A discrepancy result for Hilbert modular forms

TL;DR

This paper investigates the asymptotic behavior of the Petersson trace formula for Hilbert cusp forms over totally real fields as the weights grow large, providing a new discrepancy result for classical cusp forms.

Contribution

It derives an asymptotic formula for the Petersson trace formula for Hilbert modular forms and applies it to establish a discrepancy result for classical cusp forms.

Findings

Asymptotic formula for Petersson trace as weights increase

Discrepancy result for classical cusp forms with narrow class number one

Extension of Jung and Sardari's discrepancy results to 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.