On the enumeration of orbits of unipotent groups over finite fields
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866914887022673920 |
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| author | Rossmann, Tobias |
| author_facet | Rossmann, Tobias |
| contents | We show that the enumeration of linear orbits and conjugacy classes of $\mathbf{Z}$-defined unipotent groups over finite fields is "wild" in the following sense: given an arbitrary scheme $Y$ of finite type over $\mathbf{Z}$ and integer $n\geqslant 1$, the numbers $\# Y(\mathbf{F}_q) \bmod q^n$ can be expressed, uniformly in $q$, in terms of the numbers of linear orbits (or numbers of conjugacy classes) of finitely many $\mathbf{Z}$-defined unipotent groups over $\mathbf{F}_q$ and finitely many Laurent polynomials in $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_04646 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the enumeration of orbits of unipotent groups over finite fields Rossmann, Tobias Group Theory Algebraic Geometry Number Theory 20D15, 20E45, 05A15, 11G25 We show that the enumeration of linear orbits and conjugacy classes of $\mathbf{Z}$-defined unipotent groups over finite fields is "wild" in the following sense: given an arbitrary scheme $Y$ of finite type over $\mathbf{Z}$ and integer $n\geqslant 1$, the numbers $\# Y(\mathbf{F}_q) \bmod q^n$ can be expressed, uniformly in $q$, in terms of the numbers of linear orbits (or numbers of conjugacy classes) of finitely many $\mathbf{Z}$-defined unipotent groups over $\mathbf{F}_q$ and finitely many Laurent polynomials in $q$. |
| title | On the enumeration of orbits of unipotent groups over finite fields |
| topic | Group Theory Algebraic Geometry Number Theory 20D15, 20E45, 05A15, 11G25 |
| url | https://arxiv.org/abs/2208.04646 |