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[
, δ, ,
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,
= 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[
, 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
+ a1xy + a3y =
+ 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
= 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
gives a refinement of the
Witten genus ϕW .
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