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Is the group action on an elliptic curve reflecting the fact that it is a space with with transfers?

Through the recognition principle

$$\mathrm{Mon}_{\mathbb{E}_\infty}(\mathrm{Spc})^{gp}\simeq \mathrm{Sp}_{\geq0}\subset \mathrm{Sp}=\:\text{cohomology theories}$$

there must be a cohomology theory for every elliptic curve. Is this elliptic cohomology developed by Lurie?

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  • $\begingroup$ Crossposted from Math.SE: math.stackexchange.com/questions/4374022/…. (In the future, please note MSE crossposts in the original question to avoid duplication of effort.) $\endgroup$
    – dhy
    Commented Feb 5, 2022 at 3:24
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    $\begingroup$ I'm not sure that this question is appropriate for MO, but here is a quick answer: This is not elliptic cohomology, actually, this is a sum of two copies of ordinary singular cohomology with coefficients in $\mathbb{Z}$ (with an index shift by $1$). The reason is that the homotopy type of an elliptic curve is $S^1\times S^1$, and $S^1$ is the Eilenberg-Maclane space $K(\mathbb{Z},1)$. $\endgroup$
    – dhy
    Commented Feb 5, 2022 at 3:27
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    $\begingroup$ Note that it's grouplike $\mathbb E_\infty$ spaces that are equivalent to connective spectra. $\endgroup$
    – Maxime Ramzi
    Commented Feb 5, 2022 at 10:27

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