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started an entry cubic curve,
For the moment I wanted to record (see the entry) a pointer to Akhil Mathew’s identification of that eight-fold cover of $\mathcal{M}_{cub}$ (hence of $\mathcal{M}_{ell}$) which is analogous to the 2-fold cover of the “moduli stack of formal tori” $B \mathbb{Z}_2$ that ends up being the reason for the $\mathbb{Z}_2$-action on $KU$.
So here is the question that I am after: that cover is classified by a map $\mathcal{M}_{ell} \to B \mathbb{Z}/8\mathbb{Z}$, hence we get a double cover of the moduli space of elliptic curves $d \colon \mathcal{M}_{ell} \to B\mathbb{Z}/2\mathbb{Z}$.
Accordingly there is a spectrum $Q \coloneqq d_\ast(\mathcal{O}^{top})$ equipped with a $\mathbb{Z}_2$-action whose homotopy fixed points is $tmf$, I suppose: $tmf \simeq Q^{\mathbb{Z}_2}$. (Hm, maybe I need to worry about the compactification…).
I’d like to say that $Q$ is to $tmf$ as $KU$ is to $KO$. This is either subject to some confusion (wich one?) or else is an old hat. In the second case: what would be a reference?
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