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\section{Integrating modular forms}\label{integrating}
Let $\Gamma$ be a congruence subgroup of $SL_2(\R)$, $S$~--- a finite union of orbits of $\Gamma$ in $\HH$, $U=\HH-S$. Consider a smooth closed differential $1$-form $\omega$ on $U$ with coefficients in $V_{2k-2}$, where $V_{2k-2}$ is the space of polynomials in one variable $X$ of degree not greater than $2k-2$. $V_{2k-2}$ is equipped with the natural action of $SL_2(\R)$ (even $SL_2(\C)$). Suppose moreover, that $\omega$ is equivariant for the action of $\Gamma$ and let $A$ be a $\Gamma$-submodule of $V_{2k-2}$. We are looking for a function 
\[
I_\omega^{A,\Gamma}: U\longrightarrow V_{2k-2}/A,
\]
which satisifies the following properties:
\begin{enumerate}
\item $I_\omega^{A,\Gamma}$ is smooth, which means that for any point $z\in U$ there is a neighbourhood $W$ of $z$ and a smooth function $g:W\longrightarrow V_{2k-2}$, such that $g$ coincides with $I_\omega^{A,\Gamma}$ modulo $A$.
\item $d I_\omega^{A,\Gamma} = \omega$,
\item $I_\omega^{A,\Gamma}$ is equivariant for the action of $\Gamma$.
\end{enumerate}
Our basic example of $\omega$ is 
\[
\omega = f(z) (X-z)^{2k-2} dz,
\]
for some meromorphic modular form $f$ of weight $2k$. The module $A$ may be the submodule of $V_{2k-2}$ consisting either of polynomials with real coefficients, polynomials with imaginary coefficients or of polynomials with coefficients in some discrete subgroup of $\C$.

It is clear, that for the existense of $I_\omega^{A,\Gamma}$ the following condition is necessary:
\begin{condition}[Residue condition]
For every point $s\in S$ the integral of $\omega$ along a small loop around $s$ belongs to $A$.
\end{condition}

Choose a basepoint $a \in U$. The differential $\omega$ defines a cocycle with values in $V^{2k-2}/A$ in the following way:
\[
\sigma_a(\gamma) = \int_{a}^{\gamma a} \omega.
\]
It satisfies 
\[
\sigma_a(\gamma \gamma') = \sigma_a(\gamma) + \gamma \sigma_a(\gamma').
\]
The integral of $\omega$ is a function with values in $V_{2k-2}/A$:
\[
I_a(x) = \int_{a}^x \omega.
\]
If we change the basepoint from $a$ to $a'$ the integral and the cocycle change in the following way:
\[
I_{a'}(x) = I_a(x) + \int_{a'}^a \omega,
\]
\[
\sigma_{a'}(\gamma) = \sigma_a(\gamma) + \int_{a'}^a \omega - \int_{\gamma a'}^{\gamma a} \omega,
\]
so if we introduce an element $v_{a a'}$ of $V_{2k-2}/A$ by 
\[
v_{a a'} = \int_{a'}^a \omega,
\]
then
\[
I_{a'}(x) = I_a(x) + v_{a a'},
\]
\[
\sigma_{a'} = \sigma_a - \delta v_{a a'},
\]
where $\delta$ denotes the differential 
\[
(\delta v)(\gamma) = \gamma v - v.
\]
Therefore $\sigma_a$ defines a class in $\sigma_\omega^{A,\Gamma}\in H^1(\Gamma, V_{2k-2}/A)$, which does not depend on the base point.

If such an $I_\omega^{A,\Gamma}$ as explained in the beginning of the section exists, it must differ from $I_a$ by a constant, say $v_a\in V_{2k-2}/A$. Let us describe all $v_a\in V_{2k-2}$ such, that $I_a+v_a$ is equivariant for the action of $\Gamma$. This means that for any $\gamma\in\Gamma$
\[
I_a(\gamma x) + v_a = \gamma (I_a(x)+v_a).
\]
Splitting the path of integration and using the equivariance of $\omega$ we conclude:
\[
I_a(\gamma x) = I_{\gamma a}(\gamma x) + \sigma_a(\gamma) = \gamma I_a(x) + \sigma_a(\gamma),
\]
so the last equation is equivalent to the following:
\[
\sigma_a(\gamma) = \gamma v_a - v_a = \delta v_a.
\]
Therefore
\begin{prop}
Suppose $\omega$ satisifies the residue condition. Then, the function $I_\omega^{A,\Gamma}$ satisifying the properties (i)-(iii) exists if and only if the class $\sigma_\omega^{A,\Gamma}\in H^1(\Gamma, V_{2k-2}/A)$ is trivial. If such a function exists, then it is unique up to the addition of an element of $H^0(\Gamma,V_{2k-2}/A)$ and satisfies
\[
\gamma I_\omega^{A,\Gamma}(z) - I_\omega^{A,\Gamma}(z) = \int_z^{\gamma z} \omega \mod A \qquad \text{for all $\gamma\in\Gamma$.}
\]
\end{prop}

For any $\Gamma$-module $M$ we denote by $C^1(\Gamma, M)$ the abelian group of cocycles, i.e. maps $\sigma:\Gamma \longrightarrow M$, such that
\[
\sigma(\gamma_1\gamma_2) = \sigma(\gamma_1) + \gamma_1\sigma(\gamma_2), \qquad \text{for all $\gamma_1, \gamma_2 \in \Gamma$.}
\]
Then we have the following exact sequence:
\[
0\longrightarrow H^0(\Gamma, M) \To M \xrightarrow{\;\,d\;\,} C^1(\Gamma, M) \To H^1(\Gamma, M)\to 0,
\]
where $d$ is given by the usual formula
\[
(d m)(\gamma) = \gamma m - m, \qquad m\in M,\gamma\in\Gamma
\]
Denote by $C_1(\Gamma, M)$ the abelian group of cycles, by which we mean the quotient group of $\Z\Gamma\otimes M$ by the subgroup, generated by elements of the form
\[
\gamma_1\gamma_2\otimes m - \gamma_2\otimes m - \gamma_1\otimes\gamma_2 m, \qquad \text{for $m\in M$, $\gamma_1, \gamma_2 \in \Gamma$.}
\]
We have the corresponding exact sequence for homology:
\[
0\longrightarrow H_1(\Gamma, M) \To C_1(\Gamma, M) \xrightarrow{\;\,d\;\,} M  \To H_0(\Gamma, M)\to 0,
\]
where $d$ is given by 
\[
d (\gamma\otimes m) = \gamma m - m, \qquad m\in M,\gamma\in\Gamma.
\]
Suppose we have an invariant biadditive pairing
\[
(\cdot,\cdot): M\otimes M' \To N,
\]
where $M$ and $M'$ are $\Gamma$-modules and $N$ is an abelian group.
Denote by $C^1(\Gamma, M)_0$ the group of cocycles, which map to zero in $H^1(\Gamma, M)$, and by $M'_0$ the group of elements in $M'$, which map to zero in $H_0(\Gamma, M')$.

Note that there is a canonical pairing 
\[
C^1(\Gamma, M)\otimes C_1(\Gamma, M') \To N,
\]
given by
\[
(\sigma, \gamma\otimes m) = (\sigma(\gamma^{-1}), m),
\]
since
\begin{multline*}
(\sigma, \gamma_1\gamma_2\otimes m - \gamma_2\otimes m - \gamma_1\otimes \gamma_2 m) =\\
(\sigma(\gamma_2^{-1}\gamma_1^{-1}), m) - (\sigma(\gamma_2^{-1}), m) - (\sigma(\gamma_1^{-1}), \gamma_2 m)= 0.
\end{multline*}

This pairing has the following property, which can be easily checked:
\begin{prop}
For any $m\in M$, $c\in C_1(\Gamma, M')$
\[
(dm, c) = (m, dc).
\]
\end{prop}

This implies, that $(\cdot, \cdot)$ induces pairings
\[
H^0(\Gamma, M)\otimes H_0(\Gamma, M') \To N, \qquad 
H^1(\Gamma, M)\otimes H_1(\Gamma, M') \To N.
\]

If we have $\sigma\in C^1(\Gamma, M)_0$ and $m'\in M'_0$ we can either represent $\sigma$ as a coboundary, i.e. $\sigma = d m_0$ and then consider $(m_0, m')$, or represent $m'$ as a boundary, $m' = d c$ and then consider $(\sigma, c)$. 
\begin{prop}
Either of these two approaches defines a pairing 
\[
C^1(\Gamma, M)_0 \otimes M'_0 \To N,
\]
moreover, the resulting two pairings coincide.
\end{prop}

We put $M=V_{2k-2}/A$, $M' = B$, where $B$ is a $\Gamma$- invariant subgroup of $V_{2k-2}$, and $N=\C / (A, B)$. 

\begin{thm}\label{int_pairing}
Let $S\subset \HH$ be a finite union of orbits of $\Gamma$, $A$ and $B$ be $\Gamma$-invariant subgroups of $V_{2k-2}$, $\omega$ be a smooth closed invariant differential $1$-form on $U=\HH-S$ with coefficients in $V_{2k-2}$, whose integrals along small loops around points of $S$ belong to $A$, $z$ be a point in $U$, $v$ be an element in $B$, such that:
\begin{enumerate}
\item The class of $\omega$ in $H^1(\Gamma, V_{2k-2}/A)$ is $0$,
\item the class of $v$ in $H_0(\Gamma, B)$ is $0$.
\end{enumerate}
Then the following two approaches lead to the same element of $\C/(A, B)$, which we denote by $I^{A, B, \Gamma}(\omega, z, v)$:
\begin{enumerate}
\item First represent $\omega$ as a differential of an invariant $V_{2k-2}/A$-valued function $I^{A, \Gamma}_\omega$, and then put
\[
I^{A, B, \Gamma}(\omega, z, v) = (I_\omega^{A, \Gamma}(z), v).
\]
\item First represent $v$ as
\[
v = \sum_{i=1}^n (\gamma_i u_i - u_i),\qquad \text{for $\gamma_i\in\Gamma$, $u_i\in B$,}
\]
and then put
\[
I^{A, B, \Gamma}(\omega, z, v) = \sum_{i=1}^n (\int_z^{\gamma_i^{-1} z} \omega, u_i).
\]
\end{enumerate}
\end{thm}

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