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)).