Diophantine analysis and Arthur's trace formula
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914145551515648 |
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| author | Lee, Yuchan |
| author_facet | Lee, Yuchan |
| contents | Let $X$ be a $G$-homogeneous space over a number field $k$ such that $X\cong G_γ\backslash G$. Here, $G$ is a simply connected semisimple group over $k$ and $γ\in G(k)$ whose centralizer $G_γ$ is a maximal torus in $G$ which is anisotropic over $k$. We formulate the asymptotic for the number of integral points on $X$ bounded by a fixed norm $T>0$ as $T\rightarrow \infty$ in terms of $κ$-orbital integrals, which play a role in the stabilization of Arthur's trace formula. This formula coincides with the contribution of the stable conjugacy class of $γ$ to the geometric side of the trace formula.
As an application, we obtain an asymptotic formula for the number of $n \times n$ matrices over the ring of integers $\mathcal{O}_k$ whose characteristic polynomial equals a fixed irreducible polynomial $χ(x)$ of degree $n$. This result generalizes a case studied by Eskin-Mozes-Shah (1996). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22314 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diophantine analysis and Arthur's trace formula Lee, Yuchan Number Theory 11D45, 11F72, 11G35, 11S80, 14F22, 14G12, 20G30, 20G35 Let $X$ be a $G$-homogeneous space over a number field $k$ such that $X\cong G_γ\backslash G$. Here, $G$ is a simply connected semisimple group over $k$ and $γ\in G(k)$ whose centralizer $G_γ$ is a maximal torus in $G$ which is anisotropic over $k$. We formulate the asymptotic for the number of integral points on $X$ bounded by a fixed norm $T>0$ as $T\rightarrow \infty$ in terms of $κ$-orbital integrals, which play a role in the stabilization of Arthur's trace formula. This formula coincides with the contribution of the stable conjugacy class of $γ$ to the geometric side of the trace formula. As an application, we obtain an asymptotic formula for the number of $n \times n$ matrices over the ring of integers $\mathcal{O}_k$ whose characteristic polynomial equals a fixed irreducible polynomial $χ(x)$ of degree $n$. This result generalizes a case studied by Eskin-Mozes-Shah (1996). |
| title | Diophantine analysis and Arthur's trace formula |
| topic | Number Theory 11D45, 11F72, 11G35, 11S80, 14F22, 14G12, 20G30, 20G35 |
| url | https://arxiv.org/abs/2509.22314 |