derived smooth geometry
The ordinary moduli stack of elliptic curves when equipped with the structure sheaf of E-∞ rings which assigns to each elliptic curve its elliptic spectrum becomes a spectral Deligne-Mumford stack which as such is what modulates derived elliptic curves (see below).
By the Goerss-Hopkins-Miller theorem the structure sheaf of the moduli stack of elliptic curves (ordinary elliptic curves) lifts to a sheaf of E-∞ rings which over a given elliptic curve is the corresponding elliptic spectrum.
By (Lurie (Survey), theorem 4.1), this yields a spectral Deligne-Mumford stack refinement which is the moduli stack of derived elliptic curves, in that there is a natural equivalence in E-∞ rings of the form
This is based on the representability theorem (Lurie (Survey), prop. 4.1, Lurie (Representability)). An ingredient in the proof is the essential uniqueness of lifts of etale morphisms from commutative rings to -rings.