Notes on the multiplier systems of $\eta(\tau)$ and $\theta(\tau)$

TL;DR

This paper investigates the multiplier systems of eta and theta functions raised to integer powers, focusing on identifying their kernels, which are crucial for understanding their modular properties.

Contribution

It explicitly determines the kernels of the multiplier systems for eta and theta functions raised to integer powers, advancing the understanding of their modular characters.

Findings

Kernels of eta multiplier systems are explicitly characterized.

Kernels of theta multiplier systems are explicitly characterized.

Provides a comprehensive description of the modular character kernels for these functions.

Abstract

The multiplier systems of $\eta^{2k}(\tau)$ and $\theta^{2k}(\tau)$ $(k\in\mathbb{Z})$ are characters. In this paper, we determine their kernels, Ker$\,\nu_{\eta^{2k}}$ and Ker$\,\nu_{ \theta^{2k} } $.