xxvi INTRODUCTION

Once

Otop

has been constructed, it will turn out that Op

top

is the p-completion of

Otop,

and OQ

top

is its rationalization, so that the above diagram is the arithmetic

square for

Otop.

This thus leaves one to construct each Op

top

and OQ

top

and the gluing

map αarith. The sheaf OQ

top

is not diﬃcult to construct. Its value on an ´ etale map

Spec(R) → (Mell )Q is given by OQ

top

(Spec R) = H(R∗), the rational Eilenberg–

MacLane spectrum associated to a certain evenly graded ring R∗. This ring is

specified by R2t := Γ(ω⊗t|Spec

R

), where ω is the sheaf of invariant differentials.

The construction of Op

top

is more subtle. The first step in its construction is

to employ a natural stratification of (Mell )p. Each elliptic curve has an associated

formal group which either has height equal to 1 if the curve is ordinary, or equal to

2 if the curve is supersingular. This gives a stratification of the moduli space with

exactly two strata:

Mell

ord

ιord

−→ (Mell )p

ιss

←− Mell

ss

.

The sheaf Op top is presented by a Hasse square, gluing together a sheaf

Otop

K(1)

on

Mell ord and a sheaf

Otop

K(2)

on Mell ss . (This notation is used because the sheaves

Otop

K(i)

are also the K(i)-localizations of

Otop,

where K(i) is the ith Morava K-theory at

the prime p.)

• Op

top

is the limit

Op

top

ιss,∗OK(2)top

ιord,∗OK(1)

top

(

ιss,∗OK(2)

top

)

K(1)

for a certain ‘chromatic’ attaching map

αchrom : ιord,∗OK(1)

top

−→

(

ιss,∗OK(2)

top

)

K(1)

.

The sheaf Op

top

is thus equivalent to the following triple of data: a sheaf

OK(1)top

on Mell

ord

, a sheaf OK(2)

top

on Mell

ss

, and a gluing map αchrom as above. We have

now arrived at the core of the construction of tmf : the construction of these three

objects. This construction proceeds via Goerss–Hopkins obstruction theory.

That obstruction theory is an approach to solving the following problem: one

wants to determine the space of all E∞-ring spectra subject to some conditions,

such as having prescribed homology. More specifically, for any generalized homol-

ogy theory E∗, and any choice of E∗-algebra A in E∗E-comodules, one can calculate

the homotopy type of the moduli space of E∞-ring spectra with E∗-homology iso-

morphic to A. Goerss and Hopkins describe that moduli space as the homotopy

limit of a sequence of spaces, where the homotopy fibers are certain Andr´e–Quillen

cohomology spaces of A. As a consequence, there is a sequence of obstructions

to specifying a point of the moduli space, i.e., an

E∞-ring spectrum whose E∗-

homology is A. The obstructions lie in Andr´ e–Quillen cohomology groups of E∗-

algebras in E∗E-comodules. That obstruction theory is used to build the sheaf

Otop

K(2)

.