Hecke operators in equivariant elliptic cohomology and generalized moonshine

This paper studies connections between generalized moonshine and elliptic cohomology with a focus on the action of the Hecke correspondence and its implications for the notion of replicability.

💡 Research Summary

The paper establishes a deep link between equivariant elliptic cohomology and the phenomenon of generalized moonshine by constructing a natural action of Hecke operators on the equivariant elliptic cohomology of classifying spaces of finite groups. The authors begin by recalling that ordinary elliptic cohomology can be interpreted as sections of a sheaf of modular forms over the moduli stack of elliptic curves. When a finite group G acts, the equivariant version, denoted Ell_G, becomes a sheaf 𝒪_{Ell_G} on the same stack equipped with a G‑equivariant structure.

The central technical achievement is the definition of Hecke correspondences in this equivariant setting. For each positive integer n, the authors consider all n‑isogenies φ : E → E′ of elliptic curves. Pull‑back along φ induces a map on sections, and push‑forward (or transfer) along the finite map φ gives a composite operation φ_* ∘ φ^*. Averaging over all n‑isogenies yields an operator
\