Math Note 1: Multiplier System

2025/06/15

In this note, we are going to define multiplier systems on any congruence subgroup of $\SL_2(\Z)$, and discuss the cusps as well as the Kloosterman sums defined for cusp pairs. We use the notation $q=e(z):=e^{2\pi i z}$. For simplicity, we write $I=\lp\begin{smallmatrix} 1&0\\0&1\end{smallmatrix}\rp$, $T=\lp\begin{smallmatrix} 1&1\\0&1 \end{smallmatrix}\rp$, and $S=\lp\begin{smallmatrix} 0&-1\\1&0 \end{smallmatrix}\rp$.

We use the following well-known notations for congruence subgroups: \[ \Gamma_0(N)=\lb\gamma\in \SL_2(\Z): \gamma\equiv \lp\begin{smallmatrix} * & * \\ 0 & * \end{smallmatrix}\rp\Mod{N}\rb, \] \[ \Gamma_1(N)=\lb\gamma\in \SL_2(\Z): \gamma\equiv \lp\begin{smallmatrix} 1 & * \\ 0 & 1 \end{smallmatrix}\rp \Mod{N}\rb, \] \[ \Gamma(N)=\lb\gamma\in \SL_2(\Z): \gamma\equiv I \Mod{N}\rb, \] \[ \Gamma_{\Theta}=\lb\gamma\in \SL_2(\Z): \gamma\equiv I \text{ or } S \Mod{2}\rb. \] The last one is called the theta group, which is used in theories like Jacobi theta series.

For $k\in \R$, we fix the argument for complex numbers: $(-\pi,\pi]$.

Definition 1. We say that $\nu:\Gamma\rightarrow \C^\times$ is a multiplier system of weight $k$ if

1. $|\nu|=1$;
2. $\nu(-I)=e^{-\pi i k}$;
3. $\nu(\gamma_1\gamma_2)=w_k(\gamma_1,\gamma_2)\nu(\gamma_1)\nu(\gamma_2)$, where \[w_k(\gamma_1,\gamma_2):=j(\gamma_2,z)^k j(\gamma_1,\gamma_2 z)^k j(\gamma_1\gamma_2,z)^{-k}. \]

The weight of multiplier systems can be restricted to $[0,1)$, because a weight $k$ multiplier system is also of weight $k+2$, and its conjugation $\overline{\nu}$ is of weight $-k$. We also have the properties for $\gamma\in \Gamma$ and $n\in \Z$ such that $T^n\in \Gamma$: \[\nu(\gamma)\nu(\gamma^{-1})=1,\quad \nu(\gamma T^n)=\nu(\gamma) \nu(T)^n. \]

We want to define Kloosterman sums with multiplier systems in general. For a congruence subgroup $\Gamma$, let $\Gamma_{\mathfrak a}$ denote the stabilizer of cusp $\mathfrak a$ in $\Gamma$: \[\Gamma_{\mathfrak a}:=\lb \gamma\in \Gamma: \gamma \mathfrak a=\mathfrak a\rb. \] The first example is for the cusp $\infty$: let $h\in \Z_+$ be the smallest integer such that $T^b\in \Gamma$, then \[\Gamma_\infty = \langle T^h \rangle =\lb \pm \lp \begin{smallmatrix} 1 & hn \\ 0 & 1 \end{smallmatrix} \rp: n\in \Z\rb. \] We call $h$ the width of cusp $\infty$.

The stabilizer $\Gamma_{\infty}$ is the start point for us to talk about other cusps. Indeed, we restrict our discussion in congruence subgroup $\Gamma$ to ensure that $\infty$ is a cusp of $\Gamma$ (because $\exists N\in \Z_+$ s.t. $\Gamma(N)\subseteq \Gamma$). We use the so called "scaling matrix" to "move" conditions for another cusp to cusp $\infty$ to discuss. Specifically, we define the scaling matrix $\sigma_{\mathfrak a}$ to be any matrix in $\SL_2(\R)$ satisfying the following conditions: \[\sigma_{\mathfrak a}\infty =a\quad \text{and}\quad \sigma_{\mathfrak a}^{-1}\Gamma_{\mathfrak a}\sigma_{\mathfrak a}=\Gamma_{\infty}. \] We have the following important properties of scaling matrices:

1. If $\sigma_{\mathfrak a}$ is a scaling matrix of $\mathfrak a$, then $\pm\sigma_{\mathfrak a} T^{hn}$ is also a scaling matrix of $\mathfrak a$ for any $n\in \Z$, and vice versa. In other words, the set of all scaling matrices of $\mathfrak a$ is given by $\sigma_{\mathfrak a} \Gamma_{\infty}$.
2. We do not always have $\nu(T^h)=1$. Let us define $\kappa_\infty$ by \[\nu(T^h)=e(-\kappa_\infty),\quad \text{for }\kappa_\infty\in [0,1). \] Let $n_\infty:=n-\kappa_\infty$ for $n\in \Z$. This shows how modular form changes the phase when $z\rightarrow z+h$, i.e. the modular form with transformation formula $f(\gamma z)=\nu(\gamma)j(\gamma,z)^k f(z)$ should have the Fourier expansion \[f(z)=\sum_{n\in \Z} a_f(n) e\lp \frac{n_\infty z}{h} \rp. \]
3. The definition for width of another cusp $\ma\neq_{\Gamma}\infty$ is given by the smallest integer $h_{\ma}$ such that \[\gamma_{\ma} T^{h_{\ma}} \gamma_{\ma}^{-1} \in \Gamma_{\ma},\quad \text{where }\gamma_{\ma}\infty =\ma. \] Our choice of $\sigma_{\ma}$ here ensures that $h_{\ma}=h$. Another reason might be $\Gamma(N)\subseteq \Gamma$ and $\Gamma(N)\triangleleft \SL_2(\Z). $
4. Similarly, we can define this phase change for other cusps. Let $\kappa_{\mathfrak a}$ be defined by \[\nu(\sigma_{\mathfrak a} T^b \sigma_{\mathfrak a}^{-1}) = e(-\kappa_{\mathfrak a})\quad \text{for }\kappa_{\mathfrak a}\in [0,1)\] and define $n_{\mathfrak a}:=n-\kappa_{\mathfrak a}$ for $n\in \Z$, then the Fourier expansion of $f$ at cusp $\mathfrak a$ is given by \[(f|_k \sigma_{\ma})(z)=j(\sigma_{\mathfrak a},z)^{-k} f(\sigma_{\mathfrak a} z)= \sum_{n\in \Z} a_{f,\mathfrak a}(n) e\lp \frac{n_{\mathfrak a} z}{h}\rp\]
5. For the cusp pair $(\mathfrak a,\mathfrak b)$ of $\Gamma$, we define $\nu_{\ma\mb}(\gamma)$ for $\gamma\in \Gamma'=\sigma_{\mathfrak a}^{-1}\Gamma \sigma_{\mathfrak b}$ by \[ \begin{align*} \nu_{\mathfrak a\mathfrak b}(\gamma)&:=\nu(\sigma_{\mathfrak a}\gamma \sigma_{\mathfrak b}^{-1})w_k(\sigma_{\mathfrak a}^{-1},\sigma_{\mathfrak a}\gamma \sigma_{\mathfrak b}^{-1}) w_k(\gamma \sigma_{\mathfrak b}^{-1},\sigma_{\mathfrak b})\\ &=\nu(\sigma_{\mathfrak a} \gamma \sigma_{\mathfrak b}^{-1}) \frac{w_k(\sigma_{\mathfrak a} \gamma \sigma_{\mathfrak b}^{-1},\sigma_{\mathfrak b})}{w_k(\sigma_{\mathfrak a},\gamma)} \end{align*} \] (exercise: verify the last equation).

Lemma 2. The expression \[\overline{\nu_{\ma\mb}(\gamma)}\, e\lp\frac{-\kappa_{\ma}a-\kappa_{\mb}d}{hc}\rp\] is well-defined under \[\gamma=\lp\begin{smallmatrix} a&b\\c&d\end{smallmatrix}\rp \in \Gamma_{\infty}\setminus\sigma_{\ma}^{-1}\Gamma\sigma_{\mb}\mathrel{/}\Gamma_{\infty}.\]

With this lemma, we can define our general Kloosterman sums with multiplier system:

Definition 3. For $m,n\in \Z$ and cusps $\ma,\mb$ of $\Gamma$, if $c\in \R$ is a possible moduli (a possible "bottom-left entry") of matrices in $\sigma_{\ma}^{-1}\Gamma\sigma_{\mb}$, then we define \[S_{\ma\mb}(m,n,c,\nu):=\sum_{\gamma=\lp\begin{smallmatrix} a&b\\c&d\end{smallmatrix}\rp \in \Gamma_{\infty}\setminus\sigma_{\ma}^{-1}\Gamma\sigma_{\mb}\mathrel{/}\Gamma_{\infty}} \overline{\nu_{\ma\mb}}(\gamma) e\lp \frac{m_{\ma}a+n_{\mb}d}{hc}\rp. \]

In the next note, we will see how this general Kloosterman sum arises from Poincaré series.