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 why this should not work, or has it just not been considered?

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?

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 some reason why this should not work, or has it just not been considered?

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 why this should not work, or has it just not been considered?