ELLIPTIC GENERA AND ELLIPTIC COHOMOLOGY 15

The theory Ell was originally referred to as elliptic cohomology, but it is now

thought of as a particular elliptic cohomology theory. If we ignore the grading of

δ and , we can form an even periodic theory by MP(−) ⊗MP Z[

1

2

, δ, ,

Δ−1].

This

motivates the following definition.

Definition 4.3. An elliptic cohomology theory E consists of:

• A multiplicative cohomology theory E which is even periodic,

• An elliptic curve C over a commutative ring R,

• Isomorphisms

E0(pt)

∼

= R and an isomorphism of the formal group from

E with the formal group associated to C.

The even periodic theory associated to Ell is an elliptic cohomology theory

related to the Jacobi quartic curve over Z[

1

2

, δ, , Δ−1]. An obvious question is

whether there is a universal elliptic cohomology theory; this universal theory should

be related to a universal elliptic curve. Any elliptic curve C over R is isomorphic

to a curve given in aﬃne coordinates by the Weierstrass equation

y2

+ a1xy + a3y =

x3

+

a2x2

+ a4x + a6, ai ∈ R.

However, there is no canonical way to do this since the Weierstrass equation has

non-trivial automorphisms. There is no single universal elliptic curve, but instead

a moduli stack of elliptic curves, as seen in Chapter 4. Because of this, there is no

universal elliptic cohomology theory in the naive sense.

What one does end up with as the “universal elliptic cohomology theory” is

topological modular forms or tmf . Its mere existence is a diﬃcult and subtle the-

orem, and it will take the rest of these proceedings to construct tmf . Roughly

speaking, one uses the Landweber exact functor theorem to form a pre-sheaf of el-

liptic cohomology theories on the moduli stack of elliptic curves. One then lifts this

to a sheaf of E∞ ring-spectra and takes the global sections to obtain the spectrum

tmf . While constructed out of elliptic cohomology theories, tmf is not an elliptic

cohomology theory, as evidenced by the following properties.

There is a homomorphism from the coeﬃcients tmf

−∗

to the ring of modular

forms MF . While this map is rationally an isomorphism, it is neither injective

nor surjective integrally. In particular, tmf

−∗

contains a large number of torsion

groups, many of which are in odd degrees. Topological modular forms is therefore

not even, and the periodic version TMF has period

242

= 576 as opposed to 2 (or

as opposed to 24, the period of Ell). Furthermore, the theory tmf is not complex

orientable, but instead has an MO 8 or string orientation denoted σ. At the level

of coeﬃcients, the induced map

MString−∗

→ tmf

−∗

gives a refinement of the

Witten genus ϕW .

tmf

−∗

MString−∗

♠♠♠♠♠♠♠♠σ

ϕW

MF

While a great deal of information about tmf has already been discovered, there

are still many things not yet understood. As an example, the index of family of

(complex) elliptic operators parameterized by a space X naturally lives in K(X),

and topologically this is encoded by the complex orientation of K-theory. Because

of analytic diﬃculties, there is no good theory of elliptic operators on loop spaces.

However, it is believed that families indexes for elliptic operators on loop spaces

should naturally live in tmf and refine the Witten genus. Making mathematical