xviii INTRODUCTION

formal group of complex cobordism defines a natural isomorphism of MU

∗

with

the Lazard ring, the classifying ring for formal groups. Thus, a one-dimensional

formal group over a ring R is essentially equivalent to a complex genus, that is,

a ring homomorphism MU

∗

→ R. One important example of such a genus is

the Todd genus, a map MU

∗

→

K∗.

The Todd genus occurs in the Hirzebruch–

Riemann–Roch theorem, which calculates the index of the Dolbeault operator in

terms of the Chern character. It also determines the K-theory of a finite space

X from its complex cobordism groups, via the Conner–Floyd theorem: K∗(X)

∼

=

MU ∗(X) ⊗MU∗ K∗.

Elliptic curves form a natural source of formal groups, and hence complex

genera. An example of this is Euler’s formal group law over Z[

1

2

, δ, ] associated

to Jacobi’s quartic elliptic curve; the corresponding elliptic cohomology theory is

given on finite spaces by X → MU ∗(X) ⊗MU∗ Z[

1

2

, δ, ]. Witten defined a genus

MSpin → Z[[q]] (not a complex genus, because not a map out of MU ∗) which lands

in the ring of modular forms, provided the characteristic class

p1

2

vanishes. He also

gave an index theory interpretation of this genus, at a physical level of rigor, in

terms of Dirac operators on loop spaces. It was later shown, by Ando–Hopkins–

Rezk, that the Witten genus can be lifted to a map of ring spectra MString → tmf .

The theory of topological modular forms can therefore be seen as a solution to the

problem of finding a kind of elliptic cohomology that is connected to the Witten

genus in the same way that the Todd genus is to K-theory.

Chapter 2: Elliptic curves and modular forms. An elliptic curve is a

non-singular curve in the projective plane defined by a Weierstrass equation:

y2

+ a1xy + a3y =

x3

+

a2x2

+ a4x + a6.

Elliptic curves can also be presented abstractly, as pointed genus one curves. They

are equipped with a group structure, where one declares the sum of three points to

be zero if they are collinear in

P2.

The bundle of K¨ ahler differentials on an elliptic

curve, denoted ω, has a one-dimensional space of global sections.

When working over a field, one-dimensional group varieties can be classified into

additive groups, multiplicative groups, and elliptic curves. However, when working

over an arbitrary ring, the object defined by a Weierstrass equation will typically be

a combination of those three cases. The general fibers will be elliptic curves, some

fibers will be nodal (multiplicative groups), and some cuspidal (additive groups).

By a ‘Weierstrass curve’ we mean a curve defined by a Weierstrass equation—

there is no smoothness requirement. An integral modular form can then be defined,

abstractly, to be a law that associates to every (family of) Weierstrass curves a

section of ω⊗n, in a way compatible with base change. Integral modular forms

form a graded ring, graded by the power of ω. Here is a concrete presentation of

that ring:

Z[c4,c6, Δ] (c4

3

− c6

2

− 1728Δ).

In the context of modular forms, the degree is usually called the weight: the gen-

erators c4, c6, and Δ have weight 4, 6, and 12, respectively. As we will see, those

weights correspond to the degrees 8, 12, and 24 in the homotopy groups of tmf .

Chapter 3: The moduli stack of elliptic curves. We next describe the

geometry of the moduli stack of elliptic curves over fields of prime characteristic, and

over the integers. At large primes, the stack Mell looks rather like it does over C: