What is a topological modular form? How are they related to 'normal' (numbertheoretic) modular forms?

There is a Wikipedia entry on topological modular forms, where you can see further references. The primitive version of Topological modular forms is as a generalized cohomology theory. Like Ktheory and Cobordism theory it is a functor from spaces to graded rings, which satisfies the EilenbergSteenrod axioms, except for the dimension axiom. In the case of TMF the ground ring is complicated, but essentially known. It is periodic, like Ktheory, 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. 


If you know about classical modular forms, and you want to understand what they have to do with tmf, it is good to contemplate something more classical: the "derived functors of modular forms". Modular forms of weight $n$ are $H^0(M, \omega^n)$, where $\omega$ is a certain line bundle over $M=$compactified moduli stack of elliptic curves. There is nontrivial cohomology in $H^s(M,\omega^n)$ for $s>0$. This comes in two flavors: * free abelian group summands in $H^1(M,\omega^n)$ for $n\leq10$. These correspond by a kind of "Serre duality" to the usual modular forms in $H^0(M,\omega^{n10})$. * finite abelian groups (killed by multiplication by $24$) in $H^s(M,\omega^n)$ for arbitrarily large $s$. $H^s(M,\omega^{t/2}) \Rightarrow \pi_{ts} Tmf$, and the edge of the spectral sequence gives a map $\pi_{2n}Tmf \rightarrow H^0(M,\omega^n)$. This is essentially the map Chris describes in his answer. Slightly confusingly, the gadget I've called Tmf (which is the global sections of a sheaf of spectra over $M$) is not the same as TMF (the $576$periodic guy, which is sections on $M'=$stack of smooth elliptic curves), or tmf (the connective cover of Tmf, which I don't believe is known to be sections on anything). I recommend Goerss's Bourbaki talk for learning more about this, especially section 4.6. 


You can think of a topological modular form as a point on the tmf spectrum, but it's not clear that this is a useful view, unless you expand your notion of "point" to a large class of objects. The relation to modular forms doesn't need to go through the machinery of spectra: tmf is a multiplicative cohomology theory, so if you evaluate it at a point (or anything else), you get a commutative ring. There is a natural homomorphism from tmf(point) to the ring of modular forms with integer coefficients (taking tmf^n(point) to holomorphic level 1 forms of weight n/2), but it is neither injective nor surjective. If you invert 6 in the source and target, the map becomes an isomorphism. 

