A variant of multiplicity one theorems for half-integral weight modular forms

TL;DR

This paper demonstrates that the signs of Fourier coefficients on specific sub-families uniquely identify half-integral weight cuspidal eigenforms and explores sign change phenomena for products of Fourier coefficients of different eigenforms.

Contribution

It introduces a variant of multiplicity one theorems for half-integral weight modular forms, establishing uniqueness results based on Fourier coefficient signs.

Findings

Fourier coefficient signs determine the eigenform uniquely.

Sign change results for products of Fourier coefficients are established.

New multiplicity one variants for half-integral weight forms are proved.

Abstract

We show that signs of Fourier coefficients, on certain sub-families, determine the half-integral weight cuspidal eigenform uniquely, up to a positive constant. We also study sign change results for the product of the Fourier coefficients of two distinct half-integral weight eigenforms.