One would like an analogous E∞-ring spectrum whose homotopy groups are
rationally isomorphic to the subring
Z[c4,c6, Δ]/(c4
of integral modular forms. For that, one observes that the sheaf
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 :=
) = Γ(Mell ,
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
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 )
πn−22(Tmf )
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
= Tmf
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:
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