Mass equidistribution for Poincaré series of large index
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915089868652544 |
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| author | Kimmel, Noam |
| author_facet | Kimmel, Noam |
| contents | Let $P_{k,m}$ denote the Poincaré series of weight $k$ and index $m$ for the full modular group $\mathrm{SL}_2(\mathbb{Z})$, and let $\{P_{k,m}\}$ be a sequence of Poincaré series for which $m(k)$ satisfies $m(k) / k \rightarrow\infty$ and $m(k) \ll k^{\frac{3}{2} - ε}$. We prove that the $L^2$ mass of such a sequence equidistributes on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure as $k$ goes to infinity. As a consequence, we deduce that the zeros of such a sequence $\{P_{k,m}\}$ become uniformly distributed in $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01414 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mass equidistribution for Poincaré series of large index Kimmel, Noam Number Theory Let $P_{k,m}$ denote the Poincaré series of weight $k$ and index $m$ for the full modular group $\mathrm{SL}_2(\mathbb{Z})$, and let $\{P_{k,m}\}$ be a sequence of Poincaré series for which $m(k)$ satisfies $m(k) / k \rightarrow\infty$ and $m(k) \ll k^{\frac{3}{2} - ε}$. We prove that the $L^2$ mass of such a sequence equidistributes on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure as $k$ goes to infinity. As a consequence, we deduce that the zeros of such a sequence $\{P_{k,m}\}$ become uniformly distributed in $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ with respect to the hyperbolic measure. |
| title | Mass equidistribution for Poincaré series of large index |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.01414 |