I'm going to start with Mike Hopkins' great survey article in the ICM on topological modular forms (https://arxiv.org/abs/math/0212397). In it, he outlines a construction, for even unimodular lattices, of a "topological theta series" which is a refinement of the theta series of the lattice (living in a ring of modular forms). This forms part of his explanation for the Borcherds congruence mod 24 of theta series, which still seems magical and mysterious to me. Hopkins' analytic proof here gives me a lot more understanding, and should be generalizable to other modular lattices.
My question is whether this construction is nontrivial. In other words, are there two lattices with different topological theta series but the same theta series? Or, which torsion elements of tmf are involved in topological theta series (if any)?
It's easy to look at the 2-primary and 3-primary tables of tmf to determine the possible dimensions where tmf-torsion might be interesting for lattices here. The first interesting dimensions here seem to be 32 and 72 here, but I need to recheck this. Also, what happens when you look at level N topological modular forms? What does the map tmf(N) -> MF(N) look like, computationally for small N, ideally for all N where N+1 or N-1 divides 24.
