Conformal maps and critical points of Eisenstein series

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 $E_6$.