Sign changes of coefficients of half integral weight modular forms

TL;DR

This paper investigates the sign patterns of Fourier coefficients of half integral weight modular forms, demonstrating that for most primes, the coefficients exhibit infinitely many sign changes, especially for Hecke eigenforms.

Contribution

It establishes new results on sign changes of Fourier coefficients for half integral weight modular forms, including both Hecke eigenforms and more general cusp forms.

Findings

For Hecke eigenforms, sign changes occur infinitely often for coefficients indexed by almost all primes.

The results extend to non-eigenform cusp forms under certain conditions.

Sign change behavior is linked to properties of Fourier coefficients at square-free integers.

Abstract

For a half integral weight modular form $f$ we study the signs of the Fourier coefficients $a(n)$. If $f$ is a Hecke eigenform of level $ N$ with real Nebentypus character, and $t$ is a fixed square-free positive integer with $a(t)\neq 0$, we show that for all but finitely many primes $p$ the sequence $(a(tp^{2m}))_{m}$ has infinitely many signs changes. Moreover, we prove similar (partly conditional) results for arbitrary cusp forms $f$ which are not necessarily Hecke eigenforms.