mathscr{M}_{k}(\Gamma_{0}(2),\pm)$ is that we regard such an $f$ as a
modular form with character on $\Gamma_{0}^{+}(2)$.) For the meaning of the
notation $\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}^{+}(2),\pm 1)$, see
Section 2.2.

######  Remark 1.5.

The identity (1.15) says that the constant $e$ in (1.11) is
$\left(\frac{2}{\ell}\right)\sqrt{2}$, which depends on $\ell$. This result is
different from Theorem 1.2.

For the case $\Gamma=\Gamma_{0}^{+}(3)$, the group $\Gamma$ has one cusp
$\infty$ and two elliptic points $\rho_{1}$ and $\rho_{2}$ given by (1.14). In
this case, $\ell$ is defined by

$\ell=-1+6\kappa_{\infty}+3\left(\kappa_{\rho_{1}}-\frac{1}{2}\right)+\left(\kappa_{\rho_{2}}-\frac{1}{2}\right)+6\sum_{j=1}^{m}\left(\kappa_{j}-\frac{1}{2}\right).$

Note that by ($\text{H}_{\Gamma}$), both $\kappa_{\rho_{1}}-1/2$ and
$\kappa_{\rho_{2}}-1/2$ are integers, and $\ell$ is an integer not divisible
by $3$. Let $F(z)$ be defined by (6.3). As before, for a solution $y(z)$ of
(1.5), we let $\hat{y}(z)=F(z)y(z)$.

######  Theorem 1.6.

Let $\Gamma=\Gamma_{0}^{+}(3)$. Suppose that $Q(z)$ satisfy
($\text{H}_{\Gamma}$) with $\kappa_{\infty}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$.
Set

$\delta=\begin{cases}\chi^{0},&\text{if }\ell\equiv 1,11\ \mathrm{mod}\ 12,\\\
\chi^{1},&\text{if }\ell\equiv 2,4\ \mathrm{mod}\ 12,\\\ \chi^{2},&\text{if
}\ell\equiv 5,7\ \mathrm{mod}\ 12,\\\ \chi^{3},&\text{if }\ell\equiv 8,10\
\mathrm{mod}\ 12,\end{cases}$

where $\chi$ is the character of $\Gamma_{0}^{+}(3)$ defined by
$\chi(S)=\chi(R)=-i$. Then the following statements hold.

  1. (i)

The cusp $\infty$ is not apparent.

  2. (ii)

Let $y_{1}(z)=\left(\hat{y}_{+}|_{\ell}S\right)(z)/F(z)$ and
$y_{2}(z)=y_{+}(z)$. Then $\\{y_{1}(z),y_{2}(z)\\}$ is a fundamental solution
of (1.5). Moreover, we have

$\hat{y}_{1}(z)=\delta(S)^{-1}\sqrt{3}z\hat{y}_{2}(z)+\hat{m}_{1}(z)$

for some modular form $\hat{m}_{1}(z)$ in
$\mathscr{M}_{\ell-1}(\Gamma_{0}^{+}(3),\delta)$.

  3. (iii)

The ratio $h(z)=\delta(S)^{-1}y_{1}(z)/\sqrt{3}y_{2}(z)$ is equivariant with
respect to $\Gamma_{0}^{+}(3)$. That is, for all $\gamma\in\Gamma_{0}^{+}(3)$,
we have $h(\gamma z)=\gamma\cdot h(z)$.

  4. (iv)

Let $M_{2}^{\ast}(z)=(3E_{2}(3z)+E_{2}(z))/4$ and write $\hat{y}_{+}(z)$ as

$\hat{y}_{+}(z)=\delta(S)^{-1}\frac{\pi
i}{3\sqrt{3}}\hat{m}_{1}(z)M_{2}^{\ast}(z)+\hat{m}_{2}(z).$

Then $\hat{m}_{2}(z)$ is a modular form in
$\mathscr{M}_{\ell+1}(\Gamma_{0}^{+}(3),\delta)$. Consequently,
$\hat{y}_{+}(z)$ is a quasimodular form in
$\widetilde{\mathscr{M}}_{\ell+1}^{\leq 1}(\Gamma_{0}^{+}(3),\delta)$.

Note that $M_{2}^{\ast}(z)$ is a quasimodular form of weight $2$ and depth $1$
on $\Gamma_{0}^{+}(3)$. Also, $\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\delta)$
denote the space of modular forms of weight $k$ with character $\delta$ on
$\Gamma_{0}^{+}(3)$. See Section 2.4 for a discussion about these spaces and
the space $\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}^{+}(3),\delta)$.

The rest of the paper is organized as follows. In Section 2, we give a quick
introduction to quasimodular forms, discuss the Atkin-Lehner decomposition of
the space of quasimodular forms on $\Gamma_{0}(N)$, and then prove the
existence of extremal quasimodular forms in certain spaces, which will used to
provide examples of modular differential equations satisfied by quasimodular
forms. In Section 3, we will prove that every quasimodular form of depth $1$
will give rise to a modular differential equation of the form (1.5) and show
that such a differential equation in the case
$\Gamma=\mathrm{SL}(2,\mathbb{Z})$, $\Gamma_{0}^{+}(2)$, or
$\Gamma_{0}^{+}(3)$ will satisfy the condition ($\text{H}_{\Gamma}$). In
Sections 4–6, we consider the converse direction. We will prove that if a
modular differential equation (1.5) satisfies ($\text{H}_{\Gamma}$) with
$\Gamma=\mathrm{SL}(2,\mathbb{Z})$, $\Gamma_{0}^{+}(2)$, or
$\Gamma_{0}^{+}(3)$, then its solutions must come from quasimodular forms.
Finally, in the appendix, we discuss how to determine apparentness of a
modular differential equation and prove the existence of meromorphic modular
forms $Q(z)$ satisfying ($\text{H}_{\Gamma}$).

##  2\. Quasimodular forms

For convenience of non-specialists, we shall give a quick introduction of
quasimodular forms in this section. See [2, 18] for more detailed overview and
discussion. Note that most of the definitions and properties presented here
are valid or have analogues in the setting of general Fuchsian subgroups of
the first kind of $\mathrm{SL}(2,\mathbb{R})$, but for simplicity, here we
restrict our attention to subgroups $\Gamma$ of $\mathrm{SL}(2,\mathbb{R})$
commensurable with $\mathrm{SL}(2,\mathbb{Z})$, which we assume throughout the
section.

###  2.1. Basic definitions and properties

The notion of quasimodular forms was first introduced by Kaneko and Zagier
[9]. It was defined through the notion of nearly holomorphic modular forms
(called almost holomorphic modular forms in [9]).

######  Definition 2.1.

A function $f:\mathbb{H}\to\mathbb{C}$ is said to be _nearly holomorphic_ if
it is of the form

(2.1) $f(z)=\sum_{r=0}^{n}\frac{f_{r}(z)}{(z-\overline{z})^{r}}$

for some holomorphic functions $f_{r}$.

Let $\Gamma$ be a congruence subgroup of $\mathrm{SL}(2,\mathbb{R})$. A
function $f:\mathbb{H}\to\mathbb{C}$ is said to be a _nearly holomorphic
modular form_ on $\Gamma$ if it is nearly holomorphic, satisfies

(2.2) $f\left(\frac{az+b}{cz+d}\right)=(cz+d)^{k}f(z)$

for all $\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma$, and has a finite limit at each cusp of
$\Gamma$. The largest integer $r$ such that $f_{r}\not\equiv 0$ in (2.1) is
called the _depth_ of $f$. We let $\mathscr{M}_{k}^{\ast}(\Gamma)$ denote the
space of nearly holomorphic modular forms of weight $k$ on $\Gamma$.

On the spaces of nearly holomorphic modular forms, we have the so-called
Shimura-Maass operator.

######  Definition 2.2.

For a nearly holomorphic function $f:\mathbb{H}\to\mathbb{C}$ and an integer
$k$, define the _Shimura-Maass operator_ $\partial_{k}$ of weight $k$ by

$(\partial_{k}f)(z)=\frac{1}{2\pi
i}\left(f^{\prime}(z)+\frac{kf(z)}{z-\overline{z}}\right).$

Also, for $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathrm{GL}^{+}(2,\mathbb{R})$, the slash
operator of weight $k$ is defined by

$(f\big{|}_{k}\gamma)(z)=\frac{(\det\gamma)^{k/2}}{(cz+d)^{k}}f\left(\frac{az+b}{cz+d}\right).$

Then we have the following properties.

######  Lemma 2.3 ([15, Equations (1.5) and (1.8)]).

For any nearly holomorphic functions $f,g:\mathbb{H}\to\mathbb{C}$, any
integers $k$ and $\ell$, and any $\gamma\in\mathrm{GL}^{+}(2,\mathbb{R})$, we
have

$\partial_{k+\ell}(fg)=(\partial_{k}f)g+f(\partial_{\ell}g)$

and

$\partial_{k}\left(f\big{|}_{k}\gamma\right)=(\partial_{k}f)\big{|}_{k+2}\gamma.$

In particular, the second property in the lemma says that if $f$ is a nearly
holomorphic modular form of weight $k$ on $\Gamma$, then $\partial_{k}f$ is a
nearly holomorphic form of weight $k+2$ on $\Gamma$. That is, $\partial_{k}$
is a linear transformation from $\mathscr{M}^{\ast}_{k}(\Gamma)$ to
$\mathscr{M}^{\ast}_{k+2}(\Gamma)$.

######  Definition 2.4.

We say a holomorphic function $f:\mathbb{H}\to\mathbb{C}$ is a _quasimodular
form_ of weight $k$ and depth $r$ on $\Gamma$ if it is the holomorphic part of
a nearly holomorphic modular form of weight $k$ and depth $r$ on $\Gamma$. We
let $\widetilde{\mathscr{M}}_{k}^{\leq r}(\Gamma)$ denote the space of
quasimodular forms of weight $k$ and depth $\leq r$ on $\Gamma$. We also let

$\widetilde{\mathscr{M}}(\Gamma)=\bigoplus_{k}\bigcup_{r}\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma)$

denote the graded ring of quasimodular forms of all weights and all depths on
$\Gamma$.

Using the modular property (2.2), we can show that if
$f_{0}\in\widetilde{\mathscr{M}}_{k}(\Gamma)$ is the holomorphic part of the
nearly holomorphic modular form

$f(z)=\sum_{r=0}^{n}\frac{f_{r}(z)}{(z-\overline{z})^{r}},$

then

(2.3)
$\left(f_{0}\big{|}_{k}\gamma\right)(z)=\sum_{r=0}^{\infty}f_{r}(z)\left(\frac{c}{cz+d}\right)^{r}$

for $\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in\Gamma$ and
vice versa.

The archetypal example of a quasimodular form is $E_{2}(z)$, which is the
holomorphic part of the nearly holomorphic modular form

$E_{2}(z)+\frac{6}{\pi
i(z-\overline{z})}=E_{2}(z)-\frac{3}{\pi\,\mathrm{Im\,}z}$

of weight $2$ and depth $1$ on $\mathrm{SL}(2,\mathbb{Z})$. It can be shown
that the graded ring $\widetilde{\mathscr{M}}(\mathrm{SL}_{2}(\mathbb{Z}))$ of
quasimodular forms on $\mathrm{SL}(2,\mathbb{Z})$ is generated by $E_{2}(z)$,
$E_{4}(z)$, and $E_{6}(z)$. Also, by Ramanujan’s identities

$D_{q}E_{2}=\frac{E_{2}^{2}-E_{4}}{12},\qquad
D_{q}E_{4}=\frac{E_{2}E_{4}-E_{6}}{3},\qquad
D_{q}E_{6}=\frac{E_{2}E_{6}-E_{4}^{2}}{2},$

the ring $\widetilde{\mathscr{M}}(\mathrm{SL}(2,\mathbb{Z}))$ is closed under
the differential operator

$D_{q}:=q\frac{d}{dq}=\frac{1}{2\pi i}\frac{d}{dz}.$

More generally, we have the following description of
$\widetilde{\mathscr{M}}_{k}^{\leq r}(\Gamma)$.

######  Proposition 2.5 ([18, Proposition 20]).

Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$ commensurable with
$\mathrm{SL}(2,\mathbb{Z})$.

  1. (i)

The graded ring $\widetilde{\mathscr{M}}(\Gamma)$ of quasimodular forms on
$\Gamma$ is closed under differentiation. More precisely, we have
$D_{q}(\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma))\subseteq\widetilde{\mathscr{M}}_{k+2}^{\leq r+1}(\Gamma)$ for all
$k,r\geq 0$.

  2. (ii)

Let $\phi$ be a quasimodular form of weight $2$ and depth $1$ on $\Gamma$.
Then

$\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma)=\bigoplus_{j=0}^{r}\phi^{j}\mathscr{M}_{k-2j}(\Gamma)$

for all $k,r\geq 0$. In particular, we have

$\dim\mathscr{M}_{k}^{\leq
r}(\Gamma)=\sum_{j=0}^{r}\dim\mathscr{M}_{k-2j}(\Gamma).$

The quasimodular form $\phi$ in the proposition can be constructed by taking
the logarithmic derivative of a meromorphic modular form of nonzero weight
that is nonvanishing on $\mathbb{H}$. In general, the choice of $\phi$ is not
unique. For example, when $\Gamma=\Gamma_{0}(p)$, we can take $\phi$ to be the
logarithmic derivative of any $\Delta(z)^{m}\Delta(pz)^{n}$, as long as
$m+n\neq 0$.

More generally, we can define quasimodular forms on $\Gamma$ with character
$\chi$ to be the holomorphic parts of nearly holomorphic function
$f:\mathbb{H}\to\mathbb{C}$ satisfying

$f(\gamma z)=\chi(\gamma)(cz+d)^{k}f(z),\qquad\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma,$

and analytic conditions at cusps. (Thus, if $-I\in\Gamma$ and $\chi(-I)=-1$,
we will assume that $k$ is odd.) We similarly let
$\widetilde{\mathscr{M}}_{k}^{\leq r}(\Gamma,\chi)$ denote the space of all
quasimodular forms on $\Gamma$ of weight $k$ with character $\chi$. In this
paper, we will only consider characters of finite order.

The first examples that provides a link between quasimodular forms and modular
differential equations were due to Kaneko and Koike [7, 8]. Among other
things, they proved that if $6|k$, then there is a solution $f(z)$ of the
differential equation

(2.4)
$D_{q}^{2}y(z)-\frac{k}{6}E_{2}(z)D_{q}y(z)+\frac{k(k-1)}{12}(D_{q}E_{2}(z))y(z)=0,$

such that $f$ is an extremal quasimodular form in
$\widetilde{\mathscr{M}}^{\leq 1}_{k}(\mathrm{SL}(2,\mathbb{Z}))$, defined as
an element whose vanishing order at $\infty$ is
$\dim\widetilde{\mathscr{M}}_{k}^{\leq 1}(\mathrm{SL}(2,\mathbb{Z}))-1$. Then
by a simple transformation of the differential equation, we see that
$y(z)=f(z)/\Delta(z)^{k/12}$ is a solution of the differential equation

(2.5) $D_{q}^{2}y(z)=\left(\frac{k}{12}\right)^{2}E_{4}(z)y(z).$

Analogues of (2.4) for the groups $\Gamma_{0}^{+}(N)$, $N=2,3$, and
$\Gamma_{0}(N)$, $N=2,3,4$, were obtained in [11] and [12], respectively. In
Section 3 we will give a new proof of these results, stated in the same form
as (2.5).

###  2.2. Atkin-Lehner decomposition of $\widetilde{M}_{k}^{\leq
r}(\Gamma_{0}(N))$

Again, for convenience of non-specialists, here we shall give a quick overview
of the Atkin-Lehner decomposition of spaces of modular forms.

Let $N$ be a positive integer. It is easy to see that if
$\gamma\in\mathrm{SL}(2,\mathbb{R})$ normalizes $\Gamma_{0}(N)$, then
$f\mapsto f|_{k}\gamma$ defines an automorphism of
$\mathscr{M}_{k}(\Gamma_{0}(N))$ and $\mathscr{M}_{k}^{\ast}(\Gamma_{0}(N))$,
where $f|_{k}\gamma$ is the usual slash operator of weight $k$ acting on $f$.
Let $\mathrm{N}(\Gamma_{0}(N))$ denote the normalizer of $\Gamma_{0}(N)$ in
$\mathrm{SL}(2,\mathbb{R})$. Inside $\mathrm{N}(\Gamma_{0}(N))/\Gamma_{0}(N)$,
there is a special subgroup, called the Atkin-Lehner subgroup, whose
definition is given as follows.

Let $e$ be a positive divisor of $N$ such that $(e,N/e)=1$. Such a divisor is
often called a Hall divisor of $N$. We check that a matrix of the form

$\frac{1}{\sqrt{e}}\begin{pmatrix}ae&b\\\ cN&de\end{pmatrix},\qquad
ade^{2}+bcN=e$

normalizes $\Gamma_{0}(N)$. Let $W_{e}$ denote the set of such matrices. (In
particular, $W_{1}=\Gamma_{0}(N)$.) It is easy to see that
$W_{e}=w_{e}\Gamma_{0}(N)$ for any element $w_{e}$ of $W_{e}$. Also, if $e$
and $e^{\prime}$ are two Hall divisors of $N$, $\gamma\in W_{e}$, and
$\gamma^{\prime}\in W_{e^{\prime}}$, then $\gamma\gamma^{\prime}\in
W_{e^{\prime\prime}}$, where
$e^{\prime\prime}=ee^{\prime}/(e,e^{\prime})^{2}$. Hence, if we let

$\Gamma_{0}^{+}(N)=\bigcup_{e|N,(e,N/e)=1}W_{e},$

then $\Gamma_{0}^{+}(N)$ is a subgroup of $\mathrm{N}(\Gamma_{0}(N))$ and the
quotient group $\Gamma_{0}^{+}(N)/\Gamma_{0}(N)$ is an elementary abelian
$2$-group of order $2^{m}$, where $m$ is the number of distinct prime divisors
of $N$. This quotient group $\Gamma_{0}^{+}(N)/\Gamma_{0}(N)$ is called the
_Atkin-Lehner subgroup_ of $\mathrm{N}(\Gamma_{0}(N))/\Gamma_{0}(N)$.

Elements of the Atkin-Lehner groups acts on many things related to
$\Gamma_{0}(N)$. For example, they give rise to automorphisms of the modular
curve $X_{0}(N)$. They also act on $\mathscr{M}_{k}(\Gamma_{0}(N))$ and
$\mathscr{M}_{k}^{\ast}(\Gamma_{0}(N))$, as mentioned at the beginning of the
section. In all case, they are called _Atkin-Lehner involutions_ and denoted
by $w_{e}$. Since every $w_{e}$ has order $1$ or $2$ in
$\mathrm{N}(\Gamma_{0}(N))/\Gamma_{0}(N)$ and they commute with each other,
the space $\mathscr{M}_{k}(\Gamma_{0}(N))$ (respectively,
$\mathscr{M}^{\ast}(\Gamma_{0}(N))$) decomposes into a direct sum of $2^{m}$
subspaces that are simultaneous eigenspaces with eigenvalues $\pm 1$ for all
Atkin-Lehner involutions. For convenience, we introduce the following
notation.

######  Definition 2.6.

Let $G_{N}$ denote the group of characters of
$\Gamma_{0}^{+}(N)/\Gamma_{0}(N)$, that is, let $G_{N}$ be the group of all
homomorphisms from $\Gamma_{0}^{+}(N)/\Gamma_{0}(N)$ to $\\{\pm 1\\}$. We
shall let $\epsilon_{0}$ denote the trivial character of
$\Gamma_{0}^{+}(N)/\Gamma_{0}(N)$.

For $\epsilon\in G_{N}$, we let

$\mathscr{M}_{k}(\Gamma_{0}^{+}(N),\epsilon)=\left\\{f\in\mathscr{M}_{k}(\Gamma_{0}(N)):f|_{k}w_{e}=\epsilon(W_{e})f\text{
for all }e\right\\}.$

The notation $\mathscr{M}_{k}^{\ast}(\Gamma_{0}^{+}(N),\epsilon)$ is similarly
defined. (The choice of the notation
$\mathscr{M}_{k}(\Gamma_{0}^{+}(N),\epsilon)$ instead of
$\mathscr{M}_{k}(\Gamma_{0}(N),\epsilon)$ is to stress that we are considering
elements of $\mathscr{M}_{k}(\Gamma_{0}(N),\epsilon)$ as modular forms on
$\Gamma_{0}^{+}(N)$ with characters.)

With this notation, the Atkin-Lehner decomposition we discussed above can be
written as

$\mathscr{M}_{k}(\Gamma_{0}(N))=\bigoplus_{\epsilon\in
G_{N}}\mathscr{M}_{k}(\Gamma_{0}^{+}(N),\epsilon).$

We remark that if $\Gamma$ is a subgroup of $\Gamma_{0}^{+}(N)$ containing
$\Gamma_{0}(N)$, then

$\mathscr{M}_{k}(\Gamma)=\bigoplus_{\epsilon\in
G_{N},\Gamma/\Gamma_{0}(N)\in\ker\epsilon}\mathscr{M}_{k}(\Gamma_{0}^{+}(N),\epsilon)$

Thus, the dimension of each $\mathscr{M}_{k}(\Gamma_{0}^{+}(N),\epsilon)$ can
be derived from the following general formula and the inclusion-exclusion
principle.

######  Proposition 2.7 ([16, Theorem 2.23]).

Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$ commensurable with
$\mathrm{SL}(2,\mathbb{Z})$. Suppose that the modular curve $X(\Gamma)$ has
genus $g$, $c$ cusps, and $m$ elliptic points of order $e_{1},\ldots,e_{m}$,
respectively. Then for a nonnegative even integer $k$, we have

$\dim\mathscr{M}_{k}(\Gamma)=(k-1)(g-1)+\sum_{j=1}^{m}\left\lfloor\frac{k}{2}\left(1-\frac{1}{e_{j}}\right)\right\rfloor+\frac{ck}{2}.$

######  Example 2.8.

Consider the case of $\Gamma_{0}(2)$. Since there is only one prime divisor
here, we shall use the following more intuitive notations instead.

Let $\mathscr{M}_{k}(\Gamma_{0}^{+}(2),\pm)$ denote the Atkin-Lehner
eigenspaces with eigenvalue $\pm 1$, respectively. We observe that
$\mathscr{M}_{k}(\Gamma_{0}^{+}(2),+)$ is simply
$\mathscr{M}_{k}(\Gamma_{0}^{+}(2))$. To determine its dimension, we need to
know how many cusps and elliptic points $X_{0}(2)/w_{2}$ has.

The modular curve $X_{0}(2)$ has genus $0$, $2$ cusps, and an elliptic point
$(1+i)/2$ of order $2$. Thus,

$\dim\mathscr{M}_{k}(\Gamma_{0}(2))=1+\left\lfloor\frac{k}{4}\right\rfloor.$

To determine how many cusps and elliptic points $X_{0}(2)/w_{2}$ has, we note
that $X_{0}(2)\to X_{0}(2)/w_{2}$ is a covering of degree $2$. Thus, by the
Riemann-Hurwitz formula, the covering ramifies at $2$ points and these two
points are fixed points of $w_{2}$. Now the matrix
$\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}2&-1\\\
2&0\end{smallmatrix}\right)$ fixes $(1+i)/2$, i.e., the Atkin-Lehner $w_{2}$
fixes $(1+i)/2$. Thus, $(1+i)/2$ becomes an elliptic point of order $4$ on
$X_{0}(2)/w_{2}$. Another fixed point of $w_{2}$ is $i/\sqrt{2}$, fixed by
$\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}0&-1\\\
2&0\end{smallmatrix}\right)$. Since we have found two ramified points, all
other points are unramified. We conclude that $X_{0}(2)/w_{2}$ has one cusp,
one elliptic point of order $2$, and one elliptic point of order $4$. Hence,

$\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(2),+)=\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(2))=1-\frac{k}{2}+\left\lfloor\frac{k}{4}\right\rfloor+\left\lfloor\frac{3k}{8}\right\rfloor.$

It follows that

$\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(2),-)=\dim\mathscr{M}_{k}(\Gamma_{0}(2))-\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(2),+)=\frac{k}{2}-\left\lfloor\frac{3k}{8}\right\rfloor.$

Now we consider the Atkin-Lehner decomposition of the spaces of nearly
holomorphic modular forms and quasimodular forms. Just like
$\mathscr{M}_{k}(\Gamma_{0}(N))$, we can decompose
$\mathscr{M}_{k}^{\ast}(\Gamma_{0}(N))$ into a direct sum

$\mathscr{M}_{k}^{\ast}(\Gamma_{0}(N))=\bigoplus_{\epsilon\in
G_{N}}\mathscr{M}_{k}^{\ast}(\Gamma_{0}^{+}(N),\epsilon).$

######  Definition 2.9.

For $\epsilon\in G_{N}$, we let
$\widetilde{\mathscr{M}}_{k}(\Gamma_{0}^{+}(N),\epsilon)$ denote the space of
quasimodular forms that are holomorphic parts of functions in
$\mathscr{M}_{k}^{\ast}(\Gamma_{0}^{+}(N),\epsilon)$. Also, for a nonnegative
integer $r$, we let $\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma_{0}^{+}(N),\epsilon)$ denote the subspace of
$\widetilde{\mathscr{M}}_{k}(\Gamma_{0}^{+}(N),\epsilon)$ consisting of
quasimodular forms of depth $\leq r$. When $\epsilon=\epsilon_{0}$ is the
trivial character, we will often use the notation
$\widetilde{\mathscr{M}}_{k}^{\leq r}(\Gamma_{0}^{+}(N))$ in place of
$\widetilde{\mathscr{M}}_{k}^{\leq r}(\Gamma_{0}^{+}(N),\epsilon_{0})$.

In addition, when $N=p^{n}$ is a prime power, we will also use the more
intuitive notations $\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma_{0}^{+}(N),+)$ (or simplfy $\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma_{0}^{+}(N))$) and $\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma_{0}^{+}(N),-)$ to denote the spaces
$\widetilde{\mathscr{M}}_{k}^{\leq r}(\Gamma_{0}^{+}(N),\epsilon_{0})$ and
$\widetilde{\mathscr{M}}_{k}^{\leq r}(\Gamma_{0}^{+}(N),\epsilon)$, where
$\epsilon$ is the nontrivial element in $G_{N}$.

######  Remark 2.10.

Let $N$ be a positive integer. Let

$\phi(z)=\sum_{e|N,(e,N/e)=1}eE_{2}(ez),$

which is a scalar multiple of the logarithmic derivative of
$\prod_{e}\Delta(ez)$ and hence a quasimodular form of weight $2$ and depth
$1$ on $\Gamma_{0}^{+}(N)$. Then for $\epsilon\in G_{N}$, it is easy to see
that

$\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma_{0}^{+}(N),\epsilon)=\bigoplus_{j=0}^{r}\phi^{j}\mathscr{M}_{k-2j}(\Gamma_{0}^{+}(N),\epsilon).$

and hence

$\dim\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma_{0}^{+}(N),\epsilon)=\sum_{j=0}^{r}\dim\mathscr{M}_{k-2j}(\Gamma_{0}^{+}(N),\epsilon).$

Note that the choice of $\phi$ is not unique in general. For example, when
$N=4$, we can choose $\phi(z)=E_{2}(2z)$. Then the statement
$\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma_{0}^{+}(4),\epsilon)=\oplus_{j=0}^{r}\phi^{j}\mathscr{M}_{k-2j}(\Gamma_{0}^{+}(4),\epsilon)$
still holds.

We will need the following property in the subsequent discussion.

######  Lemma 2.11.

Let $\epsilon\in G_{N}$. We have

$D_{q}\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma_{0}^{+}(N),\epsilon)\subseteq\widetilde{\mathscr{M}}_{k+2}^{\leq
r+1}(\Gamma_{0}^{+}(N),\epsilon).$

###### Proof.

This follows from Lemma 2.3. ∎

###  2.3. Extremal quasimodular forms

Several examples given in the paper are involved with the notion of extremal
quasimodular forms, introduced first in [8], so here we shall review their
definition and discuss the existence and uniqueness of such quasimodular
forms.

######  Definition 2.12.

Given $\epsilon\in G_{N}$, let
$\widetilde{\mathscr{M}}=\widetilde{\mathscr{M}}_{k}^{\leq
r}(\Gamma_{0}^{+}(N),\epsilon)$ be the Atkin-Lehner subspace of
$\widetilde{\mathscr{M}}_{k}^{\leq r}(\Gamma_{0}(N))$ corresponding to
$\epsilon$. An element $f\in\widetilde{\mathscr{M}}$ is said to be _extremal_
if its vanishing order at $\infty$ is equal to
$\dim\widetilde{\mathscr{M}}-1$. We say $f$ is _normalized_ if its leading
Fourier coefficient is $1$.

######  Remark 2.13.

Note that in [10], Pellarin called a nonzero quasimodular form $f$
_analytically extremal_ if its vanishing order at $\infty$ is the largest
among all nonzero elements of the space $f$ belongs to. Pellarin’s definition
of extremality seems to be closer to the literal meaning of the word extremal.
However, we will stick to the definition introduced by Kaneko and Koike. It is
clear that if $f$ is analytic extremal in a space $\widetilde{\mathscr{M}}$ of
quasimodular forms and its vanishing order at $\infty$ is
$\dim\widetilde{\mathscr{M}}-1$, then it is extremal in the sense of Kaneko
and Koike.

In general, it is a complicated problem to show the existence and the
uniqueness of extremal quasimodular forms. In [10], Pellarin proved that if
$r\leq 4$, then a normalized extremal quasimodular form in
$\widetilde{\mathscr{M}}_{k}^{\leq r}(\mathrm{SL}(2,\mathbb{Z}))$ exists and
is unique. His argument also works in some other cases. For the convenience of
the reader, we reproduce the proof, adapted to the current setting.

######  Lemma 2.14.

Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$ commensurable with
$\mathrm{SL}(2,\mathbb{Z})$ and $\chi$ be a character of $\Gamma$. Let $\phi$
be a quasimodular form of weight $2$ and depth $1$ on $\Gamma$, i.e.,

$(cz+d)^{-2}\phi(\gamma z)=\phi(z)+\frac{\alpha c}{cz+d}$

for all $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma$ for some nonzero complex number
$\alpha$. For
$f(z)=f_{0}(z)+\phi(z)f_{1}(z)\in\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma,\chi)$, where $k$ is a positive integer and
$f_{j}\in\mathscr{M}_{k-2j}(\Gamma,\chi)$, $j=1,2$, let

$F_{f}(z)=\begin{pmatrix}zf(z)+\alpha f_{1}(z)\\\ f(z)\end{pmatrix}$

and define the Wronskian $W_{f}(z)$ by
$W_{f}(z)=\det(F_{f}(z),F_{f}^{\prime}(z))$. Then $W_{f}(z)$ is a modular form
of weight $2k$ with character $\chi^{2}$ on $\Gamma$. Moreover, we have

(2.6) $v_{\infty}(W_{f})=\begin{cases}2v_{\infty}(f),&\text{if
}v_{\infty}(f)\leq v_{\infty}(f_{1}),\\\
v_{\infty}(f)+v_{\infty}(f_{1}),&\text{if
}v_{\infty}(f)>v_{\infty}(f_{1}),\end{cases}$

where for a modular form or a quasimodular form $g$ we let $v_{\infty}(g)$
denote the vanishing order of $g$ at $\infty$.

Note that in the case $\Gamma=\mathrm{SL}2,\mathbb{Z})$, a proof of the fact
that $W_{f}(z)$ is a modular form of weight $2k$ can be found in [3]. (This is
also mentioned in [10], although no proof is given.) Since our setting
involves quasimodular forms with characters on general $\Gamma$, we shall
present a proof here.

###### Proof of Lemma 2.14.

First of all, by the definition of $W_{f}(z)$, we have

$W_{f}(z)=-f(z)^{2}+\alpha(f_{1}(z)f^{\prime}(z)-f_{1}^{\prime}(z)f(z)),$

from which (2.6) immediately follows. Also, by (an extension of) Proposition
2.5(i), the function $W_{f}(z)$ belongs to $\widetilde{\mathscr{M}}_{2k}^{\leq
2}(\Gamma,\chi^{2})$. On the other hand, for
$\gamma=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in\Gamma$,
we compute that

$f(\gamma z)=\chi(\gamma)(cz+d)^{k}\left(f(z)+\frac{\alpha
c}{cz+d}f_{1}(z)\right)$

and

$\begin{split}(\gamma z)f(\gamma z)+\alpha f_{1}(\gamma
z)&=\chi(\gamma)\frac{az+b}{cz+d}(cz+d)^{k}\left(f(z)+\frac{\alpha
c}{cz+d}f_{1}(z)\right)\\\
&\qquad+\chi(\gamma)\alpha(cz+d)^{k-2}f_{1}(z).\end{split}$

Using the relation $ad-bc=1$, we may simplify the last expression to

$\chi(\gamma)(cz+d)^{k-1}\left((az+b)f(z)+\alpha af_{1}(z)\right).$

Hence,

$F_{f}(\gamma z)=\chi(\gamma)(cz+d)^{k-1}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}F_{f}(z)$

and

$F_{f}^{\prime}(\gamma z)=\chi(\gamma)\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\left((k-1)(cz+d)^{k}F_{f}(z)+(cz+d)^{k+1}F_{f}^{\prime}(z)\right).$

From the computation, we see that the Wronskian $W_{f}(z)$ satisfies

$W_{f}(\gamma z)=\chi(\gamma)^{2}(cz+d)^{2k}W_{f}(z).$

That is, $W_{f}(z)$ is a quasimodular form of weight $2k$ and depth $\leq 2$
with character $\chi^{2}$ on $\Gamma$ that is actually modular. This proves
that $W_{f}(z)\in\mathscr{M}_{2k}(\Gamma,\chi^{2})$. ∎

We now prove an extension of [10, Theorem 1.3].

######  Proposition 2.15.

Suppose that $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(N),\epsilon)$ is an Atkin-Lehner subspace of
$\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}(N))$ such that

(2.7)
$\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(N))=\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(N),\epsilon)$

and the maximal vanishing order at $\infty$ of a nonzero element of
$\mathscr{M}_{2k}(\Gamma_{0}^{+}(N))$ is equal to
$\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(N))-1$. Then

  1. (i)

extremal quasimodular forms exist in $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(N),\epsilon)$ and are unique up to scalars, and

  2. (ii)

if $f$ is an extremal quasimodular form in $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(N),\epsilon)$, then we have
$v_{\infty}(f)=v_{\infty}(W_{f})$, where $W_{f}$ is the Wronskian associated
to $f$ defined in Lemma 2.14.

###### Proof.

The proof follows that of [10, Theorem 1.3]. Set
$\widetilde{\mathscr{M}}=\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(N),\epsilon)$. Let

$\phi(z)=\sum_{e|N,(e,N/e)=1}eE_{2}(ez),$

which is a scalar multiple of the logarithmic derivative of
$\prod_{e}\Delta(ez)$. It is a holomorphic quasimodular form of weight $2$ and
depth $1$ on $\Gamma_{0}^{+}(N)$ and satisfies

$\left(\phi\big{|}_{2}\gamma\right)(z)=\phi(z)+\frac{\alpha c}{cz+d}$

for all $\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma_{0}^{+}(N)$ for some nonzero complex
number $\alpha$. Assume that $f$ is an element of $\widetilde{\mathscr{M}}$.
According to Remark 2.10, we have $f=f_{0}+\phi f_{1}$ for some
$f_{j}\in\mathscr{M}_{k-2j}(\Gamma_{0}^{+}(N),\epsilon)$, $j=1,2$. Let
$W_{f}(z)$ be Wronskian associated to $f$ defined in Lemma 2.14. By the lemma,
$W_{f}(z)$ is a modular form in $\mathscr{M}_{2k}(\Gamma_{0}^{+}(N))$. (Note
that $\epsilon^{2}=\epsilon_{0}$ for any $\epsilon\in G_{N}$.) Moreover, (2.6)
implies that $W_{f}\neq 0$ whenever $f\neq 0$. Another consequence of (2.6) is
that, by the assumptions that (2.7) holds and that the maximal vanishing order
of any nonzero element of $\mathscr{M}_{2k}(\Gamma_{0}^{+}(N))$ is
$\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(N))-1$, we have

$\begin{split}\dim\widetilde{\mathscr{M}}-1&=\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(N))-1\geq
v_{\infty}(W_{f})\geq v_{\infty}(f)\end{split}$

for all nonzero elements $f$ of $\widetilde{\mathscr{M}}$. This implies that
the vanishing order at $\infty$ of an analytically extremal quasimodular form
in $\widetilde{\mathscr{M}}$ is $\dim\widetilde{\mathscr{M}}-1$ and hence
extremal quasimodular forms exists in
$\widetilde{\mathscr{M}}=\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(N),\epsilon)$ and are unique up to scalars. The argument
above also shows that if $f$ is an extremal quasimodular form in
$\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}^{+}(N),\epsilon)$, then
$v_{\infty}(f)=v_{\infty}(W_{f})$. ∎

Now we use this result to prove the existence and uniqueness of extremal
quasimodular forms in the case $N=2$ or $N=3$. Note that Sakai [11] already
established the existence in the case of
$\widetilde{\mathscr{M}}_{k}(\Gamma_{0}^{+}(N),+)$, $N=2,3$.

######  Corollary 2.16.

Let $N=2$ or $N=3$. Then for all positive even integers $k$, extremal
quasimodular forms exist in $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(N))$ and $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(N),-)$ and are unique up to scalars.

###### Proof.

In Example 2.8, we have seen that

(2.8)
$\begin{split}\dim\mathscr{M}_{k}(\Gamma_{0}(2))&=1+\left\lfloor\frac{k}{4}\right\rfloor,\\\
\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(2))&=1-\frac{k}{2}+\left\lfloor\frac{k}{4}\right\rfloor+\left\lfloor\frac{3k}{8}\right\rfloor,\\\
\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(2),-)&=\frac{k}{2}-\left\lfloor\frac{3k}{8}\right\rfloor.\end{split}$

Hence, according to Remark 2.10, we have

(2.9) $\dim\widetilde{\mathscr{M}}^{\leq
1}_{k}(\Gamma_{0}^{+}(2))=\begin{cases}1+k/4,&\text{if }k\equiv 0\
\mathrm{mod}\ 8,\\\ (k+2)/4,&\text{if }k\equiv 2,6\ \mathrm{mod}\ 8,\\\
k/4,&\text{if }k\equiv 4\ \mathrm{mod}\ 8,\end{cases}$

and

(2.10) $\dim\widetilde{\mathscr{M}}^{\leq
1}_{k}(\Gamma_{0}^{+}(2),-)=\begin{cases}k/4,&\text{if }k\equiv 0\
\mathrm{mod}\ 8,\\\ (k+2)/4,&\text{if }k\equiv 2,6\ \mathrm{mod}\ 8,\\\
1+k/4,&\text{if }k\equiv 4\ \mathrm{mod}\ 8.\end{cases}$

It follows that if $k\not\equiv 4\ \mathrm{mod}\ 8$, then

$\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(2))=\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(2))$

and if $k\not\equiv 0\ \mathrm{mod}\ 8$, then

$\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(2))=\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(2),-).$

The condition (2.7) holds for these spaces.

For the other condition of Proposition 2.15, we observe that the condition is
equivalent to that there is a basis $\\{g_{j}\\}$ for
$\mathscr{M}_{2k}(\Gamma_{0}^{+}(2))$ such that $v_{\infty}(g_{j})=j-1$ for
each $j=1,\ldots,\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(2))$. Indeed, we find
that $g_{j}(z)=M_{4}(z)^{k/2-2j}M_{8}(z)^{j}$ for $j=0,\ldots,\left\lfloor
k/4\right\rfloor$, where

$M_{4}(z)=\frac{1}{5}(4E_{4}(2z)+E_{4}(z)),\qquad
M_{8}(z)=\eta(z)^{8}\eta(2z)^{8},$

form such a basis. Therefore, by Proposition 2.15, extremal quasimodular forms
exist and are unique up to scalars in the spaces above. For the remaining two
cases $\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}^{+}(2))$ for $k\equiv
4\ \mathrm{mod}\ 8$ and $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(2),-)$ for $k\equiv 0\ \mathrm{mod}\ 8$, we observe that
when $k\equiv 4\ \mathrm{mod}\ 8$, we have

$\dim\widetilde{\mathscr{M}}_{k-4}^{\leq
1}(\Gamma_{0}^{+}(2))=\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(2)),$

and the map

$f(z)\longmapsto M_{4}(z)f(z)$

defines an isomorphism between the two spaces. It is clear that a quasimodular
form $f(z)$ is extremal in $\widetilde{\mathscr{M}}_{k-4}^{\leq
1}(\Gamma_{0}^{+}(2))$ if and only if $M_{4}(z)f(z)$ is extremal in
$\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}^{+}(2))$. Since we have seen
that extremal quasimodular forms exist and are unique up to scalars in the
former space, the same thing holds for the latter space. The same argument
works for the last case $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(2),-)$, $k\equiv 0\ \mathrm{mod}\ 8$. This proves the case
$N=2$.

For the case $N=3$, we note that $\Gamma_{0}(3)$ has two cusps and one
elliptic point of order $3$, while $\Gamma_{0}^{+}(3)$ has one cusp, one
elliptic point of order $2$, represented by $i/\sqrt{3}$ with stabilizer
subgroup generated by $\frac{1}{\sqrt{3}}\left(\begin{smallmatrix}0&-1\\\
3&0\end{smallmatrix}\right)$, and one elliptic point of order $6$, represented
by $(3+\sqrt{-3})/6$ with stabilizer subgroup generated by
$\frac{1}{\sqrt{3}}\left(\begin{smallmatrix}3&-1\\\
3&0\end{smallmatrix}\right)$. Thus, by Proposition 2.7, we have

(2.11)
$\begin{split}\dim\mathscr{M}_{k}(\Gamma_{0}(3))&=1+\left\lfloor\frac{k}{3}\right\rfloor,\\\
\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(3))&=1-\frac{k}{2}+\left\lfloor\frac{k}{4}\right\rfloor+\left\lfloor\frac{5k}{12}\right\rfloor,\\\
\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(3),-)&=\frac{k}{2}+\left\lfloor\frac{k}{3}\right\rfloor-\left\lfloor\frac{k}{4}\right\rfloor-\left\lfloor\frac{5k}{12}\right\rfloor.\end{split}$

From these formulas, we deduce that

$\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3))=\begin{cases}1+k/3,&\text{if }k\equiv 0\ \mathrm{mod}\
12,\\\ (k+1)/3,&\text{if }k\equiv 2,8\ \mathrm{mod}\ 12,\\\ (k-1)/3,&\text{if
}k\equiv 4\ \mathrm{mod}\ 12,\\\ k/3,&\text{if }k\equiv 6\ \mathrm{mod}\
12,\\\ (k+2)/3,&\text{if }k\equiv 10\ \mathrm{mod}\ 12,\end{cases}$

and

$\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),-)=\begin{cases}k/3,&\text{if }k\equiv 0\ \mathrm{mod}\
12,\\\ (k+1)/3,&\text{if }k\equiv 2,8\ \mathrm{mod}\ 12,\\\ (k+2)/3,&\text{if
}k\equiv 4\ \mathrm{mod}\ 12,\\\ 1+k/3,&\text{if }k\equiv 6\ \mathrm{mod}\
12,\\\ (k-1)/3,&\text{if }k\equiv 10\ \mathrm{mod}\ 12,\end{cases}$

Therefore, the condition (2.7) in Proposition 2.15 is met for
$\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}^{+}(3))$ with $k\equiv
0,2,8,10\ \mathrm{mod}\ 12$ and $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),-)$ with $k\equiv 2,4,6,8\ \mathrm{mod}\ 12$. For the
other condition of the proposition, we may use

$\begin{split}M_{4}(z)&=\frac{1}{10}(9E_{4}(3z)+E_{4}(z)),\\\
M_{8}(z)&=\frac{1}{2}(3E_{2}(3z)-E_{2}(z))\eta(z)^{6}\eta(3z)^{6},\\\
M_{12}(z)&=\eta(z)^{12}\eta(3z)^{12}\end{split}$

to construct a basis $\\{g_{j}\\}$ for $\mathscr{M}_{2k}(\Gamma_{0}^{+}(3))$
with the property $v_{\infty}(g_{j})=j-1$ for all
$j=1,\ldots,\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(3))$. Hence, the second
condition of Proposition 2.15 also holds and we conclude that extremal
quasimodular forms exist in $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3))$ for $k\equiv 0,2,8,10\ \mathrm{mod}\ 12$ and
$\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}^{+}(3),-)$ for $k\equiv
2,4,6,8\ \mathrm{mod}\ 12$ and are unique up to scalars.

Finally, similar to the case $N=2$, for the remaining cases, we observe that

(2.12) $\begin{split}\dim\widetilde{\mathscr{M}}_{k-4}^{\leq
1}(\Gamma_{0}^{+}(3))&=\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3)),\ \text{for }k\equiv 4\ \mathrm{mod}\ 12,\\\
\dim\widetilde{\mathscr{M}}_{k-4}^{\leq
1}(\Gamma_{0}^{+}(3),-)&=\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),-),\ \text{for }k\equiv 10\ \mathrm{mod}\ 12,\end{split}$
(2.13) $\begin{split}\dim\widetilde{\mathscr{M}}_{k-2}^{\leq
1}(\Gamma_{0}^{+}(3),-)&=\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3)),\ \text{for }k\equiv 6\ \mathrm{mod}\ 12,\\\
\dim\widetilde{\mathscr{M}}_{k-2}^{\leq
1}(\Gamma_{0}^{+}(3))&=\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),-),\ \text{for }k\equiv 0\ \mathrm{mod}\ 12,\end{split}$

and that the map

$f(z)\longmapsto(9E_{4}(3z)+E_{4}(z))f(z)$

defines an isomorphism for the pairs of spaces in (2.12), while the map

$f(z)\longmapsto(3E_{2}(3z)-E_{2}(z))f(z)$

defines an isomorphism for the pairs of spaces in (2.13). From these
observations we conclude that extremal quasimodular forms exists and are
unique up to scalars in all $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3))$ and $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),-)$. ∎

###  2.4. Quasimodular forms of odd weights on $\Gamma_{1}(3)$

In this paper, quasimodular forms of odd weights on

$\Gamma_{1}(3):=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z}):c\equiv 0\ \mathrm{mod}\
3,a,d\equiv 1\ \mathrm{mod}\ 3\right\\}$

will also appear as solutions of modular differential equations when
$\Gamma=\Gamma_{0}^{+}(3)$, so we will review their properties in this
section.

For a nonnegative integer $k$ (even or odd), let
$\mathscr{M}_{k}(\Gamma_{1}(3))$ denote the set of modular forms of weight $k$
on $\Gamma_{1}(3)$. Note that when $k$ is odd, modular forms of weight $k$ on
$\Gamma_{1}(3)$ can also be defined as modular forms on $\Gamma_{0}(3)$ with
nebentype character $\chi\left(\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\right)=\left(\frac{-3}{d}\right)$. By Theorem 2.25
of [16], we have

(2.14)
$\dim\mathscr{M}_{k}(\Gamma_{1}(3))=1+\left\lfloor\frac{k}{3}\right\rfloor.$

Hence, we have

$\sum_{k=0}^{\infty}\dim\mathscr{M}_{k}(\Gamma_{1}(3))x^{k}=\frac{1}{(1-x)(1-x^{3})},$

and the graded ring $\bigoplus_{k=0}^{\infty}\mathscr{M}(\Gamma_{1}(3))$ of
modular forms is freely generated by

(2.15) $M_{1}(z)=\sum_{(m,n)\in\mathbb{Z}^{2}}q^{m^{2}+mn+n^{2}},\qquad
q=e^{2\pi iz},$

and

(2.16)
$M_{3}(z)=\frac{9E_{4}(3z)-E_{4}(z)}{8M_{1}(z)}=\frac{\eta(z)^{9}}{\eta(3z)^{3}}-27\frac{\eta(3z)^{9}}{\eta(z)^{3}}.$

Similar to the case of $\Gamma_{0}(N)$, the Atkin-Lehner involutions normalize
$\Gamma_{1}(N)$. Hence, we may decompose $\mathscr{M}_{k}(\Gamma_{1}(3))$ into
eigenspaces of the Atkin-Lehner involution $w_{3}$. Let
$S=\frac{1}{\sqrt{3}}\left(\begin{smallmatrix}0&-1\\\
3&0\end{smallmatrix}\right)$. Because $S^{2}=-I$, the eigenvalues of $w_{3}$
in the case of odd $k$ can only be $\pm i$. For convenience, we shall consider
the eigenfunctions as modular forms on $\Gamma_{0}^{+}(3)$ with characters.
More concretely, we recall that $\Gamma_{0}^{+}(3)$ is generated by

$S=\frac{1}{\sqrt{3}}\begin{pmatrix}0&-1\\\ 3&0\end{pmatrix},\qquad
R=TS=\frac{1}{\sqrt{3}}\begin{pmatrix}3&-1\\\ 3&0\end{pmatrix},$

where $T=\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)$. They
satisfy $S^{2}=R^{6}=-I$ and there is no other relation between $S$ and $R$.
In other words, a presentation of the group $\Gamma_{0}^{+}(3)$ is $\langle
S,R:S^{4}=1,S^{2}=R^{6}\rangle$. We let $\chi$ be the character of
$\Gamma_{0}^{+}(3)$ such that

(2.17) $\chi(S)=\chi(R)=-i,$

and $\overline{\chi}$ be its complex conjugate. Then the eigenspace of $w_{3}$
with eigenvalue $-i$ (respectively, $i$) can be identified with
$\mathscr{M}(\Gamma_{0}^{+}(3),\chi)$ (respectively,
$\mathscr{M}(\Gamma_{0}^{+}(3),\overline{\chi})$). Using the transformation
formula for the Dedekind eta function (see [17, Pages 125–127]), we can show
that

$M_{3}(z)\in\mathscr{M}_{3}(\Gamma_{0}^{+}(3),\chi)$

and hence also

$M_{1}(z)\in\mathscr{M}_{1}(\Gamma_{0}^{+}(3),\chi),$

where $M_{1}(z)$ and $M_{3}(z)$ are defined by (2.15) and (2.16),
respectively.

######  Lemma 2.17.

Let $k$ be a positive odd integer. Then

$\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\chi)=\begin{cases}(k+5)/6,&\text{if
}k\equiv 1\ \mathrm{mod}\ 12,\\\ (k+3)/6,&\text{if }k\equiv 3,9\ \mathrm{mod}\
12,\\\ (k+1)/6,&\text{if }k\equiv 5,11\ \mathrm{mod}\ 12,\\\ (k-1)/6,&\text{if
}k\equiv 7\ \mathrm{mod}\ 12,\end{cases}$

and

$\dim\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\overline{\chi})=\begin{cases}(k-1)/6,&\text{if
}k\equiv 1\ \mathrm{mod}\ 12,\\\ (k+3)/6,&\text{if }k\equiv 3,9\ \mathrm{mod}\
12,\\\ (k+1)/6,&\text{if }k\equiv 5,11\ \mathrm{mod}\ 12,\\\ (k+5)/6,&\text{if
}k\equiv 7\ \mathrm{mod}\ 12.\end{cases}$

###### Proof.

Observe that
$\mathscr{M}_{k-1}(\Gamma_{0}^{+}(3))\cup\mathscr{M}_{k-1}(\Gamma_{0}^{+}(3),-)=\mathscr{M}_{k-1}(\Gamma_{0}(3))$
and
$\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\chi)\cup\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\overline{\chi})=\mathscr{M}_{k}(\Gamma_{1}(3))$.
Also,

$\begin{split}M_{1}\mathscr{M}_{k-1}(\Gamma_{0}^{+}(3))&\subseteq\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\chi),\\\
M_{1}\mathscr{M}_{k-1}(\Gamma_{0}^{+}(3),-)&\subseteq\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\overline{\chi}).\end{split}$

When $k$ is not a multiple of $3$, by (2.14), we have

$\dim\mathscr{M}_{k}(\Gamma_{1}(3))=\dim\mathscr{M}_{k-1}(\Gamma_{0}(3)).$

Hence, both inclusions above are equality. When $3|k$, we use

$\begin{split}M_{1}\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\chi)&\subseteq\mathscr{M}_{k+1}(\Gamma_{0}^{+}(3),-),\\\
M_{1}\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\overline{\chi})&\subseteq\mathscr{M}_{k+1}(\Gamma_{0}^{+}(3))\end{split}$

instead. Again, by the dimension formula (2.14), both inclusions are equality.
This proves the lemma. ∎

Passing to quasimodular forms, we have the following dimension formulas.

######  Lemma 2.18.

Let $k$ be a positive odd integer. We have

$\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),\chi)=\begin{cases}(k+2)/3,&\text{if }k\equiv 1\
\mathrm{mod}\ 12,\\\ 1+k/3,&\text{if }k\equiv 3\ \mathrm{mod}\ 12,\\\
(k+1)/3,&\text{if }k\equiv 5,11\ \mathrm{mod}\ 12,\\\ (k-1)/3,&\text{if
}k\equiv 7\ \mathrm{mod}\ 12,\\\ k/3,&\text{if }k\equiv 9\ \mathrm{mod}\
12,\end{cases}$

and

$\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),\overline{\chi})=\begin{cases}(k-1)/3,&\text{if }k\equiv
1\ \mathrm{mod}\ 12,\\\ k/3,&\text{if }k\equiv 3\ \mathrm{mod}\ 12,\\\
(k+1)/3,&\text{if }k\equiv 5,11\ \mathrm{mod}\ 12,\\\ (k+2)/3,&\text{if
}k\equiv 7\ \mathrm{mod}\ 12,\\\ 1+k/3,&\text{if }k\equiv 9\ \mathrm{mod}\
12.\end{cases}$

######  Proposition 2.19.

For each positive odd integer $k$, an extremal quasimodular form exists in
$\widetilde{\mathscr{M}}_{k}(\Gamma_{0}^{+}(3),\chi)$ or
$\widetilde{\mathscr{M}}_{k}(\Gamma_{0}^{+}(3),\overline{\chi})$ and is unique
up to scalars.

###### Proof.

For a quasimodular form $f$ in $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),\chi)$ or $\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),\overline{\chi})$, let $W_{f}$ be the Wronskian
associated $f$ defined in Lemma 2.14. By the lemma, we have
$W_{f}\in\mathscr{M}_{2k}(\Gamma_{0}^{+}(3),\chi^{2})=\mathscr{M}_{2k}(\Gamma_{0}^{+}(3),-)$.
Since

$\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(3),-)=\begin{cases}(k+2)/3,&\text{if
}k\equiv 1\ \mathrm{mod}\ 6,\\\ 1+k/3,&\text{if }k\equiv 3\ \mathrm{mod}\
6,\\\ (k+1)/3,&\text{if }k\equiv 5\ \mathrm{mod}\ 6,\end{cases}$

we find that

$\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),\chi)=\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(3),-)$

for $k\equiv 1,3,5,11\ \mathrm{mod}\ 12$, and

$\dim\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),\overline{\chi})=\dim\mathscr{M}_{2k}(\Gamma_{0}^{+}(3),-)$

for $k\equiv 5,7,9,11\ \mathrm{mod}\ 12$. Hence, extremal quasimodular forms
exist in these spaces and is unique up to scalars. For the remaining four
spaces, we use the facts that

$M_{1}\widetilde{\mathscr{M}}_{k-1}^{\leq
1}(\Gamma_{0}^{+}(3))=\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),\chi)$

for $k\equiv 1,7,9,11\ \mathrm{mod}\ 12$, and

$M_{1}\widetilde{\mathscr{M}}_{k-1}^{\leq
1}(\Gamma_{0}^{+}(3),-)=\widetilde{\mathscr{M}}_{k}^{\leq
1}(\Gamma_{0}^{+}(3),\overline{\chi})$

for $k\equiv 1,3,5,7\ \mathrm{mod}\ 12$ to obtain the same conclusion. ∎

##  3\. Modular ordinary differential equations

The purpose of this section is to prove that for any quasimodular form $f(z)$
of depth $1$, $f(z)/\sqrt{W_{f}(z)}$ is a solution of a modular differential
equation. Throughout the section, we assume that $\Gamma$ is a discrete
subgroup of $\mathrm{SL}(2,\mathbb{R})$ that is commensurable with
$\mathrm{SL}(2,\mathbb{Z})$, and $\phi$ is the quasimodular form of weight $2$
and depth $1$ appearing in Proposition 2.5, i.e.,

(3.1) $(cz+d)^{-2}\phi(\gamma z)=\phi(z)+\frac{\alpha
c}{cz+d},\qquad\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma,$

for some complex number $\alpha\neq 0$. Suppose that $f(z)$ is a quasimodular
form of weight $k$ and depth $1$ with character $\chi$ of finite order on
$\Gamma$. We have

(3.2) $f(z)=f_{1}(z)\phi(z)+f_{0}(z)$

for some $f_{0}\in\mathscr{M}_{k}(\Gamma,\chi)$ and $f_{1}\neq
0\in\mathscr{M}_{k-2}(\Gamma,\chi)$. Let $W_{f}(z)$ be the Wronskian
associated to $f$ defined in Lemma 2.14. It is a modular form of weight $2k$
with character $\chi^{2}$ on $\Gamma$. Define

(3.3) $g_{1}(z)=\frac{zf(z)+\alpha
f_{1}(z)}{\sqrt{W_{f}(z)}}\quad\text{and}\quad
g_{2}(z)=\frac{f(z)}{\sqrt{W_{f}(z)}}.$

Then we have $\det\left(\begin{smallmatrix}g_{1}&g_{2}\\\
g_{1}^{\prime}&g_{2}^{\prime}\end{smallmatrix}\right)=1$ and hence
$\det\left(\begin{smallmatrix}g_{1}&g_{2}\\\
g_{1}^{\prime\prime}&g_{2}^{\prime\prime}\end{smallmatrix}\right)=0$.
Therefore, both $g_{1}$ and $g_{2}$ are solutions of

(3.4) $y^{\prime\prime}(z)=-4\pi^{2}Q(z)y(z),$

where

(3.5)
$Q(z)=-\frac{1}{4\pi^{2}}\frac{g_{1}^{\prime\prime}(z)}{g_{1}(z)}=-\frac{1}{4\pi^{2}}\frac{g_{2}^{\prime\prime}(z)}{g_{2}(z)}.$

Clearly, $Q(z)$ is a single-valued meromorphic function not identically $0$ on
$\mathbb{H}$.

######  Proposition 3.1.

Let $Q(z)$ be defined by (3.5). Then $Q(z)$ is a meromorphic function on
$\mathbb{H}$ and (3.4) is Fuchsian. Moreover, if $z_{0}\in\mathbb{H}$ is a
pole of $Q(z)$, then $z_{0}$ is an apparent singularity for (3.4).

###### Proof.

Clearly, $g_{i}(z)=(z-z_{0})^{\alpha_{i}}(c_{i}+O(z-z_{0}))$ near $z_{0}$ for
some $c_{i}\neq 0$ and $\alpha_{i}\in\frac{1}{2}\mathbb{Z}$. Thus,

$Q(z)=\alpha_{i}(\alpha_{i}-1)(z-z_{0})^{-2}+O\left((z-z_{0})^{-1}\right)$

near $z_{0}$ and we have $\alpha_{1}(\alpha_{1}-1)=\alpha_{2}(\alpha_{2}-1)$,
which implies that the local exponents at $z_{i}$ are $\alpha_{i}$ and
$-\alpha_{i}+1$. (The two sets $\\{\alpha_{1},-\alpha_{1}+1\\}$ and
$\\{\alpha_{2},-\alpha_{2}+1\\}$ are identical.) Hence (3.4) is Fuchsian and
the difference of the local exponents are $1+2\alpha_{i}\in\mathbb{Z}$. Since
$g_{1}$ and $g_{2}$ are linearly independent and has no logarithmic
singularities near $z_{0}$, $z_{0}$ is obviously an apparent singularity. ∎

######  Remark 3.2.

Suppose that (3.4) has a solution $y(z)$ with a logarithmic singularity near
$z_{0}$. Then the local exponents $\rho_{1}$ and $\rho_{2}$ at $z_{0}$ satisfy
$\rho_{1}-\rho_{2}\in\mathbb{Z}$, and $z_{0}$ is not apparent. This remark is
used in the proof of Theorem 3.3. See below.

The following result is a generalization of Theorem 1.1.

######  Theorem 3.3.

Let $Q(z)$ be defined by (3.5). Then the following hold.

  1. (i)

The function $Q(z)$ is a meromorphic modular form of weight $4$ (with trivial
character) on $\Gamma$.

  2. (ii)

Moreover, $Q(z)$ is holomorphic at $\infty$ with
$Q(\infty)=\kappa_{\infty}^{2}/N^{2}$ for some
$\kappa_{\infty}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$, where $N$ is the width of
the cusp $\infty$ of $\Gamma$. Also, the cusp $\infty$ is not apparent for
(3.4).

  3. (iii)

For any cusp $s$, $Q(z)$ is holomorphic at $s$ with $Q(s)\geq 0$ and $s$ is
not apparent for (3.4).

###### Proof.

To prove the modularity of $Q(z)$, we consider
$y(z)=\left(g_{2}|_{-1}\gamma\right)(z):=(cz+d)g_{2}(\gamma z)$ for
$\gamma=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in\Gamma$.
By the definition of $g_{2}$, $y(z)$ is well-defined up to $\pm 1$. A direct
computation show that

$\left(g_{2}\big{|}_{-1}\gamma\right)^{\prime\prime}(z)=\left(g_{2}^{\prime\prime}\big{|}_{3}\gamma\right)(z)$

(which is a special case of Bol’s identity [1]), which implies that

$y^{\prime\prime}(z)=\left(Q\big{|}_{4}\gamma\right)(z)y(z)$

On the other hand, we have

$\begin{split}y(z)&=(cz+d)g_{2}(\gamma z)=\frac{(cz+d)f(\gamma
z)}{\sqrt{W_{f}(\gamma z)}}\\\ &=\pm\frac{\chi(\gamma)(cz+d)(f(z)+\alpha
cf_{1}(z)/(cz+d))}{\sqrt{\chi(\gamma)^{2}W_{f}(z)}}\\\
&=\pm(cg_{1}(z)+dg_{2}(z)).\end{split}$

Thus, $y(z)$ is also a solution of (3.4), which implies that
$\left(Q|_{4}\gamma\right)(z)=Q(z)$ for all $\gamma\in\Gamma$. This proves
(i).

Let $N$ be the width of the cusp $\infty$ and $q_{N}=e^{2\pi iz/N}$. By (2.6),
the order of $g_{2}$ at $\infty$ is

$\kappa_{\infty}=\frac{1}{2}v_{\infty}(f)-\frac{1}{2}\min(v_{\infty}(f),v_{\infty}(f_{1})),$

which is in $\frac{1}{2}\mathbb{Z}_{\geq 0}$. (Note that when
$\chi\left(\left(\begin{smallmatrix}1&N\\\
0&1\end{smallmatrix}\right)\right)\neq 1$, $v_{\infty}(f)$ and
$v_{\infty}(f_{1})$ are not integers, but we still have $v_{\infty}(f)\equiv
v_{\infty}(f_{1})\ \mathrm{mod}\ 1$ nonetheless.) Since

(3.6)
$Q(z)=-\frac{1}{4\pi^{2}}\frac{g_{2}^{\prime\prime}(z)}{g_{2}(z)}=\frac{1}{N^{2}}\frac{\left(q_{N}\frac{d}{dq_{N}}\right)^{2}g_{2}}{g_{2}},$

we find that $Q(\infty)=\kappa_{\infty}^{2}/N^{2}$.

Notice that $z$ appears in the expression of $g_{1}(z)$ and $g_{1}(z)$ is
another solution of (3.4). This obviously implies that $\infty$ is not
apparent. We now prove (iii).

Let $s\neq\infty$ be another cusp of $\Gamma$ and
$\sigma=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$ be an
element of $\mathrm{SL}(2,\mathbb{Z})$ such that $\sigma\infty=s$. Regarding
$f(z)$ as a quasimodular form on
$\Gamma^{\prime}=\ker\chi\cap\mathrm{SL}(2,\mathbb{Z})$, we have
$f(z)=h_{1}(z)E_{2}(z)+h_{0}(z)$ for some
$h_{j}\in\mathscr{M}_{k-2j}(\Gamma^{\prime})$. We have

$\begin{split}\left(g_{2}|_{-1}\sigma\right)(z)&=\frac{(cz+d)f(\sigma
z)}{\sqrt{W_{f}(\sigma
z)}}=\frac{(cz+d)\left(f|_{k}\sigma\right)(z)}{\sqrt{\left(W_{f}|_{2k}\sigma\right)(z)}}\\\
&=\frac{(cz+d)\left(\left(h_{1}|_{k-2}\sigma\right)E_{2}(z)+\left(h_{0}|_{k}\sigma\right)\right)+6c\left(h_{1}|_{k-2}\sigma\right)/\pi
i}{\sqrt{\left(W_{f}|_{2k}\sigma\right)(z)}}.\end{split}$

Since all terms in the above expression except for $cz+d$ has a
$q_{M}$-expansion, where $M$ is the width of the cusp $s$, we see that $z$
appears in the expression of $g_{2}|_{-1}\gamma$. (Note that $c\neq 0$ since
$s$ is assumed to be a cusp different from $\infty$.) This implies that $s$ is
not apparent and the local exponents of (3.4) at $s$ are $\pm\kappa_{s}$ for
some $\kappa_{s}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$ (see Remark 3.2). As
discussed earlier, near the cusp $s$, there is a solution $y_{+}(z)$ of (3.4)
of the form

$y_{+}(z)=q_{M}^{\kappa_{s}}\left(1+\sum_{j\geq 1}c_{j}q_{M}^{j}\right).$

By the same computation as (3.6), we see that $Q(s)=M^{-2}\kappa_{s}^{2}\geq
0$. This completes the proof of the theorem. ∎

######  Remark 3.4.

We remark that for a given point $z_{0}\in\mathbb{H}$, the local exponents of
(3.4) at $z_{0}$ and those at $\gamma z_{0}$ are the same for any
$\gamma\in\Gamma$. Also, if $z_{0}$ is apparent, then any equivalent point
$\gamma z_{0}$, $\gamma\in\Gamma$, is also apparent. See [4] for a proof of
this fact.

For the special case $\Gamma$ is one of $\mathrm{SL}(2,\mathbb{Z})$,
$\Gamma_{0}^{+}(2)$, and $\Gamma_{0}^{+}(3)$, we have the following
informations about local exponents at elliptic points. We first prove a lemma.

######  Lemma 3.5.

Let $\phi(z)$ be a quasimodular form of weight $2$ and depth $1$ on $\Gamma$.
Then $\phi(z)$ does not vanish at any elliptic point of $\Gamma$.

###### Proof.

Let $\widetilde{\phi}(z)$ be the nearly holomorphic modular form of weight $2$
corresponding to $\phi(z)$. We have

$\widetilde{\phi}(z)=\phi(z)+\frac{\alpha}{z-\overline{z}}$

for some nonzero complex number $\alpha$. Since $\widetilde{\phi}(z)$
transforms like a modular form of weight $2$ on $\Gamma$, it vanishes at all
elliptic points of $\Gamma$. It follows that $\phi(z)$ does not vanish at any
elliptic point of $\Gamma$. ∎

######  Proposition 3.6.

We have the following properties about local exponents of (3.4) at an elliptic
point.

  1. (i)

Let $f$ be a quasimodular form of depth $1$ in
$\widetilde{\mathscr{M}}_{k}^{\leq 1}(\mathrm{SL}(2,\mathbb{Z}))$. Let
$1/2\pm\kappa_{\rho}$ and $1/2\pm\kappa_{i}$,
$\kappa_{\rho},\kappa_{i}\in\frac{1}{2}\mathbb{N}$, be the local exponents of
(3.4) at the elliptic points $\rho=(1+\sqrt{-3})/2$ and $i=\sqrt{-1}$. Then
$(2\kappa_{\rho},3)=(2\kappa_{i},2)=1$.

  2. (ii)

Let $f$ be a quasimodular form of depth $1$ in
$\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}^{+}(2),\pm)$. Let
$1/2\pm\kappa_{\rho_{j}}$ with $\kappa_{\rho_{j}}\in\frac{1}{2}\mathbb{N}$,
$j=1,2$, be the local exponents of (3.4) at the elliptic points
$\rho_{1}=i/\sqrt{2}$ and $\rho_{2}=(1+i)/2$ of $\Gamma_{0}^{+}(2)$. Then
$(2\kappa_{\rho_{1}},2)=(2\kappa_{\rho_{2}},4)=1$.

  3. (iii)

Let $f$ be a quasimodular form of depth $1$ in
$\widetilde{\mathscr{M}}_{k}^{\leq 1}(\Gamma_{0}^{+}(3),\chi^{m})$ with
$m\in\\{0,1,2,3\\}$ and $k\equiv m\ \mathrm{mod}\ 2$ that is not a modular
form, where $\chi$ is the character of $\Gamma_{0}^{+}(3)$ defined by (2.17).
Let $1/2\pm\kappa_{\rho_{j}}$ with
$\kappa_{\rho_{j}}\in\frac{1}{2}\mathbb{N}$, $j=1,2$, be the local exponents
of (3.4) at the elliptic points $\rho_{1}=i/\sqrt{3}$ and
$\rho_{2}=(3+\sqrt{-3})/6$ of $\Gamma_{0}^{+}(3)$. Then
$(2\kappa_{\rho_{1}},2)=(2\kappa_{\rho_{2}},6)=1$.

###### Proof.

We first make a general remark that since the local exponents of (3.4) at a
point $z_{0}\in\mathbb{H}$ are of the form $1/2\pm\kappa_{z_{0}}$ for some
$\kappa_{z_{0}}\in\frac{1}{2}\mathbb{N}$, if the order of
$f(z)/\sqrt{W_{f}(z)}$ at $z_{0}$ is $-n_{z_{0}}$ for some
$n_{z_{0}}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$, then we must have
$1/2-\kappa_{z_{0}}=-n_{z_{0}}$, i.e., $\kappa_{z_{0}}=1/2+n_{z_{0}}$.

Assume that $f(z)\in\widetilde{\mathscr{M}}_{k}^{\leq
1}(\mathrm{SL}(2,\mathbb{Z}))$ that is not a modular form. Write it as
$f(z)=f_{1}(z)E_{2}(z)+f_{0}(z)$ with
$f_{j}\in\mathscr{M}_{k-2j}(\mathrm{SL}(2,\mathbb{Z}))$. We first observe that
if $f_{0}(z)$ and $f_{1}(z)$ have a common zero at $i$ (respectively, $\rho$),
then $f_{j}(z)/E_{6}(z)$ (respectively, $f_{1}(z)/E_{4}(z)$), $j=1,2$, are
holomorphic modular forms, and $f(z)/E_{6}(z)$ (respectively, $f(z)/E_{4}(z)$)
is a quasimodular form. Also, it is easy to check that
$W_{f}(z)=E_{6}(z)^{2}W_{f/E_{6}}(z)$ (respectively,
$W_{f}(z)=E_{4}(z)^{2}W_{f/E_{4}}(z)$), so the differential equation (3.4)
from $f(z)$ and that from $f(z)/E_{6}(z)$ (respectively, $f(z)/E_{4}(z)$) are
the same. Therefore, without loss of generality, we may assume that $f_{0}(z)$
and $f_{1}(z)$ have no common zero at $i$ or $\rho$. We note that this
assumption forces that the weight $k$ to be congruent to $0$ or $2$ modulo $6$
since any modular form on $\mathrm{SL}(2,\mathbb{Z})$ whose weight is not
divisible by $6$ must have a zero at $\rho$.

By Lemma 3.5, $E_{2}(z)$ does not vanish at $\rho$ and $i$. Then using

$\left(f|_{k}\gamma\right)(z)=f(z)+\frac{12cf_{1}(z)}{2\pi
i(cz+d)},\qquad\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$

and the assumption that $f_{0}(z)$ and $f_{1}(z)$ have no common zero at $i$
and $\rho$, we see that there exists an element $\gamma$ in
$\mathrm{SL}(2,\mathbb{Z})$ such that $f(\gamma\rho),f(\gamma i)\neq 0$. Since
the local exponents of (3.4) at $\rho$ (respectively, $i$) and at $\gamma\rho$
(respectively, $\gamma i$) are the same, by the remark we made at the
beginning of the proof, we have

$2\kappa_{\rho}=1+\operatorname{ord}_{\gamma\rho}W_{f}(z)=1+\operatorname{ord}_{\rho}W_{f}(z)$

(respectively, $2\kappa_{i}=1+\operatorname{ord}_{i}W_{f}(z)$).

Now consider first the case where the weight $k$ is congruent to $0$ modulo
$6$. In this case, the weight of $W_{f}(z)$ is a multiple of $12$ and hence

$\frac{\operatorname{ord}_{\rho}W_{f}(z)}{3}+\frac{\operatorname{ord}_{i}W_{f}(z)}{2}$

must be an integer. It follows that $3|\operatorname{ord}_{\rho}W_{f}$ and
$2|\operatorname{ord}_{i}W_{f}$ and therefore
$(2\kappa_{\rho},3)=(2\kappa_{i},2)=1$. Similarly, when the weight $k$ is
congruent to $2$ modulo $6$, we have

$\frac{\operatorname{ord}_{\rho}W_{f}(z)}{3}+\frac{\operatorname{ord}_{i}W_{f}(z)}{2}\equiv\frac{1}{3}\
\mathrm{mod}\ 1.$

Then $2|\operatorname{ord}_{i}W_{f}$ and $\operatorname{ord}_{\rho}W_{f}\equiv
1\ \mathrm{mod}\ 3$, from which we conclude that
$(2\kappa_{\rho},3)=(2\kappa_{i},2)=1$. This proves Part (i).

The proof of Part (ii) is similar. We may assume without loss of generality
that $f(z)$ does not vanish at $\gamma\rho_{1}$ and $\gamma\rho_{2}$ for some
$\gamma\in\Gamma_{0}^{+}(2)$. Then

$2\kappa_{\rho_{j}}=1+\operatorname{ord}_{\rho_{j}}W_{f}(z),\qquad j=1,2,$

and it suffices to prove that $\operatorname{ord}_{\rho_{j}}W_{f}$ is always
even. Indeed, if the weight $k$ is a multiple of $4$, then the weight of
$W_{f}\in\mathscr{M}_{2k}(\Gamma_{0}^{+}(2))$ is a multiple of $8$. The orders
of such a modular form at any elliptic point must be even. If the weight $k$
is congruent to $2$ modulo $4$ so that the weight of $W_{f}$ is congruent to
$4$ modulo $8$, then $W_{f}$ must be divisible by
$M_{4}(z)=(4E_{4}(2z)+E_{2}(z))/5$, which has a zero of order $2$ at
$\rho_{2}$ and is nonvashing at $\rho_{1}$. Since $W_{f}/M_{4}$ is again a
modular form of a weight congruent to $0$ modulo $8$, we conclude that the
order of $W_{f}$ at $\rho_{j}$ must be even for $j=1,2$. This proves Part
(ii).

The proof of (iii) is slightly more complicated. Write $f(z)$ as
$f(z)=f_{1}(z)M_{2}^{\ast}(z)+f_{0}(z)$, where
$M_{2}^{\ast}(z)=(3E_{2}(3z)+E_{2}(z))/4$ and
$f_{j}\in\mathscr{M}_{k-2j}(\Gamma_{0}^{+}(3),\chi^{m})$. Again, without loss
of generality, we assume that $f_{0}(z)$ and $f_{1}(z)$ have no common zero at
$\rho_{1}$ and $\rho_{2}$. This assumption forces the weight $k$ to be
congruent to $0$ or $2$ modulo $3$. Then, as above, we have
$2\kappa_{\rho_{j}}=1+\operatorname{ord}_{\rho_{j}}W_{f}$.

By Lemma 2.14, we have
$W_{f}\in\mathscr{M}_{2k}(\Gamma_{0}^{+}(3),\chi^{2m})=\mathscr{M}_{2k}(\Gamma_{0}^{+}(3),(-1)^{k})$.
Now recall that the ring of modular forms
$\oplus_{n=0}^{\infty}\mathscr{M}_{n}(\Gamma_{1}(3))$, where $n$ runs through
both even and odd integers, is freely generated by the modular forms
$\mathscr{M}_{1}(z)\in\mathscr{M}_{1}(\Gamma_{0}^{+}(3),\chi)$ and
$\mathscr{M}_{3}(z)\in\mathscr{M}_{3}(\Gamma_{0}^{+}(3),\chi)$ defined by
(2.15) and (2.16). Thus, $W_{f}$ is a linear combination of products of the
form $M_{1}(z)^{a}M_{3}(z)^{b}$, where $(a,b)$ are pairs of nonnegative
integers satisfying

(3.7) $a+3b=2k,\qquad a+b\equiv 2k\ \mathrm{mod}\ 4.$

(The latter condition corresponds to the requirement that
$M_{1}(z)^{a}M_{3}(z)^{b}$ must be a modular form on $\Gamma_{0}^{+}(3)$ with
character $\chi^{2k}$.) The conditions imply that $b$ must be even, while

$a\equiv\begin{cases}0\ \mathrm{mod}\ 6,&\text{if }k\equiv 0\ \mathrm{mod}\
3,\\\ 4\ \mathrm{mod}\ 6,&\text{if }k\equiv 2\ \mathrm{mod}\ 3,\end{cases}$

Since $M_{1}(z)$ and $M_{3}(z)$ have a simple zero at $\rho_{2}$ and
$\rho_{1}$, respectively, and are nonvanishing elsewhere, the above properties
of $(a,b)$ show that $\operatorname{ord}_{\rho_{1}}W_{f}\equiv 0\
\mathrm{mod}\ 2$ and

$\operatorname{ord}_{\rho_{2}}W_{f}\equiv\begin{cases}0\ \mathrm{mod}\
6,&\text{if }k\equiv 0\ \mathrm{mod}\ 3,\\\ 4\ \mathrm{mod}\ 6,&\text{if
}k\equiv 2\ \mathrm{mod}\ 3.\end{cases}$

In either case, we find that
$(2\kappa_{\rho_{1}},2)=(2\kappa_{\rho_{2}},6)=1$. The proof of the
proposition is completed. ∎

We now give several examples of quasimodular forms that are solutions of
modular differential equations. Consider first the case
$\Gamma=\Gamma_{0}^{+}(2)$. We recall that, by Corollary 2.16, extremal
quasimodular forms exist in
$\widetilde{\mathscr{M}}_{k}(\Gamma_{0}^{+}(2),\pm)$ and are unique up to
scalars.

######  Theorem 3.7.

Set

$\begin{split}M_{2}(z)&=2E_{2}(2z)-E_{2}(z),\\\
M_{4}(z)&=\frac{1}{5}(4E_{4}(2z)+E_{4}(z))=M_{2}(z)^{2},\\\
M_{8}(z)&=\eta(z)^{8}\eta(2z)^{8}.\end{split}$

Assume that $k$ is a nonnegative integer.

  1. (i)

Let $f(z)$ be an extremal quasimodular form in
$\widetilde{\mathscr{M}}_{4k}^{\leq 1}(\Gamma_{0}^{+}(2),(-1)^{k})$. Then
$f(z)/M_{8}(z)^{k/2}$ is a solution of

$y^{\prime\prime}(z)=-4\pi^{2}\left(\frac{k}{2}\right)^{2}M_{4}(z)y(z).$

  2. (ii)

Let $f(z)$ be an extremal quasimodular form in
$\widetilde{\mathscr{M}}_{4k+2}^{\leq 1}(\Gamma_{0}^{+}(2),(-1)^{k})$. Then
$f(z)/M_{8}(z)^{k/2}M_{2}(z)$ is a solution of

$y^{\prime\prime}(z)=-4\pi^{2}\left(\left(\frac{k}{2}\right)^{2}M_{4}(z)-32\frac{M_{8}(z)}{M_{4}(z)}\right)y(z).$

###### Proof.

Let $W_{f}(z)$ be the Wronskian associated to $f$ defined in Lemma 2.14. By
the lemma, $W_{f}(z)$ is a modular form of weight $8k$ on $\Gamma_{0}^{+}(2)$
(with trivial character). We then observe that, by (2.9) and (2.10),

$\dim\widetilde{\mathscr{M}}_{4k}^{\leq
1}(\Gamma_{0}^{+}(2),(-1)^{k})=1+k=\dim\mathscr{M}_{8k}(\Gamma_{0}^{+}(2)).$

It follows that, by Proposition 2.15, if $f$ is an extremal quasimodular form
in $\widetilde{\mathscr{M}}_{4k}^{\leq 1}(\Gamma_{0}^{+}(2),(-1)^{k})$, then
$v_{\infty}(f)=v_{\infty}(W_{f})=k$. However, up to scalars, $M_{8}(z)^{k}$ is
the only modular form in $\mathscr{M}_{8k}(\Gamma_{0}^{+}(2))$ whose order of
vanishing at $\infty$ is $k$ (see the proof of Corollary 2.16). In other
words, $W_{f}(z)=cM_{8}(z)^{k}$ for some nonzero complex number $c$ and the
functions $g_{1}(z)$ and $g_{2}(z)$ in (3.3) are holomorphic throughout
$\mathbb{H}$. It follows that the differential equation (3.4) has no
singularities in $\mathbb{H}$. Then the meromorphic modular form $Q(z)$ in
Theorem 3.3 must be a holomorphic modular form of $4$ and hence a multiple of
$M_{4}(z)$ due to the fact that $\dim\mathscr{M}_{4}(\Gamma_{0}^{+}(2))=1$.
Since $g_{2}(z)=cq^{k/2}+\cdots$ for some nonzero constant $c$, we conclude
that $Q(z)=(k/2)^{2}M_{4}(z)$. This proves Part (i).

The proof of Part (ii) is slightly more involved. Let $f(z)$ be an extremal
quasimodular form in
$\widetilde{\mathscr{M}}_{4k+2}(\Gamma_{0}^{+}(2),(-1)^{k})$. By the same
reasoning as in the proof of Part (i), using (2.9), (2.10), and Proposition
2.15, we see that $W_{f}(z)$ is a modular form of weight $8k+4$ such that
$v_{\infty}(W_{f})=k$ and hence must be a multiple of $M_{4}(z)M_{8}(z)^{k}$.
Now $M_{4}(z)$ has a zero of order $2$ at $\rho_{2}=(1+i)/2$ (see Section 5),
so the differential equation (3.4) has potentially a singularity at $\rho_{2}$
and we need to determine the local exponents at the point.

Write $f(z)$ as $f(z)=M_{2}^{\ast}(z)f_{1}(z)+f_{0}(z)$, where
$M_{2}^{\ast}(z)=(2E_{2}(2z)+E_{2}(z))/3$ and
$f_{j}\in\mathscr{M}_{4k+2-2j}(\Gamma_{0}^{+}(2),(-1)^{k})$, $j=0,1$. Since
$\rho_{2}=(1+i)/2$ is an elliptic point of order $2$ of $\Gamma_{0}(2)$, we
must have $f_{0}(\rho_{2})=0$. We claim that $f_{1}(\rho_{2})\neq 0$. Indeed,
since $M_{2}(z)=2E_{2}(2z)-E_{2}(z)$ has a simple zero at points equivalent to
$\rho_{2}$ and is nonvanishing elsewhere, if $f_{1}(\rho)=0$, then
$f_{0}(z)/M_{2}(z)$ and $f_{1}(z)/M_{2}(z)$ are holomorphic modular forms on
$\Gamma_{0}(2)$ with Atkin-Lehner eigenvalue $(-1)^{k-1}$. In other words,
$f(z)/M_{2}(z)\in\widetilde{\mathscr{M}}_{4k}^{\leq
1}(\Gamma_{0}^{+}(2),(-1)^{k-1})$. However, this is impossible because there
is no quasimodular form in $\widetilde{\mathscr{M}}_{4k}^{\leq
1}(\Gamma_{0}^{+}(2),(-1)^{k-1})$ whose order of vanishing at $\infty$ is $k$,
by (2.9), (2.10), and Corollary 2.16. We conclude that $f_{1}(\rho_{2})\neq
0$.

We next observe that, by Lemma 3.5, $M_{2}^{\ast}(z)$ does not vanish at any
elliptic point of $\Gamma_{0}^{+}(2)$. Therefore,
$f(z)=M^{\ast}_{2}(z)f_{1}(z)+f_{0}(z)$ does not vanish at $\rho_{2}$ and
$f(z)/\sqrt{W_{f}(z)}$ has a simple zero at $\rho_{2}$. It follows that the
local exponents of (3.4) at $\rho$ must be $-1$ and $2$, i.e., the indicial
equation at $\rho$ is $x^{2}-x-2=0$. By (A.15), we must have

$Q(z)=\left(\frac{k}{2}\right)^{2}M_{4}(z)-32\frac{M_{8}(z)}{M_{4}(z)}.$

This proves Part (ii). ∎

By the same token, we have the following analogous result for
$\Gamma_{0}^{+}(3)$.

######  Theorem 3.8.

Set

(3.8) $\begin{split}M_{2}(z)&=\frac{3E_{2}(3z)-E_{2}(z)}{2},\\\
M_{4}(z)&=\frac{9E_{4}(3z)+E_{4}(z)}{10}=M_{2}(z)^{2},\\\
M_{6}(z)&=\eta(z)^{6}\eta(3z)^{6}.\end{split}$

Let $\chi$ be the character of $\Gamma_{0}^{+}(3)$ defined by

$\chi\left(\frac{1}{\sqrt{3}}\begin{pmatrix}0&-1\\\
3&0\end{pmatrix}\right)=\chi\left(\frac{1}{\sqrt{3}}\begin{pmatrix}3&-1\\\
3&0\end{pmatrix}\right)=-i.$

Let $k$ be a nonnegative integer.

  1. (i)

Let $f(z)$ be an extremal quasimodular form in
$\widetilde{\mathscr{M}}_{3k}^{\leq 1}(\Gamma_{0}^{+}(3),\chi^{k})$. Then
$f(z)/M_{6}(z)^{k/2}$ is a solution of the differential equation

$y^{\prime\prime}(z)=-4\pi^{2}\left(\frac{k}{2}\right)^{2}M_{4}(z)y(z).$

  2. (ii)

Let $f(z)$ be an extremal quasimodular form in
$\widetilde{\mathscr{M}}_{3k+2}^{\leq 1}(\Gamma_{0}^{+}(3),\chi^{k})$. Then
$f(z)/M_{6}(z)^{k/2}M_{2}(z)$ is a solution of the differential equation

$y^{\prime\prime}(z)=-4\pi^{2}\left(\left(\frac{k}{2}\right)^{2}M_{4}(z)-18\frac{M_{6}(z)}{M_{2}(z)}\right)y(z).$

###### Proof.

The proof is similar to that of Theorem 3.7. We omit most of the details and
only mention that the Wronskian $W_{f}$ associated to $f$ in Part (ii) is a
scalar multiple of $M_{6}(z)^{k}M_{4}(z)$, where $M_{4}(z)$ has a zero of
order $4$ at $\rho_{2}=(3+\sqrt{-3})/6$. Hence, $f(z)/\sqrt{W_{f}(z)}$ has a
pole of order $2$ at $\rho_{2}$, and the indicial equation of (3.4) at
$\rho_{2}$ is $x^{2}-x-6=0$. By (A.18), we therefore have

$Q(z)=\left(\frac{k}{2}\right)^{2}M_{4}(z)-18\frac{M_{6}(z)}{M_{2}(z)},$

as claimed. ∎

######  Remark 3.9.

The proof of the two theorems also applies to quasimodular forms on
$\mathrm{SL}(2,\mathbb{Z})$. We can similarly show that if $f$ is an extremal
quasimodular form of weight $6k$, then $f/\Delta^{k/2}$ is a solution of
$y^{\prime\prime}(z)=-\pi^{2}k^{2}E_{4}(z)y(z)$. This gives a new proof of
(2.5). Also, if $f$ is an extremal quasimodular form of weight $6k+2$ and
depth $1$ on $\mathrm{SL}(2,\mathbb{Z})$, then the Wronskian $W_{f}(z)$ is a
multiple of $\Delta(z)^{k}E_{4}(z)$. Therefore, the order of
$f(z)/\sqrt{W_{f}(z)}$ at $\rho=(1+\sqrt{-3})/2$ is $-1/2$ and the indicial
equation at $\rho$ is $x^{2}-x-3/4=0$. By (A.11), $f(z)/\sqrt{W_{f}(z)}$ is a
solution of

$y^{\prime\prime}(z)=-4\pi^{2}\left(\left(\frac{k}{2}\right)^{2}E_{4}(z)-144\frac{\Delta(z)}{E_{4}(z)^{2}}\right)y(z).$

This corresponds to Theorem 2.1(2) of [8].

We remark that the reason why we do not have a corresponding statement for
extremal quasimodular forms of weight $6k+4$ and depth $1$ on
$\mathrm{SL}(2,\mathbb{Z})$ is that such a quasimodular form $f(z)$ must be a
multiple of $E_{4}(z)$, but the differential equation satisfied by
$f(z)/\sqrt{W_{f}(z)}$ is the same as that for
$f(z)/E_{4}(z)\sqrt{W_{f/E_{4}}(z)}$.

##  4\. MODE in the case of $\mathrm{SL}(2,\mathbb{Z})$

In this section, we shall consider Fuchsian differential equations of the form

(4.1) $y^{\prime\prime}(z)=-4\pi^{2}Q(z)y(z)$

on $\mathbb{H}$ for some meromorphic modular forms $Q(z)$ of weight $4$ on
$\mathrm{SL}(2,\mathbb{Z})$ satisfying the conditions in
($\text{H}_{\Gamma}$). In particular, $Q(z)$ is holomorphic at $q=0$,
$q=e^{2\pi iz}$, with $Q(\infty)\geq 0$. Let $D_{q}=qd/dq=(2\pi i)^{-1}d/dz$.
At $q=0$, (4.1) becomes

(4.2) $D_{q}^{2}y=Q(z)y.$

Thus, (4.1) is also Fuchsian at the cusp $\infty$ and its local exponents at
$\infty$ are $\pm\kappa_{\infty}$, where

(4.3) $\kappa_{\infty}=\sqrt{Q(\infty)}.$

Note that in [4, Proposition 3.6] we have shown that if (4.1) is apparent at a
point $z_{0}$, then it is apparent at all $\gamma z_{0}$,
$\gamma\in\mathrm{SL}(2,\mathbb{Z})$. Let $z_{1},\ldots,z_{m}$ be the poles of
$Q(z)$ such that $z_{j}$ and $z_{k}$ are not equivalent under
$\mathrm{SL}(2,\mathbb{Z})$ when $j\neq k$, i.e.,
$z_{k}\notin\mathrm{SL}(2,\mathbb{Z})z_{j}$, and
$z_{j}\notin\\{\rho=(1+\sqrt{-3})/2,i=\sqrt{-1}\\}$. The apparentness
condition implies that the local exponents at $\rho$, $i$, and $z_{j}$,
$j=1,\ldots,m$, are $1/2\pm\kappa_{\rho}$, $1/2\pm\kappa_{i}$, and
$1/2\pm\kappa_{j}$, $1\leq j\leq m$, where all $\kappa$’s are in
$\frac{1}{2}\mathbb{N}$.

To the modular differential equation (4.1), there associates the Bol
representation $\rho$, a homomorphism from $\mathrm{SL}(2,\mathbb{Z})$ to
$\mathrm{PSL}(2,\mathbb{C})$, that is, by choosing a fundamental solution
$Y(z)=(y_{1}(z),y_{2}(z))^{t}$, there is
$\rho(\gamma)\in\mathrm{SL}(2,\mathbb{C})$ for any
$\gamma\in\mathrm{SL}(2,\mathbb{C})$ such that

(4.4) $\begin{pmatrix}y_{1}\big{|}_{-1}\gamma\\\
y_{2}\big{|}_{-1}\gamma\end{pmatrix}=\pm\rho(\gamma)\begin{pmatrix}y_{1}\\\
y_{2}\end{pmatrix}.$

For our purpose, we would like to lift $\rho$ to a homomorphism
$\hat{\rho}:\mathrm{SL}(2,\mathbb{Z})\to\mathrm{SL}(2,\mathbb{C})$ such that
$\rho=\pi\circ\hat{\rho}$, where $\pi$ is the natural projection from
$\mathrm{SL}(2,\mathbb{C})$ to $\mathrm{PSL}(2,\mathbb{C})$. This will be done
as follows.

Define $t_{j}=E_{6}(z_{j})^{2}/E_{4}(z_{j})^{3}$ such that $F_{j}(z_{j})=0$,
where

(4.5) $F_{j}(z)=E_{6}(z)^{2}-t_{j}E_{4}(z)^{3}.$

######  Lemma 4.1.

If $t_{j}\neq 0,1$, then $E_{6}(z)^{2}-t_{j}E_{4}(z)^{3}$ has zeros only at
$z_{j}$, and $z_{j}$ is a simple zero.

The lemma can be proved by the theorem of counting zeros of modular forms
([14, p. 85, Theorem 3]).

Set

(4.6)
$F(z)=\Delta(z)^{\kappa_{\infty}}E_{4}(z)^{\kappa_{\rho}-1/2}E_{6}(z)^{\kappa_{i}-1/2}\prod_{j=1}^{m}F_{j}(z)^{\kappa_{j}-1/2}$

and

(4.7)
$\ell=-1+12\kappa_{\infty}+4\left(\kappa_{\rho}-\frac{1}{2}\right)+6\left(\kappa_{i}-\frac{1}{2}\right)+12\sum^{m}_{j=1}\left(\kappa_{j}-\frac{1}{2}\right),$

so that $F(z)^{2}$ is a modular form of weight $2(\ell+1)$ on
$\mathrm{SL}(2,\mathbb{Z})$. Note that by assumption ($\text{H}_{\Gamma}$),
$\kappa_{i}-1/2$ is an integer. Thus, $\ell$ is always an odd integer.

For any function $f(z)$ on $\mathbb{H}$, we set

$\hat{f}(z)=F(z)f(z).$

It is easy to see that if $y(z)$ is a solution of (4.1), then $\hat{y}(z)$ is
single-valued function that is holomorphic throughout $\mathbb{H}$.
Furthermore, _its order at $z_{j}$ is either $0$ or $2\kappa_{j}$_, and a
similar property holds at $\infty$, $\rho$, and $i$. Now choose a fundamental
solution $Y(z)=(y_{1}(z),y_{2}(z))^{t}$ of (4.1). Then we have a homomorphism
$\hat{\rho}:\mathrm{SL}(2,\mathbb{Z})\to\mathrm{GL}(2,\mathbb{C})$ defined by

(4.8)
$\begin{split}\left(\hat{Y}\big{|}_{\ell}\gamma\right)(z)=\hat{\rho}(\gamma)\hat{Y}(z),\quad\text{where
}\hat{Y}(z)=F(z)Y(z).\end{split}$

The next lemma shows that this homomorphism $\hat{\rho}$ has an image in
$\mathrm{SL}(2,\mathbb{C})$.

######  Lemma 4.2.

We have $\det\hat{\rho}(\gamma)=1$ for all
$\gamma\in\mathrm{SL}(2,\mathbb{Z})$. That is, $\hat{\rho}$ is a homomorphism
from $\mathrm{SL}(2,\mathbb{Z})$ to $\mathrm{SL}(2,\mathbb{C})$.

###### Proof.

Let

$W(z)=\det\begin{pmatrix}y_{1}&y_{1}^{\prime}\\\
y_{2}&y_{2}^{\prime}\end{pmatrix},\qquad\hat{W}(z)=\det\begin{pmatrix}\hat{y}_{1}&\hat{y}_{1}^{\prime}\\\
\hat{y}_{2}&\hat{y}_{2}^{\prime}\end{pmatrix}.$

On the one hand, (4.8) yields that for $\gamma\in\mathrm{SL}(2,\mathbb{Z})$,

$\hat{W}(z)\big{|}_{2(\ell+1)}\gamma=\det\hat{\rho}(\gamma)\hat{W}(z).$

On the other hand, we have $\hat{W}(z)=F(z)^{2}W(z)$. Hence,

$\begin{split}\hat{W}(z)\big{|}_{2(\ell+1)}&=\left(F(z)^{2}\big{|}_{2(\ell+1)}\gamma\right)\left(W(z)\big{|}_{0}\gamma\right)\\\
&=\left(F(z)^{2}\big{|}_{2(\ell+1)}\gamma\right)\det\rho(\gamma)W(z)=\frac{\left(F(z)^{2}\big{|}_{2(\ell+1)}\gamma\right)}{F(z)^{2}}\hat{W}(z).\end{split}$

Comparing the two expressions, we find that

$\det\hat{\rho}(\gamma)=\frac{\left(F(z)^{2}\big{|}_{2(\ell+1)}\gamma\right)}{F(z)^{2}}=1$

for all $\gamma\in\mathrm{SL}(2,\mathbb{Z})$ since $F(z)^{2}$ is a modular
form of weight $2(\ell+1)$ on $\mathrm{SL}(2,\mathbb{Z})$. This proves the
lemma. ∎

If no confusion arises, we will also call $\hat{\rho}$ the Bol representation.

Throughout the remainder of the section, we will let $y_{+}(z)$ denote the
unique solution of (4.1) of the form

(4.9) $y_{+}(z)=q^{\kappa_{\infty}}\left(1+\sum_{j\geq 1}c_{j}q^{j}\right).$

Also, we will use the standard notations $T=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)$, $S=\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right)$ and $R=TS=\left(\begin{smallmatrix}1&-1\\\
1&0\end{smallmatrix}\right)$. They satisfy

(4.10) $S^{2}=R^{3}=-I.$

Out main result of this section is the following theorem (stated as Theorem
1.2 in Section 1).

######  Theorem 4.3.

Suppose that $Q(z)$ satisfies ($\text{H}_{\Gamma}$) with
$\kappa_{\infty}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$. Then the following
statements hold true.

  1. (i)

The differential equation (4.1) is not apparent at $\infty$.

  2. (ii)

Let $y_{2}(z)=y_{+}(z)$, where $y_{+}(z)$ is the solution of (4.1) of the form
(4.9). Let $\hat{y}_{1}(z)=\left(\hat{y}_{+}|_{\ell}S\right)(z)$. Then
$\hat{y}_{1}(z)=z\hat{y}_{2}(z)+\hat{m}_{1}(z)$ for some modular form
$\hat{m}_{1}(z)$ of weight $\ell-1$ on $\mathrm{SL}(2,\mathbb{Z})$.

  3. (iii)

Using $y_{1}(z)$ and $y_{2}(z)$ as the basis, Bol’s representation satisfies

$\hat{\rho}(\gamma)=\gamma,\quad\gamma\in\mathrm{SL}(2,\mathbb{Z}).$

In particular, the ratio $h(z)=y_{1}(z)/y_{2}(z)$ is equivariant, i.e.,
$h(\gamma z)=\gamma\cdot h(z)$ for all $\gamma\in\mathrm{SL}(2,\mathbb{Z})$.

  4. (iv)

Write $\hat{y}_{+}(z)$ as $\hat{y}_{+}(z)=\frac{\pi
i}{6}\hat{m}_{1}(z)E_{2}(z)+\hat{m}_{2}(z)$. Then $\hat{m}_{2}(z)$ is a
modular form of weight $\ell+1$ with respect to $\mathrm{SL}(2,\mathbb{Z})$.
In other words, $\hat{y}_{+}(z)$ is a quasimodular form of weight $\ell+1$ and
depth $1$ on $\mathrm{SL}(2,\mathbb{Z})$.

Theorem 4.3 will be proved through a series of lemmas.

######  Lemma 4.4.

The ODE (4.1) is not apparent at $\infty$.

###### Proof.

Assume that (4.1) is apparent at $\infty$. Then for any fundamental solutions
near $\infty$, we have $\hat{\rho}(T)=I$. Now since $\ell$ is odd, we have
$\hat{\rho}(-I)=(-1)^{\ell}I=-I$. Thus, by (4.10),
$\hat{\rho}(S)^{2}=\hat{\rho}(R)^{3}=-I$. On the other hand, we have
$\hat{\rho}(R)^{2}=(\hat{\rho}(T)\hat{\rho}(S))^{2}=-I$, which is absurd since
$\hat{\rho}(R)^{2}$ and $\hat{\rho}(R)^{3}$ cannot be both equal to $-I$. This
proves that (4.1) cannot be apparent at $\infty$. ∎

######  Remark 4.5.

Observe that the parity of $\ell$ depends only on $\kappa_{i}$. If
$\kappa_{i}\in\mathbb{N}$, i.e., if $\ell$ is even, then
$\hat{\rho}(R^{3})=\hat{\rho}(S^{2})=\hat{\rho}(-I)=I$, which implies that
$\hat{\rho}(S)=\pm I$ and either $\hat{\rho}(R)=I$ or
$\hat{\rho}(R)^{2}+\hat{\rho}(R)+I=0$. In any case, there is a matrix $P$ such
that both $P\hat{\rho}(S)P^{-1}$ and $P\hat{\rho}(R)P^{-1}$ are diagonal,
which implies that $P\hat{\rho}(T)P^{-1}$ is also diagonal, and the
apparentness at $\infty$ follows. Thus, the condition
$\kappa_{i}\notin\mathbb{N}$ is necessary in order for (4.1) to be not
apparent at $\infty$.

######  Lemma 4.6.

Let $y_{2}(z)=y_{+}(z)$ and
$y_{1}(z)=\left(\hat{y}_{+}\big{|}_{\ell}S\right)(z)/F(z)$. Then the following
hold:

  1. (i)

$y_{i}$, $i=1,2$, are linearly independent, and

  2. (ii)

the ratio $h(z)=y_{1}(z)/y_{2}(z)$ is equivariant.

###### Proof.

By Lemma 4.4, (4.1) is not apparent at $\infty$. Hence, we have a solution
$y_{-}(z)$ of the form $y_{-}(z)=dzy_{+}(z)+m(z)$ with
$m(z)=q^{-\kappa_{\infty}}(1+\cdots)$ and $d\neq 0$. Then with respect to the
choice $\hat{Y}(z)=(\hat{y}_{+}(z),\hat{y}_{-}(z))^{t}$, we have
$\hat{\rho}(T)=\left(\begin{smallmatrix}1&0\\\ d&1\end{smallmatrix}\right)$.

Now suppose that $y_{+}(z)$ and
$\left(\hat{y}_{+}\big{|}_{\ell}S\right)(z)/F(z)$ are not linearly
independent, i.e.,
$\left(\hat{y}_{2}\big{|}_{\ell}S\right)(z)=\lambda\hat{y}_{2}(z)$ for some
$\lambda$. Then $\hat{\rho}(S)=\left(\begin{smallmatrix}\lambda&0\\\
c&\overline{\lambda}\end{smallmatrix}\right)$, $c\in\mathbb{C}$. Because
$\ell$ is odd, we have $\hat{\rho}(S^{2})=\hat{\rho}(-I)=(-1)^{\ell}I=-I$. It
follows that $\lambda=\pm i$ and

$\hat{\rho}(R)=\begin{pmatrix}1&0\\\
d&1\end{pmatrix}\begin{pmatrix}\lambda&0\\\
c&\bar{\lambda}\end{pmatrix}=\begin{pmatrix}\lambda&0\\\
d\lambda+c&\bar{\lambda}\end{pmatrix},$

which implies $\lambda^{3}=-1$, a contradiction because $\lambda=\pm i$. Thus
${y}_{i}$, $i=1,2$, are linearly independent.

Since (4.1) is not apparent at $\infty$ and we have shown that
$y_{2}(z)=y_{+}(z)$ and
$y_{1}(z)=\left(\hat{y}_{2}\big{|}_{\ell}S\right)(z)/F(z)$ are linearly
independent, by (1.11), there is $d\neq 0$ such that
$y_{1}(z)=dzy_{2}(z)+m_{1}(z)$ for some function
$m_{1}(z)=q^{-\kappa_{\infty}}(a_{0}+\sum_{j\geq 1}a_{j}q^{j})$ with
$a_{0}\neq 0$. We claim that

(4.11) $d=1.$

Obviously, with respect to the basis $(\hat{y}_{1}(z),\hat{y}_{2}(z))^{t}$,
$\hat{\rho}(T)=\left(\begin{smallmatrix}1&d\\\ 0&1\end{smallmatrix}\right)$.
On the other hand, since
$\hat{y}_{1}(z)=\left(\hat{y}_{2}\big{|}_{\ell}S\right)(z)$, and
$-\hat{y}_{2}(z)=\left(\hat{y}_{2}\big{|}_{\ell}S^{2}\right)(z)=\left(\hat{y}_{1}\big{|}_{\ell}S\right)(z)$,
we have

(4.12) $\hat{\rho}(S)=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$

and

$\hat{\rho}(R)=\begin{pmatrix}1&d\\\ 0&1\end{pmatrix}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}=\begin{pmatrix}d&-1\\\ 1&0\end{pmatrix}.$

By $\hat{\rho}(R^{3})=-I$, i.e. $\hat{\rho}(R)^{2}=-\hat{\rho}(R)^{-1}$, we
have

$\begin{pmatrix}d&-1\\\ 1&0\end{pmatrix}\begin{pmatrix}d&-1\\\
1&0\end{pmatrix}=\begin{pmatrix}d^{2}-1&-d\\\
d&-1\end{pmatrix}=-\begin{pmatrix}0&1\\\ -1&d\end{pmatrix},$

which yields $d=1$. This proves (4.11).

Using $h(z)=\hat{y}_{1}(z)/\hat{y}_{2}(z)$, we have
$h(z+1)=(\hat{y}_{1}(z)+\hat{y}_{2}(z))/\hat{y}_{2}(z)=h(z)+1$ and

$h(-1/z)=\frac{\hat{y}_{1}(-1/z)}{\hat{y}_{2}(-1/z)}=\frac{\left(\hat{y}_{1}\big{|}_{\ell}S\right)(z)}{\left(\hat{y}_{2}\big{|}_{\ell}S\right)(z)}=\frac{-\hat{y}_{2}(z)}{\hat{y}_{1}(z)}=-1/h(z).$

This proves that $h(z)$ is equivariant. ∎

We recall the identity
$\hat{y}_{1}(z)=\left(\hat{y}_{2}\big{|}_{\ell}S\right)(z)=z\hat{y}_{2}(z)+\hat{m}_{1}(z)$
obtained in the proof of Lemma 4.6.

######  Lemma 4.7.

We have

  1. (i)

$\hat{m}_{1}(z+1)=\hat{m}_{1}(z)$, and

  2. (ii)

$\left(\hat{m}_{1}\big{|}_{\ell-1}S\right)(z)=\hat{m}_{1}(z)$.

Hence, $\hat{m}_{1}(z)$ is a modular form of weight $\ell-1$ on
$\mathrm{SL}(2,\mathbb{Z})$.

###### Proof.

The identity (i) is obvious. By the definition, $h(z)$ of Lemma 4.6(ii) can be
written as

(4.13)
$h(z)=z+\frac{m_{1}(z)}{y_{2}(z)}=z+\frac{\hat{m}_{1}(z)}{\hat{y}_{2}(z)}.$

We have

(4.14)
$-\frac{1}{h(z)}=h\left(-\frac{1}{z}\right)=-\frac{1}{z}+\frac{\hat{m}_{1}(-1/z)}{\hat{y}_{2}(-1/z)}.$

Thus by (4.14),

$\displaystyle\frac{\hat{m}_{1}(-1/z)}{\hat{y}_{1}(z)}$
$\displaystyle=\frac{z^{\ell}\hat{m}_{1}(-1/z)}{\hat{y}_{2}(-1/z)}=z^{\ell}\left(\frac{1}{z}-\frac{1}{h(z)}\right)=\frac{(h(z)-z)z^{\ell}}{zh(z)}$
$\displaystyle=\frac{1}{z^{1-\ell}h(z)}\frac{\hat{m}_{1}(z)}{\hat{y}_{2}(z)}=\frac{\hat{m}_{1}(z)}{z^{1-\ell}\hat{y}_{1}(z)},$

which implies $\hat{m}_{1}(-1/z)=z^{\ell-1}\hat{m}_{1}(z)$. This proves (ii).
∎

######  Lemma 4.8.

Write $y_{+}(z)$ as $y_{+}(z)=\frac{\pi i}{6}m_{1}(z)E_{2}(z)+m_{2}(z)$. Then
$\hat{m}_{2}(z)$ is a modular form of weight $\ell+1$ with respect to
$\mathrm{SL}(2,\mathbb{Z})$.

###### Proof.

By the previous lemma, we have

$\displaystyle\hat{y}_{1}(z)$
$\displaystyle=\left(\hat{y}_{+}\big{|}_{\ell}S\right)(z)$
$\displaystyle=z^{-\ell}\left(\frac{\pi
i}{6}\hat{m}_{1}(-1/z)E_{2}(-1/z)+\hat{m}_{2}(-1/z)\right)$
$\displaystyle=z^{-\ell}\left(\frac{\pi
i}{6}z^{\ell-1}\hat{m}_{1}\left(z\right)\left(z^{2}E_{2}(z)+\frac{6}{\pi
i}z\right)+\hat{m}_{2}(-1/z)\right)$ $\displaystyle=\frac{\pi
i}{6}\hat{m}_{1}\left(z\right)\left(zE_{2}(z)+\frac{6}{\pi
i}\right)+z^{-\ell}\hat{m}_{2}(-1/z)$
$\displaystyle=\left(z\hat{y}_{+}(z)-z\hat{m}_{2}(z)+\hat{m}_{1}(z)\right)+z^{-\ell}m_{2}(-1/z).$

On the other hand, recall from the proof of Lemma 4.6 that the left-hand side
is equal to $z\hat{y}_{+}(z)+\hat{m}_{1}(z)$. Therefore,
$\hat{m}_{2}(-1/z)=z^{\ell+1}\hat{m}_{2}(z)$. That is,
$\left(\hat{m}_{2}\big{|}_{\ell+1}S\right)(z)=\hat{m}_{2}(z)$.

Finally, since $\hat{y}_{+}(z+1)=\hat{y}_{+}(z)$ and
$\hat{m}_{1}(z+1)=\hat{m}_{1}(z)$, we have $\hat{m}_{2}(z+1)=\hat{m}_{2}(z)$.
We conclude that $\hat{m}_{2}(z)$ is a modular form of weight $\ell+1$ on
$\mathrm{SL}(2,\mathbb{Z})$. ∎

Having proved Theorem 4.3, we now give some examples.

######  Example 4.9.

Consider the differential equation

(4.15)
$y^{\prime\prime}(z)=-4\pi^{2}\kappa_{\infty}^{2}E_{4}(z)y(z),\qquad\kappa_{\infty}\in\frac{1}{2}\mathbb{N}.$

Since $E_{4}(z)$ is a holomorphic modular form, this differential equation has
no singularities in $\mathbb{H}$. Hence, we have
$\kappa_{\rho}=\kappa_{i}=1/2$, $F(z)=\Delta(z)^{\kappa_{\infty}}$ and the
integer $\ell$ in (4.7) is $-1+12\kappa_{\infty}$. Also, the local exponents
at $\infty$ are clearly $\pm\kappa_{\infty}$. Thus, Theorem 4.3 predicts that
$\Delta(z)^{\kappa_{\infty}}y_{+}(z)$ is a quasimodular form of weight
$\ell+1$ and depth $1$ on $\mathrm{SL}(2,\mathbb{Z})$. Indeed, as mentioned in
Section 2.1, by the works of Kaneko and Koike [7, 8] and Pellarin [10,
Proposition 3.1], the differential equation (4.15) has
$y(z)=f(z)/\Delta(z)^{\kappa_{\infty}}$ as a solution, where $f(z)$ is an
extremal quasimodular form of weight $12\kappa_{\infty}=\ell+1$ and depth $1$,
agreeing with our Theorem 4.3.

######  Example 4.10.

Consider the differential equation

(4.16)
$y^{\prime\prime}(z)=-4\pi^{2}\left(\kappa_{\infty}^{2}E_{4}(z)-144\frac{\Delta(z)}{E_{4}(z)^{2}}\right)y(z),\quad\kappa_{\infty}\in\frac{1}{2}\mathbb{Z}_{\geq
0}.$

The differential equation has only singularities at points equivalent to
$\rho$ under $\mathrm{SL}(2,\mathbb{Z})$. By (A.11) (or [4, Theorem 1.7 and
Corollary 3.10]), the indicial equation as $\rho$ is $x^{2}-x-3/4=0$. Hence
the local exponents at $\rho$ are $-1/2$ and $3/2$, i.e., $\kappa_{\rho}=1$,
and the differential equation is apparent on $\mathbb{H}$, by Theorem A.8. The
integer $\ell$ in (4.7) is $1+12\kappa_{\infty}$. Theorem 4.3 then asserts
that if we let $y_{+}(z)=q^{\kappa_{\infty}}+\cdots$ be the solution with
exponent $\kappa_{\infty}$ at $\infty$, then
$\Delta(z)^{\kappa_{\infty}}(z)E_{4}(z)^{1/2}y_{+}(z)$ is a quasimodular form
of weight $\ell+1$ and depth $1$ on $\mathrm{SL}(2,\mathbb{Z})$. To check that
this is true, we recall that Kaneko and Koike [8, Theorem 2.1] has also proved
that if $k\equiv 2\ \mathrm{mod}\ 6$, then an extremal quasimodular form
$f(z)$ of weight $k$ and depth $1$ on $\mathrm{SL}(2,\mathbb{Z})$ satisfies

$D_{q}^{2}f-\left(\frac{k}{6}-\frac{1}{3}\frac{E_{6}}{E_{4}}\right)D_{q}f+\left(\frac{k(k-1)}{12}D_{q}E_{2}-\frac{k-1}{18}\frac{D_{q}E_{6}}{E_{4}}\right)f=0.$

A straightforward calculation shows that this is equivalent to the assertion
that $f(z)/\Delta(z)^{\kappa_{\infty}}E_{4}(z)^{1/2}$ is a solution of (4.16)
with $k=12\kappa_{\infty}+2=\ell+1$. This agrees with our result.

######  Example 4.11.

Consider the differential equation

$y^{\prime\prime}(z)=-4\pi^{2}\left(\frac{1}{4}E_{4}(z)+864\frac{E_{4}(z)\Delta(z)}{E_{6}(z)^{2}}\right)y(z).$

We have $\kappa_{\infty}=1/2$ and by (A.11), $\kappa_{i}=3/2$. The integer
$\ell$ in this case is $11$ and Theorem 4.3 implies that
$\Delta(z)^{1/2}E_{6}(z)y_{+}(z)$ is a quasimodular form of weight $12$ and
depth $1$ on $\mathrm{SL}(2,\mathbb{Z})$. Indeed, we compute that

$y_{+}(z)=q^{1/2}\left(1+462q+247494q^{2}+132490928q^{3}+\cdots\right)$

and find that

$11088\Delta(z)^{1/2}E_{6}(z)y_{+}(z)=E_{2}(z)E_{4}(z)E_{6}(z)+6E_{4}(z)^{3}-7E_{6}(z)^{2}.$

##  5\. MODE in the case of $\Gamma_{0}^{+}(2)$

In this section, we will obtain results analogous to Theorem 4.3 for the group
$\Gamma^{+}_{0}(2)$.

First of all, we recall that the graded ring of holomorphic modular forms with
respect to $\Gamma^{+}_{0}(2)$ is generated by three modular forms $M_{4}(z)$,
$M_{6}(z)$ and $M_{8}(z)$, whose weights are $4$, $6$ and $8$, respectively.
These modular forms can be written explicitly in terms of the Eisenstein
series on $\mathrm{SL}(2,\mathbb{Z})$ and the Dedekind eta function:

(5.1) $\begin{split}M_{4}(z)&=(4E_{4}(2z)+E_{4}(z))/5,\\\
M_{6}(z)&=(8E_{6}(2z)+E_{6}(z))/9,\\\
M_{8}(z)&=\eta(z)^{8}\eta(2z)^{8}.\end{split}$

Noticing that $\dim\mathscr{M}_{12}(\Gamma_{0}^{+}(2))=2$, there must be a
relation among $M_{4}(z)^{3}$, $M_{6}(z)^{2}$, and $M_{4}(z)M_{8}(z)$. We find
that it is $M_{6}(z)^{2}=M_{4}(z)(M_{4}(z)^{2}-256M_{8}(z))$. For our purpose,
we also need

$M_{2}(z)=2E_{2}(2z)-E_{2}(z),$

which is a modular form in $\mathscr{M}_{2}(\Gamma_{0}^{+}(2),-)$ and
satisfies $M_{2}(z)^{2}=M_{4}(z)$. Define

$M^{*}_{2}(z)=\frac{1}{2\pi
i}\frac{M^{\prime}_{8}(z)}{M_{8}(z)}=\frac{2E_{2}(2z)+E_{2}(z)}{3}.$

The holomorphic function $M^{*}_{2}(z)$ is not a modular form. Instead, it
satisfies

(5.2) $M_{2}^{*}(\gamma z)=(cz+d)^{2}M_{2}^{*}(z)+\frac{4}{\pi
i}c(cz+d),\quad\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma^{+}_{0}(2).$

We use the notations $S=\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}0&-1\\\
2&0\end{smallmatrix}\right)$, $T=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)$,
$R=TS=\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}2&-1\\\
2&0\end{smallmatrix}\right)$ such that $S^{2}=R^{4}=-I$. We collect some facts
about $\Gamma^{+}_{0}(2)$.

  1. (i)

There are only two elliptic points $\rho_{1}=i/\sqrt{2}$ and
$\rho_{2}=(1+i)/2$ of order $2$ and $4$, respectively. Their stabilizer
subgroups are generated by $S$ and $R$, respectively. Also, $\infty$ is the
only cusp and has width $1$.

  2. (ii)

The modular form $M_{2}(z)$ has only one zero at $\rho_{2}$. The zero is
simple. The modular form $M_{4}(z)$ has only one zero (a double zero) at
$\rho_{2}$ and $M_{6}(z)$ has only zeros at $\rho_{1}$ and $\rho_{2}$. Both
zeros are simple.

  3. (iii)

$M_{8}(z)\neq 0$, $\forall z\in\mathbb{H}$, and $\infty$ is the simple zero of
$M_{8}(z)$.

(Here when we say “a modular form $f(z)$ has only a zero at a point $z_{0}$”,
it should be understood as “$f(z)$ has only zeros at points that are
$\Gamma_{0}^{+}(2)$-equivalent to $z_{0}$”.)

Consider the differential equation

(5.3) $y^{\prime\prime}(z)=-4\pi^{2}Q(z)y(z),$

where $Q(z)$ is a meromorphic modular form of weight $4$ on
$\Gamma_{0}^{+}(2)$ such that the conditions ($\text{H}_{\Gamma}$) are
satisfied. Let $z_{1},\ldots,z_{m}$ be $\Gamma_{0}^{+}(2)$-inequivalent poles
of $Q(z)$ other than $\rho_{1}$ and $\rho_{2}$, and assume that the local
exponents of (5.3) at $\infty$, $\rho_{1}$, $\rho_{2}$, and
$z_{1},\ldots,z_{m}$ are $\pm\kappa_{\infty}$, $1/2\pm\kappa_{\rho_{1}}$,
$1/2\pm\kappa_{\rho_{2}}$, and $1/2\pm\kappa_{1},\ldots,1/2\pm\kappa_{m}$,
respectively, where the $\kappa$’s satisfy ($\text{H}_{\Gamma}$).

For $j=1,\ldots,m$, set $t_{j}=M_{4}(z_{j})^{2}/M_{8}(z_{j})$ and
$F_{j}(z)=M_{4}(z)^{2}-t_{j}M_{8}(z)$. This modular form $F_{j}$ has only one
simple zero at $z_{j}$. Let

(5.4)
$F(z)=M_{8}(z)^{\kappa_{\infty}}\left(\frac{M_{6}(z)}{M_{2}(z)}\right)^{\kappa_{\rho_{1}}-1/2}M_{2}(z)^{\kappa_{\rho_{2}}-1/2}\prod_{j=1}^{m}F_{j}(z)^{\kappa_{j}-1/2},$

and

(5.5)
$\ell=-1+8\kappa_{\infty}+4\left(\kappa_{\rho_{1}}-\frac{1}{2}\right)+2\left(\kappa_{\rho_{2}}-\frac{1}{2}\right)+8\sum_{j=1}^{m}\left(\kappa_{j}-\frac{1}{2}\right),$

so that $F(z)^{2}$ is a modular form of weight $2(\ell+1)$ on
$\Gamma_{0}^{+}(2)$. We note that by the condition ($\text{H}_{\Gamma}$), the
integer $\ell$ is always odd.

For any function $f(z)$ on $\mathbb{H}$, let $\hat{f}(z)$ denote the function
$F(z)f(z)$. When $f(z)=y(z)$ is a solution of (5.3), by construction,
$\hat{y}(z)$ is a single-valued function holomorphic throughout $\mathbb{H}$
and its order at $z_{j}$ is either $0$ or $2\kappa_{j}$. If a fundamental
solution $Y(z)=(y_{1}(z),y_{2}(z))^{t}$ of (5.3) is chosen, then we have a
homomorphism $\hat{\rho}:\Gamma_{0}^{+}(2)\to\mathrm{GL}(2,\mathbb{C})$ with
$\hat{\rho}(\gamma)$ defined by

$\left(\hat{Y}\big{|}_{\ell}\gamma\right)(z)=\hat{\rho}(\gamma)\hat{Y}(z).$

As in the case of $\mathrm{SL}(2,\mathbb{Z})$, we can show that the image of
$\hat{\rho}$ actually lies in $\mathrm{SL}(2,\mathbb{C})$.

######  Lemma 5.1.

We have $\det\hat{\rho}(\gamma)=1$ for all $\gamma\in\Gamma_{0}^{+}(2)$.

###### Proof.

The proof of the lemma follows exactly that of Lemma 4.2. We find that for
$\gamma\in\Gamma_{0}^{+}(2)$, we have

$\det\hat{\rho}(\gamma)=\frac{\left(F(z)^{2}\big{|}_{2(\ell+1)}\gamma\right)}{F(z)^{2}}=1$

because $F(z)^{2}$ is a modular form of weight $2(\ell+1)$ on
$\Gamma_{0}^{+}(2)$. ∎

Now let $y_{+}(z)$ be the unique solution of (5.3) of the form

$y_{+}(z)=q^{\kappa_{\infty}}\left(1+\sum_{j\geq 1}c_{j}q^{j}\right).$

Then analogous to Theorem 4.3, we have the following result for
$\Gamma_{0}^{+}(2)$ (Theorem 1.4 in Section 1).

######  Theorem 5.2.

Suppose that $Q(z)$ satisifies ($\text{H}_{\Gamma}$) with
$\kappa_{\infty}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$. Then the following
statements hold.

  1. (i)

The differential equation (5.3) is not apparent at $\infty$.

  2. (ii)

Let $y_{2}(z)=y_{+}(z)$ and
$\hat{y}_{1}(z)=\left(\hat{y}_{+}|_{\ell}S\right)(z)$. Then
$\hat{y}_{1}(z)=z\hat{y}_{2}(z)+\hat{m}_{1}(z)$ for some modular form
$\hat{m}_{1}(z)$ in
$\mathscr{M}_{\ell-1}(\Gamma_{0}^{+}(2),\left(\frac{2}{\ell}\right))$, where
$\left(\frac{2}{\cdot}\right)$ is the Legendre symbol whose values are given
by

$\left(\frac{2}{\ell}\right)=\begin{cases}1,&\text{if }\ell\equiv 1,7\
\mathrm{mod}\ 8,\\\ -1,&\text{if }\ell\equiv 3,5\ \mathrm{mod}\ 8.\end{cases}$

  3. (iii)

The ratio $h(z)=\left(\frac{2}{\ell}\right)y_{1}(z)/\sqrt{2}y_{2}(z)$ is
equivariant. That is, for all $\gamma\in\Gamma_{0}^{+}(2)$, we have $h(\gamma
z)=\gamma\cdot h(z)$.

  4. (iv)

Write $\hat{y}_{+}(z)$ as

$\hat{y}_{+}(z)=\left(\frac{2}{\ell}\right)\frac{\pi
i}{4\sqrt{2}}\hat{m}_{1}(z)M_{2}^{\ast}(z)+\hat{m}_{2}(z).$

Then $\hat{m}_{2}(z)$ is a modular form in
$\mathscr{M}_{\ell+1}(\Gamma_{0}^{+}(2),\left(\frac{2}{\ell}\right))$. Hence,
$\hat{y}_{+}(z)$ is a quasimodular form in
$\widetilde{\mathscr{M}}_{\ell+1}^{\leq
1}(\Gamma_{0}^{+}(2),\left(\frac{2}{\ell}\right))$.

The proof of (i) is the same as that of Lemma 4.4 and is omitted. We remark
that if $\kappa_{\rho_{2}}\in\mathbb{N}$, then it can be shown that the
differential equation (5.3) is apparent at $\infty$. (C.f. Remark 4.5.) Thus,
the condition that $(2\kappa_{\rho_{2}},4)=1$ is necessary in order for (5.3)
to be not apparent at $\infty$. We now prove the other parts of the theorem.

By Part (i) of the theorem, there is another solution $y_{-}(z)$ of (5.3) of
the form

$y_{-}(z)=zy_{+}(z)+q^{-\kappa_{\infty}}\left(b_{0}+\sum_{j\geq
1}b_{j}q^{j}\right),\qquad b_{0}\neq 0.$

Set

(5.6) $y_{1}(z)=\left(\hat{y}_{+}\big{|}_{\ell}S\right)(z)/F(z)\qquad
y_{2}(z)=y_{+}(z).$

We first prove that $y_{1}$ and $y_{2}$ are linearly independent. Hence, there
are $d\neq 0$ and $m_{1}(z)$ such that

(5.7) $y_{1}(z)=dzy_{2}(z)+m_{1}(z),$

where $m_{1}(z)$ has a $q$-expansion of the form
$q^{-\kappa_{\infty}}(c_{0}+\sum_{j\geq 1}c_{j}q^{j})$, $c_{0}\neq 0$.

######  Lemma 5.3.

The solutions $y_{i}$, $i=1,2$, are linearly independent, and we have
$d=\pm\sqrt{2}$.

###### Proof.

The proof of linear independence of $y_{1}$ and $y_{2}$ is similar to that of
Part (i) of Lemma 4.6 and is skipped.

To compute $d$, we use $\hat{Y}(z)=(\hat{y}_{1}(z),\hat{y}_{2}(z))^{t}$ as the
basis. We have $\hat{\rho}(T)=\left(\begin{smallmatrix}1&d\\\
0&1\end{smallmatrix}\right)$ and
$\hat{\rho}(S)=\left(\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\right)$
(see the proof of (4.12)). Hence,

$\hat{\rho}(R)=\hat{\rho}(T)\hat{\rho}(S)=\begin{pmatrix}1&d\\\
0&1\end{pmatrix}\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}=\begin{pmatrix}d&-1\\\
1&0\end{pmatrix}.$

Then $\hat{\rho}(R)^{2}=-\hat{\rho}(R)^{-2}$ yields

$\begin{pmatrix}d^{2}-1&-d\\\ d&-1\end{pmatrix}=-\begin{pmatrix}-1&d\\\
-d&d^{2}-1\end{pmatrix}.$

It follows that $d=\pm\sqrt{2}$, and the proof of the lemma is completed. ∎

######  Lemma 5.4.

Let $h(z)=y_{1}(z)/(dy_{2}(z))$. Then $h(Sz)=S\cdot h(z)$.

###### Proof.

By using the basis $\left(\hat{y}_{1},\hat{y}_{2}\right)$, we have
$\hat{\rho}(S)=\left(\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\right)$,
which implies that

$\displaystyle
h(Sz)=\frac{-y_{2}(z)}{dy_{1}(z)}=\frac{1}{d^{2}}\frac{-dy_{2}(z)}{y_{1}(z)}=\frac{-1}{2h(z)}=\frac{-1/\sqrt{2}}{2h(z)/\sqrt{2}}=S\cdot
h(z).$

This proves the lemma. ∎

Note that together with Lemma 5.8 below, this lemma yields Part (iii) of the
theorem.

######  Lemma 5.5.

Let $y_{1}(z)$ and $m_{1}(z)$ be defined as in (5.6) and (5.7), respectively.
Then $\hat{m}_{1}(z)$ satisfies

  1. (i)

$\hat{m}_{1}(z+1)=\hat{m}_{1}(z)$,

  2. (ii)

$\left(\hat{m}_{1}\big{|}_{\ell-1}S\right)(z)=\epsilon\hat{m}_{1}(z)$ if
$d=\epsilon\sqrt{2}$, $\epsilon\in\\{\pm 1\\}$, and

  3. (iii)

$\hat{m}_{1}(z)\in\mathscr{M}_{\ell-1}(\Gamma_{0}^{+}(2),\epsilon)$.

###### Proof.

Part (i) is obvious. The proof of (ii) is similar to that of Lemma 4.8. For
the convenience of readers, we repeat it here. Write

$h(z)=\frac{y_{1}(z)}{dy_{2}(z)}=z+\frac{m_{1}(z)}{dy_{2}(z)}.$

By the previous lemma,

$-\frac{1}{2h(z)}=h(Sz)=h\left(-\frac{1}{2z}\right)=-\frac{1}{2z}+\frac{m_{1}(Sz)}{dy_{2}(Sz)},$

and then

$\displaystyle\frac{m_{1}(Sz)}{dy_{2}(Sz)}=\frac{-1}{2h(z)}+\frac{1}{2z}=\frac{h(z)-z}{2h(z)z}=\frac{m_{1}(z)}{2dzh(z)y_{2}(z)}=\frac{m_{1}(z)}{2zy_{1}(z)}.$

Since

$\hat{y}_{1}(z)=\left(\hat{y}_{2}\big{|}_{\ell}S\right)(z)=(\sqrt{2}z)^{-\ell}\hat{y}_{2}(Sz),$

we deduce that

$\displaystyle\hat{m}_{1}(Sz)$
$\displaystyle=\frac{d\hat{m}_{1}(z)}{2z}\frac{\hat{y}_{2}(Sz)}{\hat{y}_{1}(z)}=\epsilon\frac{\hat{m}_{1}(z)}{\sqrt{2}z}\cdot(\sqrt{2}z)^{\ell}$
$\displaystyle={\epsilon}(\sqrt{2}z)^{\ell-1}\hat{m}_{1}(z)\quad\text{if
}d=\epsilon\sqrt{2},$

i.e., $\left(\hat{m}_{1}\big{|}_{\ell-1}S\right)(z)={\epsilon}\hat{m}_{1}(z)$.

Now we have $R=TS$. Hence, by (i) and (ii),
$\left(\hat{m}_{1}\big{|}_{\ell-1}R^{2}\right)(z)=\hat{m}_{1}(z)$. Since
$R^{2}=\left(\begin{smallmatrix}1&-1\\\ 2&-1\end{smallmatrix}\right)$ and $T$
generate the group $\Gamma_{0}(2)$, we find that $\hat{m}_{1}(z)$ is a modular
form of weight $\ell-1$ on $\Gamma_{0}(2)$. It is clear from (ii) that it
belongs to the Atkin-Lehner subspace with eigenvalue $\epsilon$. ∎

Define $m_{2}(z)$ by

(5.8) $y_{+}(z)=\frac{\pi i}{4d}m_{1}(z)M_{2}^{\ast}(z)+m_{2}(z).$

Note that since $m_{1}(z)=q^{-\kappa_{\infty}}(a_{0}+O(q))$, $a_{0}\neq 0$, we
have $m_{2}(z)=q^{-\kappa_{\infty}}(b_{0}+O(q))$ for some $b_{0}\neq 0$.

######  Lemma 5.6.

We have $\hat{m}_{2}(z)\in\mathscr{M}_{\ell+1}(\Gamma_{0}^{+}(2),\epsilon)$,
where $\epsilon\in\\{\pm 1\\}$ is determined by $d=\epsilon\sqrt{2}$.

###### Proof.

As seen in the proof of (5.5), it suffices to prove that

$\hat{m}_{2}(z+1)=\hat{m}_{2}(z),\qquad\left(\hat{m}_{2}\big{|}_{\ell+1}S\right)(z)=\epsilon\hat{m}_{2}(z).$

The first identity is obvious. To prove the second identity, by (5.2), (5.8)
and Lemma 5.5, we have

$\displaystyle\left(\hat{y}_{+}\big{|}_{\ell}S\right)(z)$
$\displaystyle=\sqrt{2}z\frac{\pi
i}{4d}\left(\hat{m}_{1}\big{|}_{\ell-1}S\right)(z)\left(M_{2}^{\ast}\big{|}_{2}S\right)(z)+\sqrt{2}z\left(\hat{m}_{2}\big{|}_{\ell+1}S\right)(z)$
$\displaystyle=\sqrt{2}z\left(\frac{\pi
i}{4d}{\epsilon}\hat{m}_{1}(z)M_{2}^{\ast}(z)+\frac{\hat{m}_{1}(z)}{\sqrt{2}z}\right)+\sqrt{2}z\left(\hat{m}_{2}\big{|}_{\ell+1}S\right)(z)$
$\displaystyle=\frac{\pi
iz}{4}\hat{m}_{1}(z)M_{2}^{\ast}(z)+\hat{m}_{1}(z)+\sqrt{2}z\left(\hat{m}_{2}\big{|}_{\ell+1}S\right)(z)$
$\displaystyle=dz\left(\hat{y}_{+}(z)-\hat{m}_{2}(z)\right)+\hat{m}_{1}(z)+\sqrt{2}z\left(\hat{m}_{2}\big{|}_{\ell+1}S\right)(z)$
$\displaystyle=\left(dz\hat{y}_{+}(z)+\hat{m}_{1}(z)\right)+\sqrt{2}z\left(\left(\hat{m}_{2}\big{|}_{\ell+1}S\right)(z)-\epsilon\hat{m}_{2}(z)\right).$

We note that the first sum of the right-hand side is equal to
$\left(\hat{y}_{+}\big{|}_{\ell}S\right)(z)$. Therefore, we have

$\left(\hat{m}_{2}\big{|}_{\ell+1}S\right)(z)={\epsilon}\hat{m}_{2}(z).$

This completes the proof. ∎

It remains to determine $\epsilon$. We first prove a lemma.

######  Lemma 5.7.

The two modular forms $\hat{m}_{1}(z)$ and $\hat{m}_{2}(z)$ cannot have a
common zero on $\mathbb{H}$.

###### Proof.

Suppose that $\hat{m}_{1}(z)$ and $\hat{m}_{2}(z)$ have a common zero at
$z_{0}\in\mathbb{H}$. Then $\hat{y}_{1}(z_{0})=\hat{y}_{2}(z_{0})=0$, which
implies that both $y_{+}$ and $y_{+}|_{-1}S$ have the behavior
$c(z-z_{0})^{1/2+\kappa}(1+O(z-z_{0}))$ for some $c\neq 0$ near $z_{0}$.
Consequently, $y_{+}$ and $y_{+}|_{-1}S$ are linearly dependent, contradicting
to Lemma 5.3. We conclude that $\hat{m}_{1}(z)$ and $\hat{m}_{2}(z)$ have no
common zero on $\mathbb{H}$. ∎

We now use this lemma to determine $\epsilon$.

######  Lemma 5.8.

We have

$\epsilon=\left(\frac{2}{\ell}\right)=\begin{cases}1,&\text{if }\ell\equiv
1,7\ \mathrm{mod}\ 8,\\\ -1,&\text{if }\ell\equiv 3,5\ \mathrm{mod}\
8.\end{cases}.$

###### Proof.

We will prove case by case that the assumption
$\epsilon=-\left(\frac{2}{\ell}\right)$ will imply that $\hat{m}_{1}(z)$ and
$\hat{m}_{2}(z)$ have a common zero at $\rho_{2}$. By the lemma above, this is
absurd. Hence, $\epsilon$ must be equal to $\left(\frac{2}{\ell}\right)$.

In general, _if $z_{0}$ is an elliptic point of order $e$ of a subgroup
$\Gamma$ of $\mathrm{SL}(2,\mathbb{R})$ commensurable with
$\mathrm{SL}(2,\mathbb{Z})$, then any modular form of even weight $k$ with
$(2e)\nmid k$ has a zero at $z_{0}$_ (see, for instance, Proposition A.7 in
the appendix). From this property, we see that

  1. (i)

If $k\equiv 2\ \mathrm{mod}\ 4$, then any modular form of weight $k$ on
$\Gamma_{0}(2)$ has a zero at $\rho_{2}$ since $\rho_{2}=(1+i)/2$ is an
elliptic point of order $2$ of $\Gamma_{0}(2)$.

  2. (ii)

If $k\equiv 4\ \mathrm{mod}\ 8$, then any modular form of weight $k$ on
$\Gamma_{0}^{+}(2)$ has a zero at $\rho_{2}$ since $\rho_{2}$ is an elliptic
point of order $4$ of $\Gamma_{0}^{+}(2)$.

Consider first the case $\ell\equiv 3,5\ \mathrm{mod}\ 8$. Suppose that
$\epsilon=1$. Then $\hat{m}_{i}(z)$ are modular forms on $\Gamma_{0}^{+}(2)$
of weight $\ell-1$ and $\ell+1$. By (i) and (ii) above, $\rho_{2}$ is a common
zero of $\hat{m}_{1}$ and $\hat{m}_{2}(z)$, a contradiction to Lemma 5.7. This
proves $\epsilon=-1$.

For the remaining cases where $\ell+1$ or $\ell-1$ is divisible by $8$, we
claim that if $k$ is divisible by $8$, then any modular form in
$\mathscr{M}_{k}(\Gamma_{0}^{+}(2),-)$ has a zero at $\rho_{2}$, which then
implies that the sign $\epsilon$ must be $1=\left(\frac{2}{\ell}\right)$.
Indeed, if $8|k$ and $f\in\mathscr{M}_{k}(\Gamma_{0}^{+}(2),-)$, then
evaluating the two sides of the identity

$f\left(\frac{2z-1}{2z}\right)=f(Rz)=-(\sqrt{2}z)^{k}f(z)$

at $\rho_{2}$, we see that $f(\rho_{2})$ must be $0$ because
$(\sqrt{2}\rho_{2})^{8}=1$. This completes the proof of the lemma. ∎

With this lemma, we conclude the proof of Theorem 5.2. We now give three
examples and show they agree with our theorem.

######  Example 5.9.

Consider

$y^{\prime\prime}(z)=-4\pi^{2}\left(\frac{k}{2}\right)^{2}M_{4}(z)y(z),$

where $k\in\mathbb{N}$. By Theorem 5.2, if $y_{+}(z)$ is the unique solution
of the form $y_{+}(z)=q^{k/2}(1+\sum_{j\geq 1}a_{j}q^{j})$, then
$M_{8}(z)^{k/2}y_{+}(z)$ is a quasimodular form in
$\widetilde{\mathscr{M}}_{4k}^{\leq 1}(\Gamma_{0}^{+}(2),(-1)^{k})$. (Note
that the function $F(z)$ and the integer $\ell$ in this case are
$M_{8}(z)^{k/2}$ and $-1+4k$, respectively.) Moreover, the vanishing order of
$M_{8}(z)^{k/2}y_{+}(z)$ at $\infty$ is
$k=\dim\widetilde{\mathscr{M}}_{4k}^{\leq 1}(\Gamma_{0}^{+}(2),(-1)^{k})-1$.
Hence, $M_{8}(z)^{k/2}y_{+}(z)$ is an extremal quasimodular form. This agrees
with Part (i) of Theorem 3.7.

######  Example 5.10.

Consider

$y^{\prime\prime}(z)=-4\pi^{2}\left(\left(\frac{k}{2}\right)^{2}M_{4}(z)-32\frac{M_{8}(z)}{M_{4}(z)}\right)y(z),$

where $k\in\mathbb{Z}_{\geq 0}$. We have $\kappa_{\infty}=k/2$,
$\kappa_{\rho_{1}}=1/2$, $\kappa_{\rho_{2}}=3/2$, and $\kappa=1/2$ elsewhere.
Hence, the function $F(z)$ and the integer $\ell$ in this case are
$F(z)=M_{8}(z)^{k/2}M_{2}(z)$ and $\ell=-1+4k+2=4k+1$. Theorem 5.2 asserts
that if $y_{+}(z)$ is the unique solution of the form
$y_{+}(z)=q^{k/2}(1+\cdots)$, then $M_{8}(z)^{k/2}M_{2}(z)y_{+}(z)$ is a
quasimodular form in $\widetilde{\mathscr{M}}_{4k+2}^{\leq
1}(\Gamma_{0}^{+}(2),(-1)^{k})$. As in the previous example, we find that it
is an extremal quasimodular form, agreeing with Part (ii) of Theorem 3.7.

######  Example 5.11.

Consider

$y^{\prime\prime}(z)=-4\pi^{2}\left(\left(\frac{k}{2}\right)^{2}M_{4}(z)+\frac{128M_{4}(z)M_{8}(z)}{M_{4}(z)^{2}-256M_{8}(z)}\right)y(z),\quad
k\in\mathbb{Z}_{\geq 0}.$

We have $\kappa_{\infty}=k/2$ and by (A.15), $\kappa_{\rho_{1}}=3/2$. The
integer $\ell$ in Theorem 5.2 is $k+3$. According to the theorem,
$M_{8}(z)^{k/2}M_{6}(z)y_{+}(z)/M_{2}(z)$ is a quasimodular form in
$\widetilde{\mathscr{M}}_{4k+4}^{\leq 1}(\Gamma_{0}^{+}(2),(-1)^{k-1})$.
Indeed, when $k=1$, we have

$y_{+}(z)=q^{1/2}\left(1+70q+5926q^{2}+503696q^{3}+42822181q^{4}+\cdots\right)$

and

$1120M_{8}(z)^{1/2}\frac{M_{6}(z)}{M_{2}(z)}y_{+}(z)=M_{2}^{\ast}(z)M_{6}(z)-M_{4}(z)^{2}+1280M_{8}(z),$

which belongs to $\widetilde{\mathscr{M}}_{8}^{\leq 1}(\Gamma_{0}^{+}(2))$,
while when $k=2$, we have

$y_{+}(z)=q+\frac{176}{3}q^{2}+\frac{13706}{3}q^{3}+\frac{1151072}{3}q^{4}+\cdots$

and

$\begin{split}&19008M_{8}(z)\frac{M_{6}(z)}{M_{2}(z)}y_{+}(z)=M_{2}^{\ast}(z)M_{2}(z)\left(-M_{4}(z)^{2}+384M_{8}(z)\right)\\\
&\qquad\qquad+M_{2}(z)M_{4}(z)M_{6}(z)-288M_{4}^{-}(z)M_{8}(z),\end{split}$

which lies in $\widetilde{\mathscr{M}}_{12}^{\leq 1}(\Gamma_{0}^{+}(2),-)$,
where
$M_{4}^{-}(z)=(4E_{4}(2z)-E_{4}(z))/3\in\mathscr{M}_{4}(\Gamma_{0}^{+}(2),-)$.

##  6\. MODE in the case of $\Gamma_{0}^{+}(3)$

We first recall the following information about $\Gamma_{0}^{+}(3)$.

  1. (i)

The group $\Gamma_{0}^{+}(3)$ is generated by
$S=\frac{1}{\sqrt{3}}\left(\begin{smallmatrix}0&-1\\\
3&0\end{smallmatrix}\right)$ and
$R=TS=\frac{1}{\sqrt{3}}\left(\begin{smallmatrix}3&-1\\\
3&0\end{smallmatrix}\right)$, where $T=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)$. They satisfy $S^{2}=R^{6}=-I$.

  2. (ii)

$\Gamma$ has only one cusp at $\infty$ with width $1$, and two elliptic points
$\rho_{1}=i/\sqrt{3}$ (order $2$) and $\rho_{2}=(3+\sqrt{-3})/6$ (order $6$).
Their stabilizer subgroups are generated by $T$, $S$, and $R$, respectively.

  3. (iii)

The ring $\oplus_{k=0}^{\infty}\mathscr{M}_{k}(\Gamma_{1}(3))$, where $k$ runs
through both even and odd integers, is freely generated by

$M_{1}(z)=\sum_{m,n\in\mathbb{Z}}q^{m^{2}+mn+n^{2}}$

and

$M_{3}(z)=\frac{9E_{4}(3z)-E_{4}(z)}{8M_{1}(z)}=\frac{\eta(z)^{9}}{\eta(3z)^{3}}-27\frac{\eta(3z)^{9}}{\eta(z)^{3}}$

of weight $1$ and $3$, respectively.

  4. (iv)

Let $\chi$ be the character of $\Gamma_{0}^{+}(3)$ determined by
$\chi(S)=\chi(R)=-i$. Then

$M_{1}(z)\in\mathscr{M}_{1}(\Gamma_{0}^{+}(3),\chi),\qquad
M_{3}(z)\in\mathscr{M}_{1}(\Gamma_{0}^{+}(3),\chi).$

For our purpose, we also need the modular forms

$\begin{split}M_{4}^{+}(z)&=\frac{9E_{4}(3z)+E_{4}(z)}{10}=M_{1}(z)^{4}\in\mathscr{M}_{4}(\Gamma_{0}^{+}(3)),\\\
M_{6}^{-}(z)&=\eta(z)^{6}\eta(3z)^{6}=\frac{M_{1}(z)^{6}-M_{3}(z)^{2}}{108}\in\mathscr{M}_{6}(\Gamma_{0}^{+}(3),-).\end{split}$

Up to $\Gamma^{+}_{0}(3)$-equivalence, $M_{1}(z)$, $M_{3}(z)$, and
$M_{6}^{-}(z)$ have simple zeros at $\rho_{2}$, $\rho_{1}$, and $\infty$,
respectively. The ratio $M_{1}(z)^{6}/M_{6}^{-}(z)$ is a Hauptmodul for
$\Gamma_{0}^{+}(3)$. Let also

$M_{2}^{\ast}(z):=\frac{1}{2\pi
i}\frac{(M_{6}^{-})^{\prime}(z)}{M_{6}^{-}(z)}=\frac{3E_{2}(3z)+E_{2}(z)}{4},$

which is a quasimodular form of weight $2$ and depth $1$ on
$\Gamma_{0}^{+}(3)$ and satisfies

(6.1) $\left(M_{2}^{\ast}|_{2}\gamma\right)(z)=M_{2}^{\ast}(z)+\frac{3c}{\pi
i(cz+d)},\qquad\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma_{0}^{+}(3).$

Consider the differential equation

(6.2) $y^{\prime\prime}(z)=-4\pi^{2}Q(z)y(z),$

where $Q(z)$ is a meromorphic modular form of weight $4$ on
$\Gamma_{0}^{+}(3)$, and assume that it satisfies the condition
($\text{H}_{\Gamma}$). Let $z_{1},\ldots,z_{m}$ be
$\Gamma_{0}^{+}(3)$-inequivalent singularities of (6.2) other than $\rho_{1}$
and $\rho_{2}$. Let $\pm\kappa_{\infty}$, $1/2\pm\kappa_{\rho_{1}}$,
$1/2\pm\kappa_{\rho_{2}}$, and $1/2\pm\kappa_{j}$, $j=1,\ldots,m$, be the
local exponents of (6.2) at $\infty$, $\rho_{1}$, $\rho_{2}$, and $z_{j}$,
respectively. For $j=1,\ldots,m$, let
$t_{j}=M_{2}^{-}(z_{j})^{3}/M_{6}^{-}(z_{j})$, and set

$F_{j}(z)=M_{2}^{-}(z)^{3}-t_{j}M_{6}^{-}(z).$

Define

(6.3)
$F(z)=M_{6}^{-}(z)^{\kappa_{\infty}}M_{3}(z)^{\kappa_{\rho_{1}}-1/2}M_{1}(z)^{\kappa_{\rho_{2}}-1/2}\prod_{j=1}^{m}F_{j}(z)^{\kappa_{j}-1/2},$

and set

(6.4)
$\ell=-1+6\kappa_{\infty}+3\left(\kappa_{\rho_{1}}-\frac{1}{2}\right)+\left(\kappa_{\rho_{2}}-\frac{1}{2}\right)+6\sum_{j=1}^{m}\left(\kappa_{j}-\frac{1}{2}\right).$

Note that by Condition ($\text{H}_{\Gamma}$), both $\kappa_{\rho_{1}}-1/2$ and
$\kappa_{\rho_{2}}-1/2$ are integers, and $(2\kappa_{\rho_{2}},3)=1$. Hence
$\ell$ is an integer not divisible by $3$. For any function $f(z)$ on
$\mathbb{H}$, we let $\hat{f}(z)$ denote

$\hat{f}(z)=F(z)f(z).$

Note that $F(z)^{2}$ is a holomorphic modular form of weight $2(\ell+1)$ on
$\Gamma_{0}^{+}(3)$ with some character depending on $\ell$. By construction,
for any solution $y(z)$ of (6.2), the function $\hat{y}(z)$ is a single-valued
function holomorphic throughout $\mathbb{H}$. Also, its order at a singularity
$z_{j}$ is either $0$ or $2\kappa_{j}$, and a similar property holds for
$\infty$, $\rho_{1}$, and $\rho_{2}$. Hence, for any fundamental solution
$(y_{1}(z),y_{2}(z))$ of (6.2), we have a representation
$\hat{\rho}:\Gamma_{0}^{+}(3)\to\mathrm{GL}(2,\mathbb{C})$ given by

$\begin{pmatrix}\hat{y}_{1}\big{|}_{\ell}\gamma\\\
\hat{y}_{2}\big{|}_{\ell}\gamma\end{pmatrix}=\hat{\rho}(\gamma)\begin{pmatrix}\hat{y}_{1}\\\
\hat{y}_{2}\end{pmatrix}.$

We now state the main result of this section (Theorem 3.8 of Section 1).

######  Theorem 6.1.

Suppose that $Q(z)$ satisify ($\text{H}_{\Gamma}$) with
$\kappa_{\infty}\in\frac{1}{2}\mathbb{Z}_{\geq 0}$. Let $\ell$ be the integer
defined by (6.4). Set

$\delta=\begin{cases}\chi^{0},&\text{if }\ell\equiv 1,11\ \mathrm{mod}\ 12,\\\
\chi^{1},&\text{if }\ell\equiv 2,4\ \mathrm{mod}\ 12,\\\ \chi^{2},&\text{if
}\ell\equiv 5,7\ \mathrm{mod}\ 12,\\\ \chi^{3},&\text{if }\ell\equiv 8,10\
\mathrm{mod}\ 12,\end{cases}$

where $\chi$ is the character of $\Gamma_{0}^{+}(3)$ defined by
$\chi(S)=\chi(R)=-i$. The following statements hold.

  1. (i)

The differential equation (6.2) is not apparent at $\infty$.

  2. (ii)

Let $y_{2}(z)=y_{+}(z)$ and
$\hat{y}_{1}(z)=\left(\hat{y}_{+}|_{\ell}S\right)(Z)$. Then
$\hat{y}_{1}(z)=z\hat{y}_{2}(z)+\hat{m}_{1}(z)$ for some modular form
$\hat{m}_{1}(z)$ in $\mathscr{M}_{\ell-1}(\Gamma_{0}^{+}(3),\delta)$.

  3. (iii)

The ratio $h(z)=\delta(S)^{-1}y_{1}(z)/\sqrt{3}y_{2}(z)$ is equivariant. That
is, for all $\gamma\in\Gamma_{0}^{+}(3)$, we have $h(\gamma z)=\gamma\cdot
h(z)$.

  4. (iv)

Write $\hat{y}_{+}(z)$ as

$\hat{y}_{+}(z)=\delta(S)^{-1}\frac{\pi
i}{3\sqrt{3}}\hat{m}_{1}(z)M_{2}^{\ast}(z)+\hat{m}_{2}(z).$

Then $\hat{m}_{2}(z)$ is a modular form in
$\mathscr{M}_{\ell+1}(\Gamma_{0}^{+}(3),\delta)$. Hence, $\hat{y}_{+}(z)$ is a
quasimodular form in $\widetilde{\mathscr{M}}_{\ell+1}^{\leq
1}(\Gamma_{0}^{+}(3),\delta)$.

Note that when $k$ is even, we have
$\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\chi^{0})$ and
$\mathscr{M}_{k}(\Gamma_{0}^{+}(3),\chi^{2})$ are simply
$\mathscr{M}_{k}(\Gamma_{0}^{+}(3))$ and
$\mathscr{M}_{k}(\Gamma_{0}^{+}(3),-)$, respectively, so
$\widetilde{\mathscr{M}}_{\ell+1}^{\leq 1}(\Gamma_{0}^{+}(3),\delta)$ in the
case of odd $\ell$ can also be written as
$\widetilde{\mathscr{M}}_{\ell+1}^{\leq
1}\left(\Gamma_{0}^{+}(3),\left(\frac{12}{\cdot}\right)\right)$.

Apart from the difference that the integer $\ell$ can be even, the proof is
very similar to those of Theorems 4.3 and 5.2.

######  Lemma 6.2.

We have $\det\hat{\rho}(T)=1$ and $\det\hat{\rho}(S)=(-1)^{\ell-1}$.

###### Proof.

Following the proof of Lemma 4.2, we have

$\det\hat{\rho}(\gamma)=\frac{\left(F(z)^{2}\big{|}_{2(\ell+1)}\gamma\right)}{F(z)^{2}}$

for $\gamma\in\Gamma_{0}^{+}(3)$. It is clear that when $\gamma=T$, we have