INTRODUCTION xxv

In order to construct that nullhomotopy, one needs to understand the homotopy

type of gl1(tmf )—this is done one prime at a time. The crucial observation is that

there is a map of spectra gl

1

(tmf )p ˆ → tmf

p

ˆ, the ‘topological logarithm’, and a

homotopy pullback square

gl

1

(tmf )

log

tmf

LK(1)(tmf )

1−Up

LK(1)(tmf )

where Up is a topological refinement of Atkin’s operator on p-adic modular forms.

The fiber of the topological logarithm is particularly intriguing: Hopkins speculates

that it is related to exotic smooth structures on free loop spaces of spheres.

Chapters 11 and 12: The sheaf of E∞ ring spectra and The construc-

tion of tmf . We outline a roadmap for the construction of tmf , the connective

spectrum of topological modular forms. The major steps in the construction are

given in reverse order.

• The spectrum tmf is the connective cover of the nonconnective spectrum

Tmf ,

tmf := τ≥0Tmf ,

and Tmf is the global sections of a sheaf of spectra,

Tmf :=

Otop(Mell

),

where Mell is the moduli stack of elliptic curves with possibly nodal sin-

gularities. This stack is the Deligne–Mumford compactification of the

moduli stack of smooth elliptic curves. Here,

Otop

is a sheaf on Mell

in the ´ etale topology. Also, TMF is the global sections of

Otop

over the

substack Mell of smooth elliptic curves,

TMF :=

Otop(Mell

).

The uppercase ‘T ’ in Tmf signifies that the spectrum is no longer connective (but

it is also not periodic). The ‘top’ stands for topological, and

Otop

can be viewed as

a kind of structure sheaf for a spectral version of Mell .

We are left now to construct the sheaf of spectra

Otop.

The first step is to isolate

the problem at every prime p and at Q. That is, one constructs Op

top,

a sheaf of

spectra on the p-completion (Mell )p and then pushes this sheaf forward along the

inclusion map ιp : (Mell )p → Mell . One then assembles these pushforwards to

obtain

Otop,

as follows.

• The sheaf

Otop

is the limit in a diagram

Otop

p

ιp,∗Optop

ιQ,∗OQ

top

p

ιp,∗Optop

Q

for a given choice of map αarith : ιQ,∗OQ

top

→

p

ιp,∗Op top

Q

.