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

Mezroui, Soufiane. "The equidistribution of Fourier coefficients of half integral weight modular forms on the plane." Czechoslovak Mathematical Journal 70.1 (2020): 235-249. <http://eudml.org/doc/297122>.

@article{Mezroui2020,
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 $k+\frac\{1\}\{2\}$ 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 $\lbrace a(t n^\{2\})\rbrace _\{n\}$, where $t$ is a squarefree integer such that $a(t)\ne 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that $\lbrace a(t n^\{2\})\rbrace _\{n\}$ is equidistributed over any arithmetic progression $n\equiv d ~\@mod \;q$.},
author = {Mezroui, Soufiane},
journal = {Czechoslovak Mathematical Journal},
keywords = {Shimura lift; Fourier coefficient; half-integral weight; Sato-Tate equidistribution},
language = {eng},
number = {1},
pages = {235-249},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The equidistribution of Fourier coefficients of half integral weight modular forms on the plane},
url = {http://eudml.org/doc/297122},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Mezroui, Soufiane
TI - The equidistribution of Fourier coefficients of half integral weight modular forms on the plane
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 1
SP - 235
EP - 249
AB - 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 $\lbrace a(t n^{2})\rbrace _{n}$, where $t$ is a squarefree integer such that $a(t)\ne 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that $\lbrace a(t n^{2})\rbrace _{n}$ is equidistributed over any arithmetic progression $n\equiv d ~\@mod \;q$.
LA - eng
KW - Shimura lift; Fourier coefficient; half-integral weight; Sato-Tate equidistribution
UR - http://eudml.org/doc/297122
ER -