Low-lying zeros of Hilbert modular $L$-functions weighted by powers of central $L$-values
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| Format: | Preprint |
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2025
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| _version_ | 1866918130547163136 |
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| author | Wei, Zhining Yang, Liyang Zhao, Shifan |
| author_facet | Wei, Zhining Yang, Liyang Zhao, Shifan |
| contents | Let $\mathcal{F}(\textbf{k},\mathfrak{q})$ be the set of primitive Hilbert modular forms of weight $\textbf{k}$ and prime level $\mathfrak{q}$, with trivial central character. We study the one-level density of low-lying zeros of $L(s,π)$ weighted by powers of central $L$-values $L(1/2,π)^r$, where $π$ runs through $\mathcal{F}(\textbf{k},\mathfrak{q})$. For $r=1,2,3$, we show that the resulting distributions $W_r$ match with predictions from Random Matrix Theory. For general $r \geq 1$, we also formulate a conjectural formula for $W_r$ based on the ``recipe'' method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18469 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low-lying zeros of Hilbert modular $L$-functions weighted by powers of central $L$-values Wei, Zhining Yang, Liyang Zhao, Shifan Number Theory 11F66, 11F30, 11F41 Let $\mathcal{F}(\textbf{k},\mathfrak{q})$ be the set of primitive Hilbert modular forms of weight $\textbf{k}$ and prime level $\mathfrak{q}$, with trivial central character. We study the one-level density of low-lying zeros of $L(s,π)$ weighted by powers of central $L$-values $L(1/2,π)^r$, where $π$ runs through $\mathcal{F}(\textbf{k},\mathfrak{q})$. For $r=1,2,3$, we show that the resulting distributions $W_r$ match with predictions from Random Matrix Theory. For general $r \geq 1$, we also formulate a conjectural formula for $W_r$ based on the ``recipe'' method. |
| title | Low-lying zeros of Hilbert modular $L$-functions weighted by powers of central $L$-values |
| topic | Number Theory 11F66, 11F30, 11F41 |
| url | https://arxiv.org/abs/2508.18469 |