derived elliptic curve
Derived elliptic curve
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.
For the notation and terminology of the following definition, see (for the time being)
Let be an E-∞ ring. A derived elliptic curve over is derived group scheme with the property that the underlying morphism is an ordinary elliptic curve.
Moduli stack of derived elliptic curves
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
where on the left we have maps of structured (∞,1)-toposes and on the right the ∞-groupoid of derived elliptic curves over .
This is based on the representability theorem (Lurie (Survey), prop. 4.1, Lurie (Representability)).
Revised on April 8, 2014 04:36:43
by Urs Schreiber