INTRODUCTION xxix

The homotopy of tmf can be computed directly using the Adams spectral

sequence. Alternatively, one can use the elliptic spectral sequence to compute the

homotopy of Tmf . The Adams spectral sequence has the form

E2

= Ext

Ap

tmf

(Fp, Fp) ⇒ π∗(tmf )ˆ,p

where Ap tmf := homtmf

−modules

(HFp,HFp) is a tmf -analog of the Steenrod algebra.

At the prime 2, the map A2

tmf

→ A ≡ A2 to the classical Steenrod algebra is

injective, and the tmf -module Adams spectral sequence can be identified with the

classical Adams spectral sequence

E2 =

ExtA(H∗(tmf

), F2) = ExtA(A/ /A(2), F2) = ExtA(2)(F2, F2) ⇒ π∗(tmf

)ˆ

2

.

The elliptic spectral sequence has the form

Hs(Mell

,

πtOtop)

⇒ πt−s(Tmf ). The

homotopy

πtOtop

is concentrated in even degrees and is the t/2-th power of a line

bundle ω; the spectral sequence thus has the form

Hq(Mell

;

ω⊗p)

⇒ π2p−q(Tmf ).

Part II

The manuscripts. The book concludes with three of the original, previously

unpublished, manuscripts on tmf : “Elliptic curves and stable homotopy I” (1996)

by Hopkins and Miller, “From elliptic curves to homotopy theory” (1998) by Hop-

kins and Mahowald, and “K(1)-local E∞ ring spectra” (1998) by Hopkins. The

first focuses primarily on the construction of the sheaf of (associative) ring spectra

on the moduli stack of elliptic curves, the second on the computation of the homo-

topy of the resulting spectrum of sections around the supersingular elliptic curve at

the prime 2, and the third on a direct cellular construction of the K(1)-localization

of tmf . These documents have been left, for the most part, in their original draft

form; they retain the attendant roughness and sometimes substantive loose ends,

but also the dense, heady insight of their original composition. The preceding chap-

ters of this book can be viewed as a communal exposition, more than fifteen years

on, of aspects of these and other primary sources about tmf .

4. Reader’s guide

This is not a textbook. Though the contents spans all the way from classical

aspects of elliptic cohomology to the construction of tmf , there are substantive

gaps of both exposition and content, and an attempt to use this book for a lecture,

seminar, or reading course will require thoughtful supplementation.

Reading straight through the book would require, among much else, some fa-

miliarity and comfort with commutative ring spectra, stacks, and spectral sequences.

Many of the chapters, though, presume knowledge of none of these topics; instead

of thinking of them as prerequisites, we suggest one simply starts reading, and as

appropriate or necessary selects from among the following as companion sources:

Commutative ring spectra:

• May, J. Peter. E∞ ring spaces and E∞ ring spectra. With contributions by

Frank Quinn, Nigel Ray, and Jorgen Tornehave. Lecture Notes in Mathemat-

ics, Vol. 577. Springer-Verlag, Berlin-New York, 1977.