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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?
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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.

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  • $\begingroup$ Your interpretation of $\Lambda$ as a set is correct. Both $\mathbb C$ and $\Lambda$ are groups under addition. Quotient here means the quotient group - i.e. addition in $\mathbb C$ where two elements are viewed as the same if they differ by a lattice element. $\endgroup$ – Mathmo123 Apr 8 '16 at 11:51
  • $\begingroup$ For (iii), it's easier to see on the modular forms side (I haven't defined what the analogue of holomorphic would be for lattices). A function $f:\mathbb H\to\mathbb C$ can't have any poles in $\mathbb H$ since it must be defined on the whole of $\mathbb H$. It can, however, have poles at the boundary (i.e. at $\infty$). However, it can have zeroes, so the quotient of two modular forms will be a meromorphic function, possibly with poles. $\endgroup$ – Mathmo123 Apr 8 '16 at 11:55
  • $\begingroup$ For (iv), yes we can start from either side. $\endgroup$ – Mathmo123 Apr 8 '16 at 11:56
  • $\begingroup$ Just to give you an example for (i), take $\Lambda = \mathbb Z \oplus\mathbb Zi = \{a+bi : a,b\in\mathbb Z\}$. This is a subgroup of $\mathbb C$. You can picture the quotient as a unit square in $\mathbb C$ where opposite edges are considered the same - topologically, this is a torus. $\endgroup$ – Mathmo123 Apr 8 '16 at 12:00
  • $\begingroup$ Thank you very much for your replies, very helpful. I'm slowly grasping the concept of taking the quotient, it is so incredibly counter intuitive :( One last thing, sorry to ask again, but I did not fully understand your answer to (iii), because if what is important is the ratio of two weight $k$ functions, then how do the cases differ $F(\lambda \Lambda)/G(\lambda \Lambda) = F(\Lambda)/G(\Lambda)$ for the first case, and $F(\lambda \Lambda)/G(\lambda \Lambda) = \lambda^{-k}F(\Lambda)/\lambda^{-k}G(\Lambda) = F(\Lambda)/G(\Lambda)$ for the 2nd case, failing to see their difference pole-wise. $\endgroup$ – user929304 Apr 8 '16 at 12:27
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(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 $\Bbb Z$.

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  • $\begingroup$ Thanks for your prompt answer. Would you be so kind to add a tad more explanation to your answers of (b) and (d)? For (b), I mean the condition 2 that $f$ has to satisfy, on the transformed $z$ (i.e. after the action of the group on $z$), looks very peculiar, is there any other insight to it other than the observation I had made? For (d), so you mean modular forms are square integrable functions that are not in Hilbert space? I'm confused with your notation of $L^2({\rm SL}_2(\Bbb Z)\backslash \cal H)$, because $\text{SL}_2$ and $\cal H$ are not spaces of same type. Thx for any clarification $\endgroup$ – user929304 Apr 6 '16 at 13:40
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    $\begingroup$ @user929304: About (b), the automorphic factor $(cz+d)^k$ is a cocycle that allows to regard modular forms as (holomorphic) global sections of certain line bundles over the Riemann surfaces obtained taking the quotient of $\cal H$ by (subgroups of) the modular group. I had formulated (d) incorrectly and edited now, but in general if you have a space $X$ acted upon by a group $\Gamma$ you may consider the space of orbits $\Gamma\backslash X$ which under some conditions on the action will be "as nice as" $X$. $\endgroup$ – Andrea Mori Apr 6 '16 at 17:23
  • $\begingroup$ Thanks for the edit and your comment. I have no doubt in the correctness of you answer, but honestly I feel it is very terse for my level of understanding, e.g. from your point (d) or "...is a cocycle that allows to regard modular forms as (holomorphic) global sections of certain line bundles over the Riemann surfaces obtained taking the quotient of..." I understand close to nothing :( $\endgroup$ – user929304 Apr 7 '16 at 9:32
  • $\begingroup$ @user929304: I understand your feelings which have been shared by everyone's first contacts with higher mathematics. Modular forms are a rich and deep theory that sits at the crossroad of many important branches of mathematics. The complex analytic approach is certainly the most elementary but, in some sense, the most obscure since many properties appear casual, somewhat artificial and overall magic. To appreciate the other points of view (where for instance the role of higher arithmetic becomes paramount) some general knowledge of other mathematical theories is required. $\endgroup$ – Andrea Mori Apr 7 '16 at 13:11
  • $\begingroup$ @user929304: I can just hope that you'll grow to see this deep richness more as a motivation than a put-off. $\endgroup$ – Andrea Mori Apr 7 '16 at 13:13

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