Isogenies and Hecke Operators on Elliptic Cohomology

Operations on tempered cohomology theories (and complex-periodic cohomology theories more generally) are known to be intimately related to the isogenies of their associated algebraic groups. The Adams operations on K-theory, for example, are associated with the $k$th power map on $\mathbb{G}_m$. Their analogues at height $2$, Hecke operators on elliptic cohomology, are not as well-understood.

In this talk, I will describe a spectral algebreo-geometric approach to studying these operators. This begins with the spectral moduli stack of isogenies of oriented elliptic curves, which I show to be a (nonconnective) spectral Deligne-Mumford stack using a connected-étale decomposition and a generalization of Lurie's theorem on the representability of derived Hilbert spaces. We will then apply this to construct stacky Hecke operators in a manner similar to their classical moduli-theoretic construction—with the notable caveat that in the spectral case, Grothendieck duality introduces a nontrivial twist.