The central basic theorem of topological modular forms states that the structure sheaf of $\widehat{\mathcal{M}}_{ell}$ lifts to a sheaf of complex-oriented $E_{\infty}$-rings whose formal groups are naturally isomorphic to the formal completions of the associated elliptic curves. I need to cite this and want to follow best practices, but despite hours of searching, I haven't been able to find the original source of this result. I have found expositions of it that attribute it to others, and papers that prove less refined or related results (e.g. Goerss-Hopkins's paper proving a lifting result for $E_{\infty}$-structures on Lubin-Tate spectra), but I have not been able to find any paper or book containing an original proof of this.
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asked May 5, 2023 at 0:43
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1$\begingroup$ My impression is that originally the GHM theorem wasn't stated as such precisely in the form it has nowadays, but rather was a "corollary" of the program outlaid by Goerss and Hopkins in "Moduli Spaces of Commutative Ring Spectra" and supplemented by their "Moduli Problems for Structured Ring Spectra". As for Lurie's version (derived elliptic curves) I suspect that the "best original source" is his survey on Elliptic Cohomology. $\endgroup$– M.G.Commented May 5, 2023 at 1:54
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4$\begingroup$ Perhaps best practice is to reference where it was first proven, by Behrens in section 12 of the tmf book, or Lurie's proof in section 7 of elliptic II. For historical context not already mentioned, Goerss has a Bourbaki seminar article which touches on the combination of the GHM theorem and Lurie's interpretation (at the time), and Hopkins' 1994 and 2002 ICM notes seem to be early summaries of the material. I'd also like a more complete history of this too... $\endgroup$– Jack DaviesCommented May 5, 2023 at 10:34
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