Let $G$ be a subgroup of $\Gamma$ of finite index $m$, $v$ a multiplier system on $G$ and $k\geq3\in\mathbb{N}$ such that $v(-I)=(-1)^k$. Let $s$ be a cusp representative of $G$ with width $w$ such that $s=\gamma_s(\infty)$ for some $\gamma_s\in\Gamma$ and let $G_s=G\cap\gamma_s\Gamma_\infty\gamma_s^{-1}$ be the *stabilizer* of the cusp $s$. Assume that $v$ is trivial on $G_s$. The **$n$th Poincaré series** for the cusp that $s$ represents is defined as
\[
P_k^n(G,v;z;\overline{s})=\sum_{g\in G_s\setminus G}\overline{v(g)}e^{2\pi
i(n/w)\gamma_s^{-1}gz}j(\gamma_s^{-1}g,z)^{-k},
\]
where $j(\gamma,z)=cz+d$ for $\gamma=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)$.

Poincaré series are *holomorphic* on the upper half-plane. Further more, for a fixed cusp $\overline{S}$, they generate the corresponding space of cusp forms. The series depends on the choice of $\gamma_s$ up to a factor of $e^{2\pi in/w}$, therefore, if $w\mid n$, it does not depend on the cusp representative.

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- Review status: beta
- Last edited by Andrew Sutherland on 2018-12-13 06:00:05

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