Journal of Differential Geometry

Critical points of the classical Eisenstein series of weight two

Zhijie Chen and Chang-Shou Lin

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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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J. Differential Geom., Volume 113, Number 2 (2019), 189-226.

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

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Chen, Zhijie; Lin, Chang-Shou. Critical points of the classical Eisenstein series of weight two. J. Differential Geom. 113 (2019), no. 2, 189--226. doi:10.4310/jdg/1571882423.

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