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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.00334 |
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| _version_ | 1866914219240194048 |
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| author | He, Zilong |
| author_facet | He, Zilong |
| contents | In the paper, we extend the ADC property to the representation of quadratic lattices by quadratic lattices, which we define as $ n $-ADC-ness. We explore the relationship between $ n$-ADC-ness, $ n $-regularity and $ n $-universality for integral quadratic lattices. Also, for $ n\ge 2 $, we give necessary and sufficient conditions for an integral quadratic lattice over arbitrary non-archimedean local fields to be $ n $-ADC. Moreover, we show that over any algebraic number field $ F $, an integral $ \mathcal{O}_{F} $-lattice with rank $ n+1 $ is $n$-ADC if and only if it is $\mathcal{O}_{F}$-maximal of class number one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00334 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $n$-ADC integral quadratic lattices over algebraic number fields He, Zilong Number Theory In the paper, we extend the ADC property to the representation of quadratic lattices by quadratic lattices, which we define as $ n $-ADC-ness. We explore the relationship between $ n$-ADC-ness, $ n $-regularity and $ n $-universality for integral quadratic lattices. Also, for $ n\ge 2 $, we give necessary and sufficient conditions for an integral quadratic lattice over arbitrary non-archimedean local fields to be $ n $-ADC. Moreover, we show that over any algebraic number field $ F $, an integral $ \mathcal{O}_{F} $-lattice with rank $ n+1 $ is $n$-ADC if and only if it is $\mathcal{O}_{F}$-maximal of class number one. |
| title | On $n$-ADC integral quadratic lattices over algebraic number fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2306.00334 |