Distribution of integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes

TL;DR

This paper extends the classification of distribution properties of Fourier coefficients of half-integer weight modular forms modulo primes, applying Rankin-Cohen brackets and exploring implications for singular moduli, Hurwitz class numbers, and overpartitions.

Contribution

It generalizes previous results to primes p ≥ 5 using Rankin-Cohen brackets and investigates distribution properties for various number-theoretic functions.

Findings

Distribution properties of Fourier coefficients modulo primes p ≥ 5.

Distribution of traces of singular moduli and Hurwitz class numbers modulo p.

Analysis of an analogue of Newman's conjecture for overpartitions.

Abstract

Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty} a_f(n)q ^n \in S_{\lambda+1/2}(\Gamma_0(N),\chi)\cap \mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p \geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $p\geq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newman's conjecture for overpartitions.