## Franklin's Notes

### Modular form

Given a natural number $k$ and a subgroup $\Gamma\subset \mathrm{SL}_2(\mathbb Z)$ of the special linear group with elements represented as mobius transformations preserving the upper half-plane $\mathbb H$, we always have that the orbit of a point $\tau\in\mathbb C$ is a discrete subset of $\mathbb C$. Given a holomorphic function $f:\mathbb C\to\mathbb C$, we say that $f$ is a modular form of weight $k$ with respect to $\Gamma$ if the following two facts hold:

1. For all $\gamma\in\Gamma$, $(c\tau+d)^{-k}f(\gamma\tau)=f(\tau)$. The quantity $(c\tau+d)^{-k}f(\gamma\tau)$ is sometimes written $(f|\gamma)(\tau)$.
2. At the cusps of $\mathbb H$, or the points of $\mathbb Q\cup{i\infty}=\mathbb P^1(\mathbb Q)$, $f$ exhibits at most polynomial growth. (If it vanishes at these points, it is called a cusp form.)

The space of all modular forms of weight $k$ with respect to $\Gamma$ is sometimes denoted $\mathscr{M}_k(\Gamma)$, and the subspace of cusp forms of weight $k$ with respect to $\Gamma$ is denoted $\mathscr{S}_k(\Gamma)$. An interesting fact is that these spaces are always finite-dimensional, a fact which gives rise to many nontrivial relationships between the modular forms of certain fixed weights.

Some examples of modular forms include the following. We have the family of Eisenstein series of weight $k$: where $q=e^{2\pi i \tau}$. We also have Ramanujan's $\Delta$ function, which is a cusp form of weight $12$: and we also have the following form of weight $2$: