Lifting all elements in $\mathrm{SL}_n(\mathbb{Z}/q\mathbb{Z})$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918407800094720 |
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| author | Kamber, Amitay Varjú, Péter P. |
| author_facet | Kamber, Amitay Varjú, Péter P. |
| contents | We show that every element of $\mathrm{SL}_{n}(\mathbb{Z}/q\mathbb{Z})$ can be lifted to an element of $\mathrm{SL}_{n}(\mathbb{Z})$ of norm at most $Cq^2\log q$, while there exists an element such that every lift of it is of norm at least $q^{2+o(1)}$. This should be compared to the recent result that almost every element has a lift of norm bounded by $q^{1+1/n+o(1)}$.
The main step in the proof is showing that for every $q$, there is a small element in $(\mathbb{Z}/q\mathbb{Z})^\times$ with a large $n$-th root, which is a result of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10269 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lifting all elements in $\mathrm{SL}_n(\mathbb{Z}/q\mathbb{Z})$ Kamber, Amitay Varjú, Péter P. Number Theory 11F06 (Primary) 11J25, 20H05 (Secondary) We show that every element of $\mathrm{SL}_{n}(\mathbb{Z}/q\mathbb{Z})$ can be lifted to an element of $\mathrm{SL}_{n}(\mathbb{Z})$ of norm at most $Cq^2\log q$, while there exists an element such that every lift of it is of norm at least $q^{2+o(1)}$. This should be compared to the recent result that almost every element has a lift of norm bounded by $q^{1+1/n+o(1)}$. The main step in the proof is showing that for every $q$, there is a small element in $(\mathbb{Z}/q\mathbb{Z})^\times$ with a large $n$-th root, which is a result of independent interest. |
| title | Lifting all elements in $\mathrm{SL}_n(\mathbb{Z}/q\mathbb{Z})$ |
| topic | Number Theory 11F06 (Primary) 11J25, 20H05 (Secondary) |
| url | https://arxiv.org/abs/2310.10269 |