A discrepancy result for Hilbert modular forms

Abstract

Let $F$ be a totally real number field and $r=[F:\mathbb{Q}].$ Let $A_k(\mathfrak{N},\ddot{I}‰)$ be the space of holomorphic Hilbert cusp forms with respect to $K_1(\mathfrak{N})$, of weight $k=(k_1,\dots ,k_r)$ such that $k_j>2$ for all $j$, and with central Hecke character $\ddot{I}‰$. For integral ideals $\mathfrak{N}$ and $\mathfrak{n}$ in $F$ such that $(\mathfrak{n},\mathfrak{N})=1$, we study the Petersson trace formula for the Hecke operator $T_{\mathfrak{n}}$ acting on the space $A_k(\mathfrak{N},\ddot{I}‰)$. We present asymptotic estimates for the terms of the Petersson formula as $k_0\to \infty ,$ where $k_0=\min (k_1,\dots ,k_r)$. As an application, we obtain a weighted discrepancy bound for the distribution of the eigenvalues of the Hecke operator $T_{\mathfrak{p}}$ (for a fixed prime ideal $\mathfrak{p}$) acting on the space $A_k(\mathfrak{N},1),$ when $F$ has narrow class number $1$, and the ideal $\mathfrak{N}$ is generated by (rational) integers. This generalizes a discrepancy result previously obtained by Jung and Sardari in the context of classical cusp forms.

23 pages