Topological Modular Forms


This will be my first and last post on this topic, since it will take us too far from the theme for this year which is arithmetic geometry. It took awhile for me to write this because something feels wrong in the last post and I wanted to correct it before doing this one. Unfortunately, I can only make a guess at what is happening. I’ll explain it when it comes up. Thus as a warning everything in the last post and in this post should be taken as approximately true (of course, this is a blog, so this warning should probably always be in place).

Recall briefly that we now have a description of (weight {2}) modular forms just as a global {1}-form on a certain moduli space of elliptic curves with level {N} structure. To get the weight {2k} modular forms we just take tensor powers, so {H^0(X_0(N), \Omega^{\otimes k})\simeq M_{2k}(\Gamma_0(N))}. It is funny to notice that a priori it is completely unclear that the collection of all modular forms of a fixed level should form a graded commutative ring, but with this description it falls right out. We define the graded ring of modular forms of level {N} to be {\displaystyle M(\Gamma_0(N))=\bigoplus_{k=1}^\infty H^0(X_0(N), \Omega^{\otimes k})=\bigoplus_{k=1}^\infty M_{2k}(\Gamma_0(N))}.

Now notice that if we take {N=1} in our moduli problem we are just taking an elliptic curve plus a cyclic subgroup of order {1}, i.e. we are marking the identity. Thus what ought to be the case is that {\overline{\mathcal{M}_{1,1}}=X_0(1)}. This is the part that confuses me. Last time I said that {X_0(N)} was a smooth Riemann surface, but {\overline{\mathcal{M}_{1,1}}} is a DM stack. My guess at what is going on is that since we only defined the moduli functor for {X_0(N)} for elliptic curves over {\mathbb{C}}, we are maybe just taking the {\mathbb{C}}-valued points. Thus maybe {\overline{\mathcal{M}_{1,1}}(\mathbb{C})\simeq X_0(1)}. In any case, there is certainly some relation between the two so it isn’t unreasonable to try to figure out what happens when we replace {X_0(1)} with {\overline{\mathcal{M}_{1,1}}}.

Now we’ll start the crazy generalizations. There is something called a derived DM stack. Since it would take a lot to define, we’ll just say that it is one of these things where “{\infty}-blah” gets thrown around. The important idea here is that we can take {\pi_0} and get back an honest DM stack. The big theorem of Hopkins, Miller, and Lurie is that there exists a derived DM stack {(\mathcal{M}, \mathcal{O})} whose underlying DM stack is {\overline{\mathcal{M}_{1,1}}} such that {\pi_{2k}\mathcal{O}\simeq \omega^k} and {\pi_{2k+1}\mathcal{O}=0}.

Now “tmf” is something called a commutative ring spectrum and it is formed by taking the derived global sections of {\mathcal{O}}. Generalities give us a descent spectral sequence {H^s(\mathcal{M}, \omega^t)\Rightarrow \pi_{2t+s}\mathbf{tmf}}. An open and interesting question is to determine which modular forms give homotopy classes in {\mathbf{tmf}} since the classical modular forms form the {0}-row of this spectral sequence. I’ll just end by pointing out how mind boggling this is. Modular forms have had such great success in number theory. Now they are successfully being used to understand homotopy groups of spheres and other extremely topological questions. The reverse has been done as well. Topological methods can transfer information back to modular forms and give number theoretic theorems such as congruence relations between {p}-adic modular forms. How amazing!

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