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

TL;DR

This paper investigates the distribution of zeros of the derivative of the Eisenstein series $E_6$, providing new methods to determine their number in fundamental domains and offering a monodromy interpretation related to complex linear ODEs.

Contribution

The paper introduces a novel approach to analyze zeros of $E_6'$ and characterizes their distribution in fundamental domains, extending previous work on Eisenstein series.

Findings

$E_6'( au)$ has 1 or 2 zeros in each fundamental domain of $ ext{Gamma}_0(2)$

A criterion for exactly 2 zeros in a fundamental domain is established

Zeros map to a dense subset of three disjoint curves in the fundamental domain

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 M\"{o}bius 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 $E_6'(\tau)$ 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 $E_6'(\tau)$ into $F_0$ via the M\"{o}bius 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 $E_6'(\tau)$ in fundamental domains of $SL(2,\mathbb{Z})$.