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The Witten genus has famously been lifted to the string orientation of tmf ("topological Witten genus"). For the Ochanine genus, I am aware of a lift to a "spin orientation of Tate K-theory", namely, to a map of ring spectra $M \mathrm{Spin}\to KO[ [q] ]$: lemma 5.4, 5.8 in

  • Matthias Kreck, Stefan Stolz, $HP^2$-bundles and elliptic homology, Acta Math, 171 (1993) 231-261

At least naively, one would hope to see a further refinement of the Ochanine genus to an orientation of something like $tmf_0(2)$. Is there any reason for why this should not work? Or has it just not been considered?

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    $\begingroup$ Is there a difference between $tmf_0(2)$ and the elliptic cohomology theory constructed by Landweber, Ravenel, Stong? They construct a (homotopy commutative) orientation on their elliptic cohomology theory as I remember. Their theory has 2 inverted, but I suspect that a construction of $tmf_0(2)$ as an $E_\infty$-ring spectrum would also have 2 inverted. I should admit that I am a bit out of my depth here. $\endgroup$
    – Justin Noel
    Commented Apr 24, 2014 at 9:49
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    $\begingroup$ Mild clarification: Since the spectrum of LRS is periodic, I should be asking if Tmf_0(2)$ is the same as their spectrum. Note any orientation lifts to their connective covers canonically up to homotopy so we could rephrase the question in terms of the periodic forms. $\endgroup$
    – Justin Noel
    Commented Apr 24, 2014 at 9:58
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    $\begingroup$ I see, thanks, I was missing the obvious. Yes, you are right (e.g. top of p. 4 in arxiv.org/abs/math/0507184). $\endgroup$
    – Urs Schreiber
    Commented Apr 24, 2014 at 10:26
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    $\begingroup$ I just stumbled across this question again. Dylan Wilson answered it in the affirmative: arxiv.org/abs/1507.05116 $\endgroup$
    – Tyler Lawson
    Commented Feb 26, 2016 at 6:29

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