Vanishing properties of Fourier coefficients of holomorphic $\eta$-quotients

TL;DR

This paper investigates the conditions under which Fourier coefficients of holomorphic eta-quotients vanish, focusing on cases involving CM newforms and eta-quotients related to sums of squares and class numbers.

Contribution

It provides new insights into the vanishing properties of Fourier coefficients for specific classes of holomorphic eta-quotients, linking them to CM forms and class number problems.

Findings

Identifies vanishing patterns for Fourier coefficients of eta-quotients with CM newforms.

Establishes connections between eta-quotients and sums of squares, Hurwitz class numbers.

Provides criteria for the vanishing of Fourier coefficients in these contexts.

Abstract

In this paper, we study vanishing of Fourier coefficients of holomorphic $\eta$-quotients. We investigate examples of two different types: the first one involves integral weight CM newforms, while the second one involves half-integral weight $\eta$-quotients associated with sums of squares and Hurwitz class numbers.