The sigma orientation is an \(H_\infty\) map.

*(English)*Zbl 1071.55003The 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 \(H_\infty\).

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 \(H_\infty\).

Reviewer: Gerd Laures (Bochum)