Neither  determinant is modular invariant or even covariant. Physicists say that there is a global anomaly, which manifests the incompatibility between maintaining holomorphicity and modular invariance. We may try a combination similar to the one we established in the non-chiral case,
\bea
\sum _{i=2}^4 { \tet _i (0|\tau)^{{N \over 2}}  \over \eta (\tau)^{{N \over 2}}  } =
{ \tet [00 ] (0|\tau)^{{N \over 2}}  \over \eta (\tau)^{{N \over 2}}  } +
{ \tet [0\half  ] (0|\tau)^{{N \over 2}}  \over \eta (\tau)^{{N \over 2}}  } +
{ \tet [\half 0 ] (0|\tau)^{{N \over 2}}  \over \eta (\tau)^{{N \over 2}}  }
\eea
The smallest value of $N$ for which we have good modular transformations is $N=16$, and more generally any multiple of $16$. In those cases, the sum of the numerators transforms as a modular form of weight $N/4$.
The first modular invariant is encountered as $N=24$ for 48 chiral fermions.

\newpage

\subsection*{$\bullet$ Bibliographical notes}

A systematic introduction to quantum field theory and more specifically  conformal field theory in two dimensions is presented in the book by Di Francesco, Mathieu, and S\'en\'echal~[47], which also has an extensive bibliography. The lecture notes by Ginsparg~[48] provide an excellent introduction.

\sm

The development of two-dimensional conformal field theory and the use of radial quantization date back to~[49]. Modern conformal field theory,  based on  the representation theory of the Virasoro algebra, was developed in [50], where minimal models were also introduced. The $bc$ system was discussed in the context of string perturbation theory in [51].  Functional determinants of Laplace and Cauchy-Riemann operators on the torus  are evaluated there as well as in the classic paper by Polchinski~[52]. The Kronecker limit formulas are discussed in detail in the lecture notes by Siegel~[53], as well as in [22].

\sm

On higher genus Riemann surfaces, functional determinants of Laplace operators  were considered in [54] in connection with Ray-Singer torsion, in [55], and in [56] in connection with Reidemeister torsion, and were evaluated in terms of the Selberg zeta function in [57, 58]. An overview is presented in the booklet by Jorgensen and Lang [59].  Functional determinants of Laplace and Cauchy-Riemann operators were evaluated in terms of Riemann $\tet$-functions via chiral bosonization in [60] and [61] by building on the properties of the holomorphic anomaly established in  [62] and [63].

\newpage

\section{Congruence subgroups and modular curves}
\setcounter{equation}{0}
\label{sec:Cong}

The modular group $SL(2,\ZZ)$ possesses an infinite number of non-Abelian subgroups. Various objects encountered in the previous sections, such as the Jacobi $\tet$-functions,  $\tet$-constants, and the Dedekind $\eta$-function,  do not transform as modular forms under $SL(2,\ZZ)$, but they are modular forms under certain subgroups of $SL(2,\ZZ)$. These subgroups belong to the general class of \textit{congruence subgroups} of $SL(2,\ZZ)$, which we shall now define and study.

\subsection{Definition of congruence subgroups of $SL(2,\ZZ)$}

The \textit{principal congruence subgroup} of level $N\geq 1$ is defined as follows,
\bea
\label{6.Gam}
\Gamma (N) =  \left \{  \left ( \begin{matrix} a & b \cr c & d \cr \end{matrix} \right ),  ~ ~ a,d \equiv 1 ~ ({\rm mod} \, N)\,,  ~~b,c \equiv 0 ~ ({\rm mod} \, N) \right \}
\eea
Clearly, we have $\Gamma (1) = SL(2,\ZZ)$ and $\Gamma(N)$ is a subgroup of $SL(2,\ZZ)$ for any $N$. Actually, $\Gamma (N)$ is a normal subgroup of $SL(2,\ZZ)$ so that the quotient is itself a group. The quotient group is isomorphic to $SL(2,\ZZ_N)$ (to physicists, isomorphism is synonymous with equality),
\bea
SL(2,\ZZ)/\Gamma(N) = SL(2,\ZZ_N)
\eea
namely the group of $2 \times 2$ matrices whose entries are integers mod $N$ and whose determinant is 1 mod $N$.
Equivalently, the group $\Gamma(N)$ may be identified with the kernel of the homomorphism $\ZZ \to \ZZ_N$ of reduction mod $N$. The index $ [SL(2,\ZZ): \Gamma(N)]$ of $\Gamma(N)$ equals the order of $SL(2,\ZZ_N)$, which is finite. It will be determined below shortly.

\sm

By definition, a group $\Gamma$  is a \textit{congruence subgroup of $SL(2,\ZZ)$  of level $N$} provided there exists an integer $N \geq 1$ such that $\Gamma (N) \subset \Gamma \subset SL(2,\ZZ)$.  Since the index of $\Gamma(N)$ is finite, it follows that the index $[SL(2,\ZZ): \Gamma]$ is  finite for any congruence subgroup $\Gamma$. Note
that $\Gamma$ need not be a normal subgroup of $SL(2,\ZZ)$ so that $SL(2,\ZZ)/\Gamma$ need not be a group.

\subsection{The classic congruence subgroups}

Besides the principal congruence subgroups $\Gamma(N)$, two other fundamental congruence subgroups of level $N$ are defined as follows,\footnote{The groups of transposes are denoted by,
$\Gamma^1(N) = \{ \gamma  | \gamma ^t \in \Gamma_1(N) \}$ and
$\Gamma^0(N) = \{ \gamma | \gamma ^t \in \Gamma_0(N) \}$. }
\bea
\label{6.Gam01}
\Gamma_1(N) &=&  \left \{  \left ( \begin{matrix} a & b \cr c & d \cr \end{matrix} \right ),  ~ ~ a,d \equiv 1 ~ ({\rm mod} \, N)\,,  ~~c \equiv 0 ~ ({\rm mod} \, N) \right \}
\no\\
\Gamma_0 (N) &=& \left \{  \left ( \begin{matrix} a & b \cr c & d \cr \end{matrix} \right ),  ~ ~ c \equiv 0 ~ ({\rm mod} \, N) \right \}
\eea
with  the following manifest subgroup  inclusions,
\bea
\Gamma (N) \subset \Gamma _1(N) \subset \Gamma_0(N) \subset SL(2,\ZZ)
\eea
For $N \geq 2$, none of the groups $\Gamma(N), \Gamma _1(N), \Gamma _0(N)$ contains the generator $S$ of $SL(2,\ZZ)$ and, while $\Gamma_1(N)$ and $\Gamma_0(N)$ contain the translation generator~$T$,  the principal congruence subgroup $\Gamma(N)$ contains $T^N$ but not $T$.

\sm

We recall that $\Gamma(N)$ is a normal subgroup of $SL(2,\ZZ)$ induced by the homomorphism $\ZZ \to \ZZ_N$ on all its entries. Similarly, $\Gamma(N)$ is a normal subgroup of $\Gamma _1(N)$ induced by $\ZZ \to \ZZ_N$ on the entry~$b$ whose quotient is isomorphic to $\ZZ_N$. Finally, $\Gamma _1(N)$ is a normal subgroup of $\Gamma_0(N)$ whose quotient is isomorphic to the group $\ZZ_N^*$ of all multiplicatively invertible elements in $\ZZ_N$. In summary, we have the following quotients and  isomorphisms,
\bea
SL(2,\ZZ)/\Gamma(N) & = & SL(2,\ZZ_N)
\no \\
\Gamma_1(N) / \Gamma(N) & = & \ZZ_N
\no \\
\Gamma_0(N)/\Gamma _1(N) & = & \ZZ_N^*
\eea
The order of $SL(2,\ZZ_N)$ is denoted $2d_N$ and will be evaluated in section \ref{5.order}.  The order of $\ZZ_N$ is $N$. The order of $\ZZ_N^*$ equals the Euler function $\phi(N)$, which counts the number of integers in $\ZZ_N$ that are relatively prime to $N$, and is given by the following formula,
\bea
|\ZZ_N^*| = \phi(N) = N \prod _i \left ( 1 - { 1 \over p_i} \right )
\hskip 1in
N = \prod _i p_i^{e_i} ~ \hbox{ for } ~ e_i \geq 1
\eea
where the products are over distinct primes $p_i$. For the special case where $N$ is prime, we have $\phi(N)=N-1$. The indices of these subgroups are related to the orders  of the quotient groups as follows,
\bea
\label{6.index1}
{} [SL(2,\ZZ):\Gamma(N)] & = &  |SL(2,\ZZ_N)| = 2d_N
\no \\
{} [\Gamma _1(N): \Gamma (N)] & = & |\ZZ_N| = N
\no \\
{} [\Gamma_0(N) : \Gamma _1(N)] & = & |\ZZ_N^*| = \phi(N)
\eea
We obtain the  remaining indices,
\bea
\label{6.index2}
{} [SL(2,\ZZ):\Gamma_0(N)]  & = &  2 d_N / (N \phi(N))
\no \\
{} [SL(2,\ZZ):\Gamma_1(N)]  & = &  2 d_N / N
\eea
using the multiplicative property $[G:H] \times [H:K]=[G:K]$ for groups $K \subset H \subset G$.

\subsection{Computing the order of $SL(2,\ZZ_N)$}
\label{5.order}

We shall now evaluate the order $2d_N$ of $SL(2,\ZZ_N)$, following [13]. The order will enter crucially in the valence formulas for congruence subgroups. To compute $d_N$, we shall make use of the Chinese remainder theorem (reviewed and proven in appendix \ref{sec:modN}), which provides a fundamental isomorphism for $\ZZ_N$ in terms of its prime decomposition factors.

\sm

Let $N$ have the following prime number decomposition $N = \prod _i p_i ^{e_i}$ where the $p_i$ are distinct primes and $e_i$ are positive integer exponents. Then by the Chinese remainder theorem, we have the following isomorphism,
\bea
SL(2,\ZZ_N) = \prod _i SL(2,\ZZ_{p_i^{e_i}})
\eea
Thus, the order $|SL(2,\ZZ_N)|$ of the group $SL(2,\ZZ_N)$ is given by,
\bea
|SL(2,\ZZ_N)| = \prod _i |SL(2,\ZZ_{p_i^{e_i}})|
\eea
To compute the order of each component $SL(2,\ZZ_{p^{e}})$ we proceed by induction on $e$. We begin by computing the order of $GL(2,\ZZ_p)$. To do so, consider the set of all matrices $\gamma$,
\bea
\gamma = \left ( \begin{matrix} a & b \cr c & d \cr \end{matrix} \right )
\eea
whose entries $a,b,c,d \in \ZZ_p$ may be parametrized by $a,b,c,d  \in \{ 0,1, \cdots , p-1\}$.  To belong to $GL(2,\ZZ_p)$, the entries must satisfy $ad-bc \not=0 ~ (\mod p)$. To enumerate all its elements we proceed by listing the number of elements in the four subsets of possible values of the doublet $(a, b)$. No matrices with  $a=b=0$ belong to $GL(2,\ZZ_p)$; there are $p-1$ doublets with $a \not=0, b=0$ and for each non-zero value of $a$ the $p-1$ non-zero values of $d$ and the $p$ arbitrary values $c$ give  $(p-1)^2p$ elements
$\gamma \in GL(2,\ZZ_p)$. For the case $a=0, b \not=0$ the roles of $c$ and $d$ are reversed and we obtain another $(p-1)^2p$ elements. Finally, for each one of the $ (p-1)^2$ doublets with $a,b \not=0$, the entry $c$ may take all $p$ values, while $d$ may take all $p$ values except the one that gives zero determinant, producing  $(p-1)^3 p$ contributions. Putting all together, we find,
\bea
|GL(2,\ZZ_p)| = (p-1)^3 p + 2 (p-1)^2 p  = (p-1) p^3 \left ( 1 - { 1 \over p^2 } \right )
\eea
Any $\gamma \in GL(2,\ZZ_p)$ satisfies $\det \gamma \not=0$, and taking the  quotient by the determinant normal subgroup isomorphic to the multiplicative group $(\ZZ_p)^*$ with $p-1$ elements, we obtain,
\bea
|SL(2,\ZZ_p)| = p^3 \left ( 1 - { 1 \over p^2 } \right )
\eea
To proceed by induction on the exponent $e$, we consider an arbitrary  matrix $\gamma '$ $(\mod p^{e+1})$  and parametrize it uniquely  $(\mod p^e)$,
\bea
\gamma' = \gamma + p^e \left ( \begin{matrix}  a'  &   b'  \cr  c'  &  d'  \cr \end{matrix} \right )
\hskip 1in
\gamma = \left ( \begin{matrix} a   & b   \cr c   & d  \cr \end{matrix} \right )
\eea
where $a,b,c,d \in \ZZ_{p^e}$ and  $a',b',c',d' \in \ZZ_p$. Requiring $\det \gamma \equiv 1 ~ (\mod p^{e+1})$
requires $ad-bc \equiv 1 ~ (\mod p^e)$ so that  $\gamma \in SL(2,\ZZ_{p^e})$. Now for an arbitrary matrix $\gamma \in SL(2,\ZZ_{p^e})$, we obtain $\gamma ' \in SL(2,\ZZ_{p^{e+1}})$ if and only if,
\bea
\label{pe}
a d' + d a' - b c' - c b' \equiv 0 ~ (\mod p)
\eea
At least two entries in $\gamma$ are non-zero $(\mod p)$ since $ad-bc \equiv 1 ~ (\mod p)$. If $a \not=0$, we let $a',b',c'$  take independent arbitrary values in $\ZZ_p$ and determine the number $d'$ uniquely by solving equation (\ref{pe}). If $a=0$, then we must have $b\not \equiv 0 ~ (\mod p)$, take the values of $a',b',d'$ arbitrary and solve uniquely for $c'$ using (\ref{pe}). In each case, to every $\gamma \in SL(2,\ZZ_{p^e})$ there correspond $p^3$ elements $\gamma ' \in SL(2,\ZZ_{p^{e+1}})$. Hence we have,
\bea
|SL(2,\ZZ_{p^e})| = p^{3e} \left ( 1 - { 1 \over p^2 } \right )
\eea
Putting all together, for any $N\geq 2$ with prime number decomposition $N = \prod _i p_i^{e_i}$ we have,
\bea
\label{6.dN}
2 d_N = |SL(2,\ZZ_N)| = N^3 \prod _i  \left ( 1 - { 1 \over p_i^2 } \right )
\eea
The order of the simplest non-trivial group  $SL(2, \ZZ_2)$ is 6 by the above formula. In fact, we can write its generators in terms of the generators $S$ and $T$ of $SL(2,\ZZ)$ and we find,
\bea
\label{5.sl2}
SL(2,\ZZ_2) = \big \{ I, \, S, \, T, \, ST, \, TS, \, STS \big \} \quad (\mod 2)
\eea
Opposite elements are not included in the set since  $I \equiv -I ~ (\mod 2)$.

\subsection{Modular curves}

Recall that the quotient $SL(2,\ZZ) \backslash \cH$ of the upper half-plane $\cH$ by $SL(2,\ZZ)$ is topologically a sphere with one puncture corresponding to the unique cusp. The quotient $SL(2,\ZZ) \backslash \cH$ may be represented  by a choice of fundamental domain in $\cH$ with boundary sides identified pairwise under generators of $SL(2,\ZZ)$. The standard choice $F$ of fundamental domain for $SL(2,\ZZ)$ was defined in (\ref{3.fund}) and depicted in Figure \ref{3.fig:2}.

\sm

For an arbitrary congruence subgroup $\Gamma$ the quotient is the \textit{modular curve} $Y(\Gamma) = \Gamma \backslash \cH$ which is  again a Riemann surface. However, $Y(\Gamma)$ may be a sphere or a surface of genus  greater than zero and may have several punctures corresponding to the several cusps of $\Gamma$.

\subsubsection{Fundamental domain for a congruence subgroup $\Gamma$}

The modular curve $Y(\Gamma)$ may again be represented by a choice of fundamental domain $F_\Gamma$  in $\cH$ with opposite sides identified pairwise under generators of $\Gamma$. Given the fundamental domain $F$ for $SL(2,\ZZ)$, the fundamental domain $F_\Gamma$ may be obtained by applying the cosets of $SL(2,\ZZ)/\Gamma$ to $F$,
\bea
SL(2,\ZZ)/\Gamma :  F  \to F_\Gamma
\eea
Therefore $F_\Gamma$ is the union of a number of copies of $F$ equal to the index $[SL(2,\ZZ): \Gamma]$.
The elliptic points of $\Gamma$ are the images of the elliptic points $i$ and $\rho$ of $SL(2,\ZZ)$ under the cosets $SL(2,\ZZ)/\Gamma$ and, similarly,  the cusps of $\Gamma$ are the images of the cusp $i \infty$ of $F$ under the cosets $SL(2,\ZZ)/\Gamma$. Thus, the number of cusps in $Y(\Gamma)$ equals the index $[SL(2,\ZZ):\Gamma]$. For a non-trivial congruence subgroup $\Gamma \not= SL(2,\ZZ)$, the index is strictly greater than 1, as may be verified for the index  for $\Gamma = \Gamma(N), \Gamma_1(N),\Gamma_0(N)$ evaluated  in (\ref{6.index1}) and (\ref{6.index2}).
Since $Y(\Gamma)$ has at least one puncture, it is always non-compact.

\sm

There is a useful alternative way of looking at the cusps of a congruence subgroup $\Gamma$ of $SL(2,\ZZ)$. Recall that the Borel subgroup $\Gamma _\infty$ of $SL(2,\ZZ)$  is defined by $\Gamma _\infty= \{ \pm T^n, n \in \ZZ\}$ and that  $SL(2,\ZZ)/\Gamma _\infty $ acts transitively on $\QQ \cup \{ \infty \}$. This property results from the fact that every rational number  $r \in \QQ$ is the image of $i \infty$ under a unique coset in $SL(2,\ZZ)/ \Gamma _\infty$.  To show this, we distinguish two cases: $r=0$ is the image of $i \infty$ under the transformation  $S \in SL(2,\ZZ)/\Gamma _\infty $, while for $r \not=0$ we write $r=a/c$ uniquely  with $a,c \in \ZZ^*$,  $\gcd (a,c)=1$, and $c >0$. By B\'ezout's theorem (reviewed in appendix \ref{sec:modN}) there exists a particular solution $(b_0,d_0)$ for $b$ and $d$ to the equation $ad-bc=1$.  The general solution is given by $b=b_0+ka$ and $d=d_0+kc$, which is equivalent to the particular solution by right multiplication by $T^k$, which proves the assertion.
The cusps of $Y(\Gamma)$ may now be viewed as those points in $\QQ \cup \{ \infty\}$ which are inequivalent under the action of the congruence group $\Gamma$, or equivalently,\footnote{Throughout, it will be convenient to use the notation $\Gamma z = \{ \gamma z \hbox{ for all } \gamma \in \Gamma\}$ for the set of all images of the point $z$ under the action of the group $\Gamma$. }
\bea
\label{5.cusps}
\{ \hbox{cusps of } \Gamma \} = \big ( \Gamma \backslash SL(2,\ZZ)/\Gamma_\infty  \big ) \, \{ \infty \}
\eea
The number of cusps, counted with multiplicity, is again given by the index $[SL(2,\ZZ):\Gamma]$.

\sm

For  the  example $\Gamma = \Gamma (2)$ with $SL(2,\ZZ)/\Gamma (2) = SL(2,\ZZ_2)$, the fundamental domain $F_{\Gamma (2)}$ is the union of the 6 images of $F$ under the elements of $SL(2,\ZZ_2)$ listed in (\ref{5.sl2}). The resulting fundamental domain $F_{\Gamma(2)}$  is depicted in Figure  \ref{5.fig:1}. Using the area formula (\ref{3.area}) we have ${\rm area}(F_{\Gamma (2)}) = 2 \pi$ or ${\rm area}(F_{\Gamma (2)}) = 6 \, {\rm area}(F)$, given by the order of $SL(2,\ZZ_2)$, as expected. The  $\Gamma(2)$-inequivalent elliptic points in $F_{\Gamma(2)}$ are $i, i', i'', i''', \rho, \rho'$ and its $\Gamma(2)$-inequivalent cusps are $0,1, i \infty$. The elliptic points $i$ and $i'$ have multiplicity 2;  $i''$ and $i'''$ have multiplicity 1; $ \rho$ and $\rho'$ have multiplicity 6, while the cusps $0,1,i \infty$ all have multiplicity 2.

\begin{figure}[htb]
\begin{center}
\tikzpicture[scale=1.05]
\scope[xshift=-5cm,yshift=0cm]
\draw[thick] (-7,0) -- (7,0);
\draw[ultra thick] (-4, 0) -- (-4, 7);
\draw[ultra thick] (4, 0) -- (4, 7);
\draw [ultra thick] (0,0) arc (0:180:2 and 2);
\draw [ultra thick] (0,0) arc (180:0:2 and 2);

\draw[thick, dotted] (-2,0) -- (-2,3.464);
\draw[thick, dotted] (2,0) -- (2,3.464);

\draw[thick, dashed] (-2, 3.464) -- (-2, 7);
\draw[thick, dashed] (2 , 3.464) -- (2, 7);

\draw [thick, dashed] (-2,3.464) arc (120:60:4 and 4);
\draw [dashed] (-4,0) arc (180:120:4 and 4);
\draw [thick, dashed] (4,0) arc (0:60:4 and 4);
%image under T
%\draw[thick, dashed] (6 , 3.464) -- (6, 10);
\draw [thick, dashed] (2,3.464) arc (120:90:4 and 4);
%image under S
\draw [thick, dashed] (0,0) arc (0:90:4 and 4);
\draw [thick, dashed] (0,0) arc (180:90:4 and 4);
%\draw [thick, dashed] (4,0) arc (180:120:4 and 4);
% labels
\draw (0,-0.5) node{$0$};
\draw (-2,-0.5) node{$-\half $};
\draw (2,-0.5) node{$\half $};
\draw (-4,-0.5) node{$-1$};
\draw (4,-0.5) node{$1$};
\draw (0,6) node{\tiny $F$};
\draw (3,6) node{\tiny $T\! F$};
\draw (-3,6) node{\tiny $T\! F$};
\draw (0,3.2) node{\tiny $S\! F$};
\draw (3.2,3.3) node{\tiny $T\! S\! F$};
\draw (-3.2,3.3) node{\tiny $T\! S\! F$};
\draw (1.4,2.4) node{\tiny $S\! T\! F$};
\draw (2.5,2.4) node{\tiny $S\! T\! S \! F$};
\draw (-1.4,2.4) node{\tiny $S\! T\! F$};
\draw (-2.5,2.4) node{\tiny $S\! T\! S \! F$};
\draw (6,6) node{$\cH$};
\draw (6,0.5) node{$\RR$};
\draw (0,4.5) node{\tiny $i$};
\draw (-4.4,4.2) node{\tiny $i'$};
\draw (4.4,4.2) node{\tiny $i'$};
\draw (2.2,1.7) node{\tiny $i''$};
\draw (-2.2,1.7) node{\tiny $i'''$};

\draw (0,4) node{$\bullet$};
\draw (4,4) node{$\bullet$};
\draw (-4,4) node{$\bullet$};
\draw (2,2) node{$\bullet$};
\draw (-2,2) node{$\bullet$};
\draw (2,3.464) node{$\bullet$};
\draw (-2,3.464) node{$\bullet$};
\draw (1.7,4) node{\tiny $\rho'$};
\draw (-1.7,4) node{\tiny $\rho$};
\draw (0,7) node{\small $i \infty$};
\endscope
\endtikzpicture
\end{center}
\caption{\textit{The fundamental domain $\Gamma _{\Gamma (2)}$ for the congruence subgroup $\Gamma(2)$, its elliptic points $i, i', i'', i''', \rho, \rho'$, and its cusps $0, \pm 1, i \infty$ are obtained  as the images of the fundamental domain $F$, the elliptic points $i, \rho$, and the cusp $i \infty$ of $SL(2,\ZZ)$, respectively,   under the action of the group of cosets $SL(2,\ZZ_2) = \{ I, S, T, ST, TS, STS \}$.} \label{5.fig:1}}
\end{figure}

\subsection{Compactification of the modular curves}

To compactify $Y(SL(2,\ZZ))$, one adjoins the point $\{i \infty \}$ to the fundamental domain $F$ of $SL(2,\ZZ)$. To compactify  $Y(\Gamma)$ for an arbitrary congruence subgroup $\Gamma$, one similarly adjoins $\{ \hbox{cusp of } \Gamma\} = \Gamma \backslash SL(2,\ZZ)/ \Gamma _\infty \, \{ i\infty \}$  to $Y(\Gamma)$. For an arbitrary congruence subgroup $\Gamma$, the resulting \textit{modular curve} $X(\Gamma)$ is a connected and compact Riemann surface.

\sm

An alternative but equivalent way to obtain the compact modular curve $X(\Gamma)$ is to start from the union $\cH^* = \cH \cup \{i \infty\} \cup \QQ$ of the upper half-plane with the rational numbers and infinity. One chooses a topology for  $\cH^*$ whose open sets include all the standard open sets of $\cH$ (which are generated by the open coordinate discs whose closure lies entirely in $\cH$),  as well as the open sets generated by discs that are open in $\cH$ and intersect $\RR$ at a rational number. The latter provide an open neighborhood for every point in $\{i \infty\} \cup \QQ$ and are depicted in green in Figure \ref{5.fig:2} for the discs at infinity $\cU_{i \infty}$ and the discs $\cU_{\pm 1}$ and $\cU_0$ intersecting the real axis at the rational points $\pm 1$ and $0$. The topological space $\cH^*$ obtained this way is not Haussdorff. Recall that a topological space is Haussdorff if for any two distinct points of $\cH^*$ there exist neighborhoods of each point that are disjoint from each other.

\begin{figure}[tp]
\begin{center}
\tikzpicture[scale=0.7]
\scope[xshift=0cm,yshift=0cm]
\draw[ fill=green!30!] (-7,6) rectangle (7,8);
\draw (0,7.5) node{\small $i \infty$};

\draw  [domain=0:360,fill=green!30!] plot ({3/4*cos(\x)},{3/4+3/4*sin(\x)});
\draw  [domain=0:360,fill=green!30!] plot ({3+3/4*cos(\x)},{3/4+3/4*sin(\x)});
\draw  [domain=0:360,fill=green!30!] plot ({-3+3/4*cos(\x)},{3/4+3/4*sin(\x)});
\draw [thick] (-7,0) -- (7,0);
\draw [thick] (0,-0.2) -- (0,7);
\draw (0,3) node{$-$};
\draw (0,0) node{$\bullet$};
\draw (-3,0) node{$\bullet$};
\draw (3,0) node{$\bullet$};
\draw (0,-0.5) node{$0$};
\draw (-3,-0.5) node{$-1$};
\draw (3,-0.5) node{$1$};
\draw (6,4) node{$\cH$};
\draw (6,0.5) node{$\RR$};
\draw (0.5,3) node{ $i$};
\draw (-6.5,5.3) node{\small  $\cU_{i \infty}$};
\draw (-3.9,1.8) node{\small $\cU_{-1}$};
\draw (-0.9,1.8) node{\small  $\cU_0$};
\draw (2.1,1.8) node{\small  $\cU_1$};
\endscope
\endtikzpicture
\end{center}
\caption{\textit{The green rectangle $\cU_{i \infty}$ represents a neighborhood of the cusp $i \infty$ of $SL(2,\ZZ)$ while the green discs $\cU_{-1}, \cU_0, \cU_1$ represent the neighborhoods of the additional cusps of $\Gamma (2)$. } \label{5.fig:2}}
\end{figure}

\sm

The modular curve $X(\Gamma)$ may be obtained from $\cH^*$ by taking the \textit{extended quotient}, which includes the action of $\Gamma$ on $\QQ \cup \{i \infty \}$,
\bea
X(\Gamma) = \Gamma \backslash \cH^*
\eea
The modular curve $X(\Gamma)$ is  Haussdorff even though $\cH^*$ is not. The compact modular curves for the classic congruence subgroups are denoted as follows,
\bea
X(N) & = & X \big (\Gamma(N) \big )
\no \\
X_1(N) & = & X \big ( \Gamma_1(N) \big )
\no \\
X_0(N) & = & X \big ( \Gamma_0(N) \big )
\eea
A modular form $f(\tau)$ of weight $k$ for a congruence subgroup $\Gamma \subset SL(2,\ZZ)$  corresponds to a $\Gamma$-invariant differential form $\mf= f(\tau) d\tau^{{k \over 2}}  $ on $\cH^*$, and thus pulls back to a well-defined differential on $X(\Gamma) = \Gamma \backslash \cH^*$.
To obtain the dimensions of the spaces $\cM_k(\Gamma)$ and $\cS_k(\Gamma)$ of modular forms and cusp forms of weight $k$, we can now use the machinery of line bundles on the compact Riemann surfaces $X(\Gamma)$, reviewed in appendix \ref{sec:LB}.

\subsection{The genus of $X(\Gamma)$}

To study $X(\Gamma)$, we shall view it as a branched covering of the sphere $X(SL(2,\ZZ))$ by a non-constant holomorphic function $f : X(\Gamma) \to X(SL(2,\ZZ))$.  We provide a summary of various relevant topological aspects of Riemann surfaces in appendix \ref{sec:RS}.

\subsubsection{The Hurwitz formula for branched coverings}

Consider a non-constant holomorphic map  $f:X \to Y$ between two compact Riemann surfaces $X,Y$. The degree of $f$ is the cardinality of $f^{-1}$ denoted $d=|f^{-1}(y)|$ for all but finitely many points $y \in Y$. A formula valid for all points in $Y$ is obtained in terms of the \textit{degree of ramification} $e_x$ of $f$ at a point $x\in X$ which,  in local coordinates, is given by $f(z)-f(x) \sim (z-x)^{e_x}$.  The degree  of $f$ may then be defined for all $y \in Y$ by the  formula,
\bea
\label{6.degreea}
d= \sum_{x \in f^{-1}(y)} e_x
\eea
valid for an arbitrary point $y \in Y$.
\label{5.thm:1}
The Riemann-Hurwitz formula relates the genera $g_X$ and $g_Y$  of  $X$ and $Y$ to the degree of $f$ and the deviations from 1 of the degree of ramification,
\bea
\label{6.RH}
2g_X -2 = d(2g_Y-2) + \sum _{x \in X} (e_x-1)
\eea}
A proof of the Riemann-Hurwitz formula is given in appendix \ref{sec:RS}.

\subsubsection{Calculating the genus of $X(\Gamma)$}

Returning now to the map $f : X(\Gamma) \to X(SL(2,\ZZ))$ we obtain an alternative formula for its degree in terms of the index of the congruence subgroup $\Gamma \subset SL(2,\ZZ)$,
\bea
\label{6.deg}
d=  \left \{ \bma [SL(2,\ZZ): \Gamma]  & \hbox{ if } & -I  \in \Gamma \cr
\thalf [SL(2,\ZZ): \Gamma]  & \hbox{ if } & -I  \not \in \Gamma \cr \ema \right .
\eea
We shall denote  the elliptic points of $SL(2,\ZZ)$ of order $2h$ by $y_h\in SL(2,\ZZ) \{e^{\pi i /h}\}$ for $h=2,3$ and the cusps by $y_\infty \in SL(2,\ZZ) \{i\infty\}$. The number of elliptic points of order $2h$  in $X(\Gamma)$ will be denoted by $\ep_h$, and the number of cusps in $X(\Gamma)$ by $\ep_\infty$.

\label{5.thm:2}
The genus $g$ of $X(\Gamma)$ is given by,
\bea
\label{6.genus}
g = 1+{d \over 12} -{\ep_2 \over 4} -{ \ep_3 \over 3} -{\ep_\infty \over 2}
\eea
where $d$ is given in (\ref{6.deg}),  $\ep_h$ denotes the number of elliptic points of order $2h$, and $\ep_\infty$ denotes the number of cusps in $X(\Gamma)$.}

To prove this result using the Riemann-Hurwitz formula (\ref{6.RH}), we evaluate the contributions from the points $x\in X(\Gamma) $ for which $e_x >1$, namely from the points $f^{-1}(y_h)$ and the cusps of $X(\Gamma)$. If $f^{-1}(y_h)$ is a ramification point of $X(\Gamma)$ then  $e_{y_h}=1$, while if it is not a ramification point of $X(\Gamma)$ then $e_{y_h}=h$. Using the degree formula (\ref{6.degree}) at the points $y_h$, we have,
\bea
\sum_{x \in f^{-1}(y_h)} e_x = \ep_h + h (|f^{-1} (y_h)| -\ep_h)
\eea
where the first term arises from those points $y_h$ that are ramification points of $X(\Gamma)$ while the second term arises from those points in $f^{-1}(y_h)$ that are not ramification points of $X(\Gamma)$, in each case weighted by their respective ramification degrees 1 and $h$. Using the degree formula again to eliminate the number $|f^{-1}(y_h)|$, one obtains,
\bea
\label{6.yh}
\sum_{x \in f^{-1} (y_h) } (e_x-1) = { h-1 \over h} (d-\ep_h)
\eea
The formula holds for the cusps by setting $h = \infty$. Using the Riemann-Hurwitz formula (\ref{6.RH}), we set $g_Y=0$, $g=g_{X(\Gamma)}$, and substitute the contributions to the sum of $(e_x-1)$ obtained in (\ref{6.yh}) to obtain (\ref{6.genus}).

\subsection{Formulas for $d$, $\ep_2, \ep_3 $ and $ \ep_\infty$ for $\Gamma(N), \Gamma_1(N)$ and $\Gamma_0(N)$}

In order to obtain formulas for the genus of $X(\Gamma)$, and later on the dimensions of the spaces of modular and cusp forms on $X(\Gamma)$, we need to obtain the degree $d$, and formulas for $\ep_2, \ep_3, \ep_\infty$ for an arbitrary congruence subgroup $\Gamma$. But first, it will be useful to obtain these formulas  for the special cases of the congruence subgroups $\Gamma(N), \Gamma_1(N)$, and $\Gamma_0(N)$. To do so, we begin by assembling all the partial results that are required.

\subsubsection{The absence of elliptic points}

Recall  that the elliptic points of $SL(2,\ZZ)$ are the points of order four, namely $SL(2,\ZZ) \{i\}$, and the points of order six, namely $SL(2,\ZZ) \{e^{ 2 \pi i /6}\}$. In this subsection and the next, we shall count the number of elliptic points of orders four and six of the congruence subgroups $\Gamma(N), \Gamma_1(N)$, and $\Gamma _0(N)$.  We begin with the following theorem.

{\thm The following congruence groups have no elliptic points,
\begin{enumerate}
\itemsep =-0.05in
\item $\Gamma(N)$ for $N \geq 2$;
\item $\Gamma_1(N)$ for $N \geq 4$;
\item $\Gamma _0(N)$ for any $N$ divisible by a prime $p$ which obeys $p \equiv -1 ~ (\mod 12)$.
\end{enumerate}}
To prove 1, we parametrize $\gamma \in \Gamma (N) $ as follows,
\bea
\gamma
= \left ( \begin{matrix} a &   b \cr c & d \cr \end{matrix} \right )
= \left ( \begin{matrix} 1+ N a' &  N b' \cr Nc' & 1+N d' \cr \end{matrix} \right )
\eea
For an elliptic $\gamma$ we must have $|a+d|<2$, following the analysis of subsection \ref{sec:3.3a}. The condition $|a+d| = |2+N(a'+d')|<2$ allows for three possibilities, namely $N(a'+d') =-1,-2,-3$, which are all ruled out for all $N \geq 4$. For both $N=2$ and $N=3$, the only solution is given by $a'+d'=-1$. Combining this condition with $\det \gamma =1$, we find that we must have $N(a'd'-b'c')=1$ which has no solutions for $N \geq 2$.  To prove 2, we parametrize $\gamma \in \Gamma _1(N) $ as follows,
\bea
\gamma = \left ( \begin{matrix} 1+ N a' &  b \cr Nc' & 1+N d' \cr \end{matrix} \right )
\eea
The condition $|a+d| = |2+N(a'+d')|<2$ for $\gamma$ to be elliptic again allows for three possibilities, namely $N(a'+d') =-1,-2,-3$, which are all ruled out for $N \geq 4$.  To prove~3, we parametrize $\gamma \in \Gamma _0(N)$ as follows,
\bea
\gamma = \left ( \begin{matrix} a &  b \cr Nc' & d \cr \end{matrix} \right )
\eea
The condition $|a+d| <2$ again leaves three possibilities, namely $a+d =0, \pm 1$. For $a+d=0$, we furthermore have
$-a^2-Nbc'=1$, so that $a^2 \equiv -1 ~ (\mod N)$. For $a+d=\pm 1$, we have $-a^2 \pm a -Nbc'=1$ so that $a^2 \pm a \equiv -1 ~ (\mod N)$. Now if $p \equiv -1 ~ (\mod 12)$ then $p \equiv -1 ~ (\mod 3)$  and $p \equiv -1 ~ (\mod 4)$. But $a^2 \equiv -1 ~ (\mod p)$ has no solutions if $p \equiv -1 ~ (\mod 4)$ in view of (\ref{quadcon.1}) while $a^2 + a \equiv -1 ~ (\mod p)$ has no solutions in view (\ref{quadcon.2}), which completes the proof of point 3 and of the theorem.

\subsubsection{Counting elliptic points for $\Gamma _0(N)$}

We now count elliptic points for $\Gamma _0(N)$, at the values of $N$ for which they exist. There are two cases, namely elliptic points of orders 4 and 6, for which the transformation in $SL(2,\ZZ)$ is conjugate to either,
\bea
S = \left ( \bma 0 & -1 \cr 1 & 0 \cr \ema \right )
\hskip 1in
ST = \left ( \bma 0 & -1 \cr 1 & 1 \cr \ema \right )
\eea
with $S^4= (ST) ^6 = I$, $\tr (S)=0$, and $\tr (ST)=1$. Given the conditions on the traces, the representatives of $S$ and $ST$  in $\Gamma _0(N)$ are of the form,
\bea
\gamma_4 = \left ( \bma a & b \cr Nc' & -a \ema \right )
\hskip 1in
\gamma_6 = \left ( \bma a & b \cr Nc' & 1-a \ema \right )
\eea
with $-a^2-bc'N=1$ and $a-a^2-bc'N=1$ respectively. We shall now assume the prime decomposition $N= \prod _i p_i ^{\a_i}$ for distinct primes $p_i$ and $\a_i \geq 1$.
For the elliptic points of order~4  the congruence equation mod $N$,
\bea
a^2 \equiv -1 ~ (\mod N)
\eea
is equivalent to the congruence equations $a^2 \equiv -1 ~ (\mod p_i ^{\a_i})$. Using the results of subsection \ref{sec:1.5.1} the number of solutions is given by (\ref{quadcon.1}),
\bea
\label{6.ep2}
\ep_2(\Gamma _0(N)) = \left \{ \bma 0 && 4 | N \cr
\prod _i \Big ( 1+ (-1|p_i) \Big ) && 4 \nmid  N \cr \ema \right .
\eea
where $(-1|p) = \pm 1$ for $p \equiv \pm 1 ~ (\mod 4)$ and vanishes for $p=2$.
For the elliptic points of order 6, the congruence equation, $a^2+a \equiv -1 ~ (\mod N)$ is equivalent to,
\bea
(2a+1)^2+3 \equiv 0 ~ (\mod 4N)
\eea
The number of solutions was obtained in subsection \ref{sec:1.5.2}, and is given by,
\bea
\label{6.ep3}
\ep_3 (\Gamma _0(N)) = \left \{ \bma 0 && 9 | N \cr
\prod _i \Big ( 1+ (-3|p_i) \Big ) && 9 \nmid  N \cr \ema \right .
\eea
where $(-3|p)=\pm 1$ for $p \equiv \pm 1 ~ (\mod 3)$ and vanishes for $p=3$.

\subsubsection{Counting the number of cusps}
\label{sec:numbercusps}

The cusps of a modular curve $X(\Gamma)$ for a congruence subgroup $\Gamma \subset SL(2,\ZZ)$ are given by the cusps $\{ \infty\} \cup \QQ = SL(2,\ZZ) \infty $ modulo equivalence under $\Gamma$. In this subsection, we shall count the number $\ep_\infty$ of cusps of $X(\Gamma)$ for $\Gamma = \Gamma (N), \Gamma _1(N)$, and $ \Gamma _0(N)$.

\sm

We begin by considering the case $\Gamma = \Gamma(N)$. Two cusps $r=m/n$ and $r'=m'/n'$ with $\gcd(m,n)=\gcd(m',n')=1$ are equivalent under $\Gamma(N)$ provided there exists an $\gamma \in \Gamma (N)$ such that $r' = \gamma r$, or more explicitly,
\bea
{m'\over n'} = { (1+Na') m + Nb'n \over Nc'm +(1+Nd') n}
\hskip 1in
\gamma = \left ( \bma 1+Na' & Nb' \cr Nc' & 1+Nd' \ema \right )
\eea
The equation on the left side is equivalent to $mn' \equiv m'n ~ (\mod N)$, whose solution is given by $(m',n') = \pm (m,n) ~ (\mod N)$ in view of the fact that both pairs $(m,n)$ and $(m',n')$ are relatively prime. Therefore, the cusps for $\Gamma (N)$ are labelled by all pairs $(m,n) ~ (\mod N)$ with $\gcd(m,n)=1$.

\sm

An alternative perspective on this problem uses \textit{double cosets} and will lend itself more directly to the cases $\Gamma_1(N)$ and $\Gamma _0(N)$. Double cosets will be used in the construction of Hecke operators in section \ref{sec:Hecke}, where a more extensive discussion will be provided. Suffice it here to say that if $G$ is a group and $H_1, H_2$ are subgroups of $G$, then a double coset is an orbit under the left action of $H_1$ and the right action of $ H_2$ containing a given element $g \in G$, and is of the form $H_1 g H_2$. The set of all double cosets is denoted,
\bea
H_1 \backslash G / H_2 = \{ H_1 \, g \, H_2, ~ g \in G \}
\eea
Applying the construction to computing the cusps of $\Gamma (N)$, we set $G=SL(2,\ZZ)$, $H_1=\Gamma (N)$, $H_2= \Gamma _\infty = \{ \pm \left ( \begin{smallmatrix} 1 & b \cr 0 & 1 \end{smallmatrix} \right ), b \in \ZZ \}$. The simple coset $SL(2,\ZZ) /\Gamma _\infty$ provides a one-to-one map from $\infty$ to all the cusps of $SL(2,\ZZ)$ while the further left cosets identified cusps that are equivalent under $\Gamma(N)$. Hence we have,
\bea
\ep_\infty (\Gamma (N)) = | \Gamma (N) \backslash SL(2,\ZZ) / \Gamma _\infty |
\eea
Since $\Gamma (N)$ is a normal subgroup of $SL(2,\ZZ)$ and $\Gamma (N) \backslash SL(2,\ZZ) = SL(2,\ZZ_N)$, we have,
\bea
\Gamma (N) \backslash SL(2,\ZZ) / \Gamma _\infty = SL(2,\ZZ_N) / \bar \Gamma _\infty
\eea
where $\bar \Gamma _\infty = \{ \pm \left ( \begin{smallmatrix} 1 & b \cr 0 & 1 \end{smallmatrix} \right ), b \in \ZZ_N \}$, and therefore the number of cusps is given by,
\bea
\label{6.cusp1}
\ep _\infty (\Gamma(N)) = |SL(2,\ZZ_N) |/ |\bar \Gamma_\infty| = { d_N \over N}
\eea
in view of (\ref{6.dN}) and the fact that  $|\bar \Gamma_\infty| = 2N$.

\sm

The case of an arbitrary congruence subgroup $\Gamma$ may similarly be treated with the double coset construction, and we have,
\bea
\ep_\infty (\Gamma) = | \Gamma \backslash SL(2,\ZZ) / \Gamma _\infty |
\eea
The difference with the case of $\Gamma(N)$ is that an arbitrary congruence subgroup $\Gamma$ is not a normal subgroup of $SL(2,\ZZ)$, and this is indeed the case for $\Gamma_1(N)$ and $\Gamma_0(N)$. The calculation of the number of cusps in these cases is more involved and we refer to [11, 13] for a detailed exposition and here only quote the result,
\bea
\label{6.cusp2}
N \geq 5 & \hskip 0.3in & \ep_\infty (\Gamma _1(N)) = \half \sum _{ad=N} \phi(a) \phi(d)
\no \\
N \geq 3 & \hskip 0.3in & \ep_\infty (\Gamma _0(N)) =  \sum _{ad=N} \phi \big ( \gcd(a,d) \big )
\eea
where the sums are over $a,d >0$ and furthermore,
\bea
\label{6.cusp3}
\ep_\infty(\Gamma_1(2)) = \ep_\infty(\Gamma_1(3)) = \ep_\infty(\Gamma_0(2)) = 2
\hskip 1in
\ep_\infty(\Gamma_1(4)) = 3
\eea
This concludes our calculation of $d, \ep_2, \ep_3$ and $ \ep_\infty$, from which the genus of $X(\Gamma)$ may be computed using (\ref{6.genus}).

\begin{table}[htp]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$N$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13  \\ \hline \hline
$d_N/N$ & ${3\over 2}$ & 4 & 6 & 12 & 12 & 24 & 24 & 36 & 36 & 60 & 48 & 84  \\ \hline
$g \big (X(N) \big )$ & 0  & 0 & 0 & 0 & 1 & 3 & 5 & 10 & 13 & 26 & 25 & 50  \\ \hline
$g \big (X_1(N) \big )$ &  0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  & 0 & 1 & 0 & 2  \\ \hline
$g \big (X_0(N) \big )$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\
$\ep_2,\ep_3$ & $1,0$ & $0,1$& $0,0$& $2,0$& $0,0$& $0,2$& $0,0$& $0,0$& $2,0$ & $0,0$& $0,0$& $2,2$ \\ \hline
\end{tabular}
\end{center}
%\caption{Genera for $X(N)$, $X_1(N)$, $X_0(N)$ for $2\leq N \leq 13$ and elliptic points in $X_0(N)$.\label{tab:6.1}}
\end{table}

\begin{table}[htp]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$N$                               & 14 	& 15 		& 16 		& 17 		& 18 		& 19 		& 20 		& 21 		& 22 		& 23 		& 24		& 25   \\ \hline \hline
$d_N/N$                       & 72  	& 96 		&  96		&  144	&  108	&  180	&  144	&  192	&  180	&  264	&  192  	& 300 \\ \hline
$g \big (X(N) \big )$      & 49  	& 73		&  81		& 133  	&  109	&  196	&  169	&  241	&  241	&  375	&  289   	& 476 \\ \hline
$g \big (X_1(N) \big )$  & 1  	&  1		&  2		&  5		&  2		&  7		&  3		&   5		&  6		&  12		&  5  		& 12 \\ \hline
$g \big (X_0(N) \big )$  & 1 	&  1		&  0		&  1		&  0		&  1		&  1		&  1		&  2		&  2		&  1  		& 0 \\
$\ep_2,\ep_3$              & $0,0$ & $0,0$	& $0,0$	& $2,0$	& $0,0$	& $0,2$	& $0,0$	& $0,2$	& $0,0$ 	& $0,0$	& $0,0$ 	& 2,0 \\ \hline
\end{tabular}
\end{center}
\caption{\textit{Genera for $X(N)$, $X_1(N)$, $X_0(N)$ for $2\leq N \leq 25$ and the number of elliptic points $\ep_2$ and $\ep_3$ in $X_0(N)$.\label{tab:6.1}}}
\label{tab:generamodcurve}
\end{table}

\subsection{Explicit expressions for the genus}

The formulas for the genus of $X(\Gamma)$ are particularly simple for the cases when no elliptic points occur, as specified in Theorem \ref{5.thm:1}. In these cases the genus is obtained by combining (\ref{6.genus}) with the formula for the degree in (\ref{6.degreea}) with the expressions of the indices given in (\ref{6.index1}), (\ref{6.index2}), and the formulas for the number of cusps in (\ref{6.cusp1}), (\ref{6.cusp2}), (\ref{6.cusp3}),
\bea
N \geq 3: & \hskip 0.2in & g (X(N)) = 1+ { d_N (N-6) \over 12 N}
\\
N \geq 5: & & g(X_1(N)) = 1 + { d_N \over 12 N} - {1  \over 4} \sum_{ad=N} \phi(a) \phi(d)
\no \\
N\geq 2: & & g(X_0(N)) = 1 + { d_N \over 6 N \phi(N)} -{\ep_2(\Gamma_0(N)) \over 4}
-{\ep_3(\Gamma_0(N)) \over 3} - {1  \over 2} \sum_{ad=N} \phi \big ( \gcd(a,d) \big )
\no
\eea
where $\ep_2(\Gamma_0(N))$ and $\ep_3(\Gamma_0(N))$ were given in (\ref{6.ep2}) and (\ref{6.ep3}) respectively. For low values of $N$, we obtain the expressions for $d_N$ and the genera given in Table \ref{tab:generamodcurve}.

\subsection*{$\bullet$ Bibliographical notes}

The books by Shimura~[11] and by Diamond and Shurman~[13] stand out for their clear and useful introductions to congruence subgroups and modular curves. Our summary here closely follows their presentation.

\newpage

\section{Modular forms for congruence subgroups}
\setcounter{equation}{0}
\label{sec:Forms}

In this section, we shall discuss the structure and the counting of modular forms  for an arbitrary congruence subgroup $\Gamma$, and present the corresponding Eisenstein series for $\Gamma$. These modular forms are maps from the modular curves $X(\Gamma)$ to spaces of meromorphic differentials. The modular curves are compact Riemann surfaces of arbitrary genus, and the spaces of meromorphic differentials may be viewed as spaces of meromorphic sections of holomorphic line bundles of the Riemann surface. We refer to appendix \ref{sec:LB} for a review of holomorphic line bundles on compact Riemann surfaces, the Riemann-Roch theorem,  various vanishing theorems, and dimension formulas for the spaces of meromorphic sections. We shall then give, without proof, the general formula for the dimensions of modular forms and cusp forms for arbitrary $\Gamma$ and provide concrete examples for the cases of the standard congruence subgroups.

\subsection{Modular forms and cusp forms with respect to $\Gamma$}

One defines modular functions, modular forms, and cusp forms  for an arbitrary  congruence subgroup $\Gamma$ by analogy with $SL(2,\ZZ)$.  Let $f(\tau)$  be a meromorphic function  on the upper half-plane $\cH$ which obeys the following transformation law,
\bea
\label{6.gamma}
f(\gamma \tau) = (c \tau+d)^k f(\tau)
\hskip 1in
\gamma = \left ( \bma a & b \cr c & d \cr \ema \right )
\eea
for some integer $k \in \ZZ$, and all $\gamma \in \Gamma$.
Since $\Gamma$ is a congruence subgroup, there exists an integer $N$ such that $\Gamma (N) \subset \Gamma$, and hence a minimum integer $h$ which divides $N$ such that $T^h \in \Gamma$. As a result, $f$ is invariant under $\tau \to \tau + h$ and  admits a Fourier expansion,
\bea
f(\tau) = \sum _{n\in \ZZ} a_n q_h^n
\hskip 1in
q_h = e^{2 \pi i \tau/h}
\eea
\begin{enumerate}
\itemsep -0.02in
\item \textit{$f$ is a modular function of weight $k$} if $f$ is meromorphic at every cusp of $\Gamma \backslash \cH$, namely if the Fourier expansion of $f(\gamma \tau)$  has only a finite number of non-vanishing Fourier coefficients $a_n$ with negative $n$ for every $\gamma \in SL(2,\ZZ)$;
\item \textit{$f$ is a modular form of weight $k$} if it is holomorphic in $\cH$ and at every cusp of $\Gamma \backslash \cH$, namely the Fourier coefficients $a_n$ of $f(\gamma \tau)$  vanish for all negative $n$ for every $\gamma \in SL(2,\ZZ)$, and thus $f$ is finite at all cups;
\item \textit{$f$ is a cusp form of weight $k$} if it is a modular form  and vanishes at all cusps.
\end{enumerate}
The spaces of modular and cusp forms of weight $k$ are denoted $\cM_k(\Gamma)$ and $\cS_k(\Gamma)$ respectively. Clearly, the smaller the subgroup, the more modular forms it will have,
\bea
\Gamma' \subset \Gamma \hskip 0.5in
\left \{ \bma \cM_k(\Gamma) \subset \cM_k (\Gamma') \cr  \cS_k(\Gamma) \subset \cS_k (\Gamma') \ema \right .
\eea
In particular, a congruence subgroup $\Gamma$ of level $N$ contains $\Gamma(N)$ so that $\cM_k(\Gamma) \subset \cM_k(\Gamma(N))$ and similarly for cusp forms.

\subsubsection{Modular forms of odd weight}

For the full modular group $SL(2,\ZZ)$ there are no non-zero modular forms of odd weight $k$ because the transformation law (\ref{6.gamma}) for the element $-I \in SL(2,\ZZ)$ shows that $f(\tau)=0$. The same conclusion holds for any congruence subgroup $\Gamma$ that contains $-I$. The congruence subgroups $\Gamma(2), \Gamma_1(2)=\Gamma_0(2)$ and $\Gamma_0(N)$ for every $N$  contain $-I$, and thus have no modular forms of odd weight $k$, while $\Gamma = \Gamma_1(N), \Gamma_0(N)$ for $N >2$ do not contain $-I$ so that the corresponding spaces $\cM_k(\Gamma)$ and $\cS_k(\Gamma)$ for $k$ odd are not necessarily trivial.

\subsubsection{Comments on terminology}
\label{sec:terminology}

The terminology for automorphic functions and automorphic forms and its relation with the terminology for modular functions and modular forms is not entirely standard in the Mathematics literature, and may lead to some confusion, which we shall eliminate here with the help of precise definitions.

\sm

Automorphic functions and forms may be defined  generally in terms of functions with specific transformation properties under a discrete group acting on a complex manifold. Here, we shall limit attention to the case where the complex manifold is the Poincar\'e upper half-plane $\cH$ and the discrete group $\Gamma $ is  a Fuchsian subgroup of $SL(2,\RR)$, acting on $\cH$ by M\"obius transformations (see appendix \ref{sec:RS} for the  definition of Fuchsian groups).

\sm

In the complex analytic category, following Shimura [11] and Diamond and Shurman [13], an \textit{automorphic function} $f$ with respect to a Fuchsian group $\Gamma$ is a meromorphic  function on $\cH$ that is invariant under all $\gamma \in \Gamma$,
\bea
f(\gamma \tau) = f(\tau)
\hskip 1in
\hbox{ for all } \gamma = \left ( \bma a & b \cr c & d \cr \ema \right )
\eea
and is meromorphic at the cusps of $\Gamma$ in $\cH$. An \textit{automorphic form of weight $k$} with respect to $\Gamma$  is a meromorphic function on $\cH$ that instead transforms under all $\gamma \in \Gamma$ with a non-trivial automorphy factor $(c \tau+d)^k$,
\bea
\label{6.Gamma}
f(\gamma \tau) = (c \tau+d)^k f(\tau)
\eea
Equivalently, we may view an automorphic form $f$ of weight $k$ as a \textit{differential $k/2$-form}
\bea
\mf = f(\tau) (d\tau)^{\tfrac{k}{2}}
\eea
which is invariant under $\Gamma$, in analogy with the approach taken in subsection \ref{3.modif}. The nomenclature of a \textit{form} is particularly suited here, and familiar to a physics audience.

\sm

In the category of non-holomorphic functions, following Terras [20], we may define an automorphic function as a complex-valued function of $\cH$ that is invariant under $\Gamma$, thereby generalizing the notion from the meromorphic case. From the point of view of a physicist, it is natural to generalize the notion of an automorphic form to the non-holomorphic category by defining an automorphic form of weight $(k,\ell)$ to transform under all $\gamma \in \Gamma $ by,
\bea
f(\gamma \tau) = (c \tau+d)^k (c\bar \tau + d)^\ell f(\tau)
\eea
corresponding to a differential form $\mf$ of weight $(\tfrac{k}{2}, \tfrac{\ell}{2})$,
\bea
\mf = f(\tau) (d\tau)^{\tfrac{k}{2}} (d \bar \tau)^{\tfrac{\ell}{2}}
\eea
which is strictly invariant under $\Gamma$. In these notes, unless the meaning is clear from the context,  we shall add qualifiers, such as \textit{holomorphic automorphic forms}, \textit{meromorphic automorphic forms}, or more generally \textit{non-holomorphic automorphic forms}.

\sm

In the special case where $\Gamma = SL(2,\ZZ)$ or a congruence subgroup thereof, one refers to modular functions and forms instead of automorphic functions or forms. Again, unless the meaning is clear from the context,  these terms will be accompanied by qualifiers, \textit{holomorphic modular forms} such as the holomorphic Eisenstein series $\HE_k(\tau)$, \textit{meromorphic modular functions} such as the $j(\tau)$-function, \textit{meromorphic modular forms} such as the inverse discriminant $\Delta (\tau)^{-1}$, and \textit{non-holomorphic modular functions} such as the \textit{non-holomorphic Eisenstein series} $E_s(\tau)$ and \textit{modular graph functions}.

\subsection{Holomorphic Eisenstein series for $\Gamma(N)$}

The holomorphic Eisenstein series of integer weight $k \geq 3$ was defined in (\ref{EisenG})  by,
\bea
G_k (\tau) = \sum  _{m,n \in \ZZ} ' { 1 \over ( m \tau + n)^k}
\eea
where the prime on the sum instructs us to omit any contribution that makes the summand singular, in this case just $(m,n) = (0,0)$. Clearly $G_k(\tau)$ vanishes  for odd $k$, and is a modular form of weight $k$ for even $k$, transforming as follows,
\bea
G_k (\gamma \tau)  = (c \tau+d)^k G_k(\tau)
\hskip 0.7in
\gamma = \left ( \bma a & b \cr c & d \cr \ema \right ) \in SL(2,\ZZ)
\eea
The ring of modular forms of $SL(2,\ZZ)$ is polynomial and generated by $G_4$ and $G_6$.

\sm

To construct modular forms for congruence subgroup $\Gamma (N)$ with $N \geq 2$ we parametrize the elements of the group $\Gamma(N) \backslash SL(2,\ZZ) = (\ZZ_N)^2$ by a row vector  $v=(\mu ~  \nu) $ whose entries satisfy $\mu, \nu \in \{ 0,1, \cdots N-1\} = \ZZ_N$,  and introduce the following Kronecker-Eisenstein sum,\footnote{When no confusion is expected to arise we shall sometimes abbreviate $(\mod N)$ as simply $(N)$.}
\bea
G _k (v | \tau) = \sum _{m,n\in \ZZ}' { 1 \over \big ( (mN+\mu)\tau + nN+\nu \big )^k}
= \sum_{(m,n) \equiv v \, (N)} ' { 1 \over (m\tau+n)^k}
\eea
For $v \not \equiv 0 ~ (\mod N)$ none of the denominators vanish and the prime on the sum may be omitted.
The double sum is absolutely convergent for integer $k \geq 3$, while for $k=1,2$ it may be defined by the Eisenstein summation convention,
\bea
G _k (v | \tau) = \sum _{n \equiv \nu \, (N)} ' { 1 \over (\mu \tau + n)^k} + \sum _{{m \not =0  \atop m \equiv \mu \, (N)}}
\sum_{n \equiv \nu \, (N)} { 1 \over (m \tau + n)^k}
\eea
Under a transformation $ \gamma \in SL(2,\ZZ)$, we have,
\bea
G_k ( v \gamma | \g \tau) = (c \tau +d)^k G_k (v | \tau)
\eea
where the ingredients are defined as follows with $a,b,c,d \in \ZZ$ and $ad-bc=1$,
\bea
\gamma = \left ( \bma a & b \cr c & d \cr \ema \right )
\hskip 0.8in
\g \tau = { a \tau + b \over c \tau +d}
\hskip 0.8in
v \gamma  = (a\mu+c \nu, b \mu + d \nu)
\eea
Under the action of $SL(2,\ZZ)$ a row vector $v$ may be mapped into itself or into different row vector. For example, $(0 ~ 0)$ maps to itself under any $\gamma \in SL(2,\ZZ)$, while the row $(0 ~ \nu_0)$ for $\nu_0 \not \equiv 0 ~ (\mod N)$  maps to $(\mu ~ \nu) = (c \nu_0 ~ d \nu_0) ~ (\mod N)$. Conversely,  given a row $v=(\mu ~ \nu)$ with $\gcd(\gcd(\mu,\nu), N) =1$, then there exists a $\g \in SL(2,\ZZ)$ which maps $v$ to $(0,\nu_0)$ with $\nu_0 = \gcd(\mu, \nu) ~ (\mod N)$.

\subsubsection{Invariance under $\Gamma (N)$}

A transformation $\gamma \in SL(2,\ZZ)$ maps  $G_k(v| \tau)$ to a multiple of itself provided $v$ satisfies $v \gamma - v \equiv 0 ~ (\mod N)$, or more explicitly,
\bea
(a-1) \mu + c \nu & \equiv & 0 ~ (\mod N)
\no \\
b \mu + (d-1) \nu & \equiv & 0 ~ (\mod N)
\eea
This condition is realized for an arbitrary $\gamma \in \Gamma(N)$ and an arbitrary pair $(\mu ~ \nu) \in (\ZZ_N)^2$.  A given pair  $v=(\mu ~ \nu)$ may be invariant under a subgroup of $SL(2,\ZZ)$ that is larger than $\Gamma(N)$. For example $(0 ~ 0)$ is invariant under all of $SL(2,\ZZ)$. But if we require that all $v \in (\ZZ_N)^2$ be invariant then the largest invariance group is $\Gamma (N)$. To see this, it suffices to choose $(\mu ~ \nu) = (1,0) ~ (\mod N)$ and $(\mu ~ \nu) = (0,1) ~ (\mod N)$ to obtain the conditions $a,d \equiv 1 ~ (\mod N)$ and $b,c \equiv 0 ~ (\mod N)$ so that $\gamma \in \Gamma (N)$.

\subsubsection{Asymptotics near the cusps}

The cusps of $\Gamma(N)$ are given by the double coset,
\bea
\Gamma(N) \backslash SL(2,\ZZ)/ \Gamma _\infty
\eea
acting on $i \infty$, namely  by all the images under $SL(2,\ZZ)$ of the cusp at infinity modulo equivalence under $\Gamma (N)$. The number of cusps is denoted $\ep_\infty (\Gamma (N))$ and was given in (\ref{6.cusp1}). Since every cusp  may be mapped to $\infty$ under  a suitable $\gamma \in SL(2,\ZZ)$, under which $G_k(v|\tau)$ maps to $G_k(v\gamma | \gamma \tau)$, it suffices to examine the asymptotics of $G_k(v|\tau)$ at the cusp $\infty$, the asymptotics at the other cusps being deduced by mapping back with $\gamma ^{-1}$.

\sm

The asymptotics of $G_k(v|\tau)$ is qualitatively different for the cases where $\mu \equiv 0 ~ (\mod N)$ or $\mu \not \equiv 0 ~ (\mod N)$.  For $\mu \not\equiv0 ~ (\mod N)$ the limit at the cusp $\tau \to i\infty$ vanishes,
\bea
\lim _{\tau \to \infty} G_k(v| \tau) = 0
\eea
Of course, $G_k(v|\tau)$ will in general be non-vanishing at cusps other than $i\infty$.
If $\mu \equiv 0 ~ (\mod N)$ and $\nu \not \equiv 0 ~ (\mod N)$, then the limit at the cusp $\tau \to i\infty$ is given by,
\bea
\lim _{\tau \to i\infty} G_k(v|\tau)  = \sum_{n \equiv \nu \, (\mod N)} { 1 \over n ^k}  = { 1 \over N^k} \Big ( \zeta (k, \tfrac{\nu}{N}) + (-)^k \zeta (k, 1-\tfrac{\nu}{N}) \Big )
\eea
It is shown in [13] that the Eisenstein series form a basis  for the Eisenstein space, which is defined to be the quotient,
\bea
\cE_k \big ( \Gamma(N) \big ) = \cM_k \big ( \Gamma(N) \big )/\cS_k \big ( \Gamma(N) \big )
\eea
where $ \cM_k(\Gamma(N))$ is the vector space of all holomorphic modular forms of weight $k$ and $\cS_k(\Gamma(N))$ is the sub-space of cusp forms. For even $k$, the dimension of the Eisenstein space is given by the number of cusps $\ep_\infty (\Gamma (N))$,
\bea
\dim \cE_k \big ( \Gamma(N) \big )  = \ep _\infty \big ( \Gamma (N) \big )
\eea
with $\ep _\infty (\Gamma (N))$ given in section \ref{sec:numbercusps}. One may consider  linear combinations of the forms $G_k(v|\tau)$ which produce  normalized Eisenstein series for $\Gamma(N)$, defined by,
\bea
\HE_k(v|\tau) = \sum _{(c,d) \equiv v \, (N) \atop \gcd(c,d)=1} { 1 \over (c \tau +d)^k}
\eea
For even $k$ the limit of $\HE_k(v|\tau)$ vanishes at all the cusps of $\Gamma(N)$ except for the cusp corresponding to $v$.

\subsection{Holomorphic Eisenstein series for $\Gamma_1(N)$ and $\Gamma _0(N)$}

The construction of holomorphic Eisenstein series for arbitrary congruence subgroups uses Dirichlet characters to symmetrize the Eisenstein series for $\Gamma(N)$. As discussed in more detail in appendix \ref{sec:Lseries}, a Dirichlet character $\chi_a$ for $a \in \NN$ is a map $\chi_a:\ZZ \rightarrow \CC$ which is periodic (i.e. $\chi_a(n) = \chi_a(n+a)$), completely multiplicative (i.e. $\chi_a(nm) = \chi_a (n) \chi_a(m)$), and which is non-vanishing only when $a$ and the argument $n$ are coprime.

\sm

For any positive integer $N$ and integer $k \geq 3$, we define the set $\cC_{N,k}$ of triples $(\chi_{a_1}, \widetilde{\chi}_{a_2}, t)$ with $\chi_{a_1}$ and $\tilde{\chi}_{a_2}$ (primitive) Dirichlet characters and $t \in \NN$,
\bea
\cC_{N,k} := \left\{(\chi_{a_1}, \widetilde{\chi}_{a_2}, t) \, \big| \, \chi_{a_1}(-1)  \widetilde\chi_{a_2}(-1) = (-1)^k \,\,\,\,\mathrm{and}\,\,\,\, t a_1 a_2 \, | N \right\}
\eea
In terms of Dirichlet characters, we also define the generalized divisor sums
\bea
\sigma_m^{\chi_{a_1}, \widetilde \chi_{a_2}}(n) = \sum_{d|n} \chi_{a_1}(n/d) \widetilde\chi_{a_2}(d) d^{m}
\eea
as well as the delta function and Dirichlet $L$-series,
\bea
\delta(\chi_{a_1}) = \left\{ \begin{matrix} 1 && \chi_{a_1} = 1~, \\ 0 && \chi_{a_1} \neq 1~, \end{matrix} \right. \hspace{0.8 in}L(\widetilde \chi_{a_2}, s) = \sum_{n=1}^\infty {\widetilde \chi_{a_2}(n)\over n^s}~
\eea
In terms of these ingredients, we may then construct the following functions
\bea
\HE_k^{\chi_{a_1}, \chi_{a_2},t}(q) = \delta(\chi_1) + {2 \over L(\widetilde{\chi}_{a_2}, 1-k)}\sum_{n=1}^\infty \sigma_{k-1}^{\chi_{a_1}, \widetilde\chi_{a_2}}(n) q^{tn}
\eea
When $\chi_{a_1}=\widetilde \chi_{a_2}= 1$ and $k \geq 4$, the generalized divisor sums reduce to the usual divisor sums, and the $L$-series reduces to the Riemann-zeta function. In this case the functions $\HE_k^{\chi_{a_1}, \chi_{a_2},t}(q)$ reduce to the usual Eisenstein series for $SL(2,\ZZ)$ at shifted values of $q$, i.e. $\HE_k^{1,1,t}(q) = \HE_k(q^t)$. It is clear that for any $t |N$ the functions $\HE_k(q^t)$ should be Eisenstein series for $\Gamma_1(N)$. What is less obvious is the following theorem, which we quote without proof,
\label{thm:Gamma1basis}
The space of weight-$k$ holomorphic Eisenstein series for $\Gamma_1(N)$ is generated by $\HE_k^{\chi_{a_1}, \chi_{a_2},t}(q)$ for $({\chi_{a_1}, \widetilde\chi_{a_2},t}) \in \cC_{N,k}$, i.e.
\bea
\cE_k(\Gamma_1(N)) = \left\langle \HE_k^{\chi_{a_1}, \widetilde \chi_{a_2},t} \,\,\,\, \mathrm{for}\,\,\mathrm{all}\,\,\,\, ({\chi_{a_1}, \widetilde\chi_{a_2},t}) \in \cC_{N,k}  \right\rangle
\eea
for $k\geq 3$. For $k=2$, we have
\bea
\label{eq:E2Gamma1Nres}
\cE_2(\Gamma_1(N)) = \left\{\begin{matrix} \left\langle \HE_2(q) - t \HE_2(q^t) \,\,\,\, \mathrm{for}\,\,\mathrm{all}\,\,\,\, t |N\right\rangle&& \chi_{a_1}= \widetilde \chi_{a_2}=1 \\  \left\langle \HE_2^{\chi_{a_1}, \widetilde \chi_{a_2},t} \,\,\,\, \mathrm{for}\,\,\mathrm{all}\,\,\,\, ({\chi_{a_1}, \widetilde\chi_{a_2},t}) \in \cC_{N,2}  \right\rangle && \mathrm{otherwise}  \end{matrix} \right.
\eea
The proof can be found in [13, 64].
Note the qualitative distinction between the cases of $k=2$ and $k>2$, which results from the fact that there is no weight-2 modular function for $SL(2,\ZZ)$. The first line of (\ref{eq:E2Gamma1Nres}) is indeed consistent with the triviality of the space of weight-2 modular forms for $N=1$ (and hence $\chi_{a_1}= \widetilde \chi_{a_2}=t=1$).

\sm

We now briefly discuss the case of $\Gamma_0(N)$. First note that if $\chi_{a_1}$ and $\widetilde \chi_{a_2}$ are two Dirichlet characters and $A$ is a common multiple of $a_1$ and $a_2$, then it is possible to define a modulo $A$ Dirichlet character $\widehat \chi_A = \chi_{a_1}^{(A)} \widetilde \chi_{a_2}^{(A)}$, where
\bea
\chi_a^{(A)}(n) := \left\{ \begin{matrix} \chi_a(n) && (n,A) = 1 \\ 0 && (n,A)>1 \end{matrix} \right.
\eea
We may now further define the following subgroups of $\cE_k(\Gamma_1(N)) $,
\bea
\cE_k(N, \widehat \chi_N) = \left\langle \HE_k^{\chi_{a_1}, \chi_{a_2},t} \,\,\,\, \mathrm{for}\,\,\mathrm{all}\,\,\,\,  ({\chi_{a_1}, \widetilde \chi_{a_2},t}) \in \cC_{N,k}\,\,\,\,\mathrm{and}\,\,\,\, \chi_{a_1}^{(N)} \widetilde \chi_{a_2}^{(N)} = \widehat \chi_N  \right\rangle
\eea
such that $\cE_k(\Gamma_1(N)) = \bigoplus_{\widehat \chi_N}\cE_k(N, \widehat \chi_N) $. The quantity $\widehat \chi_N$ is referred to as the ``nebentypus."  The space of modular forms of $\Gamma_0(N)$ is then simply obtained by restricting to the sector with trivial nebentypus, i.e.
\bea
\label{eq:Gamma0Nbasis}
\cE_k(\Gamma_0(N))  = \cE_k(N, 1)
\eea

Let us close with a note for the practical researcher. Holomorphic Eisenstein series for various congruence subgroups, along with many of their properties, can be straightforwardly implemented in the computer program SageMath [65]. For example, the simple code
\bea
&\vphantom{.}&> \mathtt{E=EisensteinForms(Gamma0(9),4,prec=10)}\no\\
\eea
can be used to output the Fourier expansions to order $O(q^{10})$ for all generators of weight-4 Eisenstein series of $\Gamma_0(9)$. The triplets $(\chi_{a_1}, \widetilde{\chi}_{a_2}, t)$ labelling each generator can be further obtained using
\bea
&\vphantom{.}&> \mathtt{e=E.eisenstein\_series()}\no\\
\eea
In practice, using such computer algebra programs to generate the spaces of modular forms at a given level can be much simpler than working through the definitions above.

\subsection{Jacobi's theorem on sums of four squares}

We now provide a classic application of modular forms for congruence subgroups, here associated with $\Gamma _0(4)$. For any positive integer $n$, we denote by $r_k(n)$ the number of ways in which $n$ can be written as the sum of $k$ positive, zero, or negative integers squared,
\bea
r_k(n) = \# \left \{ m_1 , \cdots, m_k \in \ZZ ~ \hbox{such that} ~ n =  m_1^2 +\cdots + m_k^2  \right \}
\eea
The generating function for $r_k(n)$ in powers of $q= e^{2 \pi i \tau}$  is the function $\tet (0|2\tau)^k$,
\bea
\tet (0 |2 \tau)^k = \sum _{m_1, \cdots, m_k \in \ZZ}  q^{ m_1^2 + \cdots + m_k^2}
= \sum _{n=0} ^\infty r_k(n) q^n
\eea
Note that the argument of $\tet$ has been taken to be $ 2 \tau$  for convenience, so that the expansion is in terms of $q$ raised to integer powers. The function $\tet (0 |2 \tau)^k$ is invariant under $\tau \to \tau+1$ but it is not invariant under $\tau \to -1/\tau$ for $k \not=0$ and, therefore,  is not a modular form under $SL(2,\ZZ)$. However, it is a modular form under a certain congruence subgroup of $SL(2,\ZZ)$. We specialize to the case $k=4$, and propose to prove Jacobi's theorem,
\bea
\label{Jacobi}
r_4 (n) =
8 \sum _{d \,  | \, n, ~ 4 \, \nmid \, d } d
\eea
with the help of modular forms.
The sum is over positive divisors $d$ of $n$ which are not divisible by 4. Jacobi's theorem  implies the famous theorem of Lagrange  that every positive integer can be written as the sum of four squares, but Lagrange did not evaluate the number of ways in which this can be done.

\sm

We begin by identifying the modular subgroup under which $\tet (0|2 \tau)^4$ is a modular form. One generator may be taken to be $T : \tau \to \tau+1$. Under $\tau \to -1/\tau$ we have,
\bea
\tet (0|-2/\tau)^4 = (- i \tau/2)^2 \tet (0|\tau/2)^4
\eea
so that $\tet (0|2 \tau)^4$ clearly is not a modular form under $SL(2,\ZZ)$. Instead, we look for more general modular transformations $\gamma = \left(\begin{smallmatrix}a & b \\ 0 & d \end{smallmatrix}\right) \in SL(2,\ZZ)$ under which we obtain good transformation laws. A general $\gamma \in SL(2, \ZZ)$ may be simplified by using the fact that we have invariance under $T$, and this allows us to set $b=0$. But then we have $ad=1$ which is solved by $a=d=1$ and $a=d=-1$. These cases are equivalent to one another upon reversal of the sign of $c$, so that we consider the modular transformation,
\bea
\tau ' = \gamma \tau = { \tau \over c \tau +1}
\eea
under which we have,
\bea
\tet \left ( 0 |  2 \tau'  \right )^4
=  \left ( i \, { c \tau +1 \over 2 \tau} \right )^2 \tet \left ( 0 \Big | - { c \over 2 } - {1 \over 2 \tau} \right )^4
\eea
The shift by $-c/2$ cancels in view of $T$-invariance if and only if $c \equiv 0 ~ ({\rm mod} \, 4)$, which may be established by recalling that $\tet(z|\tau+1) = \tet_2(z|\tau)$ from (\ref{tetplusone}). Assuming now that $c \equiv 0 ~ ({\rm mod} \, 4)$ the  remaining argument $-1/(2\tau)$ of $\tet$ is  then  independent of $c$. Using $\tet(0|-1/(2\tau))^4 = - 4 \tau^2 \tet (0|2 \tau)^4$, we obtain,
\bea
\tet ( 0 | 2 \tau')^4  =  ( c \tau +1)^2 \tet (0  | 2 \tau)^4
\eea
We have now identified two transformations under which $\tet (0|2\tau)^4$  transforms as a modular form of weight 2, namely,
\bea
T: \tau \to \tau +1
\hskip 1in
\tilde S : \tau \to { \tau \over 4\tau+1}
\eea
These transformations may be used to generate the group of their compositions, which is $\Gamma _0 (4)$.
Therefore $\tet(0|2 \tau)^4$ is a weight-2 modular form under the congruence subgroup $\Gamma _0(4)$. Of course there are no modular forms of weight two under the full modular group $SL(2, \ZZ)$. So this is a simple example of the fact that, the ``smaller" the modular subgroup of $SL(2,\ZZ)$ is, the larger the number of modular forms it will admit for a given weight.

\sm

Now we would like to identify $\tet(0|2 \tau)^4$ with an Eisenstein series whose Fourier expansion we can compute directly in terms  of divisor functions $\sigma _\alpha (n)$. What are the modular forms of weight two? Let's start from $\HE_2(\tau)$ which is not a modular form under $SL(2,\ZZ)$, but rather a modular connection. Since the difference of two connections is a tensor, we investigate how $\HE_2 ( n \tau)$ transforms for various values of $n$. Under $T : \tau \to \tau +1$, the form  $\HE_2(n \tau)$ is invariant for any $n\in \NN$. Next, examine its transformation properties under the transformation
\bea
\tau \to \tau' = {\tau \over N \tau+1}
\eea
Since $\HE_2$ is a derivative of $\ln \Delta$ given by (\ref{3.E2Del}), we examine the transformation law of $\Delta$,
\bea
\Delta (n \tau') = \left ( - { N \tau +1 \over n \tau} \right )^{12} \Delta \left ( - {N \over n}  - {1 \over n \tau} \right )
\eea
When $n | N$, the combination $-N/n$ in the argument of $\Delta$ cancels by $T$-symmetry. Taking the transform of this simplified result, we obtain after some simplifications,
\bea
\Delta (n \tau') = ( N \tau +1)^{12}  \Delta (n \tau)
\eea
Thus, for $n | N$, the function $\Delta (n\tau)$ is a weight 12 modular form under $\Gamma _0(N)$. Applying this general result to the sum of four squares problem we set $N=4$, so that $\Delta (\tau), \Delta (2\tau), \Delta (4\tau)$ are modular forms of $\Gamma _0(4)$. Taking the derivatives of the logarithms on both sides, and expressing the result in terms of $\HE_2$, we find,
\bea
\HE_2 (n \tau') = ( 4 \tau +1)^2 \HE_2(n\tau) + { 4 \over n} \times { 12 \over 2 \pi i} (4 \tau +1)
\eea
It follows that the two linear combinations,
\bea
\HE_2 (\tau) - 2\,  \HE_2 (2 \tau)
\hskip 1in
\HE_2 (2 \tau) - 2\,  \HE_2 (4 \tau)
\eea
are linearly independent modular forms under $\Gamma _0 (4)$ of weight 2. This matches with the results from (\ref{eq:Gamma0Nbasis}) and Theorem \ref{thm:Gamma1basis}.

\sm

We can alternatively count the number of modular forms of weight two under $\Gamma _0(4)$ by the same arguments we used for $SL(2,\ZZ)$. The difference is that the fundamental domain for $\Gamma _0(4)$ is four times larger than the fundamental domain for $SL(2,\ZZ)$, so we would need to derive an extension of our previous result (\ref{thm1a}) to $\Gamma _0(4)$. We shall do so soon, but for the time being will simply invoke Theorem \ref{thm:Gamma1basis} to claim that the space has dimension two. The above combinations generate this space completely, and we must have,
\bea
\tet (0|2 \tau)^4 = \alpha \Big ( \HE_2 (\tau) - 2\, \HE_2 (2 \tau) \Big )
+ \beta \Big ( \HE_2 (2 \tau) - 2\, \HE_2 (4 \tau) \Big )
\eea
for some rational coefficients $\alpha, \beta$. Using the $q$-series representation for $\HE_2$,
\bea
\HE_2 (n \tau) = 1 - 24 \sum _{\ell=1}^\infty \sigma _1(\ell) q^{n \ell}
\eea
and matching the $q^0$ and $q^1$ terms requires $ 1 = -\alpha - \beta$ and $ 1 =  - 3 \alpha$ respectively so that,
\bea
\tet (0|2 \tau)^4 = -{ 1 \over 3} \, \HE_2 (\tau)  + { 4 \over 3}  \, \HE_2 (4 \tau)
= 1 + 8 \sum _{\ell=1}^\infty \sigma _1(\ell) q^\ell - 32 \sum _{\ell=1}^\infty \sigma _1(\ell) q^{ 4 \ell}
\eea
Jacobi's theorem (\ref{Jacobi}) now readily follows.

\subsection*{$\bullet$ Bibliographical notes}

The books by Shimura~[11] and by Diamond and Shurman~[13] provide  useful introductions  for  modular forms for congruence subgroups. We also refer to the book by Miyake~[64] and the software data base [65]. A clear account of Jacobi's sum of four squares theorem may be found in Apostol's book~[8]. Our summary  closely follows their presentation. Essays geared towards applications to the proof of Fermat's Last Theorem are collected in [66].

\newpage

\section{Modular differential equations and vector-valued \\ modular forms}
\setcounter{equation}{0}
\label{sec:MDEsvvmfs}

In this section, we construct differential equations in the modular parameter $\tau$, and find solutions to these equations in simple cases. The solutions can generically be assembled into ``vector-valued modular forms," which have proven fruitful in recent works in Mathematics and Physics. In fact, we will see that in general the components of a vector-valued modular forms are modular forms for congruence subgroups, discussed in the previous section.

\subsection{Modular covariant derivatives}

In section \ref{E2section} we encountered the holomorphic function  $\HE_2(\tau)$, whose transformation under $SL(2,\ZZ)$ is that of a modular connection, as given in (\ref{E2transfs}). We may use this object to define a modular covariant differential operator, acting on modular forms of weight $k$,
\bea
D_k = {1 \over 2 \pi i}{d \over d \tau} - {k \over 12 } \HE_2(\tau)
\eea
The operator $D_k$ maps a modular form of weight $k$ to a modular form of weight $(k+2)$,
\bea
D_k : \cM_k \rightarrow \cM_{k+2}
\eea
as was already shown in section \ref{sec:introtodiffeq}.

\iffalse
To show this we apply the derivative to an arbitrary modular form $f(\tau)$ of weight $k$, and  apply a modular transformation using the transformations for $f$ and the derivative in $\tau$,
\bea
(D_k f) \left(  {a \tau + b \over c \tau + d}\right) = \left({1 \over 2 \pi i} (c \tau + d)^2{d \over d \tau} - {k \over 12}\HE_2\left( {a \tau + b \over c \tau + d}\right) \right) (c \tau + d)^k f(\tau)
\eea
where $a,b,c,d \in \ZZ$ with $ad-bc=1$.
Using transformation properties of $\HE_2(\tau)$ in (\ref{E2transfs}), we see that the contribution generated by the derivative of the factor $(c \tau+d)^k$ is precisely cancelled by the inhomogeneous part of the transformation rule of $E_2$. Regrouping terms after this simplification, we obtain,
\bea
(D_k f) \left(  {a \tau + b \over c \tau + d}\right)
= (c \tau + d)^{k+2} (D_k f) (\tau)
\eea
which is the transformation property of a weight-$(k+2)$ modular form.
\fi

\sm

Note that the first order differential operators introduced above can be composed to define higher-order modular covariant derivatives,
\bea
\cD^{(d)}_k = \prod_{s=1}^d D_{k+2 s -2}~,\hspace{0.6 in}\cD^{(d)}_k : \cM_k \rightarrow \cM_{k+2 d}
\eea

For example, the first order differential operators on Eisenstein series give,
\bea
\label{eq:DonEk}
D_4 \HE_4(\tau) = - {1 \over 3} \HE_6 \hspace{0.8 in} D_6 \HE_6(\tau) = - {1 \over 2} \HE_4^2
\eea
while for the second order differential operators we have
\bea
\cD^{(2)}_4 \HE_4(\tau) ={1 \over 6} \HE_4^2\hspace{0.8 in}\cD^{(2)}_6 \HE_6(\tau) ={1 \over 3} \HE_4 \HE_6
\eea
The second order derivatives follow directly from the first order ones. To prove the former, we note that the spaces of weight $6$ and $8$ modular forms are generated by $\HE_6$ and $\HE_4^2$ respectively, so the results are fixed up to normalization. To determine the normalization, one can compare the first terms in the $q$-expansion using (\ref{Ektau}).

\subsection{Modular differential equations}
\label{sec:MDEintro}

Consider now an arbitrary linear differential equation of order $d$ in $\tau$ on a modular form $f(\tau)$ of weight $k$, referred to as a modular invariant differential equation (MDE),
\bea
\label{basicMDE}
\left[ \cD^{(d)}_k + \sum_{r = 1}^d h_{r}(\tau) \cD^{(d-r)}_k\right] f(\tau) = 0
\eea
In the above equation, $h_{r}(\tau)$ is an arbitrary meromorphic modular form of weight $2 r$ and $f(\tau)$ has weight $k$. The simplest MDEs are obtained by taking the $h_{r}(\tau)$ to be holomorphic, in which case they are given by polynomials in $\HE_4(\tau)$ and $\HE_6(\tau)$.

\subsubsection{First order MDE}

The generic first order MDE for modular weight $k$ takes the form,
\bea
D_k f + h_1 f=0
\eea
for a modular function $h_1$ of weight 2 and a modular function $f$ of weight $k$. Using the relation,
\bea
\HE_2 (\tau) = { 1 \over 2 \pi i } \p_\tau \ln \Delta (\tau)
\eea
we obtain a simplified equation for the modular function $f_0$ defined by,
\bea
f(\tau) = \Delta (\tau)^{\tfrac{k}{12}} f_0(\tau)
\hskip 1in
{ 1 \over 2 \pi i } { d f _0 \over d \tau} + h_1 (\tau) f_0(\tau)=0
\eea
Furthermore, since $h_1$ is a meromorphic function of weight 2 it may be expressed as $h_1(\tau) = {\HE_6(\tau)\over \HE_4(\tau)}  h_0(j)$, where $h_0$ is a modular function of $\tau$ which may be expressed as a function of $j(\tau)$.

Using the definition of $j(\tau)$ in terms of $\HE_4$ and $\HE_6$ given by $j= 12^3 \HE_4^3/(\HE_4^3-\HE_6^2)$, together with (\ref{eq:DonEk}), we obtain the derivative of $j(\tau)$ respect to $\tau$,
\bea
\label{10.jder}
{1 \over 2 \pi i} {dj \over d \tau} = - {\HE_6 \over \HE_4} \, j
\eea
which by the chain rule also gives
\bea
{ 1 \over 2 \pi i } { d f _0 \over d \tau} = - {\HE_6 \over \HE_4} \, j \, { d f_0(j) \over dj}
\eea
Cancelling a common factor of $\HE_6/\HE_4$, the equation for $f_0$ now becomes,
\bea
j \, { d \ln f_0(j) \over d j} = h_0(j)
\eea
which may be solved by quadrature. It remains to ensure that the local solution $f(\tau)$ obtained via this procedure is a proper modular form of weight $k$.  This requirement may place conditions on the allowed values of $k$.

\subsubsection{Second order MDE}
\label{sec:secondorderMDE}

Consider now the example of a second order MDE with holomorphic coefficient functions, acting on a weight-zero modular function. The most general such MDE takes the form,
\bea
[D_2 D_0 + \g \HE_4(\tau)] f(\tau) = 0
\eea
for some free constant parameter $\g$, or more explicitly,
\bea
\label{eq:explicit2OMDE}
\left[\left({1 \over 2 \pi i}{d \over d \tau} \right)^2  - {1 \over 12 \pi i} \HE_2(\tau) {d \over d \tau} + \g \HE_4(\tau)\right] f(\tau) = 0
\eea
The general solution to this equation may be obtained as follows. We begin by trading the derivatives with respect to $\tau$ for derivatives with respect to the $j$-function, as we had done for the first order MDE, using (\ref{10.jder}). It follows that
\bea
{df \over dj} = - {\HE_4 \over j \, \HE_6} {1 \over 2 \pi i} {df \over d \tau}
\eea
and by similar means,
\bea
{d^2f \over dj^2} = \left[\left({\HE_4 \over j \, \HE_6} \right)^2 \left(\left( {1 \over 2 \pi i} {d \over d \tau} \right)^2 - {\HE_2 \over 6} {1 \over 2 \pi i} {d \over d \tau} \right) - \left({7j - 6912  \over 6 j (j-1728)} \right) {d \over d j}\right] f
\eea
With these results the equation (\ref{eq:explicit2OMDE}) can be rewritten as,
\bea
\left[j (j-1728){d^2 \over dj^2} +{1 \over 6} (7j - 6912) {d \over dj} + \g \right] f(\tau(j)) = 0
\eea
In terms of the variable $j(\tau)/1728$, this equation is of standard hypergeometric type. For generic choice of $\g$, there exist two distinct solutions given in closed form by,\footnote{When $\mu \in 6 \, \NN$, the second solution is divergent, since the third argument of the hypergeometric function is a negative integer. In this case, the second solution should be replaced by the Meijer G-function,
\bea
f_2(\tau) = G^{2,0}_{2,2}\left(\begin{matrix} {2 \over 3} & 1 \\{1- \mu \over 12} & {1+\mu \over 12}\end{matrix}\,\, \Big| 1728 \, j(\tau)^{-1}\right)
\eea
\bea
\label{eq:2dexplicitsols}
f_1(\tau) &=& j(\tau)^{- \tfrac{(1+\mu)}{12}} {}_2 F_1\left(\tfrac{1 + \mu}{12}, \tfrac{5 + \mu}{12}\,;\, 1 + \tfrac{\mu}{6}\,; \, 1728 \, j(\tau)^{-1} \right)
\no\\
f_2(\tau) &=& j(\tau)^{-{(1-\mu) \over 12}} {}_2 F_1\left( \tfrac{1 - \mu}{12}, \tfrac{5 - \mu}{12}\,;\, 1 - \tfrac{\mu}{ 6}\,; \, 1728 \, j(\tau)^{-1} \right)
\eea
where we have defined $\mu = \sqrt{1 - 144 \g}$.

\iffalse The corresponding Fourier expansions are
\bea
f_1(q) &=& q^{(1+\mu) \over 12} \left[ 1+ {2 (1+\m)(5\m-6) \over \m+6} q + {(1+ \m) (50 \m^3 - 25 \m^2 + 54 \m -72) \over (\m +6)(\m+12)}q^2+ \dots  \right]
\no\\
f_2(q) &=&q^{(1-\mu) \over 12} \left[ 1+ {2 (1-\m)(5\m+6) \over \m-6} q -{(1- \m) (50 \m^3 + 25 \m^2 + 54 \m +72) \over (\m -6)(\m-12)}q^2+\dots  \right]
\no
\eea
\fi

\subsubsection{Third order MDE}
\label{sec:thirdorderMDE}

We may similarly analyze the case of a third order modular differential equation with holomorphic coefficient functions and with $f$ of weight 0. In this case the most general modular differential equation depends on two free parameters $a$ and $b$,
\bea
\label{3rdorderMDE}
\left[D_4 D_2 D_0 + a \HE_4(\tau) D_0 + b \HE_6(\tau) \right] f(\tau) = 0
\eea
By switching to the local coordinate $j(\tau)$ as before, we can again re-express this as a hypergeometric equation. For generic choices of parameters, there are three distinct solutions given by
\bea
f_1(\tau) &=& j^{- \tfrac{\m_1 + 1}{6} } \, {}_3 F_2 \left( \tfrac{\m_1 + 1}{6}, \tfrac{\m_1 + 3}{6}, \tfrac{\m_1 + 5}{6}; \tfrac{\m_1 - \m_2 }{6}+1, \tfrac{\m_1 - \m_3}{6}+1 ; 1728 \, j^{-1} \right)
\no\\
f_2(\tau) &=& j^{-{\m_2 + 1 \over 6} } \, {}_3 F_2 \left( \tfrac{\m_2 + 1}{6}, \tfrac{\m_2 + 3}{6}, \tfrac{\m_2 + 5}{6}; \tfrac{\m_2 - \m_1 }{6}+1, \tfrac{\m_2 - \m_3}{6}+1 ; 1728\, j^{-1} \right)
\no\\
f_3(\tau) &=& j^{-{\m_3 + 1 \over 6 }} \, {}_3 F_2 \left( \tfrac{\m_3 + 1}{6}, \tfrac{\m_3 + 3}{6}, \tfrac{\m_3 + 5}{6}; \tfrac{\m_3 - \m_2 }{6}+1, \tfrac{\m_3 - \m_1}{6}+1 ; 1728 \, j^{-1} \right) \,\,\,\,\,\,\,\,\,\,\,
\eea
Here we have defined $\m_i$ as
\bea
\m_1 &=& 4 x_1 - 2 x_2 -2 x_3
\no\\
\m_2 &=& 4 x_2 - 2 x_3 -2 x_1
\no\\
\m_3 &=& 4 x_3 - 2 x_1 -2 x_2
\eea
and $x_i$, $i=1,2,3$ as the three roots of the cubic equation
\bea
x^3 - \thalf  x^2 + \left(a + \tfrac{1}{18} \right) x + b =0
\eea

\subsubsection{Example}

One interesting example is when the parameters in (\ref{3rdorderMDE}) are taken to be
\bea
a = - {107 \over 2304}\hspace{0.8 in}b = {23 \over 55296}
\eea
In this case the exact solutions given above have the following $q$-expansions
\bea
\label{Isingcharacterqexp}
f_1(\tau) &=& q^{23 \over 48}\left(1+q+q^2 + q^3 + 2 q^4 + 2 q^5+ \dots \right)
\no\\
f_2(\tau) &=& q^{-{1 \over 48}}\left(1+q^2 + q^3 + 2 q^4 +2 q^5 + \dots \right)
\no\\
f_3(\tau) &=& q^{1 \over 24}\left(1+q+q^2 + 2q^3 + 2 q^4 +3 q^5 + \dots \right)
\eea
To all order in $q$, these match with the following alternative closed-form expressions,
\bea
\label{concretedim3sols}
f_1(\tau) & = & \half\left( \sqrt{\tet_3(0|\tau) \over \eta(\tau)} - \sqrt{\tet_4(0|\tau) \over \eta(\tau)}  \right)
\no\\
f_2(\tau) & = & \half\left( \sqrt{\tet_3(0|\tau) \over \eta(\tau)} + \sqrt{\tet_4(0|\tau) \over \eta(\tau)}  \right)
\no\\
f_3(\tau) & = &\sqrt{\tet_2(0|\tau) \over \eta(\tau)}
\eea
Indeed, these can be checked to be the solutions to the MDE in question.

\subsubsection{Modular invariance of solution spaces}

Because MDEs are by definition modular invariant, so too are their solution spaces. This means that under $SL(2, \ZZ)$ the solutions must transform into linear combinations of themselves. It is useful to organize the solutions of an order-$d$ MDE into a $d$-vector, in which case the $S$ and $T$ generators of $SL(2,\ZZ)$ can be represented by $d \times d$ matrices. For example, in the concrete case of  (\ref{concretedim3sols}) we can organize the solutions into a three-dimensional vector ${\bf f}(\tau) = (f_1(\tau), \, f_2(\tau), \, f_3(\tau))^t$, and then from the known transformation properties of $\tet_i(0|\tau)$ and $\eta(\tau)$ we obtain $T$ and $S$ as
\bea
T = \left(\begin{matrix} e^{23 \pi i \over 24}&0&0 \\ 0 & e^{- {\pi i \over 24}} & 0 \\ 0 & 0 & e^{\pi i \over 12} \end{matrix} \right)\hspace{0.5 in} S =\half \left(\begin{matrix}1 & \,1 &-1 \\1 &\, 1 & \,1 \\ -2&\, 2& \,0 \end{matrix} \right)
\eea
This leads us to a fruitful generalization of the notion of modular forms, known as \textit{vector-valued modular forms}.

\subsection{Vector-valued modular forms}

A $d$-dimensional vector-valued modular form ${\bf f}(\tau)=(f_1(\tau), \dots, f_d(\tau))^t$ of weight $k$ is defined by its modular transformation properties,
\bea
{\bf f}\left(\gamma \tau \right) =(c \tau + d)^k \rho (\gamma) \, {\bf f}(\tau) \hspace{0.5 in}
\gamma = \left(\begin{matrix}a\,\, & b \\ c\,\, & d \end{matrix} \right) \in SL(2, \ZZ)
\eea
where $\rho: SL(2, \ZZ) \rightarrow GL(d,\CC)$ is a $d \times d$ matrix representation. ${\bf f}(\tau)$ is further required to be holomorphic in the upper-half plane and to have at most finite-order poles at the cusps.

\subsubsection{Examples}

We begin with some simple examples of vector-valued modular functions. We have already seen that solutions to an order-$d$ MDE form a $d$-dimensional vector-valued modular form, and in the section \ref{sec:MDEvvmf} we will show that the converse is also true, i.e. the components of a $d$-dimensional vector-valued modular form are always solutions to an order-$d$ MDE. In this subsection however we will momentarily ignore the connection to MDEs.

\sm

The Jacobi $\tet$-functions at $z=0$ form a 3-dimensional vector-valued modular form $(\tet_2^8(0|\tau), \tet_3^8(0|\tau), \tet_4^8(0|\tau))$. We have taken the eighth power to eliminate potential phases from the modular transformations given in (\ref{eq:Jactetmodtransf}). Under $S$ and $T$ transformations one has,
\bea
\tet_2^8(0|\tau+1) &=& \tet_2^8(0|\tau)  \hspace{0.8 in}\tet_2^8(0|-1/\tau) \,\,=\,\, \tau^4\tet_4^8(0|\tau)
\no\\
\tet_3^8(0|\tau+1) &=& \tet_4^8(0|\tau)  \hspace{0.8 in}\tet_3^8(0|-1/\tau) \,\,= \,\,\tau^4\tet_3^8(0|\tau)
\no\\
\tet_4^8(0|\tau+1) &=& \tet_3^8(0|\tau)  \hspace{0.8 in}\tet_4^8(0|-1/\tau) \,\,=\,\,\tau^4 \tet_2^8(0|\tau)
\eea
Clearly these act as permutations of the three elements. In the usual permutation notation, the $T$ transformation corresponds to the transformation $(132)$, while the $S$ transformation corresponds to $(321)$. Together, these generate the group   $S_3$ of permutations of three elements.  One way of saying this is that there is a surjective homomorphism $\rho: SL(2, \ZZ)\rightarrow S_3$. It is interesting to ask what the kernel of the homomorphism $\rho$ is. That is to say, we are interested in the group defined by,
\bea
{\rm Ker} (\rho) = \left\{ \gamma = \left(\begin{matrix} a & b \\ c & d \end{matrix} \right) \in SL(2, \ZZ) \,\,
\Big | \,\,\tet_i^8\left(0| \gamma \tau \right) = (c \tau + d)^4\tet_i^8(0|\tau), \quad i=2,3,4 \right\}
\eea
Recalling the notation in terms of half-integer characteristics $\tet_2 = \tet\left[\begin{smallmatrix}1/2 \\ 0 \end{smallmatrix} \right],\tet_3 = \tet\left[\begin{smallmatrix}0 \\ 0 \end{smallmatrix} \right],$ and $\tet_4 = \tet\left[\begin{smallmatrix}0 \\ 1/2 \end{smallmatrix} \right]$, from (\ref{eq:Jactetmodtransf}) we see that elements of  ${\rm Ker} (\rho)$ must have,
\bea
a \a + c \b \equiv \a \,\,\,({\rm mod}\,\,1) \hspace{0.8 in}b \a + d \b \equiv \b \,\,\,({\rm mod}\,\,1)
\eea
for any choice $(\a,\b)\in \left\{(0,0), \left(\half, 0\right), \left(0,\half\right) \right\}$. For $\a = \b = 0$ the equations are trivially satisfied. Taking $\a= \half$ and $\b= 0$, the two equations above require,
\bea
a \equiv 1 \,\,\,({\rm mod}\,\,2)\hspace{0.8 in}b \equiv 0 \,\,\,({\rm mod}\,\,2)
\eea
while taking $\a= 0$ and $\b= \half$ require,
\bea
c \equiv 0 \,\,\,({\rm mod}\,\,2)\hspace{0.8 in}d \equiv 1 \,\,\,({\rm mod}\,\,2)
\eea
We thus conclude that  ${\rm Ker}( \rho)$ is given by
\bea
{\rm Ker} (\rho) = \left\{ \left(\begin{matrix} a & b \\ c & d \end{matrix} \right) \in SL(2, \ZZ) \,\,\Big |\left(\begin{matrix} a & b \\ c & d \end{matrix} \right) \equiv  \left(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right)\,\,\,({\rm mod}\,\,2) \right\}
\eea
In other words, we find that ${\rm Ker} (\rho)$ is equal to the principal congruence subgroup $\Gamma(2)$.
Thus while the vector $(\tet_2^8(0|\tau), \tet_3^8(0|\tau), \tet_4^8(0|\tau))$ transforms as a vector-valued modular form in a non-trivial permutation representation of $SL(2,\ZZ)$, we see that each of the individual components of this vector is itself a  genuine modular form under $\Gamma(2)$.

\sm

Proceeding to a four-dimensional example, we consider the vector ${\bf f}(\tau)$ with components,
\bea
f_1(\tau) &=&3^{12}\, \eta(3 \tau)^{24}\hspace{0.98 in} f_2 (\tau)\,\, =\,\, \eta\left({\tau \over 3} \right)^{24}
\no\\
f_3(\tau) &=&\eta\left({\tau +1\over 3} \right)^{24}\hspace{0.8 in} f_4 (\tau) \,\,=\,\, \eta\left({\tau +2\over 3} \right)^{24}
\eea
Under $S$ and $T$ modular transformations, we have the following
\bea
f_1(\tau+1 ) &=& f_1(\tau)  \hspace{0.8 in} f_1(-1/\tau)\,\,=\,\, \tau^{12} f_2(\tau)
\no\\
f_2(\tau+1)&=& f_3(\tau)\hspace{0.8 in} f_2(-1/\tau)\,\,=\,\, \tau^{12} f_1(\tau)
\no\\
f_3(\tau+1)&=& f_4(\tau) \hspace{0.8 in} f_3(-1/\tau)\,\,=\,\, \tau^{12} f_4(\tau)
\no\\
f_4(\tau+1) &=& f_2(\tau)\hspace{0.8 in} f_4(-1/\tau)\,\,=\,\, \tau^{12} f_3(\tau)
\eea
The generators $S$ and $T$ again permute elements of the vector; in permutation notation, they correspond to $(1342)$ and $(2143)$, respectively. However, now these transformations do not generate the full group $S_4$ of permutations of four elements. Instead, we only obtain the even permutations, i.e. we generate the alternating group $A_4$. So in this case we obtain a surjective homomorphism $\rho: SL(2,\ZZ) \rightarrow A_4$. We may again ask about the kernel of $\rho$. Demanding invariance of $f_1(\tau)$ under $\left(\begin{smallmatrix}a& b \\ c & d \end{smallmatrix} \right)\in SL(2, \ZZ)$ is equivalent to requiring,
\bea
\left( \begin{matrix}3 & 0 \\ 0 & 1 \end{matrix}\right)\left( \begin{matrix}a & b \\ c & d \end{matrix}\right)\left( \begin{matrix}3 & 0 \\ 0 & 1 \end{matrix}\right)^{-1} = \left( \begin{matrix}a & 3b \\ c/3 & d \end{matrix}\right) \in SL(2, \ZZ)
\eea
which in particular requires that $c \in 3 \ZZ$. Similarly, demanding invariance of $f_2(\tau)$ requires,
\bea
\left( \begin{matrix}1 & 0 \\ 0 & 3 \end{matrix}\right)\left( \begin{matrix}a & b \\ c & d \end{matrix}\right)\left( \begin{matrix}1 & 0 \\ 0 & 3 \end{matrix}\right)^{-1} = \left( \begin{matrix}a & b/3 \\ 3c & d \end{matrix}\right) \in SL(2, \ZZ)
\eea
which requires $b \in 3 \ZZ$. Any permutation in $A_4$ which fixes $f_1(\tau)$ and $f_2(\tau)$ also fixes $f_3(\tau)$ and $f_4(\tau)$. We conclude that the kernel consists of all matrices for which  $b \equiv c \equiv 0\,\,\,({\rm mod}\,\,3)$. This includes in particular the element $-I \in SL(2,\ZZ)$. Hence the kernel is not quite $\Gamma(3)$, but rather $\pm\Gamma(3)$, where we use the shorthand $\pm \Gamma(N) = \Gamma(N) \times \{\pm {I}\}$. So we have found that while the vector ${\bf f}(\tau)$ transforms as a vector-valued modular form in an alternating representation of $SL(2,\ZZ)$, each of the individual components of this vector is itself modular under the group $\pm\Gamma(3)$.

\sm

For completeness, let us record here some of the isomorphisms which we have seen, as well as some we have not yet seen,
\bea
SL(2, \ZZ) / \Gamma(2) &\cong&S_3 \hspace{0.8 in} SL(2, \ZZ) /\pm \Gamma(3) \,\,\cong\,\, A_4
\no\\
SL(2, \ZZ) /\pm \Gamma(4) &\cong&S_4 \hspace{0.8 in} SL(2, \ZZ) / \pm\Gamma(5) \,\,\cong \,\,A_5
\eea
These identities can be checked to be consistent with the index $[SL(2,\ZZ); \Gamma(N)]$, or equivalently with the order $|SL(2,\ZZ_N)|$, as given in (\ref{6.index1}) and (\ref{6.dN}). Indeed, from those formulas one finds
\bea
[SL(2,\ZZ); \Gamma(2)]& = &6 \hspace{0.8 in}[SL(2,\ZZ); \Gamma(3)] \,\,= \,\,24
\no\\
\vphantom{,}[SL(2,\ZZ); \Gamma(4)] &=& 48\hspace{0.8 in}[SL(2,\ZZ); \Gamma(5)]\,\,=\,\, 120
\eea
which can be compared with the orders of the relevant permutation and alternating groups,
\bea
|S_3| = 6 \hspace{0.5 in} |A_4| = 12 \hspace{0.5 in} |S_4 | =24 \hspace{0.5 in} |A_5 | = 60
\eea
For $N=2$ we see a match, whereas for $N>2$ there is a factor of two difference, stemming from the fact that we are quotienting by $\pm \Gamma(N)$ as opposed to $\Gamma(N)$.

\subsubsection{Integrality and $\Gamma(N)$}
\label{sec:Intconj}

In both of the above examples, we found that the components of the vector-valued modular forms were themselves modular under $\Gamma(N)$ for some appropriate $N$. This is not true in general, and it is interesting to ask when it holds. The following  result has been proven for 2- and 3-dimensional vector-valued modular forms, and is conjectured to hold more generally:\footnote{In the context of vector-valued modular functions realized by the characters of RCFTs, which is the case of principal interest in Physics, this conjecture has been proven in [67].}

\begin{conj}[Integrality Conjecture]
Consider a component $f_i(\tau)$ of a vector-valued modular form with Fourier expansion
\bea
\label{vvmfFourier}
f_i(\tau) =q^{n^{(i)}_{0}} \sum_{n \geq0} a_n^{(i)} q^n
\eea
with $n^{(i)}_{0} = p_i/N_i$ and $(p_i, N_i)=1$.  If all coefficients $a_n^{(i)}$ are algebraic integers, then $f_i(\tau)$ is a modular form  for $\Gamma(N_i) $.
\end{conj}
\noindent

The Integrality Conjecture states that a vector-valued modular form transforming in a representation $\rho$ is integral (i.e. each component of the vector has integer Fourier coefficients) only if $\mathrm{Ker}( \rho)$ contains a principal congruence subgroup $\Gamma(N)$. Indeed, the two examples given above had integer Fourier coefficients, and are directly in line with this conjecture.  As another example, the three-dimensional vector-valued modular function given in (\ref{concretedim3sols}) has all integer Fourier coefficients, and indeed the components can be seen to be modular functions for $\Gamma(48)$.

\sm

One common place in Physics where vector-valued modular forms appear is as characters of CFTs. Since the Fourier coefficients in that case are interpretable as physical degeneracies, they must of course be algebraic integers (and in fact rational integers). The conjecture then tells us that CFT characters are always modular functions for some $\Gamma(N)$. Going back to the example of (\ref{concretedim3sols}), the functions in this case are actually characters for the Ising CFT,
\bea
f_1(\tau) = \chi_\eps(\tau) \hspace{0.5 in} f_2(\tau) = \chi_1(\tau)\hspace{0.5 in} f_3(\tau) = \chi_\sigma(\tau)
\eea

\subsubsection{Relation to MDEs}
\label{sec:MDEvvmf}

We now show that the components of a $d$-dimensional vector-valued modular form are always solutions to an order-$d$ MDE. To see this, we begin by constructing the Wronskians $W_r$ associated to ${\bf f}(\tau)$,
\bea
\label{Wronskian1}
W_r =\det \left(\begin{matrix}
f_1 & \dots & f_d
\\ \cD^{(1)}_k f_1 & \dots & \cD^{(1)}_k f_d
\\\vdots & & \vdots \\
\cD^{(r-1)}_k  f_1 & \dots & \cD^{(r-1)}_k f_d
\\\cD^{(r+1)}_k  f_1 & \dots & \cD^{(r+1)}_k f_d
\\\vdots & & \vdots \\
\cD^{(d-1)}_k  f_1 & \dots & \cD^{(d-1)}_k f_d
\\\cD^{(d)}_k  f_1 & \dots & \cD^{(d)}_k f_d
\end{matrix}\right)
\eea
for $r=1, \dots, d$. From the trivial identity $W_{d-1} - \left({W_{d-1} \over W_d} \right) W_d = 0$, we obtain the following equation,
\bea
\det  \left(\begin{matrix}
f_1 & \dots & f_d
\\ \cD^{(1)}_k f_1 & \dots & \cD^{(1)}_k f_d
\\\vdots & & \vdots \\
\cD^{(d-2)}_k  f_1 & \dots & \cD^{(d-2)}_k f_d
\\\cD^{(d)}_k  f_1- {W_{d-1} \over W_d} \cD^{(d-1)}_k  f_1 & \dots & \cD^{(d)}_k  f_d- {W_{d-1} \over W_d} \cD^{(d-1)}_k  f_d
\end{matrix}\right) = 0
\eea
For this determinant to vanish, the bottom row must be a linear combination of the other rows, which tells us that
\bea
\label{vvmfMDE1}
\cD^{(d)}_k  f_i- {W_{d-1} \over W_d} \cD^{(d-1)}_k  f_i + \sum_{r=2}^{d} h_r(\tau) \cD^{(d-r)}_k f_i = 0
\eea
This is of the usual MDE form (\ref{basicMDE}), where in particular we have the coefficient function
\bea
h_1(\tau) = - {W_{d-1} \over W_d}
\eea
In fact, by plugging the expression for $\cD^{(d)}_k  f_i$ obtained from (\ref{vvmfMDE1}) into (\ref{Wronskian1}), we may express all of the coefficient functions in terms of Wronskians, giving
\bea
h_r(\tau) = (-1)^{r} {W_{d-r} \over W_d}
\eea

\sm

Say that the function $f_i(\tau)$ is bounded in the interior of the fundamental domain, and has zeroes at locations $p$ of order $\mathrm{ord}_{f_i}(p)$. We define the so-called \textit{Wronskian index} by
\bea
\label{Wronskianindexdef}
\ell := 6 \left(\half \mathrm{ord}_{W_d}(i) + {1 \over 3} \mathrm{ord}_{W_d}(\rho) + \sum_{p\in F} \mathrm{ord}_{W_d}(p) \right)
\eea
so that $\ell/6$ is the number of zeroes of $W_d$. The Wronskian index can take any non-negative integer value except $1$. We note for future use that the valence formula (\ref{thm1a}) implies
\bea
\label{vvmfvalence}
\mathrm{ord}_{W_d}(i \infty) + {\ell \over 6} = {d(k+d-1) \over 12}
\eea
where we have used the fact that the Wronskian has modular weight $d(k+d-1)$.

\sm

For $\ell=0$, the coefficient functions $h_r(\tau)$ are holomorphic, which is the case analyzed in detail for $d=2,3$ in the previous section. For $\ell >0$ on the other hand, the coefficient functions will generically be singular. For example, consider the case for which $d=\ell = 2$. By definition (\ref{Wronskianindexdef}) we conclude that $ \mathrm{ord}_{W_2}(\rho)  = 1$, and hence $W_2 \sim E_4(\tau)$. The coefficient functions then take the form $h_r(\tau) \sim {1 \over E_4(\tau)} W_{2-r}$. The remaining $\tau$ dependence can be fixed by simply requiring that one has the correct modular weight, giving $h_1(\tau) \sim E_4(\tau)$ and $h_2(\tau) \sim {E_6(\tau) \over E_4(\tau)}$.

Finding a solution to an MDE with meromorphic coefficients is often significantly more difficult than for its holomorphic counterpart (though for $d=2$ and $\ell < 6$ it turns out that the solutions can again be written as hypergeometric functions). As we discuss in section \ref{sec:Heckevvmf}, Hecke operators can be used to generate such solutions.

\subsection*{$\bullet$ Bibliographical notes}

Solving modular differential equations by recasting them as hypergeometric equations has been discussed in [68, 69, 70]. Vector-valued modular forms grew out of work of Gunning [71], as well as Selberg [72], who used them as a way of studying Fourier coefficients of modular forms for non-congruence subgroups. For a modern review of various properties of vector-valued modular forms see [73], and  for a discussion of modular forms and differential operators see [74]. The integrality conjecture given in section \ref{sec:Intconj} was first put forward in [75], and has since been proven for two-dimensional  and three-dimensional   vector-valued modular forms in [76, 77] and [78, 79], respectively. It has also been proven for vector-valued modular forms whose entries can be interpreted as the characters for RCFTs [67]. Its validity in a more general context has been discussed in e.g. [69, 80].

\sm

Finally, we note that modular differential equations have been applied to the classification of rational conformal field theories, starting with the seminal work in [81, 82], together with follow-up papers [83, 84] shortly thereafter. In subsequent works modular differential techniques were applied to the classification of bosonic rational conformal fields theories with two characters at Wronskian index $\ell =2,4$ [85, 86, 87], and to the classification of theories with three characters at $\ell=0$ [88]; for a recent review of some of this progress, see [89].
As of the writing of these lecture notes, the state-of-the-art for the classification of bosonic rational conformal field theories is for theories with up to five characters (subject to restrictions on the Wronskian index), obtained in [90]. For fermionic theories on the other hand, there is a complicating factor that the relevant modular differential equations need not be invariant under $SL(2, \ZZ)$, but only an appropriate index 2 congruence subgroup. Works towards the classification of fermionic theories with small numbers of characters have appeared in [91, 92].

\newpage

\section{Modular graph functions and forms}
\setcounter{equation}{0}
\label{sec:MGF}

Modular graph functions and forms map  certain (decorated) graphs to complex-valued functions on the Poincar\'e upper half-plane $\cH$ with definite transformation properties under $SL(2,\ZZ)$. The terminology of modular graph \textit{functions} versus \textit{forms} follows the terminology adopted for automorphic functions and forms in the holomorphic and non-holomorphic categories discussed  in subsection \ref{sec:terminology}. Thus, modular graph functions may be viewed as $SL(2,\ZZ)$-invariant functions on $\cH$, while modular graph forms may be identified with $SL(2,\ZZ)$-invariant differential forms.

\sm

To be more precise, we introduce the spaces $\cM_{k,\ell}$ of complex-valued smooth functions $f: \cH \to \CC$ with the following $SL(2,\ZZ)$ transformation properties,
\bea
\label{6.sl2}
\cM_{k, \ell} = \left \{  f(\gamma \tau)  = (c \tau + d) ^k (c \bar \tau +d)^\ell f(\tau)
\hbox{ for all }
\gamma = \left ( \bma a & b \cr c & d \ema \right ) \in SL(2,\ZZ) \right \}
\eea
for $k, \ell \in \CC$, subject to the condition $k-\ell \in \ZZ$. This condition is  imposed to ensure that the functions are single-valued. The functions of interest to us here will have at most polynomial growth at the cusp $\tau _2 \to \infty$, and we shall impose this property also on the functions in $\cM_{k,\ell}$. A modular graph form of modular weight $(k,\ell)$ is then a map from a certain (decorated) graph to $\cM_{k,\ell}$. For the special case $k=\ell$, one refers to such maps as modular graph functions. For $k, \ell \in \ZZ$, we may equivalently view the functions in $\cM_{k,\ell}$ as differential forms of weight $(\tfrac{k}{2},\tfrac{\ell}{2})$.

\sm

We shall establish that the non-holomorphic Eisenstein series introduced in section \ref{sec:4.2} may be associated with simple one-loop graphs and thus represent a special class of modular graph functions. Eisenstein series and modular graph functions and forms beyond Eisenstein series occur naturally and pervasively in the study of the low energy expansion of superstring amplitudes, as will be made clear in section \ref{sec:SA}. Here we shall present a purely mathematical approach with only minimal reference to Physics.

\subsection{One-loop modular graph functions are Eisenstein series}

We begin by considering the scalar Green function $G(z-w|\tau)$ on the torus $\Sigma=\CC/\Lambda$ for a lattice $\Lambda = \ZZ + \tau \ZZ$ with modulus $\tau \in \cH$ and $z, w \in \Sigma$. The scalar Green function  was defined  as the inverse of the scalar Laplacian on the torus on the space of functions orthogonal to constant functions in (\ref{10.green3}), solved in terms of Jacobi $\tet$-functions in (\ref{10.green-theta}), and shown to be invariant under $SL(2,\ZZ)$ in (\ref{10.G-mod}). Here we shall use its expression in terms of a double Fourier series or Kronecker-Eisenstein series, given in (\ref{10.green1}) and expressed here in terms of a sum over the lattice $\Lambda ' = \Lambda \setminus \{ 0 \}$,
\bea
G(z|\tau) = \sum_{(m,n) \not = (0,0)} { \tau_2 \over \pi |m + n \tau|^2} \, e^{2 \pi i (nx-my)}
\eea
where $z=x+\tau y$ and $x, y$ are real ``co-moving coordinates" valued in $\RR/\ZZ$. By concatenating the Green function repeatedly using convolution,  we may construct an infinite family of modular invariant functions, which may be defined recursively as follows,
\bea
G_s (z|\tau) = \int _\Sigma {d^2 u \over \tau_2} \, G (z-u|\tau ) G_{s-1} (u|\tau)
\eea
where we set $G_1=G$ and $s$ is an arbitrary positive integer.  To the Green function $G$ and its generalizations $G_s$ for positive integer $s$ it is natural to associate a graph, which physicists refer to as a Feynman graph,
\bea
\tikzpicture[scale=1.5]
\scope[xshift=0cm,yshift=0cm]
\draw [thick] (0,0) -- (1,0) ;
\draw (0,0) [fill=white] circle(0.05cm) ;
\draw (1,0) [fill=white] circle(0.05cm) ;
\draw(-1,0) node{$G(z-w|\tau) ~ = ~$};
\draw (0,-0.2) node{\small $z$};
\draw (1,-0.2) node{\small $w$};
\endscope
\scope[xshift=0cm,yshift=-0.8cm]
\draw [thick] (0,0) -- (3,0) ;
\draw [thick, dotted] (3,0) -- (4,0) ;
\draw [thick]  (4,0) -- (5,0) ;
\draw (0,0) [fill=white] circle(0.05cm) ;
\draw (1,0) [fill=black] circle(0.05cm) ;
\draw (2,0) [fill=black] circle(0.05cm) ;
\draw (3,0) [fill=black] circle(0.05cm) ;
\draw (4,0) [fill=black] circle(0.05cm) ;
\draw (5,0) [fill=white] circle(0.05cm) ;
\draw(-1,0) node{$G_s(z-w|\tau) ~ = ~$};
\draw (0,-0.2) node{\small $z$};
\draw (1,-0.2) node{\small $u_1$};
\draw (2,-0.2) node{\small $u_2$};
\draw (3,-0.2) node{\small $u_3$};
\draw (4,-0.2) node{\small $u_{s-1}$};
\draw (5,-0.2) node{\small $w$};
\endscope
\endtikzpicture
\label{6.fig:1}
\eea
The black dots represent points $u_1, \cdots u_{s-1}$  that are integrated over $\Sigma$, while the white dots represent given points in $\Sigma$, that are not integrated.  Upon Fourier transformation, convolution acts by multiplication, and we obtain the following representation for $G_s$,
\bea
\label{6.Gs}
G_s(z|\tau) = \sum_{(m,n) \not= (0,0)} { \tau_2^s \over \pi^s |m + n \tau|^{2s}} \, e^{2 \pi i (nx-my)}
\eea
where $z=x+\tau y$. As long as $s \geq 2$, we may set the point $z=0$ which graphically is equivalent to closing the open chain in (\ref{6.fig:1}) into the closed loop of (\ref{6.fig:2}),
\bea
\tikzpicture[scale=1.5]
\scope[xshift=0cm,yshift=0cm]
\draw [thick]  (1,0) -- (1/2,0.866);
\draw [thick]  (1,0) -- (1/2,-0.866);
\draw [thick, dotted]  (-1,0) -- (-1/2,-0.866);
\draw [thick]  (-1,0) -- (-1/2,0.866);
\draw [thick]  (-1/2,-0.866) -- (1/2,-0.866);
\draw [thick]  (-1/2,0.866) -- (1/2,0.866);
\draw (1,0) [fill=black] circle(0.05cm) ;
\draw (-1,0) [fill=black] circle(0.05cm) ;
\draw (1/2,0.866) [fill=black] circle(0.05cm) ;
\draw (1/2,-0.866) [fill=black] circle(0.05cm) ;
\draw (-1/2,0.866) [fill=black] circle(0.05cm) ;
\draw (-1/2,-0.866)[fill=black] circle(0.05cm) ;
\draw(-2,0) node{$E_s(\tau) ~ = ~$};
\draw (1.2,0.15) node{\small $u_1$};
\draw (-1.2,0.15) node{\small $u_4$};
\draw (1/2,1.05) node{\small $u_2$};
\draw (1/2,-1.05) node{\small $u_s$};
\draw (-1/2,1.05) node{\small $u_3$};
\draw (-1/2,-1.05) node{\small $u_{s-1}$};
\endscope
\endtikzpicture
\label{6.fig:2}
\eea
Comparing to (\ref{3.nhE}), we see that $G_s(0|\tau) = E_s(\tau)$, the non-holomorphic Eisenstein series.  Actually, the double sum in (\ref{6.Gs}) converges absolutely for any $s \in \CC $ with $\Re(s) >1$, but the graphical interpretation no longer makes sense when $s$ is not a positive integer.

\sm

When $s$ is integer and satisfies $s \geq 2$, the number $s$ counts the number of edges in the graph (\ref{6.fig:2}) and is referred to as the \textit{weight} of the graph (not to be confused with the modular weight of the corresponding modular graph function). In subsection \ref{sec:12.7} we shall identify this definition of the weight with that of \textit{transcendental weight}.

\subsection{Maass operators and Laplacians}
\label{sec:9.Maass}

Before studying higher-loop modular graph functions in detail, let us give a brief intermission on differential operators on the space $\cM_{k,\ell}$. There exist several natural maps between the spaces $\cM_{k,\ell}$ which we shall now exhibit.

\sm

One noteworthy element of $\cM_{-1,-1}$ is the imaginary part $\tau_2$ of the modulus $\tau$, whose powers satisfy $\tau_2 ^s \in \cM_{-s,-s}$. Multiplication by a suitable power of $\tau_2$ therefore provides a map between the different spaces $\cM_{k,\ell}$. Since a function $f(\tau)$ and its image $\tau_2^s f(\tau)$ under this map are essentially equivalent to one another, we may  introduce the equivalence relations,
\bea
\label{6.equiv}
\cM_{k,\ell} ~ \approx ~ \cM_{k-\ell, 0} ~ \approx ~ \cM_{0,\ell-k}
\eea
This equivalence is particularly convenient when it comes to the action of partial differential operators. The Cauchy-Riemann operator $\p_{\bar \tau}$ acts covariantly on the spaces $\cM_{k,0}$ without the need for a connection and maps to the space $\cM_{k,2}$. Similarly, the operator $\p_\tau$ maps $\cM_{0,\ell}$ covariantly to the space $\cM_{2,\ell}$ without the need for a connection. One may, of course, include a $k$- and $\ell$-dependent connection to obtain the Maass operators (see appendix \ref{sec:C4}). We shall circumvent doing so here by applying  $\p_{\bar \tau} $ to $\cM_{k,0}$ and $\p_\tau$ to $\cM_{0,\ell}$ only, using the equivalence of (\ref{6.equiv}). Actually, we can multiply the Cauchy-Riemann operators by suitable powers of $\tau_2$ so that the operators $\nabla = 2 i \tau_2^2 \p_\tau$ and $\bar \nabla = - 2 i \tau_2^2 \p_{\bar \tau}$ map the spaces as follows,
\bea
\bar \nabla : \cM_{k, 0} & \to & \cM_{k-2,0}
\no \\
\nabla : \cM_{0,\ell} & \to & \cM_{0,\ell-2}
\eea
Laplace operators may be defined as follows,\footnote{Here we define the Laplacians as negative operators, while in appendix \ref{sec:C4} they are positive operators.}
\bea
\Delta _k ^+ : \cM_{k,0} \to \cM_{k,0} & \hskip 0.6in &
\Delta _k^+ = \tau_2^{-k} \, \nabla \, \tau_2^{k-2} \, \bar \nabla
\no \\
\Delta _\ell ^- : \cM_{0,\ell} \to \cM_{0,\ell} & \hskip 0.6in &
\Delta _\ell^- = \tau_2^{-\ell} \, \bar \nabla \, \tau_2^{\ell-2} \, \nabla
\eea
On modular forms of weight $(0,0)$ the Laplace operators $\Delta_0^+$ and $\Delta_0^-$ coincide and are denoted by $\Delta = 4 \tau_2^2 \p_\tau \p_{\bar \tau}$. Famously, the Eisenstein series satisfies the Laplace eigenvalue equation,
\bea
\Delta E_s  = s(s-1) E_s
\eea
Its expansion at the cusp consists of two power behaved terms, plus an infinite number of exponentially suppressed terms, given in (\ref{4.d12}), and which we recall here for convenience,
\bea
E_s(\tau)  =
2  \, { \zeta (2s) \over \pi^s} \, \tau_2^s  + 2 { \Gamma (s-\thalf ) \over \pi^{s-\half} \Gamma (s)}  \, \zeta (2s-1) \, \tau_2^{1-s} +\cO(e^{-2 \pi \tau_2})
\eea
In terms of the independent variables $\tau, \bar \tau$ and the real co-moving coordinates  $x,y$, the Laplace operator $\Delta$ also has $G_s(x+\tau y|\tau)$ as an eigenfunction with eigenvalue independent of $x,y$,
\bea
\Delta G_s(x+\tau y|\tau) = s(s-1) G_s(x+\tau y|\tau)
\eea
Application of the operator $\nabla^k$ to $E_s$ produces modular graph forms of weight $(0,-2k)$,
\bea
\nabla ^k E_s (\tau) =
{ \Gamma (s+k) \over \Gamma (s)}  \sum_{(m,n) \not= (0,0)} { \tau_2^{s+k} \over \pi^s (m + n \tau)^{s+k} (m+n \bar \tau)^{s-k}}
\eea
These functions coincide with Zagier's elliptic polylogarithm, up to an overall normalization.

\subsection{Two-loop modular graph functions}

%Before discussing modular graph functions and forms for graphs of arbitrary topology,
We now investigate the special class of modular graph functions in $\cM_{0,0}$ associated with arbitrary (decorated) two-loop graphs. A connected two-loop graph has two trivalent vertices and an arbitrary number of bivalent vertices, distributed on the three concatenated edges connecting the trivalent vertices at $z$ and $w$ according to three positive integers $a_1, a_2, a_3$. The most general such graph is  depicted in (\ref{6.fig:3}),
\bea
\tikzpicture[scale=3]
\scope[xshift=0cm,yshift=0cm]
\draw (1.9,0) node{$\bullet$} ;
\draw (2.4,0) node{$\bullet$} ;
\draw (2.9,0) node{$\bullet$} ;
\draw (3.4,0) node{$\bullet$} ;
\draw (3.9,0) node{$\bullet$} ;
\draw [thick] (1.9,0) -- (2.4,0) ;
\draw [thick]  (2.4,0) -- (2.9,0) ;
\draw[dotted, thick] (2.9,0) -- (3.4,0) ;
\draw [thick]  (3.4,0) -- (3.9,0) ;
\draw (2.4,0.25) node{$\bullet$} ;
\draw (2.9,0.30) node{$\bullet$} ;
\draw (3.4,0.25) node{$\bullet$} ;
\draw [thick]  (1.9,0) -- (2.4,0.25) ;
\draw [thick]  (2.4,0.25) -- (2.9,0.3) ;
\draw[dotted, thick] (2.9,0.3) -- (3.4,0.25) ;
\draw [thick]  (3.4,0.25) -- (3.9,0) ;
\draw (2.4,-0.25) node{$\bullet$} ;
\draw (2.9,-0.30) node{$\bullet$} ;
\draw (3.4,-0.25) node{$\bullet$} ;
\draw  [thick]  (1.9,0) -- (2.4,-0.25) ;
\draw [thick]  (2.4,-0.25) -- (2.9,-0.3) ;
\draw[dotted, thick] (2.9,-0.3) -- (3.4,-0.25) ;
\draw [thick]  (3.4,-0.25) -- (3.9,0) ;
\draw(1.3,0) node{$C_{a_1,a_2,a_3}(\tau) ~ = $};
\draw(2.9,0.39) node{$a_1$};
\draw(2.9,0.11) node{$a_2$};
\draw(2.9,-0.19) node{$a_3$};
%\draw(2.9,0.39) node{$a_1$};
%\draw(2.9,0.1) node{$a_2$};
%\draw(2.9,-0.2) node{$a_3$};

\draw(1.9,0.1) node{$z$};
\draw(3.9,0.1) node{$w$};
\endscope
\endtikzpicture
\label{6.fig:3}
\eea
The associated modular graph function may be defined in terms of a double integral over $\Sigma$ of a triple product of concatenated Green functions $G_s$,
\bea
C_{a_1, a_2, a_3} (\tau) = \int _\Sigma { d^2 z \over \tau_2} \int _\Sigma { d^2 w \over \tau_2} G_{a_1}(z-w|\tau)
G_{a_2}(z-w|\tau)  G_{a_3}(z-w|\tau)
\eea
Translation invariance on the torus renders one of the integrals redundant, and we have the following equivalent expression,
\bea
C_{a_1, a_2, a_3} (\tau) = \int _\Sigma { d^2 z \over \tau_2} G_{a_1}(z|\tau)
G_{a_2}(z|\tau)  G_{a_3}(z|\tau)
\eea
Upon Fourier transforming, the corresponding modular graph function is given by the following Kronecker-Eisenstein sums,
\bea
C_{a_1, a_2, a_3} (\tau) = \sum _{{ m_r, n_r \in \ZZ \atop r=1,2,3}} '
\delta \Big ( \sum _{r=1}^3 m_r \Big )
\delta \Big ( \sum _{r=1}^3 n_r \Big )
\prod _{r=1}^3 \left (  {\tau _2 \over \pi |m _r + n_r \tau |^2} \right )^{a_r}
\eea
The sums are absolutely convergent provided $a_r + a_s > 1$ for all pairs $r \not= s$. The weight is given by $w=a_1+a_2+a_3$ and is required to satisfy  $w \geq 3$ for convergence. The Kronecker-Eisenstein sums are absolutely convergent for complex values of $a_1, a_2, a_3$ as long as $\Re(a_1), \Re(a_2), \Re(a_3) >1$, but contrarily to the case of the Eisenstein series,  the problem of the existence of analytic continuations in these variables remains unexplored.

\sm

Two key properties of the non-holomorphic Eisenstein series, namely that they satisfy a differential equation, and have a simple expansion at the cusp, have more complicated counterparts in two-loop modular graph functions.
\label{6.thm1}
Two-loop modular graph functions $C_{a_1,a_2,a_3}$  of weight $w=a_1+a_2+a_3$ obey a system of differential equations of uniform weight $w$
\bea
2 \Delta C_{a_1, a_2, a_3} & = &
2 a_1 a_2 \, C_{a_1+1,a_2-1,a_3} + a_1a_2 \, C_{a_1+1,a_2+1,a_3-2} - 4 a_1a_2 \, C_{a_1+1,a_2, a_3-1}
\no \\ &&
+  a_1 (a_1-1) \, C_{a_1,a_2,a_3}  +  \hbox{ 5 permutations of } (a_1,a_2,a_3)
\no
\eea
where $\Delta = 4 \tau_2^2 \p_\tau \p_{\bar \tau}$. When $a_1, a_2, a_3 \in \NN$, the system of differential equations
for a given weight truncates to a finite-dimensional linear system of inhomogeneous equations where the inhomogeneous part consists of a  linear combination of $E_w$ and $E_{w_1} E_{w_2} $ with $w=w_1+w_2$ and $w_1, w_2 \geq 2$, as a consequence of the following relations,
\bea
C_{w_1,w_2,0} & = & E_{w_1} E_{w_2} - E_w
\no \\
C_{w_1+1,w_2,-1} & = & E_{w_1} E_{w_2} + E_{w_1+1} E_{w_2-1}
\eea
The eigenvalues of the linear system are of the form $s(s-1)$ where $1 \leq s \leq w-2$ and $w-s \in 2\NN$ and have multiplicity $ [ (s+2)/3 ]$. }
\sm

For example, up to weight 5, the relations of Theorem \ref{6.thm1} are given as follows,
\bea
\label{6.exC}
\Delta  C_{1,1,1} & = & 6 E_3
\no \\
(\Delta -2) C_{2,1,1} & = & 9 E_4 - E_2^2
\no \\
(\Delta -6) C_{3,1,1} - 3 C_{2,2,1} & = & 16 E_5 - 4 E_2 E_3
\no \\
\Delta C_{2,2,1} & = & 8 E_5
\eea
Note that the first and last identities imply $C_{1,1,1}-E_3$ is constant, as must be $5C_{2,2,1}-2E_5$. These constants may be determined from the asymptotic behavior of these functions at the cusp, a problem to which we now turn.

\label{6.thm2}
The expansion of $C_{a_1, a_2, a_3}(\tau)$ near the cusp $\tau \to i \infty$ is given by a Laurent polynomial in  $\tau_2$ of degree $(w,1-w)$
\bea
C_{a_1, a_2, a_3} (\tau) = c_w (-4 \pi \tau_2)^w + { c_{2-w} \over (4 \pi \tau_2)^{w-2}}
+ \sum _{k=1}^{w-1}  { c_{w-2k-1} \, \zeta (2k+1) \over (4 \pi \tau_2)^{2k+1-w}} + \cO(e^{ - 2 \pi \tau_2})
\eea
where  $c_w, c_{w-2k-1} \in \QQ$ and $c_{2-w}$ is a linear combination with integer coefficients of products of odd zeta-values,
\bea
c_{2-w} = \sum_{k=1}^{w-2} \gamma _k \, \zeta (2k+1) \zeta (2w-2k-3)
\hskip 1in \gamma _k \in \ZZ
\eea} A consequence of this theorem is that no multiple zeta-values occur in the Laurent polynomial of two-loop modular graph functions. It was shown by Zerbini that higher loop modular graph functions do involve irreducible multiple zeta values.

\sm

At every odd weight, there is one linear combination of the above differential identities in the subspace of vanishing  eigenvalue whose inhomogeneous part does not involve bilinears in the Eisenstein series. This property may be verified in the identities for $C_{1,1,1} $ and $C_{2,2,1}$ in (\ref{6.exC}), and holds for all odd weights. Each one of these identities may  be integrated up to an additive constant which may be fixed by the asymptotics near the cusp, and we find,
\begin{align}
\label{eq:MGFidentities}
C_{1,1,1}  & =  E_3 + \zeta (3) &
C_{3,3,1} + C_{3,2,2} & = \frac{3}{7} E_7 +\frac{\zeta (7)}{252}
\no \\
C_{2,2,1} & =  {2 \over 5} E_5 +{\zeta (5) \over 30} &
9 C_{4,4,1} + 18 C_{4,3,2} + 4 C_{3,3,3}  &= 4 E_7 +\frac{ \zeta (9)}{240}
\end{align}
The equation for $C_{1,1,1}$ was derived by Zagier in an unpublished note  by direct summation of the Kronecker-Eisenstein series which defines $C_{1,1,1}$.

\subsection{General modular graph functions and forms}

At higher loops, it is no longer true that the Laplace operator in $\tau$, (acting on the immediate generalization of two-loop modular graph functions to higher loops) closes onto functions defined by Kronecker-Eisenstein series in which the exponents of holomorphic momenta $m_r + n _r \tau$ and their complex conjugate coincide. It is natural to also lift the restriction to modular functions, and consider more generally  \textit{modular graph forms} which have non-trivial modular weights. The required generalization may be defined as follows.

\sm

The connectivity matrix $\Gamma$ of an arbitrary graph  has components $\Gamma _{v \, r}$ where the index $v=1,\cdots, V$ labels the vertices of the graph and the index $r=1,\cdots, R$ labels its edges. No edge is allowed to begin and end on the same vertex. When edge $r$ contains vertex $v$ we have $\Gamma _{v \, r}=\pm 1 $,  while otherwise we have $\Gamma _{v\, r} = 0$. We shall view the pair of exponents $(a_r, b_r)$ of the momenta $p_r=m_r+n_r \tau$ and its complex conjugate $ \bar p_r$ through edge $r$ as providing a \textit{decoration} of the graph. The decorated graph is then specified by the connectivity matrix $\Gamma$ and the pair of arrays  $(A,B)$ defined by,
\bea
\label{ab}
A = [a_1, \cdots, , a_R] & \hskip 1in & \ma=a_1+ \cdots + a_R
\no \\
B = [b_1, \cdots, , b_R] && \mb=b_1+ \cdots + b_R
\eea
where $a_r, b_r \in \CC$ with $a_r-b_r \in \ZZ$ for all $r = 1, \cdots , R$.  The pair of total exponents $(\ma,\mb)$ generalizes the weight to the full decorated graph.

\sm

To a decorated graph  $(\Gamma, A, B)$ we associate a complex-valued function  on  $\cH$, defined by the following Kronecker-Eisenstein sum, whenever this sum is absolutely convergent,
\bea
\label{def:Cgamma}
\cC_\Gamma \left [ \begin{matrix} A \cr B \cr \end{matrix} \right ]  (\tau)
=   \left ( { \tau_2 \over \pi} \right ) ^{ \half \ma+ \half \mb} \sum_{p_1,\dots,p_R \in \Lambda '}  ~ \prod_{r =1}^R
{ 1 \over  (p_r) ^{a_r} ~ (\bar p _r) ^{b_r} }\, \prod_{v =1}^V
\delta \left ( \sum_{s =1}^R \Gamma _{v \, s} \, p_s \right )
\eea
The Kronecker $\delta$-symbol equals 1 when its argument vanishes as an element of the lattice $\Lambda$, and equals 0  otherwise.  The number of loops $L$  is the number of independent momenta, given by  $L=R-V+1$.  For any given decorated graph $(\Gamma, A,B) $, the domain of absolute convergence of the sums in (\ref{def:Cgamma}) is given by a system of inequalities on the combinations $\Re (a_r+b_r)$. Outside of the domain of convergence, it is an open question as to whether and how the functions $\cC$ of (\ref{def:Cgamma}) may be defined by analytic continuation in the variables $a_r+ b_r$.
The function $\cC$ in (\ref{def:Cgamma}) vanishes whenever the integer $\ma-\mb$ is odd, or whenever it is  associated with a graph $\Gamma$ which becomes disconnected upon severing a single edge. A function $\cC$ associated with a graph $\Gamma$ which is the union of two graphs $\Gamma = \Gamma _1 \cup \Gamma_2$, such that the intersection of $\Gamma _ 1$ and $\Gamma_2$ consists of a single vertex, factorizes into the product of  functions $\cC$ for $\Gamma_1$ and $\Gamma_2$, with the corresponding partitions of the exponents.

\subsubsection{Modular properties}

Under $SL(2,\ZZ)$ the  functions defined in (\ref{def:Cgamma})  transform as follows,
\bea
\label{mod}
\cC_\Gamma  \left [ \begin{matrix} A \cr B \cr \end{matrix}\right ] \left  ({ a \tau + b \over c \tau + d} \right )
= \left ( { c \tau+d \over c  \bar \tau +d } \right ) ^{\half \ma - \half \mb}
\cC_\Gamma  \left [ \begin{matrix} A \cr B \cr \end{matrix} \right ] (\tau)
\eea
where $a,b,c,d \in \ZZ$ and $ad-bc =1$. One defines the  \textit{modular weight} of  $\cC$  by the pair $(\half (\ma-\mb), \half (\mb-\ma))$ which is integer-valued for $\ma-\mb$  even ($\cC$ vanishes for odd $\ma-\mb$).  As we have explained above, we refer to $\cC$ as a  \textit{modular graph form} when $\ma\not= \mb$ and a \textit{modular graph function}  when  $\ma=\mb$.
For $\ma\not= \mb$ the normalization factor of $\tau_2$ is not canonical, and we  introduce the forms $\cC^+$ and $\cC^-$ of modular weight $(0,\mb-\ma)$ and $(\ma-\mb,0)$ respectively as follows,
\bea
\label{Cplus}
\cC_\Gamma^\pm  \left [ \begin{matrix} A \cr B \cr \end{matrix}\right ]
= (\tau_2)^{\pm {\ma-\mb \over 2}} \cC_\Gamma  \left [ \bma A \cr B \ema \right ] (\tau)
\eea
$\cC^+_\Gamma $ transforms with a factor $(c \bar \tau +d)^{-\ma +\mb}$ while $\cC^-_\Gamma $ transforms with a factor $(c \tau +d)^{\ma-\mb}$.

\subsubsection{Examples: Eisenstein series and dihedral graphs}

In this newfound notation, we may represent and generalize some of the examples of modular graph functions and forms encountered earlier. The Eisenstein series and its successive derivatives take the form,
\bea
\label{9.Eisen}
E_s = \cC_\Gamma \left [ \bma s & 0 \cr s & 0 \ema \right ]
\hskip 0.8in
\nabla ^k E_s = {\Gamma (s+k) \over \Gamma (s)} \, \cC^+_\Gamma  \left [ \bma s +k & 0 \cr s-k  & 0 \ema \right ]
\eea
where $\Gamma$ is a connected one-loop graph  with only bivalent vertices. The two-loop modular graph functions $C_{a_1, a_2, a_3}$ may be generalized to $(n-1)$-loop modular graph functions,
\bea
C_{a_1, \cdots, a_n} = \cC_\Gamma \left [ \bma a_1 & \cdots & a_n  \cr a_1 & \cdots & a_n \ema \right ]
\eea
where the ``dihedral" graph $\Gamma$ contains two vertices of valence $n+1$ in addition to the remaining vertices which are all bivalent. For $n=3$ we recover the two-loop modular graph functions studied earlier.

\subsubsection{Examples: holomorphic modular graph forms}
\label{sec:9.4.7}

When all anti-holomorphic exponents $b_i$ vanish identically the corresponding modular graph form reduces to the following expression,
\bea
\cC_\Gamma^-  \left [ \bma a_1 & \cdots & a_n  \cr 0 & \cdots & 0 \ema \right ](\tau)
=   \sum_{p_1,\dots,p_R \in \Lambda '}  ~ \prod_{r =1}^R
{ 1 \over  (p_r) ^{a_r} }\, \prod_{v =1}^V
\delta \left ( \sum_{s =1}^R \Gamma _{v \, s} \, p_s \right )
\eea
The multiple sum is absolutely convergent provided $a_r \geq 2$ for all $r=1,\cdots,  R\geq 2$. In this case the result is
holomorphic in $\tau$ and a modular form of weight $(\ma,0)$ with $\ma=a_1 + \cdots, + a_R$. When $\cC^-_\Gamma$ is a modular form of modular weight $(\ma,0)$, it may be expressed as a polynomial in the holomorphic modular forms $\HE_4$ and $\HE_6$, and therefore produces a map from graphs $\Gamma$ to holomorphic modular forms, or equivalently to polynomials in $\HE_4$ and $\HE_6$. The nature of this map, which is much simpler than the map to non-holomorphic modular forms, remains to be investigated. Under weaker conditions of conditional convergence  the sum must be suitably regularized. Maintaining holomorphicity leads to a violation of full modular invariance, such as is the case with $a_1=2, a_2=0$ for $R=2$ which yields the quasi-modular form $\HE_2$.

\subsubsection{Algebraic relations}

Under complex conjugation, any of these functions have their arrays of holomorphic and anti-holomorphic exponents swapped,
\bea
\label{3d2}
\cC_\Gamma  \left [ \begin{matrix} A \cr B \cr \end{matrix}\right ] (\tau) ^*
= \cC_\Gamma  \left [ \begin{matrix} B \cr A \cr \end{matrix}\right ] (\tau)
\eea
In addition, modular graph forms obey momentum conservation identities,
\bea
\label{3d3}
\sum _{r=1}^R \Gamma _{k\, r} \, \cC_\Gamma  \left [ \begin{matrix} A - S_r \cr B \cr \end{matrix}\right ]=
\sum _{r=1}^R \Gamma _{k\, r} \, \cC_\Gamma  \left [ \begin{matrix} A \cr B - S_r \cr \end{matrix}\right ]=0
\eea
where the $R$-dimensional row-vector $S_r$ is defined to have zeroes in all slots except for the $r$-th, which instead has value 1,
\bea
\label{Sr}
S_r = [\underbrace{0,\ldots,0}_{r-1},1,\underbrace{0,\ldots,0}_{R-r}]
\eea
These identities simply express the fact that the lattice momenta entering each vertex $v$ add up to zero or, as physicists would state, that momenta is conserved at each vertex as a consequence of translation invariance on the torus.

\subsubsection{Differential relations}

The action of the Cauchy-Riemann operator $\nabla = 2 i \tau_2^2 \p_{\tau}$ on modular graph forms $\cC^+$ defined and normalized  in (\ref{Cplus})  for arbitrary exponents $A,B$ is given by,
\bea
\label{nab}
\nabla \cC_\Gamma^+  \left [ \begin{matrix} A \cr B \cr \end{matrix}\right ]
= \sum_{r=1}^R a_r \, \cC^+ \left [  \bma A + S_r \cr B - S_r \ema \right]
\eea
where  $S_r$ was defined in (\ref{Sr}).  The action of the Laplace operator $\Delta = 4 \tau _2^2 \p_{\bar \tau} \p_\tau$ on modular graph functions (for which $\ma=\mb$) is given by,
\bea
\label{Lap1}
(\Delta + \ma ) \, \cC_\Gamma  \left [ \begin{matrix} A \cr B \cr \end{matrix}\right ]
\sum _{r,s=1}^R a_r b_s \, \cC \left [ \bma A+S_r - S_s  \cr B -S_r + S_s  \ema \right ]
\eea
It is similarly possible to define the action of the Laplace operator on modular graph forms of arbitrary modular weight with $\ma \not= \mb$, but we shall not need them here.

\subsubsection{Higher loop algebraic identities}

Higher-loop modular graph functions and forms satisfy systems of algebraic and differential identities which generalize those found for two-loops. For example, one has the following algebraic identities involving modular graph functions of weight 4 and 5,
\bea
\label{9.ids}
C_{1,1,1,1} & = & 24 C_{2,1,1} + 3 E_2^2 - 18 E_4
\no \\
C_{1,1,1,1,1} & = & 60 C_{3,1,1} + 10 E_2 \, C_{1,1,1} -48 E_5 + 16 \zeta (5)
\no \\
40 C_{2,1,1,1} & = & 300 C_{3,1,1} + 120 E_2 E_3 - 276 E_5 + 7 \zeta (5)
\eea
These identities were proven  using \textit{holomorphic subgraph reduction} and a \textit{sieve algorithm}. We shall discuss the techniques used in  the proof of these identities in the next subsection for the example of the first identity. We refer to the bibliographical notes for the papers where all algebraic identities between modular graphs of weight up to six, and some selected identities of weight 7 are proven, and where a Mathematica package is now available.

\subsection{Relating modular graph functions to holomorphic forms}

The Maass operator $\nabla = 2 i \tau_2^2 \p_\tau$ maps a modular graph form of modular weight $(0,\ell)$ to a modular graph form of modular weight $(0,\ell-2)$, as was shown in subsection \ref{sec:9.Maass}. When the derivative is applied to a non-holomorphic Eisenstein series $E_n$, for example, and repeated $n$ times, one obtains the following form, using (\ref{9.Eisen}),
\bea
\label{9.Edif}
\nabla ^n E_n = {\Gamma (2n) \over \Gamma (n)} \, \cC^+_\Gamma  \left [ \bma 2n ~ 0 \cr 0  ~~\, 0 \ema \right ]
= {\Gamma (2n) \over \Gamma (n) \pi^n } \, \tau_2^{2n} \, G_{2n}(\tau)
\eea
where $G_{2n}(\tau)$ is the holomorphic Eisenstein series of modular weight $(2n,0)$.

\sm

Applying the Maass operator to more complicated modular graph functions is not as simple as in the case of non-holomorphic Eisenstein series, but nonetheless a similar pattern emerges: $\nabla$ decreases the anti-holomorphic exponents and increases the holomorphic exponents. Let's see how this works in the case of the modular graph function $C_{1,1,1,1}$. Using the rules of (\ref{nab}) for $\nabla$, we obtain for the first derivative,
\bea
\nabla C_{1,1,1,1} =
\nabla \cC  \left [ \bma 1 ~ 1 ~ 1 ~ 1 \cr 1  ~ 1 ~ 1 ~ 1  \ema \right ]
4 \, \cC^+  \left [ \bma 2 ~ 1 ~ 1 ~  1 \cr 0  ~ 1 ~ 1 ~ 1  \ema \right ]
\eea
The form on the right has modular weight $(0,-2)$. Applying $\nabla$ again gives,
\bea
\label{9.del2}
\nabla^2 C_{1,1,1,1} =
12\, \cC^+  \left [ \bma 2 ~ 2 ~ 1 ~ 1 \cr 0  ~ 0 ~ 1 ~ 1  \ema \right ]
-24 \, \cC^+  \left [ \bma 3 ~ 1 ~ 1 ~  1 \cr 0  ~ 0 ~ 1 ~ 1  \ema \right ]
\eea
where to obtain the last term we have a momentum conservation identity to convert a negative entry on the lower row,
\bea
\cC^+  \left [ \bma 3 & 1 ~ 1 ~  1 \cr -1  & 1 ~ 1 ~ 1  \ema \right ]
= -3 \, \cC^+  \left [ \bma 3 ~ 1 ~ 1 ~  1 \cr 0  ~ 0 ~ 1 ~ 1  \ema \right ]
\eea
Further application of $\nabla$ to (\ref{9.del2}), in its present form,  would yield modular graph functions with some negative anti-holomorphic exponents that cannot be removed simply by using momentum conservation identities. However, the subgraph corresponding to the first two entries forms a closed loop with only holomorphic momenta.
The momentum summations in such holomorphic subgraphs may be evaluated explicitly in terms of holomorphic modular forms. This procedure is dubbed \textit{holomorphic subgraph reduction} and, in the present case, leads to the following identity,
\bea
\label{9.holo}
\cC^+  \left [ \bma 2 ~ 2 ~ 1 ~ 1 \cr 0  ~ 0 ~ 1 ~ 1  \ema \right ]
-2 \, \cC^+  \left [ \bma 3 ~ 1 ~ 1 ~  1 \cr 0  ~ 0 ~ 1 ~ 1  \ema \right ]
2 \, \cC^+  \left [ \bma 4 ~ 1 ~ 1  \cr 0  ~ 1 ~ 1   \ema \right ]
+ 3 \, \cC^+  \left [ \bma 4 ~ 0  \cr 0  ~ 0   \ema \right ] \cC^+  \left [ \bma 2 ~ 0  \cr 2  ~ 0   \ema \right ]
\eea
Collecting all contributions to $\nabla ^2 C_{1,1,1,1}$ and proceeding analogously for the entries $C_{2,1,1}$ and $E_2^2$ in the first identity in (\ref{9.ids}), we obtain,
\bea
\label{9.del3}
\nabla^2 C_{1,1,1,1} & = &
24 \, \cC^+ \! \left [ \bma 4 ~ 1 ~ 1  \cr 0  ~ 1 ~ 1   \ema \right ]
+ 36 \, \cC^+ \!  \left [ \bma 4 ~ 0  \cr 0  ~ 0   \ema \right ] \cC^+ \! \left [ \bma 2 ~ 0  \cr 2  ~ 0   \ema \right ]
\no \\
\nabla^2 C_{2,1,1} & = &
6 \, \cC^+ \! \left [ \bma 4 ~ 1 ~ 1  \cr 0  ~ 1 ~ 1   \ema \right ]
+ 6 \, \cC^+ \! \left [ \bma 3 ~ 2 ~ 1  \cr 1  ~ 0 ~ 1   \ema \right ]
+ 4 \, \cC^+ \!  \left [ \bma 6 ~ 0  \cr 2  ~ 0   \ema \right ]
\no \\
\nabla ^2 E_2^2 & = &
8 \, \cC^+ \!  \left [ \bma 3 ~ 0  \cr 1  ~ 0   \ema \right ]^2
+ 12 \, \cC^+ \! \left [ \bma 4 ~ 0  \cr 0  ~ 0   \ema \right ] \cC^+ \! \left [ \bma 2 ~ 0  \cr 2  ~ 0   \ema \right ]
\eea
Applying $\nabla$ to either $\nabla^2 C_{1,1,1,1}$ or to $E_2^2$ produces modular graph functions with negative entries in the lower row that cannot be removed by using momentum conservation identities. This occurs when $\nabla$ is applied to one of the modular graph functions that is proportional to a holomorphic modular form, such as the first factors in the last terms on the right of $\nabla^2 C_{1,1,1,1}$ and $\nabla^2 E_2^2$. However, $C_{2,1,1}$ and the linear combination $C_{1,1,1,1}-3E_2^2$ (that cancels the last terms discussed above) produces the following $\nabla^3$ derivatives,
\bea
\label{9.nab3}
\nabla ^3 (C_{1,1,1,1}-3E_2^2) & = & 432 \, \cC^+ \!  \left [ \bma 7 ~ 0  \cr 1  ~ 0   \ema \right ]
-288 \, \cC^+ \!  \left [ \bma 4 ~ 0  \cr 0  ~ 0   \ema \right ] \cC^+ \!  \left [ \bma 3 ~ 0  \cr 1  ~ 0   \ema \right ]
\no \\
\nabla^3 C_{2,1,1} & = & 108 \, \cC^+ \!  \left [ \bma 7 ~ 0  \cr 1  ~ 0   \ema \right ]
-12 \, \cC^+ \!  \left [ \bma 4 ~ 0  \cr 0  ~ 0   \ema \right ] \cC^+ \!  \left [ \bma 3 ~ 0  \cr 1  ~ 0   \ema \right ]
\eea
Proceeding analogously for the fourth $\nabla$-derivative, we obtain,
\bea
\nabla ^4 ( C_{1,1,1,1} - 24 C_{2,1,1} - 3 E_2^2) =- 15120 \, \cC^+ \!  \left [ \bma 8 ~ 0  \cr 0  ~ 0   \ema \right ]
\eea
The right side is proportional to $\tau_2^8 G_8$, where $G_8$ is the holomorphic Eisenstein series of weight $(8,0)$.
But this form is proportional to $\nabla ^4 E_4$, so that we obtain,
\bea
\label{9.D4}
\nabla ^4\big ( C_{1,1,1,1} - 24 C_{2,1,1} - 3 E_2^2 +18 E_4 \big )=0
\eea
Next, we invoke the following lemma.

\label{9.lem1}
Let $F$ be a non-holomorphic modular function with polynomial growth near the cusp as $\tau_2 \to \infty$. If $F$ satisfies the differential equation,
\bea
\label{9.F}
\nabla ^n F=0
\eea
for an arbitrary integer $n \geq 1$, then $F$ is constant as a function of $\tau$.}

\sm

A simple proof of the lemma proceeds by making use of the following formula,
\bea
\bar \nabla ^n (\tau_2)^{-2n} \nabla ^n = \prod _{s=1}^n \Big ( \Delta -s(s-1) \Big )
\eea
This formula immediately implies that $F$ must satisfy the weaker condition,
\bea
\prod _{s=1}^n \Big ( \Delta -s(s-1) \Big ) F=0
\eea
Being a modular graph function,  $F$ has at most polynomial growth at the cusp. The solution to the equation $\Delta f_s=s(s-1)f_s$ for positive integer $s$ is either a constant for $s=1$ or an Eisenstein series $E_s$ for $s \geq 2$. This implies that $F$ must be a linear combination of a constant and the Eisenstein series $E_s$ for $s=2, \cdots, n$. Returning now to the original equation (\ref{9.F}) in the formulation of the lemma and using the asymptotics of the Eisenstein series at the cusp of (\ref{4.d12}), we conclude that the coefficients of the Eisenstein series in $F$ must all vanish so that $F$ is constant. This completes the proof of the lemma.

\sm

Applying the lemma to  (\ref{9.D4}), it follows that,
\bea
F_4 = C_{1,1,1,1} - 24 C_{2,1,1} - 3 E_2^2 +18 E_4
\eea
must be constant. To calculate the constant, we investigate the asymptotics at the cusp.
Clearly, all $\tau$-dependent terms must cancel on the right side as one may verify using the expansions of $C_{1,1,1,1}$ and $C_{2,1,1}$ in terms of the variable $y = \pi \tau_2$,
\bea
C_{1,1,1,1}(\tau) & = & { y^4 \over 945} + { 2 \zeta(3) y \over 3} + {10 \zeta(5) \over y}  -{ 3 \zeta(3)^2 \over y^2} +{9 \zeta(7) \over 4 y^3} + \cO(e^{-2y})
\no \\
C_{2,1,1} (\tau) & = & { 2y^4 \over 14175} +{\zeta(3) y \over 45} +{5 \zeta(5) \over 12 y} -{ \zeta (3)^2 \over 4 y^2} +{9 \zeta (7) \over 16 y^3} + \cO(e^{-2y})
\eea
and the expansion of the Eisenstein series in (\ref{4.d12}). The constant $y^0$ term has vanishing coefficient in each of the the modular graph functions in $F_4$, so that $F_4=0$. This completes the proof of the first identity in (\ref{9.ids}). The other identities in (\ref{9.ids}) may be established with the same methods, and holomorphic subgraph reduction formulas that generalize (\ref{9.holo}).

\subsection{Modular graph functions and iterated modular integrals}

The Fourier expansion of non-holomorphic Eisenstein series $E_k(\tau)$ for integer $k \geq 2$, presented in (\ref{Eas}), shows that the Fourier non-zero modes are of the form $(q^N+\bar q^N)$ times a polynomial  in $1/\tau_2$.