INTRODUCTION xiii

One would like an analogous E∞-ring spectrum whose homotopy groups are

rationally isomorphic to the subring

Z[c4,c6, Δ]/(c4

3

− c6

2

− 1728Δ)

of integral modular forms. For that, one observes that the sheaf

Otop

is defined

not only on the moduli stack of elliptic curves, but also on the Deligne–Mumford

compactification Mell of the moduli stack—this compactification is the moduli

stack of elliptic curves possibly with nodal singularities. The spectrum of global

sections over Mell is denoted

Tmf :=

Otop(Mell

) = Γ(Mell ,

Otop).

The element Δ24 ∈ π242 (Tmf ) is no longer invertible in the homotopy ring, and

so the spectrum Tmf is not periodic. This spectrum is not connective either, and

the mixed capitalization reflects its intermediate state between the periodic version

TMF and the connective version tmf , described below, of topological modular

forms.

In positive degrees, the homotopy groups of Tmf are rationally isomorphic to

the ring Z[c4,c6, Δ]/(c4 3 − c6 2 − 1728Δ). The homotopy groups π−1,...,π−20 are all

zero, and the remaining negative homotopy groups are given by:

π−n(Tmf )

∼

=

πn−21(Tmf )

torsion-free

⊕ πn−22(Tmf )

torsion

.

This structure in the homotopy groups is a kind of Serre duality reflecting the

properness (compactness) of the moduli stack Mell .

If we take the (−1)-connected cover of the spectrum Tmf , that is, if we kill all

its negative homotopy groups, then we get

tmf := Tmf 0 ,

the connective version of the spectrum of topological modular forms. This spectrum

is now, as desired, a topological refinement of the classical ring of integral modular

forms. Note that one can recover TMF from either of the other versions by inverting

the element Δ24 in the 576th homotopy group:

TMF = tmf

[Δ−24]

= Tmf

[Δ−24].

There is another moduli stack worth mentioning here, the stack Mell

+

of elliptic

curves with possibly nodal or cuspidal singularities. There does not seem to be

an extension of Otop to that stack. However, if there were one, then a formal

computation, namely an elliptic spectral sequence for that hypothetical sheaf, shows

that the global sections of the sheaf over Mell

+

would be the spectrum tmf . That

hypothetical spectral sequence is the picture that appears before the preface. It is

also, more concretely, the Adams–Novikov spectral sequence for the spectrum tmf .

So far, we have only mentioned the connection between tmf and modular forms.

The connection of tmf to the stable homotopy groups of spheres is equally strong

and the unit map from the sphere spectrum to tmf detects an astounding amount

of the 2- and 3-primary parts of the homotopy π∗(S) of the sphere.

The homotopy groups of tmf are as follows at the prime 2: