The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

TL;DR

This paper proves that the signs of Fourier coefficients of certain half-integer weight modular forms are evenly distributed across all arithmetic progressions, extending previous results on their equidistribution.

Contribution

It establishes the equidistribution of Fourier coefficients over any arithmetic progression for a broad class of half-integer weight modular forms, generalizing prior specific cases.

Findings

Signs of Fourier coefficients are equidistributed in arithmetic progressions.

Extends previous results from specific subfamilies to all arithmetic progressions.

Supports the conjecture of Bruinier and Kohnen on sign distribution.

Abstract

Let $f=\sum_{n=1}^{\infty}a(n)q^{n}\in S_{k+1/2}(N,\chi_{0})$ be a non-zero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus $\chi_{0}$ where the Fourier coefficients $a(n)$ are real. Bruinier and Kohnen conjectured that the signs of $a(n)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies $\{a(t n^{2})\}_{n}$ where $t$ is a squarefree integer such that $a(t)\neq 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that $\{a(t n^{2})\}_{n}$ is equidistributed over any arithmetic progression $n\equiv d\text{ mod }q$.