# On $m$–fold Holomorphic Differentials and Modular Forms

Damir Mikoč and Goran Muić  Department of Teacher Education Studies in Gospić,
University of Zadar, dr. Ante Starčevića 12, 53000 Gospić, Croatia
<EMAIL_ADDRESS>Department of Mathematics, Faculty of Science,
University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia<EMAIL_ADDRESS>
###### Abstract.

Let $\Gamma$ be the Fuchsian group of the first kind. For an even integer
$m\geq 4$, we study $m/2$–holomorphic differentials in terms of space of
(holomorphic) cuspidal modular forms $S_{m}(\Gamma)$. We also give in depth
study of Wronskians of cuspidal modular forms and their divisors.

###### Key words and phrases:

Wronskians, holomorphic differentials, cuspidal modular forms

###### 2000 Mathematics Subject Classification:

11F11

The author acknowledges Croatian Science Foundation grant IP-2018-01-3628.

##  1\. Introduction

Let $\Gamma$ be the Fuchsian group of the first kind [7, Section 1.7, page
28]. Examples of such groups are the important modular groups such as
$SL_{2}(\mathbb{Z})$ and its congruence subgroups $\Gamma_{0}(N)$,
$\Gamma_{1}(N)$, and $\Gamma(N)$ [7, Section 4.2]. Let $\mathbb{H}$ be the
complex upper half-plane. The quotient $\Gamma\backslash\mathbb{H}$ can be
compactified by adding a finite number of $\Gamma$-orbits of points in
$\mathbb{R}\cup\\{\infty\\}$ called cusps of $\Gamma$ and we obtain a compact
Riemann surface which will be denoted by $\mathfrak{R}_{\Gamma}$. For $l\geq
1$, let $H^{l}\left(\mathfrak{R}_{\Gamma}\right)$ be the space of all
holomorphic differentials on $\mathfrak{R}_{\Gamma}$ (see [5], or Section 3)
in this paper).

Let $m\geq 2$ be an even integer. Let $S_{m}(\Gamma)$ be the space of
(holomorphic) cusp forms of weight $m$ (see Section 2). It is well–known that
$S_{2}(\Gamma)$ is naturally isomorphic to the vector space
$H^{1}\left(\mathfrak{R}_{\Gamma}\right)$ (see [7, Theorem 2.3.2]). This is
employed on many instances in studying various properties of modular curves
(see for example [19, Chapter 6]). In this paper we study the generalization
of this concept to the holomorphic differentials of higher order.

For an even integer $m\geq 4$, in general, the space $S_{m}(\Gamma)$ is too
big to be isomorphic to $H^{m/2}\left(\mathfrak{R}_{\Gamma}\right)$ due to
presence of cusps and elliptic points. So, in general we define a subspace

$S^{H}_{m}(\Gamma)=\left\\{f\in S_{m}(\Gamma);\ \ \text{$f=0$ or $f$ satisfies
(\ref{int-mhd-8})}\right\\},$

where

(1-1) $\mathfrak{c}_{f}\geq\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
elliptic}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}+\left(\frac{m}{2}-1\right)\sum_{\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp}\mathfrak{b}.$

The integral divisor $\mathfrak{c}_{f}$ is defined in Lemma 2-2 while the
multiplicities $e_{\mathfrak{a}}$ are defined in Section 2. Now, we have the
following result (see Section 4):

######  Theorem 1-2.

The usual map $f\longmapsto\omega_{f}$ from the space of all cuspidal modular
form into space of meromorphic differentials (see [7, Theorem 2.3.3]) induces
the isomorphism of $S^{H}_{m}(\Gamma)$ onto
$H^{m/2}\left(\mathfrak{R}_{\Gamma}\right)$.

We study the space $S^{H}_{m}(\Gamma)$ in detail in Section (see Section 4).
The main results are contained in a very detailed Lemma 4-5 and Theorem 4-16.
We recall (see [5, III.5.9] or Definition 3-6 that
$\mathfrak{a}\in\mathfrak{R}_{\Gamma}$ is a $m/2$-Weierstrass point if there
exists a non–zero $\omega\in H^{m/2}\left(\mathfrak{R}_{\Gamma}\right)$ such
that

$\nu_{\mathfrak{a}}(\omega)\geq\dim
H^{m/2}\left(\mathfrak{R}_{\Gamma}\right).$

Equivalently [5, Proposition III.5.10] , if

$\nu_{\mathfrak{a}}\left(W\left(\omega_{1},\ldots,\omega_{t}\right)\right)\geq
1,$

where $W\left(\omega_{1},\ldots,\omega_{t}\right)$ is the Wronskian of
holomorphic differential forms $\omega_{1},\ldots,\omega_{t}$ (see Section 3).

When $m=2$ we speak about classical Weierstrass points. So, $1$-Weierstrass
points are simply Weierstrass points. Weierstrass points on modular curves are
very-well studied (see for example [19, Chapter 6], [16], [17], [20], [21],
[22]). Higher–order Weierstrass points has not been not studied much (see for
example [16]).

The case $m\geq 4$ is more complex. We recall that $\mathfrak{R}_{\Gamma}$ is
hyperelliptic if $g(\Gamma)\geq 2$, and there is a degree two map onto
$\mathbb{P}^{1}$. By general theory [9, Chapter VII, Proposition 1.10], if
$g(\Gamma)=2$, then $\mathfrak{R}_{\Gamma}$ is hyperelliptic. If
$\mathfrak{R}_{\Gamma}$ is not hyperelliptic, then $\dim
S_{2}(\Gamma)=g(\Gamma)\geq 3$, and the regular map
$\mathfrak{R}_{\Gamma}\longrightarrow\mathbb{P}^{g(\Gamma)-1}$ attached to a
canonical divisor $K$ is an isomorphism onto its image [9, Chapter VII,
Proposition 2.1].

Let $\Gamma=\Gamma_{0}(N)$, $N\geq 1$. Put
$X_{0}(N)=\mathfrak{R}_{\Gamma_{0}(N)}$. We recall that $g(\Gamma_{0}(N))\geq
2$ unless

$\begin{cases}N\in\\{1-10,12,13,16,18,25\\}\ \ \text{when
$g(\Gamma_{0}(N))=0$, and}\\\ N\in\\{11,14,15,17,19-21,24,27,32,36,49\\}\ \
\text{when $g(\Gamma_{0}(N))=1$.}\end{cases}$

Let $g(\Gamma_{0}(N))\geq 2$. Then, we remark that Ogg [17] has determined all
$X_{0}(N)$ which are hyperelliptic curves. In view of Ogg’s paper, we see that
$X_{0}(N)$ is not hyperelliptic for $N\in\\{34,38,42,43,44,45,51-58,60-70\\}$
or $N\geq 72$. This implies $g(\Gamma_{0}(N))\geq 3$.

We prove the following result (see Theorem 4-12)

######  Theorem 1-3.

Let $m\geq 4$ be an even integer. Assume that $\mathfrak{R}_{\Gamma}$ is not
hyperelliptic. Then, we have

$S_{m,2}^{H}(\Gamma)=S_{m}^{H}(\Gamma),$

where we denote the subspace $S_{m,2}^{H}(\Gamma)$ of $S_{m}^{H}(\Gamma)$
spanned by all monomials

$f_{0}^{\alpha_{0}}f_{1}^{\alpha_{1}}\cdots f_{g-1}^{\alpha_{g-1}},\ \
\alpha_{i}\in\mathbb{Z}_{\geq 0},\ \sum_{i=0}^{g-1}\alpha_{i}=\frac{m}{2}.$

Here $f_{0},\ldots,f_{g-1}$, $g=g(\Gamma)$, is a basis of $S_{2}(\Gamma)$

The criterion is given by the following corollary (see Corollary 4-14):

######  Corollary 1-4.

Let $m\geq 4$ be an even integer. Assume that $\mathfrak{R}_{\Gamma}$ is not
hyperelliptic. Assume that $\mathfrak{a}_{\infty}$ is a cusp for $\Gamma$. Let
us select a basis $f_{0},\ldots,f_{g-1}$, $g=g(\Gamma)$, of $S_{2}(\Gamma)$.
Compute $q$–expansions of all monomials

$f_{0}^{\alpha_{0}}f_{1}^{\alpha_{1}}\cdots f_{g-1}^{\alpha_{g-1}},\ \
\alpha_{i}\in\mathbb{Z}_{\geq 0},\ \sum_{i=0}^{g-1}\alpha_{i}=\frac{m}{2}.$

Then, $\mathfrak{a}_{\infty}$ is not a $\frac{m}{2}$–Weierstrass point if and
only if there exist a basis of the space of all such monomials, $F_{1},\ldots
F_{t}$, $t=\dim{S_{m}^{H}(\Gamma)}=(m-1)(g-1)$ (see Lemma 4-5 (v)), such that
their $q$–expansions are of the form

$F_{u}=a_{u}q^{u+m/2-1}+\text{higher order terms in $q$},\ \ 1\leq u\leq t,$

where

$a_{u}\in\mathbb{C},\ \ a_{u}\neq 0.$

This is useful for explicit computations in SAGE at least when
$\Gamma=\Gamma_{0}(N)$. We give examples in Section 5 (see Propositions 5-1
and 5-2). A different more theoretical criterion is contained in Theorem 4-16.

Various other aspects of modular curves has been studied in [1], [2], [3],
[4], [10], [14], [15] and [24]. We continue the approach presented in [11],
[12], and [13]. In the proof of Theorem 4-12 we give an explicit construction
of a higher order canonical map i.e., a map attached to divisor
$\frac{m}{2}K$, where $K$ is a canonical divisor of $\mathfrak{R}_{\Gamma}$.
The case $m=2$ is studied in depth in many papers (see for example [3]).

In Section 6 we deal with a generalization of the usual notion of the
Wronskian of cuspidal modular forms [22], ([19], 6.3.1), ([10], the proof of
Theorem 4-5), and ([12], Lemma 4-1). The main result of the section is
Proposition 6-2 which in the most important case has the following form:

######  Proposition 1-5.

Let $m\geq 1$. Then, for any sequence $f_{1},\ldots,f_{k}\in M_{m}(\Gamma)$,
the Wronskian

$W\left(f_{1},\ldots,f_{k}\right)(z)\overset{def}{=}\left|\begin{matrix}f_{1}(z)&\cdots&f_{k}(z)\\\
\frac{df_{1}(z)}{dz}&\cdots&\frac{df_{k}(z)}{dz}\\\ &\cdots&\\\
\frac{d^{k-1}f_{1}(z)}{dz^{k-1}}&\cdots&\frac{d^{k-1}f_{k}(z)}{dz^{k-1}}\\\
\end{matrix}\right|$

is a cuspidal modular form in $S_{k(m+k-1)}(\Gamma)$ if $k\geq 2$. If
$f_{1},\ldots,f_{k}$ are linearly independent, then
$W\left(f_{1},\ldots,f_{k}\right)\neq 0$.

What is new and deep is the computation of the divisor of
$W\left(f_{1},\ldots,f_{k}\right)$ (see Section 7). The main results are
Proposition 7-2 and Theorem 7-3. A substantial example has been given in
Section 8 in the case of $\Gamma=SL_{2}(\mathbb{Z})$ (see Proposition 8-4).

We would like to thank I. Kodrnja for her help with the SAGE system. Also we
would like to thank M. Kazalicki and F. Najman for some useful discussions
about modular forms and curves in general.

##  2\. Preliminaries

In this section we recall necessary facts about modular forms and their
divisors [7]. We follow the exposition in ([12], Section 2).

Let $\mathbb{H}$ be the upper half–plane. Then the group
$SL_{2}({\mathbb{R}})$ acts on $\mathbb{H}$ as follows:

$g.z=\frac{az+b}{cz+d},\ \ g=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in SL_{2}({\mathbb{R}}).$

We let $j(g,z)=cz+d$. The function $j$ satisfies the cocycle identity:

(2-1) $j(gg^{\prime},z)=j(g,g^{\prime}.z)j(g^{\prime},z).$

Next, $SL_{2}({\mathbb{R}})$–invariant measure on $\mathbb{H}$ is defined by
$dxdy/y^{2}$, where the coordinates on $\mathbb{H}$ are written in a usual way
$z=x+\sqrt{-1}y$, $y>0$. A discrete subgroup $\Gamma\subset
SL_{2}({\mathbb{R}})$ is called a Fuchsian group of the first kind if

$\iint_{\Gamma\backslash\mathbb{H}}\frac{dxdy}{y^{2}}<\infty.$

Then, adding a finite number of points in ${\mathbb{R}}\cup\\{\infty\\}$
called cusps, $\mathcal{F}_{\Gamma}$ can be compactified. In this way we
obtain a compact Riemann surface $\mathfrak{R}_{\Gamma}$. One of the most
important examples are the groups

$\Gamma_{0}(N)=\left\\{\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)\in
SL_{2}(\mathbb{Z});\ \ c\equiv 0\ (\mathrm{mod}\ N)\right\\},\ \ N\geq 1.$

We write $X_{0}(N)$ for $\mathfrak{R}_{\Gamma_{0}(N)}$.

Let $\Gamma$ be a Fuchsian group of the first kind. We consider the space
$M_{m}(\Gamma)$ (resp., $S_{m}(\Gamma)$) of all modular (resp., cuspidal)
forms of weight $m$; this is the space of all holomorphic functions
$f:\mathbb{H}\rightarrow{\mathbb{C}}$ such that
$f(\gamma.z)=j(\gamma,z)^{m}f(z)$ ($z\in\mathbb{H}$, $\gamma\in\Gamma$) which
are holomorphic (resp., holomorphic and vanish) at every cusp for $\Gamma$. We
also need the following obvious property: for $f,g\in M_{m}(\Gamma,\chi)$,
$g\neq 0$, the quotient $f/g$ is a meromorphic function on
$\mathfrak{R}_{\Gamma}$.

Next, we recall from ([7], 2.3) some notions related to the theory of divisors
of modular forms of even weight $m\geq 2$ and state a preliminary result.

Let $m\geq 2$ be an even integer and $f\in M_{m}(\Gamma)-\\{0\\}$. Then,
$\nu_{z-\xi}(f)$ denotes the order of the holomorphic function $f$ at $\xi$.
For each $\gamma\in\Gamma$, the functional equation
$f(\gamma.z)=j(\gamma,z)^{m}f(z)$, $z\in\mathbb{H}$, shows that
$\nu_{z-\xi}(f)=\nu_{z-\xi^{\prime}}(f)$ where $\xi^{\prime}=\gamma.\xi$.
Also, if we let

$e_{\xi}=\\#\left(\Gamma_{\xi}/\Gamma\cap\\{\pm 1\\}\right),$

then $e_{\xi}=e_{\xi^{\prime}}$. The point $\xi\in\mathbb{H}$ is elliptic if
$e_{\xi}>1$. Next, following ([7], 2.3), we define

$\nu_{\xi}(f)=\nu_{z-\xi}(f)/e_{\xi}.$

Clearly, $\nu_{\xi}=\nu_{\xi^{\prime}}$, and we may let

$\nu_{\mathfrak{a}_{\xi}}(f)=\nu_{\xi}(f),$

where

$\mathfrak{a}_{\xi}\in\mathfrak{R}_{\Gamma}$ is the projection of $\xi$ to
$\mathfrak{R}_{\Gamma}$,

a notation we use throughout this paper.

If $x\in{\mathbb{R}}\cup\\{\infty\\}$ is a cusp for $\Gamma$, then we define
$\nu_{x}(f)$ as follows. Let $\sigma\in SL_{2}({\mathbb{R}})$ such that
$\sigma.x=\infty$. We write

$\\{\pm 1\\}\sigma\Gamma_{x}\sigma^{-1}=\\{\pm
1\\}\left\\{\left(\begin{matrix}1&lh^{\prime}\\\ 0&1\end{matrix}\right);\ \
l\in{\mathbb{Z}}\right\\},$

where $h^{\prime}>0$. Then we write the Fourier expansion of $f$ at $x$ as
follows:

$(f|_{m}\sigma^{-1})(\sigma.z)=\sum_{n=1}^{\infty}a_{n}e^{2\pi\sqrt{-1}n\sigma.z/h^{\prime}}.$

We let

$\nu_{x}(f)=l\geq 0,$

where $l$ is defined by $a_{0}=a_{1}=\cdots=a_{l-1}=0$, $a_{l}\neq 0$. One
easily see that this definition does not depend on $\sigma$. Also, if
$x^{\prime}=\gamma.x$, then $\nu_{x^{\prime}}(f)=\nu_{x}(f)$. Hence, if
$\mathfrak{b}_{x}\in\mathfrak{R}_{\Gamma}$ is a cusp corresponding to $x$,
then we may define

$\nu_{\mathfrak{b}_{x}}=\nu_{x}(f).$

Put

$\mathrm{div}{(f)}=\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma}}\nu_{\mathfrak{a}}(f)\mathfrak{a}\in\
\ {\mathbb{Q}}\otimes\mathrm{Div}(\mathfrak{R}_{\Gamma}),$

where $\mathrm{Div}(\mathfrak{R}_{\Gamma})$ is the group of (integral)
divisors on $\mathfrak{R}_{\Gamma}$.

Using ([7], 2.3), this sum is finite i.e., $\nu_{\mathfrak{a}}(f)\neq 0$ for
only a finitely many points. We let

$\mathrm{deg}(\mathrm{div}{(f)})=\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma}}\nu_{\mathfrak{a}}(f).$

Let
$\mathfrak{d}_{i}\in{\mathbb{Q}}\otimes\mathrm{Div}(\mathfrak{R}_{\Gamma})$,
$i=1,2$. Then we say that $\mathfrak{d}_{1}\geq\mathfrak{d}_{2}$ if their
difference $\mathfrak{d}_{1}-\mathfrak{d}_{2}$ belongs to
$\mathrm{Div}(\mathfrak{R}_{\Gamma})$ and is non–negative in the usual sense.

######  Lemma 2-2.

Assume that $m\geq 2$ is an even integer. Assume that $f\in M_{m}(\Gamma)$,
$f\neq 0$. Let $t$ be the number of inequivalent cusps for $\Gamma$. Then we
have the following:

  * (i)

For $\mathfrak{a}\in\mathfrak{R}_{\Gamma}$, we have $\nu_{\mathfrak{a}}(f)\geq
0$.

  * (ii)

For a cusp $\mathfrak{a}\in\mathfrak{R}_{\Gamma}$, we have that
$\nu_{\mathfrak{a}}(f)\geq 0$ is an integer.

  * (iii)

If $\mathfrak{a}\in\mathfrak{R}_{\Gamma}$ is not an elliptic point or a cusp,
then $\nu_{\mathfrak{a}}(f)\geq 0$ is an integer. If
$\mathfrak{a}\in\mathfrak{R}_{\Gamma}$ is an elliptic point, then
$\nu_{\mathfrak{a}}(f)-\frac{m}{2}(1-1/e_{\mathfrak{a}})$ is an integer.

  * (iv)

Let $g(\Gamma)$ be the genus of $\mathfrak{R}_{\Gamma}$. Then

$\displaystyle\mathrm{deg}(\mathrm{div}{(f)})$
$\displaystyle=m(g(\Gamma)-1)+\frac{m}{2}\left(t+\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
\ elliptic}(1-1/e_{\mathfrak{a}})\right)$
$\displaystyle=\frac{m}{4\pi}\iint_{\Gamma\backslash\mathbb{H}}\frac{dxdy}{y^{2}}.$

  * (v)

Let $[x]$ denote the largest integer $\leq x$ for $x\in{\mathbb{R}}$. Then

$\displaystyle\dim S_{m}(\Gamma)$
$\displaystyle=\begin{cases}(m-1)(g(\Gamma)-1)+(\frac{m}{2}-1)t+\sum_{\begin{subarray}{c}\mathfrak{a}\in\mathfrak{R}_{\Gamma},\\\
elliptic\end{subarray}}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right],\ \
\text{if $m\geq 4$,}\\\ g(\Gamma),\ \ \text{if $m=2$.}\\\ \end{cases}$
$\displaystyle\dim M_{m}(\Gamma)$ $\displaystyle=\begin{cases}\dim
S_{m}(\Gamma)+t,\ \ \text{if $m\geq 4$, or $m=2$ and $t=0$,}\\\ \dim
S_{m}(\Gamma)+t-1=g(\Gamma)+t-1,\ \ \text{if $m=2$ and $t\geq 1$.}\\\
\end{cases}$

  * (vi)

There exists an integral divisor $\mathfrak{c}^{\prime}_{f}\geq 0$ of degree

$\begin{cases}\dim M_{m}(\Gamma)+g(\Gamma)-1,\ \ \text{if $m\geq 4$, or $m=2$
and $t\geq 1$,}\\\ 2(g(\Gamma)-1),\ \ \text{if $m=2$ and $t=0$}\end{cases}$

such that

$\displaystyle\mathrm{div}{(f)}=$
$\displaystyle\mathfrak{c}^{\prime}_{f}+\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
\
elliptic}\left(\frac{m}{2}(1-1/e_{\mathfrak{a}})-\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\right)\mathfrak{a}.$

  * (vii)

Assume that $f\in S_{m}(\Gamma)$. Then, the integral divisor defined by
$\mathfrak{c}_{f}\overset{def}{=}\mathfrak{c}^{\prime}_{f}-\sum_{\begin{subarray}{c}\mathfrak{b}\in\mathfrak{R}_{\Gamma},\\\
cusp\end{subarray}}\mathfrak{b}$ satisfies $\mathfrak{c}_{f}\geq 0$ and its
degree is given by

$\begin{cases}\dim S_{m}(\Gamma)+g(\Gamma)-1;\ \ \text{if $m\geq 4$,}\\\
2(g(\Gamma)-1);\ \ \text{if $m=2$.}\end{cases}$

###### Proof.

The claims (i)–(v) are standard ([7], 2.3, 2.4, 2.5). The claim (vi) follows
from (iii), (iv), and (v) (see Lemma 4-1 in [10]). Finally, (vii) follows from
(vi). ∎

##  3\. Holomorhic Differentials and $m$–Weierstrass Points on
$\mathfrak{R}_{\Gamma}$

Let $\Gamma$ be a Fuchsian group of the first kind. We let
$D^{m}\left(\mathfrak{R}_{\Gamma}\right)$ (resp.,
$H^{m}\left(\mathfrak{R}_{\Gamma}\right)$)be the space of meromorphic (resp.,
holomorphic) differential of degree $m$ on $\mathfrak{R}_{\Gamma}$ for each
$m\in\mathbb{Z}$. We recall that
$D^{0}\left(\mathfrak{R}_{\Gamma}\right)=\mathbb{C}\left(\mathfrak{R}_{\Gamma}\right)$,
and $D^{m}\left(\mathfrak{R}_{\Gamma}\right)\neq 0$ for all other
$m\in\mathbb{Z}$. In fact, if we fix a non–zero $\omega\in
D^{1}\left(\mathfrak{R}_{\Gamma}\right)$, then
$D^{m}\left(\mathfrak{R}_{\Gamma}\right)=\mathbb{C}\left(\mathfrak{R}_{\Gamma}\right)\omega^{n}$.
We have the following:

(3-1) $\deg{\left(\mathrm{div}{(\omega)}\right)}=2m(g(\Gamma)-1),\ \ \omega\in
D^{m}\left(\mathfrak{R}_{\Gamma}\right),\ \omega\neq 0.$

We shall be interested in the case $m\geq 1$, and in holomorphic
differentials. We recall [5, Proposition III.5.2] that

(3-2) $\dim H^{m}\left(\mathfrak{R}_{\Gamma}\right)=\begin{cases}0\ \
&\text{if}\ \ m\geq 1,g(\Gamma)=0;\\\ g(\Gamma)\ \ &\text{if}\ \ m=1,\
g(\Gamma)\geq 1;\\\ g(\Gamma)\ \ &\text{if}\ \ m\geq 2,\ g(\Gamma)=1;\\\
(2m-1)\left(g\left(\mathfrak{R}_{\Gamma}\right)-1\right)\ \ &\text{if}\ \
m\geq 2,\ g(\Gamma)\geq 2.\\\ \end{cases}$

This follows easily from Riemann-Roch theorem. Recall that a canonical class
$K$ is simply a divisior on any non–zero meromorphic form $\omega$ on
$\mathfrak{R}_{\Gamma}$. Different choices of a $\omega$ differ by a divisor
of a non–zero function $f\in\mathbb{C}\left(\mathfrak{R}_{\Gamma}\right)$

$\mathrm{div}{(f\omega)}=\mathrm{div}{(f)}+\mathrm{div}{(\omega)}.$

Different choices of $\omega$ have the same degree since
$\deg{\left(\mathrm{div}{(f)}\right)}=0$.

For a divisor $\mathfrak{a}$, we let

$L(\mathfrak{a})=\left\\{f\in\mathbb{C}\left(\mathfrak{R}_{\Gamma}\right);\ \
f=0\ \text{or}\ \mathrm{div}{(f)}+\mathfrak{a}\geq 0\right\\}.$

We have the following three facts:

  * (1)

for $\mathfrak{a}=0$, we have $L(\mathfrak{a})=\mathbb{C}$;

  * (2)

if $\deg{(\mathfrak{a})}<0$, then $L(\mathfrak{a})=0$;

  * (3)

the Riemann-Roch theorem: $\dim
L(\mathfrak{a})=\deg{(\mathfrak{a})}-g(\Gamma)+1+\dim L(K-\mathfrak{a})$.

Now, it is obvious that $f\omega^{m}\in
H^{m}\left(\mathfrak{R}_{\Gamma}\right)$ if and only if

$\mathrm{div}{(f\omega^{m})}=\mathrm{div}{(f)}+m\mathrm{div}{(\omega)}=\mathrm{div}{(f)}+mK\geq
0.$

Equivalently, $f\in L(mK)$. Thus, we have that $\dim
H^{m}\left(\mathfrak{R}_{\Gamma}\right)=\dim L(mK)$. Finally, by the Riemann-
Roch theorem, we have the following:

$\dim L(mK)=\deg{(mK)}-g(\Gamma)+1+\dim
L((1-m)K)=(2m-1)(g\left(\mathfrak{R}_{\Gamma}\right)-1)+\dim L((1-m)K).$

Now, if $g(\Gamma)\geq 2$, then $\deg{(K)}=2(g(\Gamma)-1)>0$, and the claim
easily follows from (1) and (2) above. Next, assume that $g(\Gamma)=1$. If
$\omega\in\dim H^{1}\left(\mathfrak{R}_{\Gamma}\right)$ s non–zero, then it
has a degree zero. Thus, it has no zeroes. This means that $\omega
H^{l-1}\left(\mathfrak{R}_{\Gamma}\right)=H^{l}\left(\mathfrak{R}_{\Gamma}\right)$
for all $l\in\mathbb{Z}$. But since obviously
$H^{0}\left(\mathfrak{R}_{\Gamma}\right)$ consists of constants only, we
obtain the claim. Finally, the case $g(\Gamma)=0$ is obvious from (2) since
the degree of $mK$ is $2m(g(\Gamma)-1)<0$ for all $m\geq 1$.

Assume that $g(\Gamma)\geq 1$ and $m\geq 1$. Then, $\dim
H^{m}\left(\mathfrak{R}_{\Gamma}\right)\neq 0$. Let $t=\dim
H^{m}\left(\mathfrak{R}_{\Gamma}\right)$. We fix the basis
$\omega_{1},\ldots,\omega_{t}$ of $H^{m}\left(\mathfrak{R}_{\Gamma}\right)$.
Let $z$ be any local coordinate on $\mathfrak{R}_{\Gamma}$. Then, locally
there exists unique holomorphic functions $\varphi_{1},\ldots,\varphi_{t}$
such that $\omega_{i}=\varphi_{i}\left(dz\right)^{m}$, for all $i$. Then,
again locally, we can consider the Wronskian $W_{z}$ defined by

(3-3)
$W_{z}\left(\omega_{1},\ldots,\omega_{t}\right)\overset{def}{=}\left|\begin{matrix}\varphi_{1}(z)&\cdots&\varphi_{t}(z)\\\
\frac{d\varphi_{1}(z)}{dz}&\cdots&\frac{d\varphi_{t}(z)}{dz}\\\ &\cdots&\\\
\frac{d^{t-1}\varphi_{1}(z)}{dz^{k-1}}&\cdots&\frac{d^{t-1}\varphi_{t}(z)}{dz^{t-1}}\\\
\end{matrix}\right|.$

As proved in [5, Proposition III.5.10], collection of all

(3-4)
$W_{z}\left(\omega_{1},\ldots,\omega_{t}\right)\left(dz\right)^{\frac{t}{2}\left(2m-1+t\right)},$

defines a non–zero holomorphic differential form

$W\left(\omega_{1},\ldots,\omega_{t}\right)\in
H^{\frac{t}{2}\left(2m-1+t\right)}\left(\mathfrak{R}_{\Gamma}\right).$

We call this form the Wronskian of the basis $\omega_{1},\ldots,\omega_{t}$.
It is obvious that a different choice of a basis of
$H^{m}\left(\mathfrak{R}_{\Gamma}\right)$ results in a Wronskian which differ
from $W\left(\omega_{1},\ldots,\omega_{t}\right)$ by a multiplication by a
non–zero complex number. Also, the degree is given by

(3-5)
$\deg{\left(\mathrm{div}{\left(W\left(\omega_{1},\ldots,\omega_{t}\right)\right)}\right)}=t\left(2m-1+t\right)(g\left(\mathfrak{R}_{\Gamma}\right)-1).$

Following [5, III.5.9], we make the following definition:

######  Definition 3-6.

Let $m\geq 1$ be an integer. We say that
$\mathfrak{a}\in\mathfrak{R}_{\Gamma}$ is a $m$-Weierstrass point if there
exists a non–zero $\omega\in H^{m}\left(\mathfrak{R}_{\Gamma}\right)$ such
that

$\nu_{\mathfrak{a}}(\omega)\geq\dim H^{m}\left(\mathfrak{R}_{\Gamma}\right).$

Equivalently [5, Proposition III.5.10] , if

$\nu_{\mathfrak{a}}\left(W\left(\omega_{1},\ldots,\omega_{t}\right)\right)\geq
1.$

When $m=1$ we speak about classical Wierstrass points. So, $1$-Weierstrass
points are simply Weierstrass points.

##  4\. Interpretation in Terms of Modular Forms

In this section we give interpretation of results of Section 3 in terms of
modular forms. Again, $\Gamma$ stand for a Fuschsian group of the first kind.
Let $m\geq 2$ be an even integer. We consider the space
$\mathcal{A}_{m}(\Gamma)$ be the space of all all meromorphic functions
$f:\mathbb{H}\rightarrow{\mathbb{C}}$ such that
$f(\gamma.z)=j(\gamma,z)^{m}f(z)$ ($z\in\mathbb{H}$, $\gamma\in\Gamma$) which
are meromorphic at every cusp for $\Gamma$. By [7, Theorem 2.3.1], there
exists isomorphism of vector spaces $\mathcal{A}_{m}(\Gamma)\longrightarrow
D^{m/2}\left(\mathfrak{R}_{\Gamma}\right)$, denoted by
$f\longmapsto\omega_{f}$ such that the following holds (see Section 2 for
notation, and [7, Theorem 2.3.3]):

(4-1)
$\displaystyle\nu_{\mathfrak{a}_{\xi}}(f)=\nu_{\mathfrak{a}_{\xi}}(\omega_{f})+\frac{m}{2}\left(1-\frac{1}{e_{\mathfrak{a}_{\xi}}}\right)\
\ \text{if}\ \xi\in\mathbb{H}$
$\displaystyle\nu_{\mathfrak{a}}(f)=\nu_{\mathfrak{a}}(\omega_{f})+\frac{m}{2}\
\ \text{for $\Gamma$---cusp $\mathfrak{a}$.}$
$\displaystyle\mathrm{div}{(f)}=\mathrm{div}{(\omega_{f})}+\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma}}\frac{m}{2}\left(1-\frac{1}{e_{\mathfrak{a}}}\right)\mathfrak{a},$

where $1/e_{\mathfrak{a}}=0$ if $\mathfrak{a}$ is a cusp. Let $f\in
M_{m}(\Gamma)$. Then, combining Lemma 2-2 (vi) and (4-1), we obtain

(4-2)
$\mathrm{div}{(\omega_{f})}=\mathfrak{c}^{\prime}_{f}-\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
elliptic}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}-\frac{m}{2}\sum_{\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp}\mathfrak{b}.$

This shows that $\omega_{f}$ is holomorphic everywhere except maybe at cusps
and elliptic points. Moreover, if $f\in S_{m}(\Gamma)$, then (see Lemma 2-2
(vii))

(4-3)
$\mathrm{div}{(\omega_{f})}=\mathfrak{c}_{f}-\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
elliptic}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}-\left(\frac{m}{2}-1\right)\sum_{\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp}\mathfrak{b}.$

Next, we determine all $f\in M_{m}(\Gamma)$ such that $\omega_{f}\in
H^{m/2}\left(\mathfrak{R}_{\Gamma}\right)$. From (4-2) we see that such $f$
must belong to $S_{m}(\Gamma)$, and from (4-3)

(4-4) $\mathfrak{c}_{f}\geq\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
elliptic}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}+\left(\frac{m}{2}-1\right)\sum_{\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp}\mathfrak{b}.$

Now, we define the subspace of $S_{m}(\Gamma)$ by

$S^{H}_{m}(\Gamma)=\left\\{f\in S_{m}(\Gamma);\ \ \text{$f=0$ or $f$ satisfies
(\ref{mhd-8})}\right\\}.$

It is mapped via $f\longmapsto\omega_{f}$ isomorphically onto
$H^{m/2}\left(\mathfrak{R}_{\Gamma}\right)$.

We remark that when $m=2$, (4-4) and reduces to obvious $\mathfrak{c}_{f}\geq
0$. Hence, $S^{H}_{2}(\Gamma)=S_{2}(\Gamma)$ recovering the standard
isomorphism of $S_{2}(\Gamma)$ and $H^{1}(\mathfrak{R}_{\Gamma})$ (see [7,
Theorem 2.3.2]). We have the following result:

######  Lemma 4-5.

Assume that $m,n\geq 2$ are even integers. Let $\Gamma$ be a Fuchsian group of
the first kind. Then, we have the following:

  * (i)

$S^{H}_{2}(\Gamma)=S_{2}(\Gamma)$.

  * (ii)

$S^{H}_{m}(\Gamma)$ is isomorphic to
$H^{m/2}\left(\mathfrak{R}_{\Gamma}\right)$.

  * (iii)

$S^{H}_{m}(\Gamma)=\\{0\\}$ if $g(\Gamma)=0$.

  * (iv)

Assume that $g(\Gamma)=1$. Let us write $S_{2}(\Gamma)=\mathbb{C}\cdot f$, for
some non–zero cuspidal form $f$. Then, we have
$S^{H}_{m}(\Gamma)=\mathbb{C}\cdot f^{m/2}$.

  * (v)

$\dim{S^{H}_{m}(\Gamma)}=(m-1)\left(g(\Gamma)-1\right)$ if $g(\Gamma)\geq 2$.

  * (vi)

$S^{H}_{m}(\Gamma)\cdot S^{H}_{n}(\Gamma)\subset S^{H}_{m+n}(\Gamma)$.

  * (vii)

There are no $m/2$–Weierstrass points on $\mathfrak{R}_{\Gamma}$ for
$g(\Gamma)\in\\{0,1\\}$.

  * (viii)

Assume that $g(\Gamma)\geq 2$, and $\mathfrak{a}_{\infty}$ is a $\Gamma$-cusp.
Then, $\mathfrak{a}_{\infty}$ is a $\frac{m}{2}$–Weierstrass point if and only
if there exists $f\in S^{H}_{m}(\Gamma)$, $f\neq 0$, such that

$\mathfrak{c}^{\prime}_{f}(\mathfrak{a}_{\infty})\geq\begin{cases}\frac{m}{2}+g(\Gamma)\
\ \text{if}\ \ m=2;\\\ \frac{m}{2}+(m-1)(g(\Gamma)-1)\ \ \text{if}\ \ m\geq
4.\\\ \end{cases}$

  * (ix)

Assume that $g(\Gamma)\geq 1$, and $\mathfrak{a}_{\infty}$ is a $\Gamma$-cusp.
Then, there exists a basis $f_{1},\ldots f_{t}$ of $S_{m}^{H}(\Gamma)$ such
that their $q$–expansions are of the form

$f_{u}=a_{u}q^{i_{u}}+\text{higher order terms in $q$},\ \ 1\leq u\leq t,$

where

$\frac{m}{2}\leq
i_{1}<i_{2}<\cdots<i_{t}\leq\frac{m}{2}+m\left(g(\Gamma)-1\right),$

and

$a_{u}\in\mathbb{C},\ \ a_{u}\neq 0.$

  * (x)

Assume that $g(\Gamma)\geq 1$, and $\mathfrak{a}_{\infty}$ is a $\Gamma$-cusp.
Then, $\mathfrak{a}_{\infty}$ is not a $\frac{m}{2}$–Weierstrass point if and
only if there exists a basis $f_{1},\ldots f_{t}$ of $S_{m}^{H}(\Gamma)$ such
that their $q$–expansions are of the form

$f_{u}=a_{u}q^{u+m/2-1}+\text{higher order terms in $q$},\ \ 1\leq u\leq t,$

where

$a_{u}\in\mathbb{C},\ \ a_{u}\neq 0.$

###### Proof.

(i) and (ii) follow from above discussion. Next, using above discussion and
(3-2) we obtain

$\dim{S^{H}_{m}(\Gamma)}=\dim
H^{m/2}\left(\mathfrak{R}_{\Gamma}\right)=\begin{cases}0\ \ &\text{if}\ \
m\geq 2,g(\Gamma)=0;\\\ g(\Gamma)\ \ &\text{if}\ \ m=2,\ g(\Gamma)\geq 1;\\\
g(\Gamma)\ \ &\text{if}\ \ m\geq 4,\ g(\Gamma)=1;\\\
(m-1)\left(g(\Gamma)-1\right)\ \ &\text{if}\ \ m\geq 4,\ g(\Gamma)\geq 2.\\\
\end{cases}$

This immediately implies (iii) and (v). Next, assume that $g(\Gamma)=1$. Then,
we see that $\dim{S^{H}_{m}(\Gamma)}\leq 1$ for all even integers $m\geq 4$.
It is well known that $f^{m/2}\in S_{m}(\Gamma)$. Next, (4-4) for $m=2$
implies $\mathrm{div}{(\omega_{f})}=\mathfrak{c}_{f}$. Also, the degree of
$\mathfrak{c}_{f}$ is zero by Lemma 2-2 (vii). Hence,

$\mathrm{div}{(\omega_{f})}=\mathfrak{c}_{f}=0.$

Using [7, Theorem 2.3.2], we obtain

$\omega_{f^{m/2}}=\omega^{m/2}_{f}.$

Hence

$\mathrm{div}{(\omega_{f^{m/2}})}=\frac{m}{2}\mathrm{div}{(\omega_{f})}=0.$

Then, applying (4-3) with $f^{m/2}$ in place of $f$, we obtain

$\mathfrak{c}_{f^{m/2}}=\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
elliptic}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}+\left(\frac{m}{2}-1\right)\sum_{\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp}\mathfrak{b}.$

This shows that $f^{m/2}\in S^{H}_{m}(\Gamma)$ proving (iv). Finally, (vi)
follows from [7, Theorem 2.3.1]. We can also see that directly as follows. Let
$0\neq f\in S^{H}_{m}(\Gamma)$ and $0\neq g\in S^{H}_{n}(\Gamma)$. Then,
$fg\in S_{m+n}(\Gamma)$ since $f\in S^{H}_{m}(\Gamma)\subset S_{m}(\Gamma)$
and $g\in S^{H}_{n}(\Gamma)\subset S_{n}(\Gamma)$. We have the following:

$\mathrm{div}{(f\cdot g)}=\mathrm{div}{(f)}+\mathrm{div}{(g)}.$

Using Lemma 2-2 (vi) we can rewrite this identity as follows:

$\displaystyle\mathfrak{c}^{\prime}_{f\cdot
g}-\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\ \
elliptic}\left[\frac{m+n}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}=$
$\displaystyle\mathfrak{c}^{\prime}_{f}-\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
\
elliptic}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}+\mathfrak{c}^{\prime}_{g}-\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
\ elliptic}\left[\frac{n}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}.$

By Lemma 2-2 (vii) we obtain:

$\displaystyle\mathfrak{c}_{f\cdot
g}-\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\ \
elliptic}\left[\frac{m+n}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}-\left(\frac{m+n}{2}-1\right)\sum_{\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp}\mathfrak{b}=$
$\displaystyle\left(\mathfrak{c}_{f}-\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
\
elliptic}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}-\left(\frac{m}{2}-1\right)\sum_{\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp}\mathfrak{b}\right)+$
$\displaystyle\left(\mathfrak{c}_{g}-\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
\
elliptic}\left[\frac{n}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}-\left(\frac{n}{2}-1\right)\sum_{\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp}\mathfrak{b}\right).$

Finally, (vi) follows applying (4-4) since both terms on the right hand of
equality are $\geq 0$.

Next, (vii) follows immediately form the discussion in Section 3, and it is
well–known. (viii) is a reinterpretation of Definition 3-6. The details are
left to the reader as an easy exercise.

Finally, (ix) and (x) in the case of $g(\Gamma)=1$ are obvious since by Lemma
2-2 we have $S_{2}(\Gamma)=\mathbb{C}\cdot f$ where

$\mathfrak{c}^{\prime}_{f}=\mathfrak{a}_{\infty}+\sum_{\begin{subarray}{c}\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp\\\ \mathfrak{b}\neq\mathfrak{a}_{\infty}\end{subarray}}\mathfrak{b}.$

We prove (ix) and (x) in the case of $g(\Gamma)\geq 2$. Let $f\in
S_{m}^{H}(\Gamma)$, $f\neq 0$. Then, by the definition of $S_{m}^{H}(\Gamma)$,
we obtain

(4-6)
$\mathfrak{c}^{\prime}_{f}(\mathfrak{a}_{\infty})=1+\mathfrak{c}_{f}(\mathfrak{a}_{\infty})\geq
1+\left(\frac{m}{2}-1\right)=\frac{m}{2}.$

On the other hand, again by the definition of $S_{m}^{H}(\Gamma)$ (see (4-4))
and the fact that $\mathfrak{c}^{\prime}_{f}\geq 0$, we obtain

$\displaystyle\deg{(\mathfrak{c}^{\prime}_{f})}$
$\displaystyle=\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma}}\
\mathfrak{c}^{\prime}_{f}(\mathfrak{a})\geq$
$\displaystyle\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
elliptic}\mathfrak{c}^{\prime}_{f}(\mathfrak{a})+\sum_{\begin{subarray}{c}\mathfrak{b}\in\mathfrak{R}_{\Gamma},\
cusp\\\
\mathfrak{b}\neq\mathfrak{a}_{\infty}\end{subarray}}\mathfrak{c}^{\prime}_{f}(\mathfrak{b})+\mathfrak{c}^{\prime}_{f}(\mathfrak{a}_{\infty})\geq$
$\displaystyle\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
elliptic}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]+\frac{m}{2}\left(t-1\right)+\mathfrak{c}^{\prime}_{f}(\mathfrak{a}_{\infty})$

where $t$ is the number of inequivalent $\Gamma$–cusps. The degree
$\deg{(\mathfrak{c}^{\prime}_{f})}$ is given by Lemma 2-2 (vi)

$\displaystyle\deg{(\mathfrak{c}^{\prime}_{f})}$ $\displaystyle=\dim
M_{m}(\Gamma)+g(\Gamma)-1$ $\displaystyle=\begin{cases}2(g(\Gamma)-1)+t\ \
\text{if}\ \ m=2;\\\
m(g(\Gamma)-1)+\frac{m}{2}t+\sum_{\begin{subarray}{c}\mathfrak{a}\in\mathfrak{R}_{\Gamma},\\\
elliptic\end{subarray}}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\ \
\text{if}\ \ m\geq 4.\\\ \end{cases}$

Combining with the previous inequality, we obtain

$\mathfrak{c}^{\prime}_{f}(\mathfrak{a}_{\infty})\leq\frac{m}{2}+m(g(\Gamma)-1)\
\ \text{if}\ \ m\geq 2.$

Having in mind (4-6), the rest of (ix) has standard argument (see for example
[10, Lemma 4.3]). Finally, (x) follows (viii) and (ix). ∎

The criterion in Lemma 4-5 (x) is a quite good criterion to check whether or
not $\mathfrak{a}_{\infty}$ is a Weierstrass points (the case $m=2$) using
computer systems such as SAGE since we need just to list the basis. This case
is well-known (see [19, Definition 6.1]). This criterion has been used in
practical computations in combination with SAGE in [13] for
$\Gamma=\Gamma_{0}(N)$.

But it is not good when $m\geq 4$, regarding the bound for $S^{H}_{m}(\Gamma)$
given by Lemma 4-5 (ix), since then a basis of $S_{m}(\Gamma)$ contains
properly normalized cusp forms having leading terms
$q^{m/2},\ldots,q^{m/2+m(g(\Gamma)-1)}$ at least when $\Gamma$ has elliptic
points for $m$ large enough and we do not know which of then belong to
$S^{H}_{m}(\Gamma)$. We explain that in Corollary 4-10 bellow.

First, we recall the following result [12, Lemma 2.9] which is well-known in a
slightly different notation ([20], [21]):

######  Lemma 4-7.

Let $m\geq 4$ be an even integer such that $\dim S_{m}(\Gamma)\geq
g(\Gamma)+1$. Then, for all $1\leq i\leq t_{m}-g$, there exists $f_{i}\in
S_{m}(\Gamma)$ such that
$\mathfrak{c}^{\prime}_{f_{i}}(\mathfrak{a}_{\infty})=i$.

######  Lemma 4-8.

Assume that $\Gamma$ has elliptic points. (For example,
$\Gamma=\Gamma_{0}(N)$.) Then, for a sufficiently large even integer $m$, we
have

(4-9) $\frac{m}{2}+m(g(\Gamma)-1)\leq\dim S_{m}(\Gamma)-g(\Gamma).$

###### Proof.

Assume that $m\geq 4$ is an even integer. Then, by Lemma 2-2 (v), we obtain

$\displaystyle\dim
S_{m}(\Gamma)-g(\Gamma)-\left(\frac{m}{2}+m(g(\Gamma)-1)\right)$
$\displaystyle=\left(\frac{m}{2}-1\right)t-\frac{m}{2}+\sum_{\begin{subarray}{c}\mathfrak{a}\in\mathfrak{R}_{\Gamma},\\\
elliptic\end{subarray}}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]-2g(\Gamma)+1$
$\displaystyle\geq\sum_{\begin{subarray}{c}\mathfrak{a}\in\mathfrak{R}_{\Gamma},\\\
elliptic\end{subarray}}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]-2g(\Gamma)-t+1.$

Since $\Gamma$ has elliptic points, the last term is $\geq 0$ for $m$
sufficiently large even integer. ∎

######  Corollary 4-10.

Assume that (4-9) holds. Then, given a basis $f_{1},\ldots f_{t}$ of
$S_{m}^{H}(\Gamma)$ such that
$\mathfrak{c}^{\prime}_{f_{j}}(\mathfrak{a}_{\infty})=i_{j}$, $1\leq j\leq t$,
where

$\frac{m}{2}\leq
i_{1}<i_{2}<\cdots<i_{t}\leq\frac{m}{2}+m\left(g(\Gamma)-1\right)$

can be extended by additional $g(\Gamma)$ cuspidal modular forms in
$S_{m}(\Gamma)$ to obtain the collection $F_{k}$, $\frac{m}{2}\leq
k\leq\frac{m}{2}+m\left(g(\Gamma)-1\right)$ such that
$\mathfrak{c}^{\prime}_{F_{k}}(\mathfrak{a}_{\infty})=k$ for all $k$.

###### Proof.

This follows directly from Lemmas 4-7 and 4-8. ∎

Now, explain the algorithm for testing that $\mathfrak{a}_{\infty}$ is a
$\frac{m}{2}$–Weierstrass point for $m\geq 6$. It requires some geometry. We
recall that $\mathfrak{R}_{\Gamma}$ is hyperelliptic if $g(\Gamma)\geq 2$, and
there is a degree two map onto $\mathbb{P}^{1}$. By general theory [9, Chapter
VII, Proposition 1.10], if $g(\Gamma)=2$, then $\mathfrak{R}_{\Gamma}$ is
hyperelliptic. If $\mathfrak{R}_{\Gamma}$ is not hyperelliptic, then $\dim
S_{2}(\Gamma)=g(\Gamma)\geq 3$, and the regular map
$\mathfrak{R}_{\Gamma}\longrightarrow\mathbb{P}^{g(\Gamma)-1}$ attached to a
canonical divisor $K$ is an isomorphism onto its image [9, Chapter VII,
Proposition 2.1].

Let $\Gamma=\Gamma_{0}(N)$, $N\geq 1$. Put
$X_{0}(N)=\mathfrak{R}_{\Gamma_{0}(N)}$. We recall that $g(\Gamma_{0}(N))\geq
2$ unless

$\begin{cases}N\in\\{1-10,12,13,16,18,25\\}\ \ \text{when
$g(\Gamma_{0}(N))=0$, and}\\\ N\in\\{11,14,15,17,19-21,24,27,32,36,49\\}\ \
\text{when $g(\Gamma_{0}(N))=1$.}\end{cases}$

Let $g(\Gamma_{0}(N))\geq 2$. Then, we remark that Ogg [17] has determined all
$X_{0}(N)$ which are hyperelliptic curves. In view of Ogg’s paper, we see that
$X_{0}(N)$ is not hyperelliptic for $N\in\\{34,38,42,43,44,45,51-58,60-70\\}$
or $N\geq 72$. This implies $g(\Gamma_{0}(N))\geq 3$.

Before we begin the study of spaces $S_{m}^{H}(\Gamma)$ we give the following
lemma.

######  Lemma 4-11.

Let $m\geq 4$ be an even integer. Let us select a basis
$f_{0},\ldots,f_{g-1}$, $g=g(\Gamma)$, of $S_{2}(\Gamma)$. Then, all of
$\binom{g+\frac{m}{2}-1}{\frac{m}{2}}$ monomials
$f_{0}^{\alpha_{0}}f_{1}^{\alpha_{1}}\cdots f_{g-1}^{\alpha_{g-1}}$,
$\alpha_{i}\in\mathbb{Z}_{\geq 0}$, $\sum_{i=0}^{g-1}\alpha_{i}=\frac{m}{2}$,
belong to $S_{m}^{H}(\Gamma)$. We denote this subspace of $S_{m}^{H}(\Gamma)$
by $S_{m,2}^{H}(\Gamma)$.

###### Proof.

This follows from Lemma 4-5 (vi) since $S_{2}(\Gamma)=S_{2}^{H}(\Gamma)$ (see
Lemma 4-5 (i)). ∎

######  Theorem 4-12.

Let $m\geq 4$ be an even integer. Assume that $\mathfrak{R}_{\Gamma}$ is not
hyperelliptic. Then, we have

$S_{m,2}^{H}(\Gamma)=S_{m}^{H}(\Gamma).$

###### Proof.

We use notation of Section 3 freely. The reader should review Lemma 4-5. Let
$F\in S_{2}(\Gamma)$, $F\neq 0$. We define a holomorphic differential form
$\omega\in H\left(\mathfrak{R}_{\Gamma}\right)$ by $\omega=\omega_{F}$. Define
a canonical class $K$ by $K=\mathrm{div}{(\omega)}$. We prove the following:

(4-13) $L\left(\frac{m}{2}K\right)=\left\\{\frac{f}{F^{m/2}};\ \ f\in
S_{m}^{H}(\Gamma)\right\\}.$

The case $m=2$ is of course well–known. By the Riemann-Roch theorem and
standard results recalled in Section 3 we have

$\displaystyle\dim L\left(\frac{m}{2}K\right)$
$\displaystyle=\deg{\left(\frac{m}{2}K\right)}-g(\Gamma)+1+\dim
L\left(\left(1-\frac{m}{2}\right)K\right)$
$\displaystyle=(m-1)(g(\Gamma)-1)+\begin{cases}1\ \ \text{if}\ \ m=2;\\\ 0\ \
\text{if}\ \ m\geq 4.\\\ \end{cases}$

Next, we recall that $S_{2}(\Gamma)=S_{2}^{H}(\Gamma)$ (see Lemma 4-5 (i)).
Then, Lemma 4-5 (vi) we obtain $F^{\frac{m}{2}}\in S_{m}^{H}(\Gamma)$.
Therefore, $f/F^{\frac{m}{2}}\in\mathbb{C}\left(\mathfrak{R}_{\Gamma}\right)$
for all $f\in S_{m}^{H}(\Gamma)$.

By the correspondence described in (4-1) we have

$\displaystyle\mathrm{div}{(F)}=\mathrm{div}{(\omega_{F})}+\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma}}\left(1-\frac{1}{e_{\mathfrak{a}}}\right)\mathfrak{a}$
$\displaystyle=K+\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma}}\left(1-\frac{1}{e_{\mathfrak{a}}}\right)\mathfrak{a}$
$\displaystyle=K+\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\ \
elliptic}(1-1/e_{\mathfrak{a}})\mathfrak{a}+\sum_{\begin{subarray}{c}\mathfrak{b}\in\mathfrak{R}_{\Gamma},\\\
cusp\end{subarray}}\mathfrak{b}.$

Thus, for $f\in S_{m}^{H}(\Gamma)$, we have the following:

$\displaystyle\mathrm{div}{\left(\frac{f}{F^{\frac{m}{2}}}\right)}+\frac{m}{2}K$
$\displaystyle=\mathrm{div}{(f)}-\frac{m}{2}\mathrm{div}{(F)}+\frac{m}{2}K$
$\displaystyle=\mathrm{div}{(f)}-\frac{m}{2}\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\
\
elliptic}(1-1/e_{\mathfrak{a}})\mathfrak{a}-\frac{m}{2}\sum_{\begin{subarray}{c}\mathfrak{b}\in\mathfrak{R}_{\Gamma},\\\
cusp\end{subarray}}\mathfrak{b}$

Next, using Lemma 2-2 (vi), the right–hand side becomes

$\mathfrak{c}^{\prime}_{f}-\sum_{\mathfrak{a}\in\mathfrak{R}_{\Gamma},\ \
elliptic}\left[\frac{m}{2}(1-1/e_{\mathfrak{a}})\right]\mathfrak{a}-\frac{m}{2}\sum_{\begin{subarray}{c}\mathfrak{b}\in\mathfrak{R}_{\Gamma},\\\
cusp\end{subarray}}\mathfrak{b}\geq 0$

by the definition of $S_{m}^{H}(\Gamma)$. Hence, $f/F^{\frac{m}{2}}\in
L\left(\frac{m}{2}K\right)$. Now, comparing the dimensions of the right-hand
and left-hand side in (4-13), we obtain their equality. This proves (4-13).

Let $W$ be any finite dimensional $\mathbb{C}$–vector space. Let
$\text{Symm}^{k}(W)$ be symmetric tensors of degree $k\geq 1$. Then, by Max
Noether theorem ([9], Chapter VII, Corollary 3.27) the multiplication induces
surjective map

$\text{Symm}^{k}{\left(L\left(K\right)\right)}\twoheadrightarrow
L\left(\frac{m}{2}K\right).$

The theorem follows. ∎

Now, we combine Theorem 4-12 with Lemma 4-5 (x) to obtain a good criterion in
the case $m\geq 4$ for testing that $\mathfrak{a}_{\infty}$ is a
$\frac{m}{2}$–Weierstrass point. We give examples in Section 5 (see
Propositions 5-1 and 5-2).

######  Corollary 4-14.

Let $m\geq 4$ be an even integer. Assume that $\mathfrak{R}_{\Gamma}$ is not
hyperelliptic. Assume that $\mathfrak{a}_{\infty}$ is a cusp for $\Gamma$. Let
us select a basis $f_{0},\ldots,f_{g-1}$, $g=g(\Gamma)$, of $S_{2}(\Gamma)$.
Compute $q$–expansions of all monomials

$f_{0}^{\alpha_{0}}f_{1}^{\alpha_{1}}\cdots f_{g-1}^{\alpha_{g-1}},\ \
\alpha_{i}\in\mathbb{Z}_{\geq 0},\ \sum_{i=0}^{g-1}\alpha_{i}=\frac{m}{2}.$

Then, $\mathfrak{a}_{\infty}$ is not a $\frac{m}{2}$–Weierstrass point if and
only if there exist a basis of the space of all such monomials, $F_{1},\ldots
F_{t}$, $t=\dim{S_{m}^{H}(\Gamma)}=(m-1)(g-1)$ (see Lemma 4-5 (v)), such that
their $q$–expansions are of the form

$F_{u}=a_{u}q^{u+m/2-1}+\text{higher order terms in $q$},\ \ 1\leq u\leq t,$

where

$a_{u}\in\mathbb{C},\ \ a_{u}\neq 0.$

When $\mathfrak{R}_{\Gamma}$ is hyperelliptic, for example if $g(\Gamma)=2$,
the space $S_{m,2}^{H}(\Gamma)$ could be proper subspace of
$S_{m}^{H}(\Gamma)$. For example, if $N=35$, then $g(\Gamma_{0}(N))=3$ and
$X_{0}(N)$ is hyperelliptic. For $m=4,6,8,8,10,12,14$ we checked that
$\dim{S_{m,2}^{H}(\Gamma)}=m+1$ while by general theory
$\dim{S_{m}^{H}(\Gamma)}=(m-1)(g(\Gamma_{0}(N))-1)=2(m-1)$. We see that

$\dim{S_{m}^{H}(\Gamma)}-\dim{S_{m,2}^{H}(\Gamma)}=m-3\geq 1,\ \ \text{for}\
m=4,6,8,8,10,12,14.$

In fact, the case of $g(\Gamma)=2$ could be covered in full generality. We
leave easy proof of the following proposition to the reader.

######  Proposition 4-15.

Assume that $g(\Gamma)=2$. Let $f_{0},f_{1}$ be a basis of $S_{2}(\Gamma)$.
Then, for any even integer $m\geq 4$, $f_{0}^{u}f_{1}^{\frac{m}{2}-u}$, $0\leq
u\leq m$ is a basis of $S_{m,2}^{H}(\Gamma)$. Therefore,

$\dim{S_{m}^{H}(\Gamma)}=(m-1)(g(\Gamma)-1)=m-1>\frac{m}{2}+1,\ \ \text{for}\
m\geq 6,$

and $S_{4,2}^{H}(\Gamma)=S_{4}^{H}(\Gamma)$.

We end this section with the standard yoga.

######  Theorem 4-16.

Let $m\geq 2$ be an even integer. Let $\Gamma$ be a Fuchsian group of the
first kind such that $g(\Gamma)\geq 1$. Let $t=\dim S^{H}_{m}(\Gamma)=\dim
H^{m/2}\left(\mathfrak{R}_{\Gamma}\right)$. Let us fix a basis
$f_{1},\ldots,f_{t}$ of $S^{H}_{m}(\Gamma)$, and let
$\omega_{1},\ldots,\omega_{t}$ be the corresponding basis of
$H^{m/2}\left(\mathfrak{R}_{\Gamma}\right)$. As above, we construct
holomorphic differential $W\left(\omega_{1},\ldots,\omega_{t}\right)\in
H^{\frac{t}{2}\left(m-1+t\right)}\left(\mathfrak{R}_{\Gamma}\right)$. We also
construct the Wronskian $W(f_{1},\ldots,f_{t})\in S_{t(m+t-1)}(\Gamma)$ (see
Proposition 6-2). Then, we have the following equality:

$\omega_{W(f_{1},\ldots,f_{t})}=W\left(\omega_{1},\ldots,\omega_{t}\right).$

In particular, we obtain the following:

$W(f_{1},\ldots,f_{t})\in S^{H}_{t(m+t-1)}(\Gamma).$

Moreover, assume that $\mathfrak{a}_{\infty}$ is a $\Gamma$-cusp. Then,
$\mathfrak{a}_{\infty}$ is a $\frac{m}{2}$–Weierstrass point if and only if

$\mathfrak{c}^{\prime}_{W(f_{1},\ldots,f_{t})}(\mathfrak{a}_{\infty})\geq
1+\frac{t}{2}\left(m-1+t\right)\ \ \text{i.e.,}\ \
\mathfrak{c}_{W(f_{1},\ldots,f_{t})}(\mathfrak{a}_{\infty})\geq\frac{t}{2}\left(m-1+t\right)$

(See also Lemma 4-5 (x) for more effective formulation of the criterion.)

###### Proof.

Since this is a equality of two meromorphic differentials, it is enough to
check the identity locally. Let $z\in\mathbb{H}$ be a non–elliptic point, and
$z\in U\subset\mathbb{H}$ a chart of $\mathfrak{a}_{z}$ such that $U$ does not
contain any elliptic point. Then, one can use [7, Section 2.3] to check the
equality directly. Indeed, we have the following argument.

Let $z_{0}\in\mathbb{H}$ be a non–elliptic point, and $z\in
U\subset\mathbb{H}$ a chart of $\mathfrak{a}_{z_{0}}$ such that $U$ does not
contain any elliptic point.

Let $t_{\mathfrak{a}}$ be a local coordinate on a neighborhood
$V_{\mathfrak{a}}=\pi(U)$. By the [7, Section 1.8] if $U$ is small enough,
projection $\pi$ gives homeomorphism of $U$ to $V_{\mathfrak{a}}$ such that

(4-17) $t_{\mathfrak{a}}\circ\pi(z)=z\ \text{ for }z\in U.$

Let $f\in\mathcal{A}_{m}(\Gamma)$ and let $\omega_{f}$ be the corresponding
differential. Locally there exist unique meromorphic function $\varphi$ such
that $\omega_{f}=\varphi\left(dz\right)^{m/2}$.

By [7, Section 2.3], local correspondence $f\mapsto\omega_{f}$ is given by

$\varphi(t_{\mathfrak{a}}\circ\pi(z))=f(z)\left(d(t_{\mathfrak{a}}\circ\pi)/dz\right)^{-m/2},$

which by the choice of local chart 4-17 become

$\varphi(z)=f(z)\ \text{ for }\ z\in U.$

So, in the neighborhood of non-elliptic point $z\in{\mathbb{H}}$ we have

$\omega_{f}=f\left(dz\right)^{m/2}.$

This gives us local identity

$W_{z}\left(\omega_{1},\ldots,\omega_{t}\right)=W(f_{1},\ldots,f_{t}).$

Since above is valid for any even $m\geq 2$, we get local identity of two
meromorphic differentials

$\displaystyle\omega_{W(f_{1},\ldots,f_{t})}$
$\displaystyle=W(f_{1},\ldots,f_{t})\left(dz\right)^{\frac{t}{2}\left(m-1+t\right)}$
$\displaystyle=W_{z}\left(\omega_{1},\ldots,\omega_{t}\right)\left(dz\right)^{\frac{t}{2}\left(m-1+t\right)}$
$\displaystyle=W\left(\omega_{1},\ldots,\omega_{t}\right)$

Now, assume that $\mathfrak{a}_{\infty}$ is a $\Gamma$-cusp. Then,
$\mathfrak{a}_{\infty}$ is a $\frac{m}{2}$–Weierstrass point if and only if

$\nu_{\mathfrak{a}_{\infty}}\left(W\left(\omega_{1},\ldots,\omega_{t}\right)\right)\geq
1$

i.e.,

$\nu_{\mathfrak{a}_{\infty}}\left(\omega_{W(f_{1},\ldots,f_{t})}\right)\geq 1$

by the first part of the proof. Finally, by (4-1), this is equivalent to

$\mathfrak{c}^{\prime}_{W(f_{1},\ldots,f_{t})}(\mathfrak{a}_{\infty})=\nu_{\mathfrak{a}_{\infty}}\left(\omega_{W(f_{1},\ldots,f_{t})}\right)+\frac{t}{2}\left(m-1+t\right)\geq
1+\frac{t}{2}\left(m-1+t\right).$

This completes the proof of the theorem. ∎

##  5\. Explicit computations based on Corollary 4-14 for
$\Gamma=\Gamma_{0}(N)$

In this section we apply the algorithm in Corollary 4-14 combined with SAGE.
The method is the following. We take $q$-expansions of the base elements of
$S_{2}(\Gamma_{0}(N))$:

$f_{0},\ldots,f_{g-1},$

where $g=g(\Gamma_{0}(N))$. For even $m\geq 4$, we compute $q$–expansions of
all monomials of degree $m/2$:

$f_{0}^{\alpha_{0}}f_{1}^{\alpha_{1}}\cdots f_{g-1}^{\alpha_{g-1}},\ \
\alpha_{i}\in\mathbb{Z}_{\geq 0},\ \sum_{i=0}^{g-1}\alpha_{i}=\frac{m}{2}.$

The number of monomials is

$\binom{g+m/2-1}{m/2}$

By selecting first $m/2+m\cdot(g-1)$ terms from $q$–expansions of the
monomials (see Lemma 4-5 (ix)), we can create matrix of size

$\binom{g+m/2-1}{m/2}\times\left(\frac{m}{2}+m\cdot(g-1)\right).$

Then, we perform suitable integral Gaussian elimination method to transform
the matrix into row echelon form. The procedure is as follows. We successively
sort and transform the row matrices to cancel the leading row coefficients
with the same number of leading zeros as their predecessor. We use the
Quicksort algorithm for sorting. We obtain the transformed matrix and the
transformation matrix. The non-null rows of the transformed matrix give the
$q$-expansions of the basis elements, and the corresponding rows of the
transformation matrix give the corresponding linear combinations of monomials.

Using above described method we perform various computations mentioned below.
For example, we can easily verify particular cases Theorem 4-12.

######  Proposition 5-1.

For $m=4,6,8,10,12$ and $N=34,38,44,55$, and for $m=4,6,8,10$ and $N=54,60$,
we have $S_{m}^{H}(\Gamma_{0}(N))=S_{m,2}^{H}(\Gamma_{0}(N))$. (We remark that
all curves $X_{0}(N)$, $N\in\\{34,38,44,54,55,60\\}$ are not hyperelliptic
(see the paragraph after Corollary 4-10.)

We can also deal with generalized Weierstrass points. For example, we can
check the following result:

######  Proposition 5-2.

For $m=2,4,6,8,10$, $\mathfrak{a}_{\infty}$ is not $\frac{m}{2}$–Weierstrass
point for $X_{0}(34)$. Next, $\mathfrak{a}_{\infty}$ is not ($1$–)Weierstrass
point for $X_{0}(55)$, but it is $\frac{m}{2}$–Weierstrass point for
$X_{0}(55)$ and $m=4,6,8,10$.

For example, let $m=4$. Then, for $X_{0}(34)$ the monomials are

$\displaystyle f_{0}^{2}$
$\displaystyle=q^{2}-4q^{5}-4q^{6}+12q^{8}+12q^{9}-2q^{10}$ $\displaystyle
f_{0}f_{1}$ $\displaystyle=q^{3}-q^{5}-2q^{6}-2q^{7}+2q^{8}+5q^{9}+2q^{10}$
$\displaystyle f_{0}f_{2}$
$\displaystyle=q^{4}-2q^{5}-q^{6}-q^{7}+6q^{8}+6q^{9}+2q^{10}$ $\displaystyle-
f_{1}^{2}+f_{0}f_{2}$
$\displaystyle=-2q^{5}+q^{6}-q^{7}+5q^{8}+6q^{9}+4q^{10}$ $\displaystyle-
f_{1}^{2}+f_{0}f_{2}+2f_{1}f_{2}$
$\displaystyle=-3q^{6}-5q^{7}+11q^{8}+16q^{9}+2q^{10}$ $\displaystyle-
f_{1}^{2}+f_{0}f_{2}+2f_{1}f_{2}+3f_{2}^{2}$
$\displaystyle=-17q^{7}+17q^{8}+34q^{9}+17q^{10}$

Their first exponents are $\frac{m}{2}=4,5,6,\frac{m}{2}+(m-1)(g-1)-1=7$ which
shows that $\mathfrak{a}_{\infty}$ is not $2$–Weierstrass point for
$X_{0}(34)$.

For $X_{0}(55)$ the monomials are

$\displaystyle f_{0}^{2}$ $\displaystyle f_{0}f_{1}$ $\displaystyle
f_{0}f_{2}$ $\displaystyle f_{0}f_{3}$ $\displaystyle f_{0}f_{4}$
$\displaystyle-f_{1}f_{2}+f_{0}f_{3}$ $\displaystyle-
f_{1}f_{2}+f_{0}f_{3}+2f_{2}f_{3}$ $\displaystyle-
f_{1}f_{2}+f_{0}f_{3}+2f_{2}f_{3}-f_{3}^{2}$ $\displaystyle-
f_{1}f_{2}+f_{0}f_{3}+2f_{2}f_{3}-f_{3}^{2}-2f_{3}f_{4}$ $\displaystyle-
f_{1}f_{2}+f_{0}f_{3}+2f_{2}f_{3}-f_{3}^{2}-2f_{3}f_{4}+f_{4}^{2}$
$\displaystyle-
f_{1}f_{2}-f_{2}^{2}+f_{0}f_{3}+2f_{2}f_{3}-f_{3}^{2}+f_{0}f_{4}-6f_{3}f_{4}-f_{4}^{2}$
$\displaystyle-
f_{2}^{2}+f_{3}^{2}+f_{0}f_{4}-f_{2}f_{4}-4f_{3}f_{4}+2f_{4}^{2}.$

Their $q$–expansions are given by the following expressions:

$\displaystyle
q^{2}-2q^{8}-2q^{9}-2q^{10}+2q^{11}-4q^{13}+3q^{14}+4q^{15}+3q^{16}-2q^{17}+5q^{18}$
$\displaystyle q^{3}-2q^{7}+q^{10}-2q^{11}+q^{12}-2q^{14}-4q^{16}+5q^{18}$
$\displaystyle
q^{4}-2q^{7}-q^{8}+3q^{9}+4q^{10}-4q^{11}-q^{13}-2q^{14}-3q^{15}-10q^{16}-2q^{17}+3q^{18}$
$\displaystyle
q^{5}-2q^{7}-q^{8}+3q^{9}+4q^{10}-4q^{11}-3q^{13}+q^{14}-q^{15}-11q^{16}-2q^{17}+5q^{18}$
$\displaystyle q^{6}-2q^{11}-q^{12}-q^{13}-q^{14}+q^{15}-q^{16}+3q^{18}$
$\displaystyle-2q^{7}+q^{8}+6q^{9}+q^{10}-10q^{11}-3q^{12}-5q^{13}+13q^{14}+21q^{15}-17q^{16}-8q^{17}-14q^{18}$
$\displaystyle
q^{8}+2q^{9}-5q^{10}-6q^{11}+19q^{12}+7q^{13}-13q^{14}-33q^{15}-7q^{16}+38q^{17}+14q^{18}$
$\displaystyle
2q^{9}-q^{10}-4q^{11}+9q^{12}-5q^{13}+4q^{14}-13q^{15}-12q^{16}+18q^{17}+4q^{18}$
$\displaystyle-q^{10}+11q^{12}-11q^{13}-7q^{15}-22q^{16}+22q^{17}+22q^{18}$
$\displaystyle 11q^{12}-11q^{13}-11q^{15}-22q^{16}+22q^{17}+22q^{18}$
$\displaystyle-22q^{13}+44q^{15}-44q^{16}+44q^{18}$
$\displaystyle-22q^{14}+22q^{15}-22q^{16}+44q^{18}$

The last exponent is $14>\frac{m}{2}+(m-1)(g-1)-1=13$. So,
$\mathfrak{a}_{\infty}$ is not $2$–Weierstrass point for $X_{0}(55)$.

##  6\.  Wronskians of Modular Forms

In this section we deal with a generalization of the usual notion of the
Wronskian of cuspidal modular forms [22], ([19], 6.3.1), ([10], the proof of
Theorem 4-5), and ([12], Lemma 4-1).

######  Lemma 6-1.

Let $f\in M_{m}(\Gamma,\chi)$. Let $\gamma\in\Gamma$. Then, for $k\geq 0$,
$k$–th derivative of the function $f(\sigma.z)$ is given by

$\frac{d^{k}}{dz^{k}}f(\gamma.z)=\chi(\gamma)j(\gamma,z)^{m+2k}\cdot\frac{d^{k}f(z)}{dz^{k}}+\chi(\gamma)\sum_{i=0}^{k-1}D_{ik}\cdot
j(\gamma,z)^{m+k+i}\cdot\frac{d^{i}f(z)}{dz^{i}}.$

where $D_{ik}$ are some constants depending on $m$, $k$, and $\gamma$. If
$\Gamma\subset SL_{2}(\mathbb{Z})$, then the constants can be taken to be from
$\mathbb{Z}$.

###### Proof.

This follows by an easy induction on $k$ using the fact that

$\frac{d}{dz}\gamma.z=j(\gamma,z)^{-2}.$

See also the proof of Theorem 4-5 in[10], the text between the lines (4-6) and
(4-8). ∎

The following proposition is the main result of the present section:

######  Proposition 6-2.

Let $m\geq 1$. Then, for any sequence $f_{1},\ldots,f_{k}\in
M_{m}(\Gamma,\chi)$, the Wronskian

$W\left(f_{1},\ldots,f_{k}\right)(z)\overset{def}{=}\left|\begin{matrix}f_{1}(z)&\cdots&f_{k}(z)\\\
\frac{df_{1}(z)}{dz}&\cdots&\frac{df_{k}(z)}{dz}\\\ &\cdots&\\\
\frac{d^{k-1}f_{1}(z)}{dz^{k-1}}&\cdots&\frac{d^{k-1}f_{k}(z)}{dz^{k-1}}\\\
\end{matrix}\right|$

is a cuspidal modular form in $S_{k(m+k-1)}(\Gamma,\chi^{k})$ if $k\geq 2$. If
$f_{1},\ldots,f_{k}$ are linearly independent, then
$W\left(f_{1},\ldots,f_{k}\right)\neq 0$.

###### Proof.

This is a standard fact. We apply Lemma 6-1 to conclude

$W\left(f_{1},\ldots,f_{k}\right)(\gamma.z)=\chi^{k}(\gamma)j(\gamma,z)^{k(m+k-1)}W\left(f_{1},\ldots,f_{k}\right)(z),\
\ \gamma\in\Gamma,\ z\in\mathbb{H}.$

Let $x\in{\mathbb{R}}\cup\\{\infty\\}$ be a cusp for $\Gamma$. Let $\sigma\in
SL_{2}({\mathbb{R}})$ such that $\sigma.x=\infty$. We write

$\\{\pm 1\\}\sigma\Gamma_{x}\sigma^{-1}=\\{\pm
1\\}\left\\{\left(\begin{matrix}1&lh^{\prime}\\\ 0&1\end{matrix}\right);\ \
l\in{\mathbb{Z}}\right\\},$

where $h^{\prime}>0$ is the width of the cusp. Then we write the Fourier
expansion of each $f_{i}$ at $x$ as follows:

$(f_{i}|_{m}\sigma^{-1})(\sigma.z)=\sum_{n=0}^{\infty}a_{n,i}\exp{\frac{2\pi\sqrt{-1}n\sigma.z}{h^{\prime}}}.$

Using the cocycle identity

$1=j(\sigma^{-1}\sigma,z)=j(\sigma^{-1},\sigma.z)j(\sigma,z),$

this implies the following:

$j(\sigma,z)^{m}\cdot
f_{i}(z)=\sum_{n=0}^{\infty}a_{n,i}\exp{\frac{2\pi\sqrt{-1}n\sigma.z}{h^{\prime}}}.$

By induction on $t\geq 0$, using

$\frac{d}{dz}\sigma.z=j(\sigma,z)^{-2},$

we have the following:

(6-3)
$j(\sigma,z)^{m+2t}\frac{d^{t}f_{i}(z)}{dz^{t}}+\sum_{u=0}^{t-1}D_{i,t}j(\sigma,z)^{m+t+u}\frac{d^{u}f_{i}(z)}{dz^{u}}=\sum_{n=0}^{\infty}a_{n,i,t}\exp{\frac{2\pi\sqrt{-1}n\sigma.z}{h^{\prime}}},$

for some complex numbers $D_{i,t}$ and $a_{n,i,t}$, where

$a_{0,i,t}=0,\ \ t\geq 1.$

Now, by above considerations, using (6-3), we have

$\displaystyle\left(W\left(f_{1},\ldots,f_{k}\right)|_{k(m+k-1)}\sigma^{-1}\right)(\sigma.z)$
$\displaystyle=j(\sigma,z)^{k(m+k-1)}W\left(f_{1},\ldots,f_{k}\right)(z)$
$\displaystyle=\left|\begin{matrix}j(\sigma,z)^{m}f_{1}(z)&\cdots&j(\sigma,z)^{m}f_{k}(z)\\\
j(\sigma,z)^{m+2}\frac{df_{1}(z)}{dz}&\cdots&j(\sigma,z)^{m+2}\frac{df_{k}(z)}{dz}\\\
&\cdots&\\\
j(\sigma,z)^{m+2(k-1)}\frac{d^{k-1}f_{1}(z)}{dz^{k-1}}&\cdots&j(\sigma,z)^{m+2(k-1)}\frac{d^{k-1}f_{k}(z)}{dz^{k-1}}\\\
\end{matrix}\right|$
$\displaystyle=\det{\left(\sum_{n=0}^{\infty}a_{n,i+1,t}\exp{\frac{2\pi\sqrt{-1}n\sigma.z}{h^{\prime}}}\right)_{0\leq
i,t\leq k-1}}$

Now, we see that the Wronskian is holomorphic at each cup of $\Gamma$ and
vanishes at the order at least $k-1$. In particular, it belongs to
$S_{k(m+k-1)}(\Gamma,\chi^{k})$ if $k\geq 2$.

The claim that linear independence is equivalent to the fact that Wronskian is
not identically zero is standard ([9], Chapter VII, Lemma 4.4). ∎

We end this section with an elementary remark regarding Wronskians. In the
case when $\Gamma$ has a cusp at the infinity $\mathfrak{a}_{\infty}$, it is
more convenient to use the derivative with respect to

$q=\exp{\frac{2\pi\sqrt{-1}z}{h}},$

where $h>0$ is the width of the cusp since all modular forms have
$q$–expansions. Using the notation from Proposition 6-2. It is easy to see

$\frac{d}{dz}=\frac{2\pi\sqrt{-1}}{h}\cdot q\frac{d}{dq}.$

This implies that

$\frac{d^{k}}{dz^{k}}=\left(\frac{2\pi\sqrt{-1}}{h}\right)^{k}\cdot\left(q\frac{d}{dq}\right)^{k},\
\ k\geq 0.$

Thus, we may define the $q$–Wronskian as follows:

(6-4)
$W_{q}\left(f_{1},\ldots,f_{k}\right)\overset{def}{=}\left|\begin{matrix}f_{1}&\cdots&f_{k}\\\
q\frac{d}{dq}f_{1}&\cdots&q\frac{d}{dq}f_{k}\\\ &\cdots&\\\
\left(q\frac{d}{dq}\right)^{k-1}f_{1}&\cdots&\left(q\frac{d}{dq}\right)^{k-1}f_{k}\\\
\end{matrix}\right|,$

considering $q$–expansions of $f_{1},\ldots,f_{k}$.

We obtain

(6-5)
$W\left(f_{1},\ldots,f_{k}\right)=\left(\frac{2\pi\sqrt{-1}}{h}\right)^{k(k-1)/2}W_{q}\left(f_{1},\ldots,f_{k}\right).$

##  7\. On a Divisor of a Wronskian

In this section we discuss the divisor of cuspidal modulars forms constructed
via Wronskians (see Proposition 6-2). We start with necessary preliminary
results.

######  Lemma 7-1.

Let $\varphi_{1},\cdots,\varphi_{l}$ be a sequence of linearly independent
meromorphic functions on some open set $U\subset\mathbb{C}$. We define their
Wronskian as usual
$W(\varphi_{1},\cdots,\varphi_{k})=\det{\left(\frac{d^{i-1}\varphi_{j}}{dz^{i-1}}\right)_{i,j=1,\ldots,k}}$.
Then, we have the following:

  * (i)

The Wronskian $W(\varphi_{1},\cdots,\varphi_{k})$ is a non–zero meromorphic
function on $U$.

  * (ii)

We have
$W(\varphi_{1},\cdots,\varphi_{k})=\varphi^{k}W(\varphi_{1}/\varphi,\cdots,\varphi_{k}/\varphi)$
for all non–zero meromorphic functions $\varphi$ on $U$.

  * (iii)

Let $\xi\in U$ be such that all $\varphi_{i}$ are holomorphic. Let $A$ be the
$\mathbb{C}$–span of all $\varphi_{i}$. Then, all $\varphi\in A$ are
holomorphic at $\xi$, and the set $\\{\nu_{z-\xi}(\varphi);\ \ \varphi\in A,\
\varphi\neq 0\\}$ has exactly $k=\dim A$ different elements (Here as in
Section 2, $\nu_{z-\xi}$ stands for the order at $\xi$.). Let
$\nu_{z-\xi}(\varphi_{1},\ldots,\varphi_{k})$ be the sum of all $\dim
A$–values of that set. Then, $W(\varphi_{1},\cdots,\varphi_{k})$ is
holomorphic at $\xi$, and the corresponding order is

$\nu_{z-\xi}(\varphi_{1},\ldots,\varphi_{k})-\frac{k(k-1)}{2}$

###### Proof.

(i) is well–known. See for example ([9], Chapter VII, Lemma 4.4) or it is a
consequence of [5, Proposition III.5.8]. (ii) is a consequence of the proof of
[5, Proposition III.5.8] (see formula (5.8.4)). Finally, we prove (iii). Then,
by the text before the statement of [5, Proposition III.5.8], we see that we
can select another basis $\psi_{1},\ldots,\psi_{k}$ of $A$ such that

$\nu_{z-\xi}(\psi_{1})<\nu_{z-\xi}(\psi_{2})<\ldots<\nu_{z-\xi}(\psi_{k}).$

Then, by [5, Proposition III.5.8], we have that the order of
$W(\psi_{1},\ldots,\psi_{k})$ at $z$ is equal to

$\sum_{i=1}^{k}\left(\nu_{z-\xi}(\psi_{i})-i+1\right)=\nu_{z-\xi}(\varphi_{1},\ldots,\varphi_{k})-\frac{k(k-1)}{2}.$

But $\varphi_{1},\ldots,\varphi_{k}$ is also a basis of $A$. Thus, we see that
we can write

$\left(\begin{matrix}\varphi_{1}\\\ \varphi_{2}\\\ \vdots\\\
\varphi_{k}\end{matrix}\right)=A\cdot\left(\begin{matrix}\psi_{1}\\\
\psi_{2}\\\ \vdots\\\ \psi_{k}\end{matrix}\right)$

for some $A\in GL_{k}(\mathbb{C})$. This implies

$\left(\begin{matrix}\varphi_{1}&d\varphi_{1}/dz&\cdots&d^{k-1}\varphi_{1}/dz^{k-1}\\\
\varphi_{2}&d\varphi_{2}/dz&\cdots&d^{k-1}\varphi_{2}/dz^{k-1}\\\ \vdots\\\
\varphi_{k}&d\varphi_{k}/dz&\cdots&d^{k-1}\varphi_{k}/dz^{k-1}\end{matrix}\right)=A\cdot\left(\begin{matrix}\psi_{1}&d\psi_{1}/dz&\cdots&d^{k-1}\psi_{1}/dz^{k-1}\\\
\psi_{2}&d\psi_{2}/dz&\cdots&d^{k-1}\psi_{2}/dz^{k-1}\\\ \vdots\\\
\psi_{k}&d\psi_{k}/dz&\cdots&d^{k-1}\psi_{k}/dz^{k-1}\end{matrix}\right).$

Hence

$W(\varphi_{1},\ldots,\varphi_{k})=\det A\cdot W(\psi_{1},\ldots,\psi_{k})$

has the same order at $z$ as $W(\psi_{1},\ldots,\psi_{k})$. ∎

As a direct consequence of Lemma 7-1, we obtain the following result. At this
point the reader should review the text in Section 2 before the statement of
Lemma 2-2 as well as Proposition 6-2. The proof is left to the reader as an
exercise.

######  Proposition 7-2.

Assume that $m\geq 2$ is even. Let $f_{1},\ldots,f_{k}\in M_{m}(\Gamma)$ be a
sequence of linearly independent modular forms. Let $\xi\in\mathbb{H}$. Then,
we have the following:

$\nu_{\mathfrak{a}_{\xi}}\left(W(f_{1},\ldots,f_{k})\right)=\frac{1}{e_{\xi}}\cdot\left(\nu_{z-\xi}(\varphi_{1},\ldots,\varphi_{k})-\frac{k(k-1)}{2}\right).$

The case of a cusp requires a different technique but final result is similar:

######  Theorem 7-3.

Assume that $m\geq 2$ is even. Suppose that $\mathfrak{a}_{\infty}$ is a cusp
for $\Gamma$. Let $f_{1},\ldots,f_{k}\in M_{m}(\Gamma)$ be a sequence of
linearly independent modular forms. Let $i\in\\{1,\ldots,k\\}$. Consider
$f_{1},\ldots,f_{k}$ as meromorphic functions in a variable $q$ in a
neighborhood of $q=0$, and define $\nu_{q-0}\left(f_{1},\ldots,f_{k}\right)$
as in Lemma 7-1 (iii). Then, we have the following identity:

$\nu_{\mathfrak{a}_{\infty}}\left(W\left(f_{1},\ldots,f_{k}\right)\right)=\nu_{q-0}\left(f_{1},\ldots,f_{k}\right).$

###### Proof.

By Lemma 7-1 (ii), we can write

(7-4) $W\left(f_{1},\ldots,f_{k}\right)=f_{1}^{k}\cdot
W(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})$

as meromorphic functions on $\mathbb{H}$. But the key fact that
$1,f_{2}/f_{1},\ldots,f_{k}/f_{1}$ can be regarded as meromorphic (rational)
functions on $\mathfrak{R}_{\Gamma}$ i.e., they are elements of
$\mathbb{C}\left(\mathfrak{R}_{\Gamma}\right)$.

The key point now is that these meromorphic functions, and their Wronskian
define meromorphic $k(k-1)/2$–differential form, denoted by $W_{\Gamma}$.
Details are contained in [9, Section 4, Lemma 4.9]. We recall the following.

Let $w\in\mathbb{H}$ be a non–elliptic point for $\Gamma$, such that
$f_{1}\neq 0$, and $U\subset\mathbb{H}$ small neighborhood of $w$ giving a
chart of $\mathfrak{a}_{w}$ on the curve $\mathfrak{R}_{\Gamma}$. Then, in the
chart $U$ we have:

$W_{\Gamma}=W(1,f_{2}(z)/f_{1}(z),\ldots,f_{k}(z)/f_{1}(z))\left(dz\right)^{k(k-1)/2}.$

On the other hand, in a chart of $\mathfrak{a}_{\infty}$, $W_{\Gamma}$ is
given by the usual Wronskian, denoted by
$W_{\Gamma,q}(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})$, of
$1,f_{2}/f_{1},\ldots,f_{k}/f_{1}$ presented by $q$–expansions with respect to
the derivatives $d^{i}/dq^{i}$, $0\leq i\leq k-1$, multiplied by
$\left(dq\right)^{k(k-1)/2}$ i.e.,

$W_{\Gamma}=W(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})\left(dq\right)^{k(k-1)/2}.$

Next, we insert $q$-expansions of $f_{1},\ldots,f_{k}$ into
$W(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})$. So, we can express

$W(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})=c_{m}q^{m}+c_{m+1}q^{m+1}+\cdots,$

where $c_{m}\neq 0$, $c_{m+1},c_{m+2},\ldots$ are complex numbers. Hence,

(7-5)
$\nu_{\mathfrak{a}_{\infty}}\left(W(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})\right)=\nu_{q-0}\left(W(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})\right)=m.$

Let us fix a neighborhood $U$ of $\infty$ such that it is a chart for
$\mathfrak{a}_{\infty}$, and there is no elliptic points in it. Then, we fix
$w\in U$, $w\neq\infty$, and a chart $V$ of $w$ such that
$V\subset\mathbb{H}\cap U$. Now, on $V$, we have the following expression for
$W_{\Gamma}$:

$W(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})\left(dz\right)^{k(k-1)/2}=\left(c_{m}q^{m}+c_{m+1}q^{m+1}+\cdots\right)\left(dz\right)^{k(k-1)/2},\
\ q=\exp{\frac{2\pi\sqrt{-1}z}{h}}.$

On the other hand, on $U$, we must have the expression for $W_{\Gamma}$ of the
form

$W_{\Gamma,q}(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})\left(dq\right)^{k(k-1)/2}=\left(d_{n}q^{n}+d_{n+1}q^{m+1}+\cdots\right)\left(dq\right)^{k(k-1)/2},$

where $d_{n}\neq 0$, $d_{n+1},d_{n+2},\ldots$ are complex numbers. We have

(7-6)
$\nu_{\mathfrak{a}_{\infty}}\left(W_{\Gamma}\right)=\nu_{q-0}\left(W_{\Gamma,q}(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})\right)=n.$

By definition of meromorphic $k(k-1)/2$–differential, on $V$ these expressions
must be related by

$c_{m}q^{m}+c_{m+1}q^{m+1}+\cdots=\left(d_{n}q^{n}+d_{n+1}q^{m+1}+\cdots\right)\cdot\left(\frac{dq}{dz}\right)^{k(k-1)/2}.$

Hence, we obtain

$n=m-k(k-1)/2.$

Using (7-5) and (7-6) this can be written as follows:

$\nu_{q-0}\left(W_{\Gamma,q}(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})\right)=\nu_{q-0}\left(W(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})\right)-k(k-1)/2.$

Consider again $f_{1},\ldots,f_{k}$ as meromorphic functions in a variable $q$
in a neighborhood of $q=0$, and define the Wronskian
$W_{\Gamma,q}(f_{1},\ldots,f_{k})$ using derivatives with respect to $q$.
Then, Lemma 7-1 (ii) implies

$\displaystyle\nu_{\mathfrak{a}_{\infty}}\left(W\left(f_{1},\ldots,f_{k}\right)\right)$
$\displaystyle=\nu_{\mathfrak{a}_{\infty}}\left(f^{k}_{1}\right)+\nu_{\mathfrak{a}_{\infty}}\left(W\left(1,f_{2}/f_{1},\ldots,f_{k}/f_{1}\right)\right)$
$\displaystyle=\nu_{q-0}\left(f^{k}_{1}\right)+\nu_{q-0}\left(W\left(1,f_{2}/f_{1},\ldots,f_{k}/f_{1}\right)\right)$
$\displaystyle=\nu_{q-0}\left(f^{k}_{1}\right)+\nu_{q-0}\left(W_{\Gamma,q}(1,f_{2}/f_{1},\ldots,f_{k}/f_{1})\right)+k(k-1)/2$
$\displaystyle=\nu_{q-0}\left(W_{\Gamma,q}(f_{1},f_{2},\ldots,f_{k})\right)+k(k-1)/2.$

Finally, we apply Lemma 7-1 (iii). ∎

##  8\. Computation of Wronskians for $\Gamma=SL_{2}(\mathbb{Z})$

Assume that $m\geq 4$ is an even integer. Let $M_{m}$ be the space of all
modular forms of weight $m$ for $SL_{2}(\mathbb{Z})$. We introduce the two
Eisenstein series

$\displaystyle E_{4}(z)=1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n}$
$\displaystyle E_{6}(z)=1-504\sum_{n=1}^{\infty}\sigma_{5}(n)q^{n}$

of weight $4$ and $6$, where $q=\exp{(2\pi iz)}$. Then, for any even integer
$m\geq 4$, we have

(8-1) $M_{m}=\oplus_{\begin{subarray}{c}\alpha,\beta\geq 0\\\
4\alpha+6\beta=m\end{subarray}}\mathbb{C}E^{\alpha}_{4}E^{\beta}_{6}.$

We have

(8-2) $k=k_{m}\overset{def}{=}\dim M_{m}=\begin{cases}\left[m/12\right]+1,\ \
m\not\equiv\ 2\text{(mod $12$)};\\\ \left[m/12\right],\ \ m\equiv\ 2\text{(mod
$12$)}.\\\ \end{cases}$

We let

$\Delta(z)=q+\sum_{n=2}\tau(n)q^{n}=q-24q^{2}+252q^{3}+\cdots=\frac{E_{4}^{3}(z)-E_{6}^{2}(z)}{1728}.$

be the Ramanujan delta function.

It is well–known that the map $f\longmapsto f\cdot\Delta$ is an ismorphism
between the vector space of modular form $M_{m}$ and the space of all cuspidal
modular forms $S_{m+12}$ inside $M_{m+12}$. In general, we have the following:

$\dim S_{m}=\dim M_{m}-1,\\\ $

for all even integers $m\geq 4$.

Now, we are ready to compute our first Wronskian (see (6-4) for notation).

######  Proposition 8-3.

We have the following:

  * (i)

$W_{q}\left(E^{3}_{4},E^{2}_{6}\right)=-1728\cdot\Delta\cdot E^{2}_{4}E_{6}$.

  * (ii)

$2E_{4}\frac{d}{dq}E_{6}-3E_{6}\frac{d}{dq}E_{4}=-1728\cdot\Delta\cdot
q^{-1}$.

###### Proof.

We compute

$\displaystyle W_{q}\left(E_{4},E_{6}\right)$
$\displaystyle=\left|\begin{matrix}E^{3}_{4}&E^{2}_{6}\\\
q\frac{d}{dq}E^{3}_{4}&q\frac{d}{dq}E^{2}_{6}\end{matrix}\right|$
$\displaystyle=2E^{3}_{4}E_{6}\cdot
q\frac{d}{dq}E_{6}-3E^{2}_{4}E^{2}_{6}\cdot q\frac{d}{dq}E_{4}$
$\displaystyle=E^{2}_{4}E_{6}\cdot
q\cdot\left(2E_{4}\frac{d}{dq}E_{6}-3E_{6}\frac{d}{dq}E_{4}\right).$

But we know that $W_{q}\left(E_{4},E_{6}\right)$ is a cusp form of weigth
$2\cdot(12+2-1)=26$. Thus, we must have that is equal to

$W_{q}\left(E_{4},E_{6}\right)=\lambda\cdot\Delta\cdot E^{2}_{4}E_{6},$

for some non–zero constant $\lambda$. This implies that

$2E_{4}\frac{d}{dq}E_{6}-3E_{6}\frac{d}{dq}E_{4}=\lambda\cdot\Delta\cdot
q^{-1}$

Considering explicit $q$–expansions, we find that

$\lambda=-1728.$

This proves both (i) and (ii). ∎

The general case requires a different proof based on results of Section 7.

######  Proposition 8-4.

Assume that $m=12t$ for some $t\geq 1$. Then, we write the basis of $M_{m}$ as
follows: $\left(E^{3}_{4}\right)^{u}\left(E^{2}_{6}\right)^{t-u}$, $0\leq
u\leq t$. Then, we have the following

$W_{q}\left(\left(E^{3}_{4}\right)^{u}\left(E^{2}_{6}\right)^{t-u},\ \ 0\leq
u\leq
t\right)=\lambda\cdot\Delta^{\frac{t(t+1)}{2}}E_{4}^{t(t+1)}E_{6}^{\frac{t(t+1)}{2}},$

for some non–zero constant $\lambda$.

###### Proof.

We can select another basis $f_{0},\ldots,f_{t}$ of $M_{m}$ such that
$f_{i}=c_{i}q^{i}+d_{i}q^{i+1}+\cdots$, $0\leq i\leq t$, where $c_{i}\neq
0,d_{i},\ldots$ are some complex constants. An easy application of Theorem 7-3
gives

$\nu_{\mathfrak{a}_{\infty}}\left(W_{q}\left(\left(E^{3}_{4}\right)^{u}\left(E^{2}_{6}\right)^{t-u},\
\ 0\leq u\leq t\right)\right)=\frac{t(t+1)}{2}.$

But since $\mathrm{div}{(\Delta)}=\mathfrak{a}_{\infty}$, we obtain that

$f\overset{def}{=}W_{q}\left(\left(E^{3}_{4}\right)^{u}\left(E^{2}_{6}\right)^{t-u},\
\ 0\leq u\leq t\right)/\Delta^{\frac{t(t+1)}{2}}$

is a non–cuspidal modular form of weight

$l=k\cdot\left(m+k-1\right)-12\frac{t(t+1)}{2}=(t+1)(12t+t)-12t=7t(t+1).$

It remains to determine $f$. In order to do that, we use Proposition 7-2, and
consider the order of vanishing of
$W_{q}\left(\left(E^{3}_{4}\right)^{u}\left(E^{2}_{6}\right)^{t-u},\ \ 0\leq
u\leq t\right)$ at elliptic points $i$ and $e^{\pi i/3}=(1+i\sqrt{3})/2$, of
order $2$ and $3$, respectively. We recall (see [12], Lemma 4-1) that

$\mathrm{div}{(E_{4})}=\frac{1}{3}\mathfrak{a}_{(1+i\sqrt{3})/2}.$

Similarly we show that

$\mathrm{div}{(E_{6})}=\frac{1}{2}\mathfrak{a}_{i}.$

This implies that $\left(E^{3}_{4}\right)^{u}\left(E^{2}_{6}\right)^{t-u}$ has
order $3u$ and $2(t-u)$ at $(1+i\sqrt{3})/2$ and $i$, respectively. Hence,
$W_{q}\left(\left(E^{3}_{4}\right)^{u}\left(E^{2}_{6}\right)^{t-u},\ \ 0\leq
u\leq t\right)$ has orders

$\nu_{\mathfrak{a}_{(1+i\sqrt{3})/2}}\left(W_{q}\left(\left(E^{3}_{4}\right)^{u}\left(E^{2}_{6}\right)^{t-u},\
\ 0\leq u\leq t\right)\right)=\frac{1}{3}t(t+1),$

and

$\nu_{\mathfrak{a}_{(1+i\sqrt{3})/2}}\left(W_{q}\left(\left(E^{3}_{4}\right)^{u}\left(E^{2}_{6}\right)^{t-u},\
\ 0\leq u\leq t\right)\right)=\frac{1}{4}t(t+1).$

This implies the following:

$\nu_{\mathfrak{a}_{(1+i\sqrt{3})/2}}\left(f\right)=\frac{1}{3}\cdot t(t+1),$

and

$\nu_{\mathfrak{a}_{i}}\left(f\right)=\frac{1}{4}\cdot t(t+1),$

Since, $f\in M_{7t(t+1)}$, comparing divisors as before, we conclude that

$f=\lambda\cdot E_{4}^{t(t+1)}E_{6}^{\frac{t(t+1)}{2}},$

for some non–zero constant $\lambda$. ∎

We are not able to determine constant $\lambda$ in Proposition 8-4 for all
$t\geq 1$. It should come out of comparison of $q$–expansions of left and
right sides of the identity in Proposition 8-4. For $t=1$, Proposition 8-3
implies that $\lambda=-1728$. Experiments in SAGE shows that $\lambda=-2\cdot
1728^{3}$ for $t=2$, and $\lambda=12\cdot 1728^{6}$ for $t=3$.

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