I am really struggling to understand what modular forms are and how I should think of them. Unfortunately I often see others being in the same shoes as me when it comes to modular forms, I imagine because the amount of background knowledge needed to fully appreciate and grasp the constructions and methods is rather large, so hopefully with this post some clarity can be offered, also for future readers.. The usual definitions one comes across are often of the form: (here taken from wikipedia)

A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.

A modular form of weight $k$ for the modular group $$ \text{SL}(2,\mathbb{Z})=\left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}| a,b,c,d \in \mathbb{Z} , ad-bc = 1 \right\} $$ is a complex-valued function $f$ on the upper half-plane $\mathbf{H}=\{z \in \mathbb{C},\text{Im}(z)>0 \}$, satisfying the following three conditions:

- $f$ is a holomorphic function on $\mathbf{H}.$
- For any $z \in \mathbf{H}$ and any matrix in $\text{SL}(2,\mathbb{Z})$ as above, we have: $$ f\left(\frac{az+b}{cz+d}\right)=(cz+d)^k f(z) $$
- $f$ is required to be holomorphic as $z\to i\infty.$

**Questions**:

- (a): I guess what I'm having least familiarity with is the
*modular group*part. My interpretation of $\text{SL}(2,\mathbb{Z}):$*The set of all $2$ by $2$ matrices, with integer components, having their determinant equal to $1.$*But where does the name come from, as in why do we call this set a group and what modular entails? - (b): If I understand correctly, the group operation here is function composition, of type: $\begin{pmatrix}a & b \\ c & d\end{pmatrix}z = \frac{az+b}{cz+d}$ which is also called a linear fractional transformation. How should one interpret the condition $2.$ that $f$ has to satisfy? My observation is that, as a result of the group operation of $\text{SL}$ on a given integer $z,$ the corresponding image is multiplied by a polynomial of order $k$ (which is the weight of the modular form).
- (c) The condition $3.$ I interpret as: $f$ should not exhibit any poles in the upper half plane, not even at infinity. About right?
- (d) A more general question: Given the definition above, it is tempting to see modular forms as particular classes of functions, much like the Schwartz class of functions, or $L^p$ functions and so on. Is this an acceptable assessment of modular forms?
- (e) Last question: It is often said that modular forms have interesting Fourier transforms, as in their Fourier coefficients are often interesting (or known) sequences. Is there an intuitive way of seeing, from the definition of modular forms, the above expectation of their Fourier transforms?