Conformal maps and critical points of Eisenstein series

TL;DR

This paper analyzes the critical points of Eisenstein series by describing associated polymorphic functions as conformal maps, providing insights into their locations and behavior.

Contribution

It offers an explicit conformal map-based description of polymorphic functions related to Eisenstein series to analyze their critical points.

Findings

Critical points correspond to poles of associated polymorphic functions

Explicit conformal maps enable qualitative analysis of critical point locations

Provides a detailed map-based understanding of Eisenstein series behavior

Abstract

We investigate the critical points of the basic (quasi-)modular forms $E_2$, $E_4$, and $E_6$. They occur where some associated polymorphic functions have poles. By an explicit description of these polymorphic functions as conformal maps, one can give an accurate qualitative analysis of the locations of the critical points of $E_2$, $E_4$, and $E_6$.