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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.11110 |
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| _version_ | 1866912156839051264 |
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| author | Soofiani, Amin |
| author_facet | Soofiani, Amin |
| contents | Let $K$ be a complete discretely valued field whose residue field has characteristic different from $2$. Let $(D,σ)$ be a $K-$division algebra with involution of the first kind, and $h$ be a $K-$anisotropic $ε$-hermitian form over $(D,σ)$. By a theorem due to Larmour, there is a decomposition $h=h_0 \perp h_1$ such that the elements in a diagonalization of $h_0$ are units, the elements in a diagonalization of $h_1$ are uniformizers, and $h_0$, $h_1$ are determined uniquely up to $K-$isometry. In this paper, we give an explicit description of the elements in the diagonalization of $h_0$ and $h_1$ in the case of quaternion algebras. Then we will derive explicit formulas for Larmour's isomorphism of Witt groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11110 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An explicit formula for Larmour's decomposition of hermitian forms Soofiani, Amin Rings and Algebras 16K20 Let $K$ be a complete discretely valued field whose residue field has characteristic different from $2$. Let $(D,σ)$ be a $K-$division algebra with involution of the first kind, and $h$ be a $K-$anisotropic $ε$-hermitian form over $(D,σ)$. By a theorem due to Larmour, there is a decomposition $h=h_0 \perp h_1$ such that the elements in a diagonalization of $h_0$ are units, the elements in a diagonalization of $h_1$ are uniformizers, and $h_0$, $h_1$ are determined uniquely up to $K-$isometry. In this paper, we give an explicit description of the elements in the diagonalization of $h_0$ and $h_1$ in the case of quaternion algebras. Then we will derive explicit formulas for Larmour's isomorphism of Witt groups. |
| title | An explicit formula for Larmour's decomposition of hermitian forms |
| topic | Rings and Algebras 16K20 |
| url | https://arxiv.org/abs/2412.11110 |