Czechoslovak Mathematical Journal, first online, pp. 1-15


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

Soufiane Mezroui

Received May 3, 2018.   Published online November 18, 2019.

Abstract:  Let $f=\sum_{n=1}^{\infty}a(n)q^n\in S_{k+1/2}(N,\chi_0)$ be a nonzero cuspidal Hecke eigenform of weight $k+\frac12$ 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 $a(t)\neq0$. 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 $n\equiv d \mod q$.
Keywords:  Shimura lift; Fourier coefficient; half-integral weight; Sato-Tate equidistribution
Classification MSC:  11F30, 11F37
DOI:  10.21136/CMJ.2019.0223-18

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Affiliations:   Soufiane Mezroui, Laboratory of Information and Communication Technologies, Department of Information and Communication Systems, National School of Applied Sciences, Abdelmalek Essaadi University, Tangier, Morocco e-mail: mezroui.soufiane@yahoo.fr


 
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