Filling the gaps in my knowledge to understand the tmf question.
So, what is the analogue of elliptic curve over the category of spectra?
Filling the gaps in my knowledge to understand the tmf question.
So, what is the analogue of elliptic curve over the category of spectra?
First you should know what a "derived scheme" (or "spectral scheme") is. Roughly, this is the same as an ordinary scheme, except instead of locally being the Spec of an ordinary commutative ring, it's locally the Spec of an E-infinity ring spectrum, or just E-infinity ring for short. An E-infinity ring is not the same as a ring spectrum; a ring spectrum is something that is "a ring up to homotopy", and an E-infinity ring is something that is "a ring up to coherent homotopy".
As a topological space, Spec of an E-infinity ring A is defined to be the Spec of pi_{0}(A), which is an ordinary commutative ring. The difference is that the sheaf of functions is no longer a sheaf of rings but a sheaf of E-infinity rings. This sheaf of E-infinity rings is (analogously to the structure sheaf for ordinary affine schemes) defined by U_{f} -> A[f^{-1}], where U_{f} is a distinguished open subset, and the localization A[f^{-1}] is now taken in the category (or rather infinity-category) of E-infinity rings (see section 2.2 of Lurie's survey for a characterization of this localization).
There is a natural functor from derived schemes to ordinary schemes. It is (X, O_{X}) -> (X, pi_{0}(O_{X})), where pi_{0}(O_{X}) denotes the sheafification of the presheaf U -> pi_{0}(O_{X}(U)).
Then Definition 4.1 of Lurie's survey article defines an elliptic curve over an E-infinity ring A: it is a commutative A-group E -> Spec A such that (E, pi_{0}(O_{E})) -> Spec pi_{0}(A) is an ordinary elliptic curve over pi_{0}(A).
I think one of the punchlines is that there is a derived Deligne-Mumford moduli stack of oriented derived elliptic curves, which becomes the ordinary Deligne-Mumford moduli stack of ordinary elliptic curves after hitting it with pi_{0}. Taking global sections on this derived moduli stack is more or less tmf. I don't have a good short explanation of what an orientation is, but it's in the beginning of section 3 of the survey.
Unfortunately I think a lot of the details of this stuff are still not available. (I think it is supposed to be in "DAG VII: Spectral Schemes"; there may be some hints and related material in "DAG V: Structured Spaces".)
Jacob Lurie has notes on this question: (pdf). The short answer is "an oriented spectral (or derived) elliptic curve." A more primitive answer is given by an elliptic spectrum (due to Hopkins and Miller), which is an even periodic complex-oriented cohomology theory, an elliptic curve, and an isomorphism between the formal group corresponding to the cohomology theory and the formal group of the elliptic curve.