Contents

cohomology

# Contents

## 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).