Demystifying modular forms

Question

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:

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

Answer 1

The definition of a modular form seems extremely unmotivated, and as @AndreaMori has pointed out, whilst the complex analytic approach gives us the quickest route to a definition, it also clouds some of what is really going on.

A good place to start is with the theory of elliptic curves, which have long been objects of geometric and arithmetic interest. One definition of an elliptic curve (over $\mathbb C$) is a quotient of $\mathbb C$ by a lattice $\Lambda = \mathbb Z\tau_1\oplus\mathbb Z\tau_2$, where $\tau_1,\tau_2\in\mathbb C$ are linearly independent over $\mathbb R$ ($\mathbb C$ and $\Lambda$ are viewed as additive groups): i.e. $$E\cong \mathbb C/\Lambda.$$

In this viewpoint, one can study elliptic curves by studying lattices $\Lambda\subset\mathbb C$. Modular forms will correspond to certain functions of lattices, and by extension, to certain functions of elliptic curves.

Why the upper half plane?

For simplicity, since $\mathbb Z\tau_1 = \mathbb Z(-\tau_1)$, there's no harm in assuming that $\frac{\tau_1}{\tau_2}\in \mathbb H$.

What about $\mathrm{SL}_2(\mathbb Z)$?

When do $(\tau_1,\tau_2)$ and $(\tau_1',\tau_2')$ define the same lattice? Exactly when $$(\tau_1',\tau_2')=(a\tau_1+b\tau_2,c\tau_1+d\tau_2)$$where $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb Z)$. Hence, if we want to consider functions on lattices, they had better be invariant under $\mathrm{SL}_2(\mathbb Z)$.

Functions on lattices:

Suppose we have a function $$F:\{\text{Lattices}\}\to\mathbb C.$$ First observe that multiplying a lattice by a non-zero scalar (i.e. $\lambda\Lambda$ for $\lambda\in\mathbb C^\times$) amounts to rotating and rescaling the lattice. So our function shouldn't do anything crazy to rescaled lattices.

In fact, since we really care about elliptic curves, and $\mathbb C/\Lambda\cong\mathbb C/\lambda\Lambda$ under the isomorphism $z\mapsto \lambda z$, $F$ should be completely invariant under such rescalings - i.e. we should insist that

$$F(\lambda \Lambda) = F(\Lambda).$$ However, if we define $F$ like this, we are forced to insist that $F$ has no poles. This is needlessly restrictive. So what we do instead is require that $$F(\lambda\Lambda) = \lambda^{-k}F(\Lambda)$$ for some integer $k$; the quotient $F/G$ of two weight $k$ functions gives a fully invariant function, this time with poles allowed.

Where do modular forms come in?

If $\Lambda = \mathbb Z\tau\oplus\mathbb Z$ with $\tau\in\mathbb H$, define a function $f:\mathbb H\to\mathbb C$ by $f(\tau)=F(\Lambda)$. For a general lattice, we have

$$\begin{align}F(\mathbb Z\tau_1\oplus\mathbb Z\tau_2)&=F\left(\tau_2(\mathbb Z({\tau_1}/{\tau_2})\oplus\mathbb Z)\right)\\ &=\tau_2^{-k}f({\tau_1}/{\tau_2}) \end{align}$$ and in particular, $$\begin{align}f(\tau) &= F(\mathbb Z\tau\oplus\mathbb Z) \\&=F(\mathbb Z(a\tau+b)\oplus\mathbb Z(c\tau+d)) &\text{by }\mathrm{SL}_2(\mathbb Z)\text{ invariance}\\&= (c\tau+d)^{-k} f\left(\frac{a\tau+b}{c\tau+d}\right).\end{align}$$

This answers your first two questions.

At this point, there's no reason to assume that condition (3) holds, and one can study such functions without assuming condition (3). However, imposing cusp conditions is a useful thing to do, as it ensures that the space of weight $k$ modular forms is finite dimensional.

To answer your fourth question, yes, and this is exactly the viewpoint taken in most research done on modular forms and their generalisations, where one considers automorphic representations.

Answer 2

(a) ${\rm SL}_2(\Bbb Z)$ is a group in the sense that is an example of the algebraic structure called group. :)

(b) That's not the group operation. The group operation in ${\rm SL}_2(\Bbb Z)$ (and in fact in any linear group) is matrix multiplication. What you describe is the action of the group ${\rm SL}_2(\Bbb Z)$ on the upper halfplane $\cal H$.

(c) Exactly.

(d) Modular forms lift to functions in $L^2({\rm GL}_2(\Bbb Q)\backslash{\rm GL}_2(\Bbb A))$ where $\Bbb A$ is the ring of rational adeles.

(e) I would say no. The importance of their Fourier coefficients becomes evident only after the Hecke operators are introduced. The analysis of the way Hecke operators act on modular forms is the main introductory link between the theory of modular forms and arithmetic since, for instance, it allows to show that the space of modular forms of fixed weight (which is finite dimensional) has a basis of modular forms with Fourier coefficients in $\Bbb Z$.