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

Mezroui, Soufiane

Czechoslovak Mathematical Journal, Tome 70 (2020) no. 1, pp. 235-249.

Résumé

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 $k+\frac {1}{2}$ 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 $n\equiv d \mod q$.

Détail

DOI : 10.21136/CMJ.2019.0223-18

Classification : 11F30, 11F37
Keywords: Shimura lift; Fourier coefficient; half-integral weight; Sato-Tate equidistribution