Roots of elements for groups over local fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915208145928192 |
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| author | Kumar, Parteek Mandal, Arunava |
| author_facet | Kumar, Parteek Mandal, Arunava |
| contents | Let $\mathbb F$ be a local field and $G$ be a linear algebraic group defined over $\mathbb F$. For $k\in\mathbb N$, let $g\to g^k$ be the $k$-th power map $P_k$ on $G(\mathbb F)$. The purpose of this article is two-fold. First, we study the power map on real algebraic group. We characterise the density of the images of the power map $P_k$ on $G(\mathbb R)$ in terms of Cartan subgroups. Next we consider the linear algebraic group $G$ over non-Archimedean local field $\mathbb F$ with any characteristic. If the residual characteristic of $\mathbb F$ is $p$, and an element admits $p^k$-th root in $G(\mathbb F)$ for each $k$, then we prove that some power of the element is unipotent. In particular, we prove that an element $g\in G(\mathbb F)$ admits roots of all orders if and only if $g$ is contained in a one-parameter subgroup in $G(\mathbb F)$. Also, we extend these results to all linear algebraic groups over global fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_16987 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Roots of elements for groups over local fields Kumar, Parteek Mandal, Arunava Number Theory Group Theory 20G25, 22E20 Let $\mathbb F$ be a local field and $G$ be a linear algebraic group defined over $\mathbb F$. For $k\in\mathbb N$, let $g\to g^k$ be the $k$-th power map $P_k$ on $G(\mathbb F)$. The purpose of this article is two-fold. First, we study the power map on real algebraic group. We characterise the density of the images of the power map $P_k$ on $G(\mathbb R)$ in terms of Cartan subgroups. Next we consider the linear algebraic group $G$ over non-Archimedean local field $\mathbb F$ with any characteristic. If the residual characteristic of $\mathbb F$ is $p$, and an element admits $p^k$-th root in $G(\mathbb F)$ for each $k$, then we prove that some power of the element is unipotent. In particular, we prove that an element $g\in G(\mathbb F)$ admits roots of all orders if and only if $g$ is contained in a one-parameter subgroup in $G(\mathbb F)$. Also, we extend these results to all linear algebraic groups over global fields. |
| title | Roots of elements for groups over local fields |
| topic | Number Theory Group Theory 20G25, 22E20 |
| url | https://arxiv.org/abs/2503.16987 |