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derived elliptic curve

Derived elliptic curve

Idea

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).

Definition

Let AA be an E-∞ ring. A derived elliptic curve over AA is derived group scheme ESpecAE \to Spec A with the property that the underlying morphism E¯Specπ 0A\bar E \to Spec \pi_0 A is an ordinary elliptic curve.

Properties

Moduli stack of derived elliptic curves

By the Goerss-Hopkins-Miller theorem the structure sheaf 𝒪\mathcal{O} of the moduli stack of elliptic curves (ordinary elliptic curves) lifts to a sheaf 𝒪 top\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 ( ell,𝒪 top)(\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 AA of the form

Hom(Spec(A),( ell,𝒪 top))E(A), Hom(Spec(A), (\mathcal{M}_{ell},\mathcal{O}^{top})) \simeq E(A) \,,

where on the left we have maps of structured (∞,1)-toposes and on the right the ∞-groupoid of derived elliptic curves over AA.

This is based on the representability theorem (Lurie (Survey), prop. 4.1, Lurie (Representability)).

References

Revised on July 16, 2014 08:47:55 by Urs Schreiber (89.204.138.8)