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 $\Gamma_{0}(2)$. We also give a criteria for a fundamental domain containing a critical point of $E_2(\tau)$. Furthermore, under the M\"obius transformation of $\Gamma_{0}(2)$ action, all critical points can be mapped into the basic fundamental domain $F_0$ and their images are contained densely on three smooth curves. A geometric interpretation of these smooth curves is also given. It turns out that these smooth curves coincide with the degeneracy curves of trivial critical points of a multiple Green function related to flat tori.

中文翻译：

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经典爱森斯坦重量二级数的临界点

在本文中，我们完全确定了权重为 $2$ 的归一化 Eisenstein 级数 $E_2(\tau)$ 的临界点。尽管 $E_2(\tau)$ 不是模形式，但我们的结果表明 $E_2(\tau)$ 在 $\Gamma_{0}(2)$ 的每个基本域中最多只有一个临界点。我们还给出了包含临界点 $E_2(\tau)$ 的基本域的标准。此外，在 $\Gamma_{0}(2)$ 动作的 M\"obius 变换下，所有临界点都可以映射到基本基本域 $F_0$ 并且它们的图像密集地包含在三个平滑曲线上。几何解释还给出了这些平滑曲线的数量。结果表明，这些平滑曲线与与平坦环面相关的多重格林函数的平凡临界点的简并曲线重合。