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 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 case the ground ring is complicated, but essentially known. It is periodic, like K-theory, with period 242 = 576$24^2= 576$. There is a map of graded rings:

TMF(pt) ---> MF

$TMF(pt) \rightarrow MF$

where MF $MF$ 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-infinity $E_\infty$ 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 arrises 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.

2 typo

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 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 case the ground ring is complicated, but essentially known. It is periodic, like K-theory, with period 242 242 = 576. There is a map of graded rings:

TMF(pt) ---> 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-infinity 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 arrises 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.

1

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 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 case the ground ring is complicated, but essentially known. It is periodic, like K-theory, with period 242 = 576. There is a map of graded rings:

TMF(pt) ---> 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-infinity 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 arrises 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.