2025/06/22
Last time we talked about defining a general multiplier system (allowing non-singular cusps) and Kloosterman sum on cusp pairs. In this note, we compute the Fourier expansion of Poincaré series and the Fourier coefficients will show as sums of Kloosterman sums.
Let us fix a congruence subgroup $\Gamma$ of $\SL_2(\Z)$ and a weight $k$ multiplier system on $\Gamma$. Recall the definition of multiplier systems in Note 1. Especially, we need the notation $w_k(\gamma_1,\gamma_2)$ and for $\Gamma_{\infty}=\langle \pm T^h \rangle$, \[\nu(\sigma_{\ma} T^h \sigma_{\ma}^{-1}) = e(-\kappa_{\ma}) \quad \text{for }\kappa_{\ma}\in [0,1). \]
Exercise 1. Verify that $\pi:\gamma\times \H\rightarrow \C$ given by \[\pi_{\ma}(\gamma,z):=\overline{\nu}(\gamma) \overline{w_k}(\sigma_{\ma}^{-1},\gamma)j(\sigma_{\ma}^{-1}\gamma,z)^{-k} e(m_{\ma}\sigma_{\ma}^{-1}\gamma z/h)\] depends only on the coset $\Gamma_{\ma}\gamma$, i.e. it is invariant if we change $\gamma$ to $\gamma_{\ma}\gamma$ for any $\gamma_{\ma}\in \Gamma_{\ma}$. Here $m_{\ma}=m-\kappa_{\ma}$ for $m\in \Z$.
I would like to call $e(z/h)$ the "seed function". Depending on the question we are facing to, there might be different choices, and may change the form of $\pi_{\ma}(\gamma,z)$ and may involve other parameters. Here we discuss about the basic one $e(z/h)$.Definition 2. The Poincaré series on cusp $\ma$ is defined by \[P_{\ma}(z):=\sum_{\gamma\in \Gamma_{\ma}\setminus \Gamma} \pi_{\ma}(\gamma,z)=\sum_{\gamma\in \Gamma_{\ma}\setminus \Gamma} \overline{\nu}(\gamma)\overline{w_k}(\sigma_{\ma}^{-1},\gamma) j(\sigma_{\ma}^{-1}\gamma,z)^{-k} e(m_{\ma}\sigma_{\ma}^{-1}\gamma z/h). \]
Next we compute the Fourier expansion of $P_{\ma}(z)$ at cusp $\mb$.
Exercise 3. Using the notation from the previous note, show that \[(P_{\ma}|_k\sigma_{\mb})(z):=j(\sigma_{\mb},z)^{-k} P_{\ma}(\sigma_{\mb} z)=\sum_{\gamma\in \Gamma_{\infty}\setminus \sigma_{\ma}^{-1}\Gamma \sigma_{\mb}} \overline{\nu_{\ma\mb}}(\gamma) j(\gamma,z)^{-k} e(m_{\ma}\gamma z/h). \]
Hint: note that \[ \begin{align} (\Gamma_{\ma}\setminus \Gamma)\sigma_\mb&=\lb\Gamma_{\ma}\gamma \sigma_{\mb}:\gamma\in \Gamma \rb=\lb \sigma_{\ma}\Gamma_{\infty}\sigma_{\ma}^{-1}\gamma\sigma_{\mb}:\gamma\in \Gamma \rb\\ &=\sigma_{\ma}(\Gamma_{\infty}\setminus \sigma_{\ma}^{-1}\Gamma \sigma_{\mb}). \end{align} \]
Lemma 4. We have \[\nu_{\ma\mb}(\gamma T^h) = \nu_{\ma\mb}(\gamma)e(-\kappa_{\mb}). \]
With the help of the above exercise and lemma, we are ready to compute the expansion now. Since $\ma =_{\Gamma}\mb$ if and only if $I\in \sigma_{\ma}^{-1}\Gamma \sigma_{\mb}$, the expansion begins with \[(P_{\ma}|_k\sigma_{\mb})(z)=e(m_{\ma}z/h)\delta_{\ma\mb} +\sum_{\substack{\gamma\in \Gamma_{\infty}\setminus \sigma_{\ma}^{-1}\Gamma \sigma_{\mb}/\Gamma_{\infty } \\ \gamma=\lp\begin{smallmatrix} a&*\\c&d \end{smallmatrix}\rp \neq I}}\frac{\overline{\nu_{\ma\mb}}(\gamma)}{c^k} e\lp \frac{m_{\ma}a}{hc}\rp f\lp z+\frac dc\rp, \] where \[f(z):=\sum_{t\in \Z} \frac{e(t \kappa_{\mb})}{(z+th)^k} e\lp \frac{-m_{\ma}}{hc^2(z+th)}\rp \] has the property that $f(z)e(\kappa_{\mb} z/h)$ has period $h$.
So $f(z)e(\kappa_b z/h)=\sum_{n\in \Z} a(n,y) e(nx/h)$ with \[a(n,y)=\frac{e^{-2\pi ny/h}}{h}\int_{-\infty+iy}^{\infty+iy} z^{-k} e\lp \frac {-m_{\ma}}{hc^2z}-\frac{n_{\mb}z}h \rp dz. \]
We get $a(n,y)=0$ if $n_{\mb}\leq 0$.
When $n_{\mb}>0$ and $m_{\ma}=0$, ... gives \[\int_{-\infty}^\infty (y+ix)^{-k} e^{2\pi i n_{\mb} x/h }dx = 2\pi e^{-2\pi n_{\mb} y/h} (2\pi n_{\mb}/h)^{k-1}\Gamma(k)^{-1}\] and hence \[a(n,y)=e(-\tfrac k4)(2\pi)^{k} n_{\mb}^{k-1} h^{-k}e^{-2\pi n_{\mb}y/h} \Gamma(k)^{-1}. \]
For $n_{\mb}>0$, $m_{\ma}>0$, \[\begin{align} a(n,y)&= \frac {e(-\frac k4) }h \sum_{j=0}^\infty \frac 1{j!}\int_{-\infty}^{\infty} \frac{\lp-\frac{2\pi m_{\ma}}{hc^2}\rp^j}{(y+ix)^{k+j}} e^{2\pi i n_{\mb}x/h} dx\\ &=e(-\tfrac k4)(2\pi)^k n_{\mb}^{k-1} h^{-k} \sum_{j=0}^\infty \frac{\lp -\frac{16\pi^2 m_{\ma}n_{\mb}}{4h^2c^2}\rp^j}{j!\Gamma(j+k-1+1)}\\ &=\frac{2\pi e(-\tfrac k4)}{c^{1-k} h}\lp \frac nm \rp^{\frac{k-1}2} J_{k-1}\lp\frac{4\pi \sqrt{m_{\ma}n_{\mb}}}{hc}\rp. \end{align} \]
Finally we get that for $m_{\ma}>0$, \[(P_{\ma}|_k\sigma_{\mb})(z)=e\lp \frac{m_{\ma}z}h\rp\delta_{\ma\mb}+\frac {2\pi e(-\frac k4)}h\sum_{n_{\mb}>0}\lp\frac nm\rp^{\frac{k-1}2}\sum_{c>0}\frac{S_{\ma\mb}(m,n,c,\nu)}{c} J_{k-1} \lp\frac{4\pi \sqrt{m_{\ma}n_{\mb}}}{hc}\rp, \] where $n_{\mb}>0$ is for all $n\in \Z$ such that $n-\kappa_{\mb}>0$, and $c>0$ is for $c\in \R_+$ admissible for $\gamma=\lp\begin{smallmatrix} a&*\\c&d \end{smallmatrix}\rp \in \sigma_{\ma}^{-1}\Gamma \sigma_{\mb}$.