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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.13383 |
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| _version_ | 1866916167349698560 |
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| author | Maucourant, François |
| author_facet | Maucourant, François |
| contents | Pick a random matrix $γ$ in $Γ={\rm SL}(2,\mathbb{Z})$. Denote by $\mathcal{O}_K$ the Dedekind ring generated by its eigenvalues, and let $Δ_K$, $Δ_γ$ and $Δ= {\rm Tr}(γ)^2-4$ be the respective discriminant of the rings $\mathcal{O}_K$, the multiplier ring $M(2,\mathbb{Z})\cap \mathbb{Q}[γ]$ and $\mathbb{Z}[γ]$. We show that their ratios admit probability limit distributions. In particular, 42% of the elements of $Γ$ have a fundamental discriminant, and $\mathbb{Z}[γ]$ is a ring of integers with probability 32%. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Size of discriminants of periodic geodesics of the modular surface Maucourant, François Dynamical Systems Number Theory Pick a random matrix $γ$ in $Γ={\rm SL}(2,\mathbb{Z})$. Denote by $\mathcal{O}_K$ the Dedekind ring generated by its eigenvalues, and let $Δ_K$, $Δ_γ$ and $Δ= {\rm Tr}(γ)^2-4$ be the respective discriminant of the rings $\mathcal{O}_K$, the multiplier ring $M(2,\mathbb{Z})\cap \mathbb{Q}[γ]$ and $\mathbb{Z}[γ]$. We show that their ratios admit probability limit distributions. In particular, 42% of the elements of $Γ$ have a fundamental discriminant, and $\mathbb{Z}[γ]$ is a ring of integers with probability 32%. |
| title | Size of discriminants of periodic geodesics of the modular surface |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2403.13383 |