nLab
Tmf(n)

Contents

Context

Elliptic cohomology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

After localization at primes dividing nn \in \mathbb{N} the covering of the moduli stack of elliptic curves ell\mathcal{M}_{{ell}} by that of elliptic curves with level-n structure ell[n] ell\mathcal{M}_{{ell}}[n] \to \mathcal{M}_{{ell}} is sufficiently good that the Goerss-Hopkins-Miller-Lurie theorem may be applied to produce a homomorphism of E-∞ rings

(TMFTMF(n))=Γ(( ell¯[n] ell¯),𝒪 top) (TMF \to TMF(n)) = \Gamma\left( \left( \mathcal{M}_{\overline{ell}}[n] \to \mathcal{M}_{\overline{ell}} \right), \mathcal{O}^{top} \right)

exhibiting TMF (after localization at nn) as the homotopy fixed points of a modular group action by SL 2(/n)SL_2(\mathbb{Z}/n\mathbb{Z}) (Hill-Lawson 13, p.3).

With a bit more work one obtains analogous statements for the compactified moduli stack of elliptic curves and TmfTmf instead of TMFTMF (Hill-Lawson 13, theorem 9.1)

This is directly analogous (Lawson-Naumann 12, Hill-Lawson 13) to how KO \to KU exhibits the inclusion of the homotopy fixed points of the 2\mathbb{Z}_2-action on complex K-theory (which defines KR-theory, see there for more).

See also at modular equivariant elliptic cohomology.

References

Created on April 11, 2014 at 06:22:29. See the history of this page for a list of all contributions to it.