Critical points of the classical Eisenstein series of weight two

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 ${\mathrm{\Gamma }}_0(2)$. We also give a criteria for a fundamental domain containing a critical point of $E_2(\tau )$. Furthermore, under the M"obius transformation of ${\mathrm{\Gamma }}_0(2)$ 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.