Elliptic Genera of Level N and Elliptic Cohomology

Elliptic genera of level N have been defined by F. Hirzebruch, generalising the earlier notion of elliptic genus due to S. Ochanine. We show that there are corresponding elliptic cohomology theories which are naturally associated to such genera and that these are obtained from the level 1 case by algebraic extension of the coefficient rings from level 1 to level N modular forms. Introduction. In [8], F. Hirzebruch has introduced elliptic genera of level N which are (multiplicative) genera ρα : MU∗ −→ Z[1/N, ζ N ]((q N )). Here ζ N = e i /N , q = e i τ for τ ∈ H (the upper half plane), q N = e i τ/N , and α ∈ (1/N)Lτ/Lτ ⊆ C/Lτ is required to have order N as an element of the torus C/Lτ associated to the lattice Lτ = 〈τ, 1〉 ⊆ C. For any ring R, we denote by R((X)) = R[[X]][X−1] the ring of Laurent series in X over R with finitely many negative degree terms. The main purpose of the present work is to fit such genera into the framework of elliptic cohomology in a manner which generalises the original level 2 constructions. When earlier versions of this paper were written the author was unaware of the work of J.-L. Brylinski [4], who constructs higher level theories in many respects similar to ours, although he does not invert N in his level N theory. However, he does make use of deep facts from the theory of moduli schemes and in some cases this allows him to prove stronger results than ours, which only use standard facts from the theory of complex cobordism comodules. We hope to investigate further the precise relationships between these approaches in future 1991 Mathematics Subject Classification. 55N22 14L05 10D05.