formal group of complex cobordism defines a natural isomorphism of MU
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
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[
, δ, ] associated
to Jacobi’s quartic elliptic curve; the corresponding elliptic cohomology theory is
given on finite spaces by X → MU ∗(X) ⊗MU∗ Z[
, 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
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:
+ a1xy + a3y =
+ 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
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
Z[c4,c6, Δ] (c4
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: