A central limit theorem for Hilbert modular forms

Abstract

For a prime ideal $\mathfrak{p}$ in a totally real number field $L$ with the adele ring $\mathbb{A}$, we study the distribution of angles $\theta_\pi(\mathfrak{p})$ coming from Satake parameters corresponding to unramified $\pi_\mathfrak{p}$ where $\pi_\mathfrak{p}$ comes from a global $\pi$ ranging over a certain finite set $\Pi_{\underline{k}}(\mathfrak{n})$ of cuspidal automorphic representations of GL$_2(\mathbb{A})$ with trivial central character. For such a representation $\pi$, it is known that the angles $\theta_\pi(\mathfrak{p})$ follow the Sato-Tate distribution. Fixing an interval $I\subseteq [0,\pi]$, we prove a central limit theorem for the number of angles $\theta_\pi(\mathfrak{p})$ that lie in $I$, as $\mathrm{N}(\mathfrak{p})\to\infty$. The result assumes $\mathfrak{n}$ to be a squarefree integral ideal, and that the components in the weight vector $\underline{k}$ grow suitably fast as a function of $x$.