Kato-Kuzumaki's properties for function fields over higher local fields
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917988815339520 |
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| author | Gambardella, Felipe |
| author_facet | Gambardella, Felipe |
| contents | Let $k$ be a $d$-local field such that the corresponding $1$-local field $k^{(d-1)}$ is a $p$-adic field and $C$ a curve over $k$. Let $K$ be the function field of $C$. We prove that for each $n,m \in \mathbf{N}$, and hypersurface $Z$ of $\mathbf{P}^n_K$ with degree $m$ such that $m^{d+1} \leq n$, the $(d+1)$-th Milnor $\mathrm{K}$-theory group is generated by the images norms of finite extension $L$ of $K$ such that $Z$ admits an $L$-point. Let $j \in \{1,\cdots , d\}$. When $C$ admits a point in an extension $l/k$ that is not $i$-ramified for every $i \in \{1, \cdots, d-j\}$ we generalise this result to hypersurfaces $Z$ of $\mathbf{P}_K^n$ with degree $m$ such that $m^{j+1} \leq n$. \par
In order to prove these results we give a description of the Tate-Shafarevich group $\Sha^{d+2}(K,\mathbf{Q}/\mathbf{Z}(d+1))$ in terms of the combinatorics of the special fibre of certain models of the curve $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kato-Kuzumaki's properties for function fields over higher local fields Gambardella, Felipe Algebraic Geometry K-Theory and Homology Number Theory 11E76, 11G20, 12G05, 12G10, 14G05, 14G27, 14J45, 14J70, 19C99, 19F05 Let $k$ be a $d$-local field such that the corresponding $1$-local field $k^{(d-1)}$ is a $p$-adic field and $C$ a curve over $k$. Let $K$ be the function field of $C$. We prove that for each $n,m \in \mathbf{N}$, and hypersurface $Z$ of $\mathbf{P}^n_K$ with degree $m$ such that $m^{d+1} \leq n$, the $(d+1)$-th Milnor $\mathrm{K}$-theory group is generated by the images norms of finite extension $L$ of $K$ such that $Z$ admits an $L$-point. Let $j \in \{1,\cdots , d\}$. When $C$ admits a point in an extension $l/k$ that is not $i$-ramified for every $i \in \{1, \cdots, d-j\}$ we generalise this result to hypersurfaces $Z$ of $\mathbf{P}_K^n$ with degree $m$ such that $m^{j+1} \leq n$. \par In order to prove these results we give a description of the Tate-Shafarevich group $\Sha^{d+2}(K,\mathbf{Q}/\mathbf{Z}(d+1))$ in terms of the combinatorics of the special fibre of certain models of the curve $C$. |
| title | Kato-Kuzumaki's properties for function fields over higher local fields |
| topic | Algebraic Geometry K-Theory and Homology Number Theory 11E76, 11G20, 12G05, 12G10, 14G05, 14G27, 14J45, 14J70, 19C99, 19F05 |
| url | https://arxiv.org/abs/2504.13100 |