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},ω) $ 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 $ω$. 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 $A_k(\mathfrak{N},ω)$. We present asymptotic estimates for the terms of the Petersson formula as $k_0\rightarrow\infty,$ 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.