On the distribution of critical points of the Eisenstein series $E_6$ and monodromy interpretation

Abstract

In previous works joint with Lin, we proved that the Eisenstein series $E_4$ (resp. $E_2$) has at most one critical point in every fundamental domain $\gamma (F_0)$ of ${\Gamma }_0(2)$, where $\gamma (F_0)$ are translates of the basic fundamental domain $F_0$ via the Möbius transformation of $\gamma \in {\Gamma }_0(2)$. But the method can not work for the Eisenstein series $E_6$. In this paper, we develop a new approach to show that $E_6^{\prime }(\tau )$ has exactly either $1$ or $2$ zeros in every fundamental domain $\gamma (F_0)$ of ${\Gamma }_0(2)$. A criterion for $\gamma (F_0)$ containing exactly $2$ zeros is also given. Furthermore, by mapping all zeros of $E_6^{\prime }(\tau )$ into $F_0$ via the Möbius transformations of ${\Gamma }_0(2)$ action, the images give rise to a dense subset of the union of three disjoint smooth curves in $F_0$. A monodromy interpretation of these curves from a complex linear ODE is also given. As a consequence, we give a complete description of the distribution of the zeros of $E_6^{\prime }(\tau )$ in fundamental domains of $SL(2,\mathbb{Z})$.