On the local $L^2$-Bound of the Eisenstein Series
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917770447290368 |
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| author | Jana, Subhajit Kamber, Amitay |
| author_facet | Jana, Subhajit Kamber, Amitay |
| contents | We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a \emph{poly-logarithmic} bound on an average, for a large class of reductive groups. The method is based on Arthur's development of the spectral side of the trace formula, and ideas of Finis, Lapid, and Müller. As applications of our method, we prove the optimal lifting property for $\mathrm{SL}_n(\mathbb{Z}/q\mathbb{Z})$ for square-free $q$, as well as the Sarnak--Xue counting property for the principal congruence subgroup of $\mathrm{SL}_n(\mathbb{Z})$ of square-free level. This makes the recent results of Assing--Blomer unconditional. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_16291 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the local $L^2$-Bound of the Eisenstein Series Jana, Subhajit Kamber, Amitay Number Theory Representation Theory 11F70, 11F72, 22E55 We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a \emph{poly-logarithmic} bound on an average, for a large class of reductive groups. The method is based on Arthur's development of the spectral side of the trace formula, and ideas of Finis, Lapid, and Müller. As applications of our method, we prove the optimal lifting property for $\mathrm{SL}_n(\mathbb{Z}/q\mathbb{Z})$ for square-free $q$, as well as the Sarnak--Xue counting property for the principal congruence subgroup of $\mathrm{SL}_n(\mathbb{Z})$ of square-free level. This makes the recent results of Assing--Blomer unconditional. |
| title | On the local $L^2$-Bound of the Eisenstein Series |
| topic | Number Theory Representation Theory 11F70, 11F72, 22E55 |
| url | https://arxiv.org/abs/2210.16291 |