Large Fourier coefficients of half-integer weight modular forms

TL;DR

This paper investigates the Fourier coefficients of half-integer weight cusp forms, proving infinitely many are non-zero at fundamental discriminants and that these coefficients can attain large values.

Contribution

It provides a new soft proof of the infinitude of non-zero Fourier coefficients at fundamental discriminants and adapts the resonance method to show these coefficients can be large.

Findings

Infinitely many fundamental discriminants with non-zero Fourier coefficients.

Fourier coefficients at these discriminants can be quite large.

The results apply to cusp forms not necessarily eigenforms.

Abstract

This article is concerned with the Fourier coefficients of cusp forms (not necessarily eigenforms) of half-integer weight lying in the plus space. We give a soft proof that there are infinitely many fundamental discriminants $D$ such that the Fourier coefficients evaluated at $|D|$ are non-zero. By adapting the resonance method, we also demonstrate that such Fourier coefficients must take quite large values.