WTF Are Modular Forms - graeme. hello
WTF Are Modular Forms
March 25, 2024
maths
This post is part of the HMN Learning Jam. One weekend to fill your head with stuff and one weekend to dump it back out. I’ve spent a few more days here and there (that convinced me I could learn the rest of what I needed to know in a weekend) but otherwise that’s about how much time I’ve got in knowing about modular forms. This is a bit out of step from the rest of the submissions which are more programming focused but look: someone had to.
As always happens I probably learned the most while I was writing this post and not when I was reading stuff.
I’m gonna take a very non-standard approach that I’ve seen fragments of here and there in the Literature, so experts on this stuff know it all, but the path it takes is so irrelevant to what working with modular forms is actually like that a mathematician would never be so cruel as to present it this way.
However, I’m not a mathematician and am completely new to all this. From what I can tell, this path I’ll take here is closer to what was going on when modular forms first appeared in mathematics 200 years ago. It’s pretty far removed from modern day modular forms. But this means we get to see what modular forms are in a certain naive sense that hopefully doesn’t require a degree in mathematics to understand.
To get the most out of this, you want to know a a little linear algebra and be comfortable with complex numbers. You should also probably read the first half with a graphing calculator open. On the other hand, I’ve deliberately avoided some convenient mathematics notations/definitions that make maths inaccessible to many people, which makes life harder for me and probably people that know that stuff.
The two books I found most useful for this are:
Development of Mathematics in the 19th Century, Felix Klein,1928
Complex Analysis, Elias M. Stein and Rami Shakarchi, 2003
The first is by the Klein bottle guy. He was pretty much there when modular forms became modular forms and is where I got this general approach from. He’s more brisk about it. Cool book, could not have understood any of this without it.
Complex Analysis is a textbook for undergraduates. Goes slow in the right places. Really well written if you know how to read maths textbooks.
Another book I’d like to recommend that you’ll probably have trouble discovering on your own is
Elliptic Curves, Henry McKean & Victor Moll, 1999
This is in my preferred style for maths textbooks, which I have heard called “breezy” before. Some people figure out they’ve got a tiny audience and write conversationally directly to them, it’s great. No sweating on proofs, just big picture ideas from people fully in the post-rigorous phase of life, and they trust you have dryer references to fall back on if you need them. But it is squarely aimed at real deal mathematicians so beyond me for the most part.
A cool resource is Richard E Borcherds lecture series on youtube if you want to see what it’s really about to a mathematician. He’s got a Fields medal in this stuff.
Another cool video is this MathKiwi video which in some sense starts where I leave off and has many good visualisations.
To round this list out here’s a bunch of mathematicians saying what they find interesting about modular forms. Check out how many different angles on them there are.
I can’t motivate why you’d want to know about modular forms. If you want to know this stuff, you want to know, and if you don’t, you don’t. Instead, I want to start at a basic number theory question and make a bee line for modular forms. Modular forms come from a few related ideas hanging together, so we’ll have to gather up those ideas first.
Here’s our number theory question: given an integer nnn, how many solutions does<br>n2=a2+b2n^2 = a^2 + b^2n2=a2+b2have? I start here because it makes the link to questions like these and Fermat’s Last Theorem clear, but I am going to say well clear of any weird factorisations and algebra that usually immediately follow this question, don’t worry.
What a lot of number theorists like to do when encountering something new is to compute large tables of numbers, and here since you never have to look at aaa and bbb above nnn you don’t have that many (a,b)(a,b)(a,b) pairs to check. Well, for a few nnn, anyway. You can get pretty far along crunching numbers. Then you stare at the numbers and hope for insight. This works surprisingly well.
But instead the way to modular forms is to draw the sucker. This turns it into the following question: on how many places does the circle<br>n2=x2+y2n^2 = x^2 + y^2n2=x2+y2lie on integer coordinates? Now instead of crunching numbers, you just draw a circle on a grid. Solutions are where the circle lies on a point where the grid lines intersect....