Elliptic curve over 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 pi0(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 Uf -> A[f-1], where Uf 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, OX) -> (X, pi0(OX)), where pi0(OX) denotes the sheafification of the presheaf U -> pi0(OX(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, pi0(OE)) -> Spec pi0(A) is an ordinary elliptic curve over pi0(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 pi0. 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".)

• I should note that "homotopy coherence" here means essentially the same thing as the "homotopies of homotopies of homotopies of ..." stuff that I described here: mathematics.stackexchange.com/questions/181/… – Kevin H. Lin Oct 11 '09 at 21:06
• Yeah, I read the Lurie up to the IV I think, so VII will be welcome. – Ilya Nikokoshev Oct 11 '09 at 21:25
• kevin, i am curious about possible other approaches to derived schemes. Jeff Smith has a theory of ideals of spectra, i am not sure what happened to it, have you heard of this? – Sean Tilson Mar 31 '10 at 3:44
• No, I have no idea, sorry. – Kevin H. Lin Mar 31 '10 at 3:46
• @Kevin Lin: The link in your top comment doesn't work for me. It takes me to answer 181 rather than question 181. And answer 181 has something to do with why the volume of a sphere is $4\pi r^3/3$. Maybe question 181 got deleted somehow? – David White Apr 26 '12 at 13:53

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.