What is a **topological modular form**? How are they related to 'normal' (number-theoretic) modular forms?



What is a TMF in topology?
There is a Wikipedia entry on [topological modular forms](http://en.wikipedia.org/wiki/Topological_modular_forms), where you can see further references. The primitive version of Topological modular forms is as a generalized cohomology theory. Like K-theory and Cobordism theory it is a functor from spaces to graded rings, which satisfies the Eilenberg-Steenrod axioms, except for the dimension axiom. 


In the case of TMF the ground ring is complicated, but essentially known. It is periodic, like K-theory, with period $24^2= 576$. There is a map of graded rings:


$TMF(pt) \rightarrow MF$


where $MF$ is the graded ring of integral modular forms (with appropriate grading based on the weight of the modular form). This map is NOT an isomorphism. It is neither surjective nor injective, but the kernel and cokernel are both torsion. In fact this torsion is only at the primes 2 and 3. 


The more sophisticated version, and the one which makes makes the connection to modular forms more clear, is to view TMF as an $E\_\infty$ ring spectrum. The category of spectra is similar to the category of topological spaces, except the suspension functor is invertible. The things which represent cohomology theories live in spectra, and TMF is a ring object in spectra.


The connection to modular forms arises when you try to extend many constructions from algebraic geometry to this larger world of spectra. It is impossible to do the subject justice in a single post, but roughly you can look at the analog of elliptic curves over spectra. These have a moduli stack and the ``ring'' (i.e. ring spectrum) of functions on this stack is TMF. 


