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The sigma orientation is an $$H_\infty$$ map. (English) Zbl 1071.55003
The paper is part of a series of papers by the authors which aims at the generalization of the Witten genus to an $$E_\infty$$-ring map from $$MO \left< 8\right>$$ to the spectrum $$tmf$$ of topological modular forms [M. J. Hopkins, Proceedings of the international congress of mathematicians, ICM ’94, 554–565 (1995; Zbl 0848.55002)].
Let $$MU \left< 6\right>$$ be the Thom spectrum associated to $$BU \left<6\right>$$, the 5-connected cover of $$BU$$. An elliptic spectrum $$(E, C, t)$$ is a homogeneous ring spectrum $$E$$, an elliptic curve $$C$$ over $$\pi_0 E$$ and an isomorphism $$t$$ from the formal group $$E^0 C P^{\infty}$$ to the formal completion of $$C$$. In [M. Ando, M. J. Hopkins and N. P. Strickland, Invent. Math. 146, No. 3, 595–687 (2001; Zbl 1031.55005)], the authors showed that any such admits a canonical orientation $$\sigma: MU \left< 6\right> \longrightarrow E$$ by identifying the set of orientations with the set of cubical structures.
In this paper the authors show that the orientation is an $$H_\infty$$-ring map if $$E$$ is an $$H_\infty$$ elliptic spectrum. Here, an $$H_\infty$$ elliptic spectrum is not just an elliptic spectrum whose underlying spectrum is $$H_\infty$$ but, roughly speaking, the isogenies associated to closed finite subgroups $$A$$ of the formal group constructed with the $$H_\infty$$ structure should extend to isogenies of the elliptic curve.
The behaviour of this structure with respect to variations in the subgroup $$A$$ gives descent data for level structures. The discussion of level structures is done in a quite general setting and takes a large part of the paper.
The commutativity of the diagram which gives the $$H_\infty$$ property is reduced via character theory [M. J. Hopkins, N. J. Kuhn and D. C. Ravenel, J. Am. Math. Soc. 13, No. 3, 553–594 (2000; Zbl 1007.55004)] to the equality of two coordinates on $$\psi^* G$$, the range of the isogeny with kernel $$A$$. These agree by the uniqueness of the cubical structures if $$p$$ is regular in $$\pi_0 E$$.
In particular, if $$E$$ is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of positive characteristic then the orientation $$\sigma$$ is $$H_\infty$$.

##### MSC:
 55N34 Elliptic cohomology 14H52 Elliptic curves 58J26 Elliptic genera 55P42 Stable homotopy theory, spectra 55P43 Spectra with additional structure ($$E_\infty$$, $$A_\infty$$, ring spectra, etc.)
##### Keywords:
elliptic cohomology; ring spectra; level structures
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