Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2307.16736 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914619774205952 |
|---|---|
| author | Balasubramanyam, Baskar Das, Jishu Sinha, Kaneenika |
| author_facet | Balasubramanyam, Baskar Das, Jishu Sinha, Kaneenika |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16736 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A discrepancy result for Hilbert modular forms Balasubramanyam, Baskar Das, Jishu Sinha, Kaneenika Number Theory 11F41, 11F60, 11K06 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. |
| title | A discrepancy result for Hilbert modular forms |
| topic | Number Theory 11F41, 11F60, 11K06 |
| url | https://arxiv.org/abs/2307.16736 |