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Transactions of the American Mathematical Society

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


Author: Yifei Zhu
Journal: Trans. Amer. Math. Soc. 373 (2020), 3733-3764
MSC (2010): Primary 55S25
DOI: https://doi.org/10.1090/tran/8032
Published electronically: February 11, 2020
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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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Additional Information

Yifei Zhu
Affiliation: Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, People’s Republic of China
Email: zhuyf@sustech.edu.cn

DOI: https://doi.org/10.1090/tran/8032
Received by editor(s): June 3, 2019
Received by editor(s) in revised form: October 3, 2019, and October 5, 2019
Published electronically: February 11, 2020
Additional Notes: This work was partly supported by the National Natural Science Foundation of China grant 11701263.
Article copyright: © Copyright 2020 American Mathematical Society