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 $x$.

12 pages