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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.05966 |
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| _version_ | 1866912624929669120 |
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| author | Pelizzari, Giulia Punch, James |
| author_facet | Pelizzari, Giulia Punch, James |
| contents | In 1887, Minkowski determined the least common multiple of the orders of all finite subgroups of $GL_n(\mathbb{Q})$; we refer to this number as $M(n)$. In (Katznelson, 1994), Katznelson provides the asymptotic behaviour of $M(n)$, with a small error term. In this paper, we use elementary techniques to find explicit upper and lower bounds on $M(n)$ that improve on Katznelson's results; we also recover his asymptotic result. Our results immediately imply explicit bounds on functions closely related to $M(n)$, which appear in the study of abelian varieties (see, for example, (Silverberg, 1992), (Guralnick and Kedlaya, 2017) and (Ozeki, 2024)). Finally, we examine the function $Φ(n)$, which also appears in (Ozeki, 2024), defined as the greatest positive integer $m$ for which $φ(m)$ divides $2n$. We provide explicit upper bounds on $Φ(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05966 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds on the Minkowski constants and a function involving $φ$ Pelizzari, Giulia Punch, James Number Theory 11N45, 11G10 In 1887, Minkowski determined the least common multiple of the orders of all finite subgroups of $GL_n(\mathbb{Q})$; we refer to this number as $M(n)$. In (Katznelson, 1994), Katznelson provides the asymptotic behaviour of $M(n)$, with a small error term. In this paper, we use elementary techniques to find explicit upper and lower bounds on $M(n)$ that improve on Katznelson's results; we also recover his asymptotic result. Our results immediately imply explicit bounds on functions closely related to $M(n)$, which appear in the study of abelian varieties (see, for example, (Silverberg, 1992), (Guralnick and Kedlaya, 2017) and (Ozeki, 2024)). Finally, we examine the function $Φ(n)$, which also appears in (Ozeki, 2024), defined as the greatest positive integer $m$ for which $φ(m)$ divides $2n$. We provide explicit upper bounds on $Φ(n)$. |
| title | Bounds on the Minkowski constants and a function involving $φ$ |
| topic | Number Theory 11N45, 11G10 |
| url | https://arxiv.org/abs/2508.05966 |