A central limit theorem for Hilbert modular forms
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910879278170112 |
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| author | Das, Jishu Prabhu, Neha |
| author_facet | Das, Jishu Prabhu, Neha |
| contents | For a prime ideal $\mathfrak{p}$ in a totally real number field $L$ with the adele ring $\mathbb{A}$, we study the distribution of angles $θ_π(\mathfrak{p})$ coming from Satake parameters corresponding to unramified $π_\mathfrak{p}$ where $π_\mathfrak{p}$ comes from a global $π$ ranging over a certain finite set $Π_{\underline{k}}(\mathfrak{n})$ of cuspidal automorphic representations of GL$_2(\mathbb{A})$ with trivial central character. For such a representation $π$, it is known that the angles $θ_π(\mathfrak{p})$ follow the Sato-Tate distribution. Fixing an interval $I\subseteq [0,π]$, we prove a central limit theorem for the number of angles $θ_π(\mathfrak{p})$ that lie in $I$, as $\mathrm{N}(\mathfrak{p})\to\infty$. The result assumes $\mathfrak{n}$ to be a squarefree integral ideal, and that the components in the weight vector $\underline{k}$ grow suitably fast as a function of $x$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_19154 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A central limit theorem for Hilbert modular forms Das, Jishu Prabhu, Neha Number Theory Primary: 11F41, 11F72, Secondary: 11F30 For a prime ideal $\mathfrak{p}$ in a totally real number field $L$ with the adele ring $\mathbb{A}$, we study the distribution of angles $θ_π(\mathfrak{p})$ coming from Satake parameters corresponding to unramified $π_\mathfrak{p}$ where $π_\mathfrak{p}$ comes from a global $π$ ranging over a certain finite set $Π_{\underline{k}}(\mathfrak{n})$ of cuspidal automorphic representations of GL$_2(\mathbb{A})$ with trivial central character. For such a representation $π$, it is known that the angles $θ_π(\mathfrak{p})$ follow the Sato-Tate distribution. Fixing an interval $I\subseteq [0,π]$, we prove a central limit theorem for the number of angles $θ_π(\mathfrak{p})$ that lie in $I$, as $\mathrm{N}(\mathfrak{p})\to\infty$. The result assumes $\mathfrak{n}$ to be a squarefree integral ideal, and that the components in the weight vector $\underline{k}$ grow suitably fast as a function of $x$. |
| title | A central limit theorem for Hilbert modular forms |
| topic | Number Theory Primary: 11F41, 11F72, Secondary: 11F30 |
| url | https://arxiv.org/abs/2310.19154 |