Equidistribution of Signs for Modular Eigenforms of Half Integral Weight

TL;DR

This paper proves an equidistribution conjecture for the signs of Fourier coefficients of half-integer weight modular forms, using the Sato-Tate theorem and addressing both prime and natural number cases.

Contribution

It provides the first unconditional proof for prime-indexed coefficients and extends the result to all natural numbers under a weaker error term assumption.

Findings

Unconditional proof for coefficients {a(tp^2)} with t squarefree.

Results in terms of natural density for prime-related coefficients.

Extension to all natural numbers under a conjectured error term.

Abstract

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato-Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp^2)}_p where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn^2)}_n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato-Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind-Dirichlet density.