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\section{Local study of Green's functions}

Let $k$ be an integer, $k>1$.
We consider the Green's function for the upper half plane of weight $2 k$, which we denote by $G_{k}^{\HH}$. It is the unique function which satisfies the following properties:

\begin{enumerate}
\item $G_k^\HH$ is a smooth function on $\HH\times\HH-\{\z_1=\z_2\}$ with values in $\R$.
\item $G_k^\HH(\gamma \z_1, \gamma \z_2) = G_k^\HH(\z_1, \z_2)$ for all $\gamma\in SL_2(\R)$.
\item $\Delta_i G_k^\HH = k(1-k) G_k^\HH$, where $\Delta_i$ denotes the Laplace operator with respect to $\z_i$.
\item $G_k^\HH = \log|\z_1-\z_2|^2 + O(1)$ when $\z_1$ tends to $\z_2$.
\item $G_k^\HH$ tends to $0$ when $\z_1$ tends to infinity.
\end{enumerate}

In this section we obtain two formulae. The first formula is for $\delta_1^n\delta_2^m G_k^\HH$, and it involves the hypergeometric series. The second formula is a particular case of the first for $n=k$, and the resulting expression for this case is a rational function of $\z_1, \z_2, \zc_2$. Note that because of the symmetry between $\z_1$ and $\z_2$ this case is similar to the case $m=k$.

Because of the second property the function $G_k^\HH$ is the function of the hyperbolic distance. Denote by $t(\z_1, \z_2)$ the hyperbolic cosine of the hyperbolic distance, i.e.
\[
\begin{split}
t(\z_1, \z_2) = 1 + 2\frac{(\z_1-\z_2)(\zc_2-\zc_1)}{(\z_1-\zc_1)(\z_2-\zc_2)} = -1 + 2\frac{(\z_1-\zc_2)(\z_2-\zc_1)}{(\z_1-\zc_1)(\z_2-\zc_2)}\\
=\frac{-2\z_1\zc_1 + (\z_1+\zc_1)(\z_2+\zc_2) - 2\z_2\zc_2}{(\z_1-\zc_1)(\z_2-\zc_2)}.
\end{split}
\]
Then 
\[
G_k^\HH(\z_1, \z_2) = -2 \calQ_{k-1}(t),
\]
where $\calQ_{k-1}$ is the Legendre's function of the second kind.

The function $\calQ_{k-1}$ has the following two expansions at infinity (see [???]):
\[
\begin{split}
\calQ_{k-1}(t) = \frac{2^{k-1} (k-1)!^2}{(2k-1)!} t^{-k} F(\frac{k}2, \frac{k+1}2; k+\frac12; t^{-2}) \\
= \frac{2^{k-1} (k-1)!^2}{(2k-1)!} (t+1)^{-k} F(k, k; 2k; \frac2{1+t}),
\end{split}
\]
here $F$ denotes the hypergeometric series.
We are going to compute various derivatives of $G_k^\HH$ using the second expansion. For this purpose we first compute:
\[
\delta_1 t = \frac{\partial t}{\partial\z_1} = 2\frac{(\zc_1-\z_2)(\zc_1-\zc_2)}{(\z_1-\zc_1)^2(\z_2-\zc_2)},
\]
\[
\delta_2 t = \frac{\partial t}{\partial\z_2} = 2\frac{(\z_1-\zc_2)(\zc_1-\zc_2)}{(\z_1-\zc_1)(\z_2-\zc_2)^2},
\]
noting that $\delta_1 t$ has weight $2$ in $\z_1$ and weight $0$ in $\z_2$ we compute:
\[
\delta_1^2 t = \delta_2^2 t = 0,
\]
\[
\delta_1 \delta_2 t = -2\left(\frac{\zc_1-\zc_2}{(\z_1-\zc_1)(\z_2-\zc_2)}\right)^2 = \frac{\delta_1 t \delta_2 t}{t+1}.
\]
We will use the following formula for the derivative of the hypergeometric series:
\[
\frac{\partial F(a,b;c;x)}{\partial x} = a \frac{F(a+1,b;c;x)-F(a,b;c;x)}{x}.
\]
We find that
\begin{equation*}
\frac{\partial ((t+1)^{-m} F(m, n; c; \frac2{t+1}))}{\partial t} = 
-m (t+1)^{-m-1} F(m + 1, n; c; \frac2{t+1}), 
\end{equation*}
so
\begin{multline*}
\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{n+1} 2^k \frac{(k-1)! (k+n-1)!}{(2k-1)!} \times \\ 
(t+1)^{-k-n} F(k+n,k;2k;\frac2{t+1}) (\delta_1 t)^n.
\end{multline*}
To apply $\delta_2^m$ we rewrite the last expression as
\begin{multline*}
\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{n+1} 2^k \frac{(k-1)! (k+n-1)!}{(2k-1)!} \times \\
(t+1)^{-k} F(k,k+n;2k;\frac2{t+1}) (\delta_1\delta_2 t)^n (\delta_2 t)^{-n},
\end{multline*}
so we can again apply the same formula, since $\delta_2$ of $\delta_1\delta_2 t$ and $\delta_2 t$ is zero:
\begin{multline*}
\delta_2^m\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{m+n+1} 2^k \frac{(k+m-1)!(k+n-1)!}{(2k-1)!} \times \\
(t+1)^{-k-m} F(k+m,k+n;2k;\frac2{t+1}) (\delta_1\delta_2 t)^n (\delta_2 t)^{m-n}.
\end{multline*}
To make the formula symmetric in $m$ and $n$ we rewrite it as
\begin{multline*}
\delta_2^m\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{m+n+1} 2^k \frac{(k+m-1)!(k+n-1)!}{(2k-1)!} \times \\
(t+1)^{-k-m-n} F(k+m,k+n;2k;\frac2{t+1}) (\delta_1 t)^n (\delta_2 t)^m.
\end{multline*}
Let us introduce the following function of weight $-2$ in $\z_1$ and $0$ in $\z_2$:
\[
Q_{\z_2}(\z_1) = \frac{(\z_1 - \z_2)(\z_1 - \zc_2)}{\z_2-\zc_2},
\]
there is a corresponding function $Q_{\z_1}(\z_2)$. One can check:
\[
\delta_1 t = \frac{t^2 - 1}2 Q_{\z_2}(\z_1)^{-1},
\]
\[
\delta_2 t = \frac{t^2 - 1}2 Q_{\z_1}(\z_2)^{-1},
\]
so our first formula is
\begin{multline}
\delta_2^m\delta_1^n G_k^\HH(\z_1, \z_2) = (-1)^{m+n+1} \frac{(k+m-1)!(k+n-1)!}{(2k-1)!} \times \\
\left(\frac{t+1}2\right)^{-k} \left(\frac{t-1}2\right)^{m+n} F(k+m,k+n;2k;\frac2{t+1}) Q_{\z_2}(\z_1)^{-n} Q_{\z_1}(\z_2)^{-m}.
\end{multline}

In particular, when $n=k$ we obtain
\begin{multline*}
\delta_2^m\delta_1^k G_k^\HH(\z_1, \z_2) = (-1)^{m+k+1} 2^{-m} (k+m-1)! (t+1)^{-k} (t-1)^{k+m} \times\\
F(k+m, 2k; 2k; \frac2{t+1}) Q_{\z_2}(\z_1)^{-k} Q_{\z_1}(\z_2)^{-m},
\end{multline*}
and using the identity
\[
F(a, b; b; x) = (1-x)^{-a}
\]
we get the second formula
\begin{multline}
\delta_1^k \delta_2^m G_k^\HH(\z_1, \z_2) = (-1)^{m+k+1} (k+m-1)! \left(\frac{t+1}2\right)^m Q_{\z_1}(\z_2)^{-m} Q_{\z_2}(\z_1)^{-k}\\
= (-1)^{k-1} (k+m-1)! \frac{(\z_2-\zc_2)^{k-m}}{(\z_1-\z_2)^{k+m}(\z_1-\zc_2)^{k-m}}.
\end{multline}

\section{Global study of Green's functions}
Let $\Gamma$ be a congruence subgroup of $SL_2(\Z)$ and $k>1$. The Green's function on $\HH/\Gamma$ of weight $2k$ is the unique function $G_k^{\HH/\Gamma}$ with the following properties:

\begin{enumerate}
\item $G_k^{\HH/\Gamma}$ is a smooth function on $\HH\times\HH-\{\z_1=\gamma\z_2\ \,|\, \gamma\in\Gamma\}$ with values in $\R$.
\item $G_k^{\HH/\Gamma}(\gamma_1 \z_1, \gamma_2 \z_2) = G_k^{\HH/\Gamma}(\z_1, \z_2)$ for all $\gamma_1, \gamma_2\in \Gamma$.
\item $\Delta_i G_k^{\HH/\Gamma} = k(1-k) G_k^{\HH/\Gamma}$.
\item $G_k^{\HH/\Gamma} = \log|\z_1-\z_2|^2 + O(1)$ when $\z_1$ tends to $\z_2$.
\item $G_k^{\HH/\Gamma}$ tends to $0$ when $\z_1$ tends to a cusp.
\end{enumerate}

The series
\[
\sum_{\gamma\in\Gamma} G_k^\HH(z_1, \gamma z_2)
\]
is convergent and satisfies the properties above, so
\[
G_k^{\HH/\Gamma}(z_1, z_2) = \sum_{\gamma\in\Gamma} G_k^\HH(z_1, \gamma z_2).
\]

Consider the function $G_k^{\HH/\Gamma}(z, z_0)$ for a fixed $z_0\in\HH$. We put 
\begin{multline*}
\G_{k, z_0}^{\HH/\Gamma}(z, X) = \wt{G_k^{\HH/\Gamma}(z, z_0)} \\
= (-1)^{k-1} \binom{2k-2}{k-1}\sum_{l=1-k}^{k-1} (-1)^l \delta^{-l}(Q_z(X)^{k-1}) \delta^l G_k^{\HH/\Gamma}(z, z_0),
\end{multline*}
recall that negative powers of $\delta$ can be defined for eigenfunctions of the Laplacian. Note, that since $G_k^{\HH/\Gamma}(z, z_0)$ has real values and $Q_z(X)$ has imaginary values (we let complex conjugation act on $X$ identically), we have the following
\begin{prop}
The function $i^{k-1} \G_{k, z_0}^{\HH/\Gamma}(z, X)$ has real values.
\end{prop}
Since 
\[
\delta^k(Q_z(X)^{k-1}) = 0,
\]
it is easy to compute, that
\begin{multline*}
\frac{\partial \G_{k, z_0}^{\HH/\Gamma}(z, X)}{\partial z} = \delta \G_{k, z_0}^{\HH/\Gamma}(z, X) = \binom{2k-2}{k-1} \delta^{1-k}(Q_z(X)^{k-1}) \delta^k G_k^{\HH/\Gamma}(z, z_0) \\
= (X-z)^{2k-2} \frac{(-1)^{k-1}\delta^k G_k^{\HH/\Gamma}(z, z_0)}{(k-1)!}.
\end{multline*}
On the other hand, because of the proposition above
\[
\frac{\partial \G_{k, z_0}^{\HH/\Gamma}(z, X)}{\partial \zc} = (X-\zc)^{2k-2} \frac{\overline{\delta^k G_k^{\HH/\Gamma}(z, z_0)}}{(k-1)!}.
\]

Consider the function 
\[
g_{k, z_0}^{\HH/\Gamma}(z) = \frac{(-1)^{k-1} \delta^k G_k^{\HH/\Gamma}(z, z_0)}{(k-1)!}.
\]
This is a meromorphic modular form in $z$. There is a corresponding differential $1$-form with coefficients in $V_{2k-2}$
\[
(X-z)^{2k-2} g_{k, z_0}^{\HH/\Gamma}(z) dz.
\]
Let $V_{2k-2}^\R$ denote the space of polynomials in $V_{2k-2}$ which have real coefficients.
\begin{prop}\label{cohProp}
The class of the differential form
\[
(X-z)^{2k-2} g_{k, z_0}^{\HH/\Gamma}(z) dz
\]
in the cohomology group
\[
H^1(\Gamma, i^{k-1} V_{2k-2}^\R) = H^1(\Gamma, V_{2k-2}/i^k V_{2k-2}^\R)
\]
is trivial and the function 
\[
\frac12 \G_{k, z_0}^{\HH/\Gamma}(z)
\]
is an integral of $\omega$.
\end{prop}
\begin{proof}
Let us denote
\[
\omega = (X-z)^{2k-2} g_{k, z_0}^{\HH/\Gamma}(z) dz.
\]
We have proved before, that
\[
d \G_{k, z_0}^{\HH/\Gamma}(z, X) = \omega + (-1)^{k-1} \bar\omega,
\]
so, that
\[
i^{k-1} d \G_{k, z_0}^{\HH/\Gamma}(z, X) = i^{k-1}\omega + \overline{i^{k-1}\omega}.
\]
It implies, that the integral of $i^{k-1}\omega$ around a pole of $\omega$ is in $i V_{2k-2}^\R$, so the integral of $\omega$ around a pole is in $i^k V_{2k-2}^\R$, that is why $\omega$ satisifes the residue condition of section \ref{integrating} for $A=i^k V_{2k-2}^\R$. Hence the class of $\omega$ in $H^1(\Gamma, V_{2k-2}/i^k V_{2k-2}^\R)$ is correctly defined. Moreover, 
\[
\frac12 d \G_{k, z_0}^{\HH/\Gamma}(z, X) = \omega - \frac{(-1)^{k} \bar\omega + \omega}2,
\]
so
\[
\frac{\G_{k, z_0}^{\HH/\Gamma}(a, X) - \G_{k, z_0}^{\HH/\Gamma}(b, X)}2 \equiv
\int_{b}^a \omega \mod{i^k V_{2k-2}^\R},
\]
which implies that $\omega$ is integrable and
\[
\sigma_\omega^{i^k V_{2k-2}^\R, \Gamma} \equiv 0
\].
\end{proof}

\begin{thm}
For any $z_0\in\HH$ which is not an elliptic point the function $g_{k, z_0}^{\HH/\Gamma}(z)$ is the unique function, which satisfies the following properties:
\begin{enumerate}
\item It is a meromorphic modular form of weight $2k$ in $z$, whose set of poles is $\Gamma z_0$, which is zero at the cusps.
\item In a neighbourhood of $z_0$
\[
g_{k, z_0}^{\HH/\Gamma}(z) = Q_{z_0}(z)^{-k} + O(1)
\]
\item The class of the corresponding differential form in the cohomology group 
\[
H^1(\Gamma, V_{2k-2}/i^{k} V_{2k-2}^\R)
\]
is trivial as in Proposition \ref{cohProp}.
\end{enumerate}
\end{thm}
\begin{proof}
First we prove that the function $g_{k, z_0}^{\HH/\Gamma}(z)$ actually satisifes these conditions. The first two conditions follow from the local study. In fact, we have
\[
\delta^k G_k^{\HH/\Gamma}(z, z_0) = \sum_{\gamma\in\Gamma} \delta^k G_k^{\HH}(z, \gamma z_0),
\]
because for all $0\le l \le k$ 
\[
\delta^l G_k^{\HH/\Gamma}(z, z_0) = O(t(z, z_0)^{l-k} |Q_{z_0}(z)|^{-l}),
\]
and using the inequality
\[
|Q_{z_0}(z)| \geq \frac{t(z, z_0)-1}2 |z-\zc|
\]
we obtain that the series
\[
\sum_{\gamma\in\Gamma} \delta^l G_k^{\HH}(z, \gamma z_0)
\]
is locally uniformly majorated by the series
\[
\sum_{\gamma\in\Gamma} t(z, \gamma z_0)^{-k},
\]
which converges locally uniformly in $z$.

This already implies that the function $g_{k, z_0}^{\HH/\Gamma}(z)$ is meromorphic and has poles of the specified type. The transformation property follows from the invariance of $G_k^{\HH/\Gamma}(z, z_0)$. Since we can move any cusp to $\infty$ by an element of $SL_2(\Z)$ it is enough to check the cuspidality at $\infty$. This is clear from the following:
\begin{multline*}
\delta^k G_k^{\HH/\Gamma}(z, z_0) =\\
(-1)^{k-1} (k-1)! (z_0-\zc_0)^k \sum_{\gamma\in\Gamma} \frac1{(\gamma z- z_0)^k (\gamma z - \zc_0)^k (cz+d)^{2k}}.
\end{multline*}

The last condition is precisely the Proposition \ref{cohProp}.

Now suppose, there are two different functions, which satisfy the conditiones. Then their difference is a cusp form of weight $2k$, which has either purely real or purely imaginary cohomology depending on whether $k$ is odd or even. This contradicts the Eichler-Shimura theorem, which says, that
\[
H_{parabolic}^1(\Gamma, V_{2k-2}) \cong S_{2k} \oplus \overline{S_{2k}},
\]
so any nontrivial parabolic cohomology class which is either purely real or purely imaginary cannot be represented by a cusp form.
\end{proof}

The value of the Green's function can be recovered in the following way:
\begin{thm}
Let $k>1$, $z_0\in\HH$ and $g_{k, z_0}^{\HH/\Gamma}(z)$ be the function that satisfies the conditions of the theorem above. Put
\[
w = (X-z)^{2k-2} g_{k, z_0}^{\HH/\Gamma}(z)
\]
and apply one of the two approaches, formulated in the theorem \ref{int_pairing} for $A = i^k V_{2k-2}^\R$, $B = i^{k-1} V_{2k-2}^\R$, $v=Q_z(X)^{k-1}$ to get an element 
\[
I^{i^k V_{2k-2}^\R, i^{k-1} V_{2k-2}^\R, \Gamma}(\omega, z, Q_z(X)^{k-1}) \in \C/i\R,
\]
for some $z\in \HH$.
Then the real part of this element equals to the number
\[
\frac12 G_k^{\HH/\Gamma}(z, z_0).
\]
\end{thm}
\begin{proof}
We note that the theorem \ref{int_pairing} can be applied since
\begin{enumerate}
\item The class of $\omega$ in $H^1(\Gamma, V_{2k-2}/A)$ is trivial by the third property of the function $g_{k, z_0}^{\HH/\Gamma}(z)$.
\item The whole homology group $H_0(\Gamma, B)$ is trivial because $k>1$.
\end{enumerate}

We can put 
\[
I^{A, \Gamma}(z) = \frac12 \G_{k,z_0}^{\HH/\Gamma}(z)
\]
because of the proposition \ref{cohProp}. This implies that
\[
I^{i^k V_{2k-2}^\R, i^{k-1} V_{2k-2}^\R, \Gamma}(\omega, z, Q_z(X)^{k-1}) \equiv (\frac12 \G_{k,z_0}^{\HH/\Gamma}(z), Q_z(X)^{k-1}) \mod \C/i\R,
\]
so the statement follows from the identity
\[
(\G_{k,z_0}^{\HH/\Gamma}(z), Q_z(X)^{k-1}) = G_k^{\HH/\Gamma}(z, z_0),
\]
which was proved in the section \ref{eigenvalues}.
\end{proof}

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