The Hecke algebra action and the Rezk logarithm on Morava E-theory of height 2
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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.References
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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
- MR Author ID: 1050537
- Email: zhuyf@sustech.edu.cn
- 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.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 3733-3764
- MSC (2010): Primary 55S25
- DOI: https://doi.org/10.1090/tran/8032
- MathSciNet review: 4082255