The Hecke algebra action on Morava E-theory of height 2

TL;DR

This paper explicitly describes the action of Hecke operators on Morava E-theory at height 2, revealing connections to modular forms, logarithmic cohomology, and modular units, with implications for the structure of E-cohomology.

Contribution

It provides an explicit computation of Hecke algebra actions on Morava E-theory of height 2 using elliptic curves and modular forms, extending classical Hecke actions.

Findings

Vanishing of Rezk's logarithmic cohomology operation on units of E

Identification of elements in the kernel related to modular forms with zero Serre derivative

Extension of classical Hecke action to certain logarithmic q-series

Abstract

Given a one-dimensional formal group of height 2, let E be the Morava E-theory spectrum associated to its universal deformation over the Lubin-Tate ring. By computing with moduli spaces of elliptic curves, we give an explicitation for an algebra of Hecke operators acting on E-cohomology. This leads to a vanishing result for Rezk's logarithmic cohomology operation on the units of E. It identifies a family of elements in the kernel with meromorphic modular forms whose Serre derivative is zero. Our calculation finds a connection to logarithms of modular units. In particular, we work out an action of Hecke operators on certain "logarithmic" q-series, in the sense of Knopp and Mason, that agrees with our vanishing result and extends the classical Hecke action on modular forms.