Fluctuations in the distribution of Hecke eigenvalues about the Sato-Tate measure

Abstract

We study fluctuations in the distribution of families of $p$-th Fourier coefficients $a_f(p)$ of normalised holomorphic Hecke eigenforms $f$ of weight $k$ with respect to $SL_2(\mathbb{Z})$ as $k \to \infty$ and primes $p \to \infty.$ These families are known to be equidistributed with respect to the Sato-Tate measure. We consider a fixed interval $I \subset [-2,2]$ and derive the variance of the number of $a_f(p)$'s lying in $I$ as $p \to \infty$ and $k \to \infty$ (at a suitably fast rate). The number of $a_f(p)$'s lying in $I$ is shown to asymptotically follow a Gaussian distribution when appropriately normalised. A similar theorem is obtained for primitive Maass cusp forms.