xii INTRODUCTION

Ordinary cohomology is complex orientable, and its associated formal group is

the formal completion of the additive formal group. Topological K-theory is also

complex orientable, and its formal group is the formal completion of the multiplica-

tive formal group. This situation naturally leads one to search for ‘elliptic’ coho-

mology theories whose formal groups are the formal completions of elliptic curves.

These elliptic cohomology theories should, ideally, be functorial for morphisms of

elliptic curves.

Complex bordism MU is complex orientable and the resulting formal group

law is the universal formal group law; this means that ring maps from MU∗ to R

are in natural bijective correspondence with formal group laws over R. Given a

commutative ring R and a map MU∗ → R that classifies a formal group law over

R, the functor

X → MU∗(X) ⊗MU∗ R

is a homology theory if and only if the corresponding map from Spec(R) to the

moduli stack MFG of formal groups is flat. There is a map

Mell → MFG

from the moduli stack of elliptic curves to that of formal groups, sending an elliptic

curve to its completion at the identity; this map is flat. Any flat map Spec(R) →

Mell therefore provides a flat map Spec(R) → MFG and thus a homology theory, or

equivalently, a cohomology theory (a priori only defined on finite CW -complexes).

In other words, to any aﬃne scheme with a flat map to the moduli stack of elliptic

curves, there is a functorially associated cohomology theory.

The main theorem of Goerss–Hopkins–Miller is that this functor (that is,

presheaf)

flat maps from aﬃne schemes to Mell → multiplicative cohomology theories ,

when restricted to maps that are ´ etale, lifts to a sheaf

Otop

: ´ etale maps to Mell → E∞-ring spectra .

(Here the subscript ‘top’ refers to it being a kind of ‘topological’, rather than

discrete, structure sheaf.) The value of this sheaf on Mell itself, that is the E∞-ring

spectrum of global sections, is the periodic version of the spectrum of topological

modular forms:

TMF :=

Otop(Mell

) = Γ(Mell ,

Otop).

The spectrum TMF owes its name to the fact that its ring of homotopy groups

is rationally isomorphic to the ring

Z[c4,c6,

Δ±1]/(c4 3

− c6

2

− 1728Δ)

∼

=

n≥0

Γ

(

Mell ,

ω⊗n

)

of weakly holomorphic integral modular forms. Here, the elements c4, c6, and Δ

have degrees 8, 12, and 24 respectively, and ω is the sheaf of invariant differentials

(the restriction to Mell of the (vertical) cotangent bundle of the universal elliptic

curve E → Mell ). That ring of modular forms is periodic with period 24, and the

periodicity is given by multiplication by the discriminant Δ. The discriminant is

not an element in the homotopy groups of TMF , but its twenty-fourth power

Δ24

∈

π242 (TMF) is, and, as a result, π∗(TMF) has a periodicity of order

242

= 576.