What are Modular Forms?

Modular forms are some of the most bizarre and wonderful objects in mathematics. They are one of the most esoteric entities in mathematics, and yet the twentieth-century number theorist Martin Eichler rated them as one of the five fundamental operations: addition, subtraction, multiplication, division, and modular forms. - Simon Singh, Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem

Many people have heard the term “modular form” because of its crucial role in solving Fermat’s Last Theorem.

If you’ve never read Simon Singh’s book on the history and solving of it, or seen the BBC special on it, I highly recommend checking them out. They present some of the most realistic depictions of the mathematical process: from years of frustrations to correcting errors to overwhelming emotion at finally pinning it down.

Still, everyone will come away from those accountings wondering what on Earth a modular form is. You’ll merely get some vague references to how “symmetric” it is.

And there’s a good reason for that! There’s no way they could present such an abstract concept to a general audience.

But we’re at Cantor’s Paradise, so let’s give this a shot.

The Formula

This is going to be the least interesting, but most rigorous, section. Modular forms can be fully defined by a simple formula.

First, I’ll define ℍ to be the (complex) upper half-plane. If you write z=x+yi then it is all such imaginary numbers where y>0.

A modular form is a (complex) analytic function f: ℍ→ℍ that satisfies the following formula:

for all integers a,b,c, and d such that ad-bc=1. There’s another technical condition, but it’s not important for the concept in this article.

The number k, which we’ll also say is an integer greater than 0, is called the weight.

I’m not going to leave you hanging here. I get that it looks really weird if that’s your first time seeing it. Let’s unravel what we can and save a better description for the part.

First off, that’s a lot of symmetry.

For any a,b,c,d we get a new condition the function has to satisfy. For example, plug in a=1, b=1, c=0, d=1. You get f(z+1)=f(z). This shows that the function repeats every shift by 1 (aka it’s periodic).

This is what is meant by a symmetry condition.

The definition above shows that modular forms have infinitely many symmetry conditions they must meet simultaneously because there are infinitely many a,b,c,d we could plug in.

It’s a nice exercise to prove that there are infinitely many a,b,c,d that satisfy ad-bc=1 if that’s not obvious to you (remember they must be integers!).

Next, the fraction that occurs is of the form (az+b)/(cz+d). These are called Linear Fractional Transformations (LFTs).

These are just combinations of inversion f(z)=-1/z and shifting/rotating (linear functions), f(z)=az+b. Again, you could even try to prove this yourself, but it’s a bit harder.

The condition that ad-bc=1 is just making sure we can define things since we only required f to be defined on the upper half-plane, ℍ.

Secretly, there’s a matrix determinant at work, as you might have guessed. Requiring a,b,c,d to be integers and considering the matrix

tells us we are working with elements of a group called SL(2, ℤ), the special linear group of 2x2 matrices with integer entries and determinant 1.

I don’t want to get too far off-topic here, but we can associate matrices to these LFT’s. Matrix multiplication corresponds to composing two of the LFT’s. That’s the main reason for the condition to have integers and the condition ad-bc=1.

Of course, this doesn’t help us see where such a thing would come from. So, let’s move on to a more conceptual understanding.

The Concept

Why are they called “modular forms?” I know someone must be asking that.

Well, that brings us to this second description. This one will be more abstract, but it’s also the essence of where those conditions come from. It will help us understand the naming better, too.

Let’s think about modular arithmetic, sometimes called “clock arithmetic” in popular math articles. In modular arithmetic, numbers wrap around like on a clock.

One way to think about this is that you use all normal integers and then “reduce mod 12” to get a number on the clock.

If you wonder what time it will be two hours after 11 o’clock, you do 11+2=13. Then 13 mod 12 = 1. So it will be 1 o’clock.

Under this system 13 and 1 are treated as the same. Also, -1 is 11 and 320 is 8. Infinitely many usual numbers collapse onto the same number mod 12.

The way to construct this “geometrically” is exactly like a clock. Take the number line and just wrap it in a circle so that 0 and 12 line up and keep wrapping infinitely.

Observation: This space isn’t “weird” at all since it’s just a circle. But we could imagine generalizing this process of folding/wrapping something “easy” and “well-understood” like the number line or ℝ² or ℂ or ℍ in a weird way and getting something we don’t understand.

If we wanted to define a function on this clock space, we could easily define it directly. But a better way to do it would be to define it on all numbers (the easy, understandable thing) so that the places where it folds onto itself matches up.

This might be confusing if you’ve never seen it before. The point is that on the unfolded thing, we can use normal formulas and functions we’ve seen before.

For instance, the function f: ℤ→ℝ given by f(x)=x makes sense.

But that function doesn’t “descend” to the clock space because when it wraps around to 13 (which “is” 1 on the clock), we don’t know if f(1)=1 or if f(1)=f(13)=13.

So the functions on all of ℤ have to have an extra symmetry conditionf(x+12)=f(x) to make sense on the clock space.

Now hopefully you see where this is going. It’s far beyond the scope of this article to actually construct “modular curves,” but they are made by folding ℍ in a complicated way.

The symmetry conditions in the definition of a modular form are just saying that since we wrote the function using all of ℍ, it has to make sense as a function after “descending” to the folded space.

Technically, if the weight is k, then it’s a differential k-form on the folded space. This is where the name comes from. The folded space is a modular curve, and the function is a differential form. Hence: modular form.

Do Modular Forms Exist?

There’s a joke, or real story, I never remember which, where a grad student writes a Ph.D. thesis on a special class of functions with lots of symmetry and possible applications to mechanical engineering.

It contains really great technical work. At the end of the thesis defense, one of the committee members asks if these functions even exist.

The student points out that the 0 function satisfies the definition. The committee member then produces a proof that the 0 function is the only function satisfying the conditions.

In math, it’s always possible to define something, but it’s good to check that there are actually examples or you’re just proving results about the 0 function.

So, as a serious question: do modular forms exist?

If your immediate answer is yes (other than the 0 function), then you might not quite have grasped just how incredibly symmetric these things are.

Constant functions don’t work (if k>0). 
Polynomials don’t work.

Just plug in a=0, b=-1, c=1, d=0 to get f(-1/z)=zᵏ f(z). That’s pretty weird and that’s only one of the infinitely many conditions to be satisfied.

It turns out that if k is odd, there are none (this is actually a straightforward exercise from things said in this article if you want to try it)!

If k < 12, there are none!

Hopefully, these facts alone help you understand just how symmetric these things have to be when someone says they “have a lot of symmetry.”

It turns out that infinitely many do exist, but they aren’t easy to write down. The space of modular forms of weight k is a finite-dimensional vector space, and the dimension of this space has an explicit formula.

So, that’s pretty cool.

If you want to see examples, you can easily look them up. My guess is that writing down a confusing infinite series would not actually help your understanding any more than what I’ve already written.

That’s it. Modular forms exist, but it’s kind of surprising that they do.