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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.10311 |
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| _version_ | 1866911545401802752 |
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| author | Hu, Yong Liu, Jing Xu, Fei |
| author_facet | Hu, Yong Liu, Jing Xu, Fei |
| contents | A quadratic lattice $M$ over a Dedekind domain $R$ with fraction field $F$ is defined to be a finitely generated torsion-free $R$-module equipped with a non-degenerate quadratic form on the $F$-vector space $F\otimes_{R}M$. Assuming that $F\otimes_{R}M$ is isotropic of dimension $\geq 3$ and that $2$ is invertible in $R$, we prove that a quadratic lattice $N$ can be embedded into a quadratic lattice $M$ over $R$ if and only if $S\otimes_{R}N$ can be embedded into $S\otimes_{R}M$ over $S$, where $S$ is the integral closure of $R$ in a finite extension of odd degree of $F$. As a key step in the proof, we establish several versions of the norm principle for integral spinor norms, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10311 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral Springer Theorem for Quadratic Lattices under Base Change of Odd Degree Hu, Yong Liu, Jing Xu, Fei Number Theory 11E04, 11E12, 11E25, 11E57, 20G35 A quadratic lattice $M$ over a Dedekind domain $R$ with fraction field $F$ is defined to be a finitely generated torsion-free $R$-module equipped with a non-degenerate quadratic form on the $F$-vector space $F\otimes_{R}M$. Assuming that $F\otimes_{R}M$ is isotropic of dimension $\geq 3$ and that $2$ is invertible in $R$, we prove that a quadratic lattice $N$ can be embedded into a quadratic lattice $M$ over $R$ if and only if $S\otimes_{R}N$ can be embedded into $S\otimes_{R}M$ over $S$, where $S$ is the integral closure of $R$ in a finite extension of odd degree of $F$. As a key step in the proof, we establish several versions of the norm principle for integral spinor norms, which may be of independent interest. |
| title | Integral Springer Theorem for Quadratic Lattices under Base Change of Odd Degree |
| topic | Number Theory 11E04, 11E12, 11E25, 11E57, 20G35 |
| url | https://arxiv.org/abs/2410.10311 |