Strong Approximation and Hasse Principle for Integral Quadratic Forms over Affine Curves
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908461266108416 |
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| author | Hu, Yong Liu, Jing Tian, Yisheng |
| author_facet | Hu, Yong Liu, Jing Tian, Yisheng |
| contents | We extend some parts of the representation theory for integral quadratic forms over the ring of integers of a number field to the case over the coordinate ring $k[C]$ of an affine curve $C$ over a general base field $k$. By using the genus theory, we link the strong approximation property of certain spin groups to the Hasse principle for representations of integral quadratic forms over $k[C]$ and derive several applications. In particular, we give an example where a spin group does not satisfy strong approximation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_08849 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strong Approximation and Hasse Principle for Integral Quadratic Forms over Affine Curves Hu, Yong Liu, Jing Tian, Yisheng Number Theory 11E04 11E25 11E57 20G35 We extend some parts of the representation theory for integral quadratic forms over the ring of integers of a number field to the case over the coordinate ring $k[C]$ of an affine curve $C$ over a general base field $k$. By using the genus theory, we link the strong approximation property of certain spin groups to the Hasse principle for representations of integral quadratic forms over $k[C]$ and derive several applications. In particular, we give an example where a spin group does not satisfy strong approximation. |
| title | Strong Approximation and Hasse Principle for Integral Quadratic Forms over Affine Curves |
| topic | Number Theory 11E04 11E25 11E57 20G35 |
| url | https://arxiv.org/abs/2312.08849 |