The distribution of lattices arising from orders in low degree number fields
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915372609830912 |
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| author | Vemulapalli, Sameera |
| author_facet | Vemulapalli, Sameera |
| contents | Orders in number fields provide natural examples of lattices. We ask: what can the successive minima of lattices arising from orders in number fields be? Given an order $\mathcal{O}$ of absolute discriminant $Δ$ in a degree $n$ number field, let $1=λ_0,\dots,λ_{n-1}$ denote the successive minima. For $3 \leq n \leq 5$ and many groups $G \subseteq S_n$, we compute asymptotics of the points $(\log_{ Δ}λ_{1},\dots,\log_{ Δ}λ_{n-1}) \in \mathbb{R}^{n-1}$ as $\mathcal{O}$ ranges across orders in degree $n$ fields with Galois group $G$ as $Δ\rightarrow \infty$. In many cases, we find that the asymptotics, normalized appropriately, are given by a piecewise linear expression and are supported on a finite union of polytopes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The distribution of lattices arising from orders in low degree number fields Vemulapalli, Sameera Number Theory Orders in number fields provide natural examples of lattices. We ask: what can the successive minima of lattices arising from orders in number fields be? Given an order $\mathcal{O}$ of absolute discriminant $Δ$ in a degree $n$ number field, let $1=λ_0,\dots,λ_{n-1}$ denote the successive minima. For $3 \leq n \leq 5$ and many groups $G \subseteq S_n$, we compute asymptotics of the points $(\log_{ Δ}λ_{1},\dots,\log_{ Δ}λ_{n-1}) \in \mathbb{R}^{n-1}$ as $\mathcal{O}$ ranges across orders in degree $n$ fields with Galois group $G$ as $Δ\rightarrow \infty$. In many cases, we find that the asymptotics, normalized appropriately, are given by a piecewise linear expression and are supported on a finite union of polytopes. |
| title | The distribution of lattices arising from orders in low degree number fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.18985 |