# What can topological modular forms do for number theory?

Topological modular forms ($TMF$) have in the recent years made an impact in algebraic topology. Roughly, the spectrum $tmf$ is the (derived) global sections of the sheaf of $E_\infty$ ring spectra over $\bar{\mathcal M}_\text{ell}$, the (derived) compactified moduli stack of elliptic curves. Further, Behrens and Lawson have generalized this construction to 'topological automorphic forms' on PEL Shimura varieties.

It is clear, e.g., from this post, that $tmf$ has important applications in algebraic topology (Witten genus, exotic spheres..)

What I would like to know is: What are the applications of $tmf$ to number theory, specifically to modular forms? Hopkins has a report titled 'Topological Methods in Automorphic Forms', but I think it is more the opposite—'automorphic methods in topology'. For example, Behrens has some results in this regards on congruences of $p$-adic modular forms.

Background: in contrast to the OP of the linked post, my research area is in number theory, hence the question.

Here are two applications that I know of which involve direct crossover.

In section 5 of the Hopkins ICM address referenced in Drew's answer to the other question, he gives a topological proof of the following congruence originally due to Borcherds. Suppose that $L$ is a positive definite, even unimodular lattice of dimension $24k$. The theta function $\theta_L$ is the generating function $$\theta_L(q) = \sum_{\ell \in L} q^{\tfrac{1}{2} \langle \ell, \ell \rangle},$$ a modular form of weight $12k$ over $\Bbb Z$. If we write $$\theta_L(q) = c_4^{3k} + x_1 c_4^{3(k-1)} \Delta + \cdots + x_k \Delta^k,$$ in terms of the standard basis elements $c_4$ and $\Delta$, then $x_k \equiv 0$ mod 24.

(However, later in the same paper Hopkins translated the input from topology into a proof in analytic terms.)

There is also the following, due to work of Ando-Hopkins-Rezk and alluded to in the final section of the ICM address. Let's suppose that we fix a prime $p$ and a level for modular forms which is relatively prime to $p$, and for simplicity either that $p > 3$ or that the modular curve is actually a scheme. We have an associated (graded) ring of integral modular forms $MF_*$. Then there is a commutative square: $$\require{AMScd} \begin{CD} (MF_{> 0})^\wedge_p @>>> (MF_*)^\wedge_{(p,E_{p-1})}\\ @VVV @VVV\\ (E_{p-1}^{-1} MF_*)^\wedge_p @>>> (E_{p-1}^{-1} (MF_*)^\wedge_{E_{p-1}})^\wedge_p \end{CD}$$ Here $E_{p-1}$ is an appropriately normalized Eisenstein series. In degree $k$:

• the top map is the Hecke operator $1 − T(p) + p^{k−1}$, which has the Eisenstein series in its kernel;

• the left map is the operator $(1 − p^{k−1}V)$ where $V$ is the Verschiebung operator $V(\sum a_n q^n = \sum a_{pn} q^n)$;

• the right map is just the localize-then-complete map, which restricts a function near the supersingular locus to the punctured neighborhood of the supersingular locus;

• the bottom map is $1 - U_p$ where $U_p$ is the Atkin operator $U_p(f(q)) = f(q^p)$.

What Ando-Hopkins-Rezk showed is that for $k \geq 2$, this square is biCartesian: it becomes a short exact sequence. I'm not aware of a purely number-theoretic proof of this result or its generalizations, though I'd love to hear one.

Having said all this, I don't know of that many results where we know something purely from topology that wasn't already known in number theory. As you say, there are certainly results from number theory applied to topology. Even more common is for the work on the topology side to lead to a question in number theory which is interesting to us but possibly outside mainstream research questions. After all, most of our concerns translate into questions about integral modular forms, operators on them, their congruences, torsion goodies, etc. for a fixed level; rational or analytic information, automorphic representations, etc. have had far less topological impact.

Sometimes topologists prove number-theoretic results as part of this search, but rarely has topological input been required for the proofs.

• Thanks so much! The second application is particularly interesting in its interpretation of $V$ and $U_p$. A follow up question: the concrete definition of a modular form kind of gets lost, e.g., the congruence statement of Behrens is more of an existence statement. Is it possible to say, pick out an honest modular form from a tmf? – Tian An Jun 3 '15 at 12:53