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},\omega) $ be the space of holomorphic Hilbert cusp forms with respect to $K_1(\mathfrak{N})$, weight $k=(k_1,\,...\,,k_r)$ with $k_j>2,$ for all $j$ and central Hecke character $\omega$. For a fixed level $\mathfrak{N}, $ we study the behavior of the Petersson trace formula for $A_k(\mathfrak{N},\omega)$ as $k_0\rightarrow\infty$ where $k_0=\min(k_1,\,...\,,k_r)$. We give an asymptotic formula for the Petersson formula. As an application, we obtain a variant of a discrepancy result for classical cusp forms by Jung and Sardari for the space $A_k(\mathfrak{N},1),$ where the ring of integers $\mathcal{O}$ has narrow class number $1$, and the ideal $\mathfrak{N}$ is generated by integers.