Zeros of replicable functions

TL;DR

This paper proves that the zeros of specific modular functions related to low-level genus zero groups are all located on the boundary of their natural fundamental domains, extending previous partial results.

Contribution

It generalizes the understanding of zero distributions of modular functions for multiple low-level genus zero groups beyond prior partial proofs.

Findings

Zeros lie on the boundary of fundamental domains for the considered groups.

Extension of previous results to new groups such as _0^*(3), _0(2 2), etc.

Provides a unified approach for multiple low-level genus zero groups.

Abstract

Following the work of Asai, Kaneko, and Ninomiya for Faber polynomials associated to $\mathrm{PSL}_2(\mathbb{Z})$, and Bannai, Kojima, and Miezaki's partial proof for the case of $\Gamma_0^*(2)$, we show that the zeros of certain modular functions associated to some low-level genus zero groups are all located on the boundary of certain natural fundamental domains for $\Gamma$. The groups considered are $\Gamma_0^*(2)$, $\Gamma_0^*(3)$, $\Gamma_0(2\Vert 2)$, $\Gamma_0^*(5)$, $\Gamma_0(6)+$, $\Gamma_0^*(7)$, $\Gamma_0(4\Vert 2)+$, $\Gamma_0(3\Vert 3)$, and $\Gamma_0(10)+$.