A Note on the Fourier coefficients of half-integral weight modular forms

TL;DR

This paper establishes criteria to determine the algebraicity of Fourier coefficients of half-integral weight modular forms and characterizes membership in Kohnen's +-subspace using finitely many coefficients.

Contribution

It provides a finite check for algebraicity and membership in Kohnen's +-subspace, simplifying analysis of these modular forms.

Findings

Algebraicity of Fourier coefficients can be checked finitely.

A necessary and sufficient condition for Kohnen's +-subspace membership.

Finite criteria streamline the study of half-integral weight modular forms.

Abstract

In this note, we show that the algebraicity of the Fourier coefficients of half-integral weight modular forms can be determined by checking the algebraicity of the first few of them. We also give a necessary and sufficient condition for a half-integral weight modular form to be in Kohnen's +-subspace by considering only finitely many terms.