nLab Tmf(n)

Idea

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

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

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

With a bit more work one obtains analogous statements for the compactified moduli stack of elliptic curves and $Tmf$ instead of $TMF$ (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 $\mathbb{Z}_2$-action on complex K-theory (which defines KR-theory, see there for more).

See also at modular equivariant elliptic cohomology.

References

• Mark MahowaldCharles Rezk, Topological modular forms of level 3, Pure Appl. Math. Quar. 5 (2009) 853-872 (pdf)

• Donald Davis, Mark Mahowald, Connective versions of $TMF(3)$ (arXiv:1005.3752)

• Vesna Stojanoska, Duality for Topological Modular Forms (arXiv:1105.3968)

• Tyler Lawson, Niko Naumann, Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2, Int. Math.

  Res. Not. (2013) (arXiv:1203.1696)

• Michael Hill, Tyler Lawson, Topological modular forms with level structure (arXiv:1312.7394)