On $p$-adic Minkowski's Theorems
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866909082812678144 |
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| author | Deng, Yingpu |
| author_facet | Deng, Yingpu |
| contents | Dual lattice is an important concept of Euclidean lattices. In this paper, we first give the right definition of the concept of the dual lattice of a $p$-adic lattice from the duality theory of locally compact abelian groups. The concrete constructions of ``basic characters'' of local fields given in Weil's famous book ``Basic Number Theory'' help us to do so. We then prove some important properties of the dual lattice of a $p$-adic lattice, which can be viewed as $p$-adic analogues of the famous Minkowski's first, second theorems for Euclidean lattices. We do this simultaneously for local fields $\mathbb{Q}_p$ (the field of $p$-adic numbers) and $\mathbb{F}_p((T))$ (the field of formal power-series of one indeterminate with coefficients in the finite field with $p$ elements). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14023 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $p$-adic Minkowski's Theorems Deng, Yingpu Number Theory Dual lattice is an important concept of Euclidean lattices. In this paper, we first give the right definition of the concept of the dual lattice of a $p$-adic lattice from the duality theory of locally compact abelian groups. The concrete constructions of ``basic characters'' of local fields given in Weil's famous book ``Basic Number Theory'' help us to do so. We then prove some important properties of the dual lattice of a $p$-adic lattice, which can be viewed as $p$-adic analogues of the famous Minkowski's first, second theorems for Euclidean lattices. We do this simultaneously for local fields $\mathbb{Q}_p$ (the field of $p$-adic numbers) and $\mathbb{F}_p((T))$ (the field of formal power-series of one indeterminate with coefficients in the finite field with $p$ elements). |
| title | On $p$-adic Minkowski's Theorems |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.14023 |