A **quasi-modular form** of weight $k$ and multiplier system $v$ for a fuchsian group $G$ is a function $f$ from $\mathfrak{H}$ to $\mathbb{C}$ which is
holomorphic and satisfies a certain growth condition and, for any $\gamma=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\in G$ and
$z\in\mathfrak{H}$, we have:
\[
(f|_k\gamma)(z)=v(\gamma)P\left(z,\frac{c}{cz+d}\right),
\]
where $P$ is a polynomial in two variables whose coefficients depend only on $f$
and not on $\gamma$ and $f|_k\gamma$ is the weight $k$ slash-action. The *degree* of $P$ is called the *depth* of $f$.

An example of a quasi-modular form is the Eisenstein series $E_2(z)=1-24\sum_{n\geq1}\sigma_1(n)q^n$, where $q=e^{2\pi iz}$ and $\sigma_1$ is a divisor function. It has weight $2$ and depth $1$ on $\Gamma$ and it has the following transformation property: \[ E_2(\gamma z)=(cz+d)^2E_2(z)-\frac{6i}{\pi}c(cz+d). \]

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**Knowl status:**

- Review status: beta
- Last edited by Andrew Sutherland on 2018-12-13 05:58:26

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