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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2014
    • (edited Apr 4th 2014)

    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 cub\mathcal{M}_{cub} (hence of ell\mathcal{M}_{ell}) which is analogous to the 2-fold cover of the “moduli stack of formal tori” B 2B \mathbb{Z}_2 that ends up being the reason for the 2\mathbb{Z}_2-action on KUKU.

    So here is the question that I am after: that cover is classified by a map ellB/8\mathcal{M}_{ell} \to B \mathbb{Z}/8\mathbb{Z}, hence we get a double cover of the moduli space of elliptic curves d: ellB/2d \colon \mathcal{M}_{ell} \to B\mathbb{Z}/2\mathbb{Z}.

    Accordingly there is a spectrum Qd *(𝒪 top)Q \coloneqq d_\ast(\mathcal{O}^{top}) equipped with a 2\mathbb{Z}_2-action whose homotopy fixed points is tmftmf, I suppose: tmfQ 2tmf \simeq Q^{\mathbb{Z}_2}. (Hm, maybe I need to worry about the compactification…).

    I’d like to say that QQ is to tmftmf as KUKU is to KOKO. 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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