higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A derived elliptic curve is an object in higher geometry that is to an E-∞ ring as an ordinary elliptic curve is to an ordinary ring.
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).
Let $A$ be an E-∞ ring. A derived elliptic curve over $A$ is derived group scheme $E \to Spec A$ with the property that the underlying morphism $\bar E \to Spec \pi_0 A$ is an ordinary elliptic curve.
By the Goerss-Hopkins-Miller theorem the structure sheaf $\mathcal{O}$ of the moduli stack of elliptic curves (ordinary elliptic curves) lifts to a sheaf $\mathcal{O}^{top}$ 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 $(\mathcal{M}_{ell}, \mathcal{O}^{top})$ which is the moduli stack of derived elliptic curves, in that there is a natural equivalence in E-∞ rings $A$ of the form
where on the left we have maps of structured (∞,1)-toposes and on the right the ∞-groupoid of derived elliptic curves over $A$.
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 $E_\infty$-rings.