The Hecke algebra action and the Rezk logarithm on Morava E-theory of height 2

Zhu, Yifei

doi:10.1090/tran/8032

The Hecke algebra action and the Rezk logarithm on Morava E-theory of height 2
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Trans. Amer. Math. Soc. 373 (2020), 3733-3764 Request permission

Abstract:

Given a one-dimensional formal group of height $2$, let $E$ be the Morava E-theory spectrum associated to the Lubin–Tate universal deformation of this formal group. By computing with moduli spaces of elliptic curves, we provide an explicit description for an algebra of Hecke operators acting on $E$-cohomology. As an application, we obtain 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 connects to logarithms of modular units. In particular, we define an action of Hecke operators on certain logarithmic $q$-series, in the sense of Knopp and Mason, which agrees with our vanishing result and extends the classical Hecke action on modular forms.

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