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October 2019 Critical points of the classical Eisenstein series of weight two
Zhijie Chen, Chang-Shou Lin
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J. Differential Geom. 113(2): 189-226 (October 2019). DOI: 10.4310/jdg/1571882423


In this paper, we completely determine the critical points of the normalized Eisenstein series $E_{2}(\tau)$ of weight 2. Although $E_{2}(\tau)$ is not a modular form, our result shows that $E_{2}(\tau)$ has at most one critical point in every fundamental domain of the form $\gamma (F_{0})$ of $\Gamma_{0}(2)$, where $\gamma (F_{0})$ are translates of the basic fundamental domain $F_{0}$ via the Möbius transformation of $\gamma \in \Gamma_{0}(2)$. We also give a criteria for such fundamental domain containing a critical point of $E_{2}(\tau)$. Furthermore, under the Möbius transformations of $\Gamma_{0}(2)$ action, all critical points can be mapped into the basic fundamental domain $F_{0}$ and their images in $F_{0}$ give rise to a dense subset of the union of three connected smooth curves in $F_{0}$. A geometric interpretation of these smooth curves is also given. It turns out that these curves coincide with the degeneracy curves of trivial critical points of a multiple Green function related to flat tori.


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Zhijie Chen. Chang-Shou Lin. "Critical points of the classical Eisenstein series of weight two." J. Differential Geom. 113 (2) 189 - 226, October 2019.


Received: 17 July 2017; Published: October 2019
First available in Project Euclid: 24 October 2019

zbMATH: 07122207
MathSciNet: MR4023291
Digital Object Identifier: 10.4310/jdg/1571882423

Rights: Copyright © 2019 Lehigh University


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Vol.113 • No. 2 • October 2019
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