@MarcHoyois sure: i just mean that they give a "naive" description of the $\infty$-category of cyclotomic spectra whose underlying spectra are bounded-below, and we extend this to give a similarly "naive" description of the $\infty$-category of all cyclotomic spectra. it turns out to be quite a nontrivial task. the short version is that there's a lax action on $Fun(BT,Sp)$ that becomes strict when you restrict to the bounded-below spectra, and this accounts for the relative simplicity.
An elliptic cohomology theory only knows about the formal group of its elliptic curve, in the sense that the sheaf of cohomology theories on $\mathcal M_{ell}$ is pulled back from $\mathcal M_{fg}$. Does the $E_\infty$ structure on an elliptic spectrum use the full structure of the elliptic curve (or at least more than just the formal group)?
@YuriSulyma It must at least know about the p-divisible group of the elliptic curve, probably for all p. and about interactions between those things at different primes.
I'm not entirely sure how to make that precise, or where a statement like that might be in the literature (I'm no expert on elliptic cohomology). Probably the names/papers to mention are Ando-Hopkins-Strickland, general works of Ganter, maybe some things by Rezk, maybe the thesis of Zhen Huan ("Quasi-Elliptic Cohomology")
The idea I guess is that the power operation structure you see from the E_\infty structure should also be coming from isogenies of your curve.
