On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels I

TL;DR

This paper investigates the zeros of Eisenstein series and modular functions related to normalizers of low-level congruence subgroups, focusing on genus zero cases up to level twelve, providing foundational theory for these modular functions.

Contribution

It introduces a general theory of modular functions for normalizers of low-level congruence subgroups, emphasizing their zeros and structural properties.

Findings

Zeros of Eisenstein series are located within specific regions.

Modular functions exhibit predictable zero distributions for genus zero normalizers.

Foundational theory supports further analysis of modular functions for these groups.

Abstract

We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In Part I, we will consider the general theory of modular functions for the normalizers.