A central limit theorem for Hilbert modular forms

TL;DR

This work proves a central limit theorem for the distribution of Satake-angle parameters $\theta_\pi(\mathfrak{p})$ arising from Hilbert modular cusp forms, by averaging over the finite family $\Pi_{\underline{k}}(\mathfrak{n})$ with squarefree level and suitably growing weights. The authors combine Beurling–Selberg approximations, Chebyshev polynomial expansions, and an Arthur/Lau–Li–Wang trace formula to analyze the fluctuations of the Sato–Tate counting function $N_I(\pi,x)$ around its mean $\pi_L(x)\mu_\infty(I)$, deriving precise moment asymptotics. The main result shows that the normalized error converges in distribution to a Gaussian with variance $\pi_L(x)(\mu_\infty(I)-\mu_\infty(I)^2)$, under growth conditions on $\underline{k}$; a higher-moments analysis confirms the Gaussian limit, and a smooth-test-function variant relaxes these growth assumptions. The findings extend probabilistic understandings of eigenvalue statistics from classical modular forms to the Hilbert modular setting and provide a framework for smooth generalizations with weaker weight-growth requirements.

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 $θ_π(\mathfrak{p})$ coming from Satake parameters corresponding to unramified $π_\mathfrak{p}$ where $π_\mathfrak{p}$ comes from a global $π$ ranging over a certain finite set $Π_{\underline{k}}(\mathfrak{n})$ of cuspidal automorphic representations of GL$_2(\mathbb{A})$ with trivial central character. For such a representation $π$, it is known that the angles $θ_π(\mathfrak{p})$ follow the Sato-Tate distribution. Fixing an interval $I\subseteq [0,π]$, we prove a central limit theorem for the number of angles $θ_π(\mathfrak{p})$ 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 $x$.

Theorems & Definitions (18)

• Theorem 1.1
• Proposition 2.1: BB-KS, Proposition 18
• Remark
• Lemma 2.1
• proof
• Lemma 2.2: RT2017
• Proposition 2.2
• proof
• Proposition 3.1
• Theorem 4.1