I was going to move on to deformations of Galois representations to try to talk about some of the ideas involved in proving Taniyama-Shimura, but after a random conversation I decided it might be more fun to try to talk a little about topological modular forms. I know nothing about these things, but it is a pretty big current research interest right now and they are somehow related to the modular forms I just talked about, so let’s give it a try.
This post is filling in more information about modular forms. Rather than make you look up this definition again, I’ll copy it back here and give some motivation for it. A modular form of weight and level is an element of the vector space which consists of holomorphic functions on the upper half plane satisfying the additional transformation property that
for all matrices such that (plus the “cusp” condition). The ones that vanish at the cusp are called cusp forms and they form a vector space denoted .
Now all of this looks extremely strange (especially this extending to be holomorphic across these bizarrely defined cusps). But now I’ll show you why it isn’t strange at all. The matrices that become upper triangular mod (the ones we defined the transformation law for) are a finite index subgroup of notated . The group acts naturally on the upper half plane via linear fractional transformations, and we can form the quotient .
This is a nice space, but we’d really like it to be a nice compact Riemann surface, so we compactify. Equivalently we have to add in finitely many points. There we have it! These are the cusps. So if we take the extended upper half plane I defined before and then take the quotient we get a smooth Riemann surface . Suppose I have some global -form on , i.e. a global section of . First note that this will be holomorphic across the cusps automatically. The condition that seemed superficial before happens for free when we think in these terms.
Second, this weird transformation law is saying exactly that it is something that can descend to be a -form on the surface. Thus a modular form is actually a -form on the modular curve . Or more precisely if we take the quotient map , then a -form can be pulled back , and is the “form” satisfying these properties. All of our conditions come out naturally. The cusp forms we were considering were just the ones that vanished at the cusps (the points we threw in to compactify). In algebraic geometry symbols we have . We have to do something a little more subtle to get all our weights, but we’ll come back to that.
Probably the hardest part of breaking into the modular forms literature is that there are so many equivalent ways to think about all these things. We actually need one more interpretation of in order to motivate the types of generalizations that happen in the definition of topological modular forms. It turns out that represents a moduli functor, i.e. it is a “moduli space”. If you’ve read Mumford’s early stuff on , then you may be able to guess what types of things it parametrizes. It turns out that is the moduli space of elliptic curves with certain “level ” structure. It parametrizes pairs where is a an elliptic curve and is a cyclic subgroup of order . Two pairs are isomorphic if the isomorphism preserves the subgroup. Surprisingly this moduli space is as nice as can be .
We could spend months talking about all these modular curves and their properties, but for now we’ll have to leave it at this. Next time we’ll start working towards the generalization.