On Kato and Kuzumaki's properties for the Milnor $K_2$ of function fields of $p$-adic curves
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910344403746816 |
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| author | Izquierdo, Diego Arteche, Giancarlo Lucchini |
| author_facet | Izquierdo, Diego Arteche, Giancarlo Lucchini |
| contents | Let $K$ be the function field of a curve $C$ over a $p$-adic field $k$. We prove that, for each $n, d \geq 1$ and for each hypersurface $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d^2 \leq n$, the second Milnor $K$-theory group of $K$ is spanned by the images of the norms coming from finite extensions $L$ of $K$ over which $Z$ has a rational point. When the curve $C$ has a point in the maximal unramified extension of $k$, we generalize this result to hypersurfaces $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d \leq n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_04214 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Kato and Kuzumaki's properties for the Milnor $K_2$ of function fields of $p$-adic curves Izquierdo, Diego Arteche, Giancarlo Lucchini Algebraic Geometry K-Theory and Homology Number Theory 11E76, 11G20, 12G05, 12G10, 14G05, 14G27, 14J45, 14J70, 19C99, 19F05 Let $K$ be the function field of a curve $C$ over a $p$-adic field $k$. We prove that, for each $n, d \geq 1$ and for each hypersurface $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d^2 \leq n$, the second Milnor $K$-theory group of $K$ is spanned by the images of the norms coming from finite extensions $L$ of $K$ over which $Z$ has a rational point. When the curve $C$ has a point in the maximal unramified extension of $k$, we generalize this result to hypersurfaces $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d \leq n$. |
| title | On Kato and Kuzumaki's properties for the Milnor $K_2$ of function fields of $p$-adic curves |
| topic | Algebraic Geometry K-Theory and Homology Number Theory 11E76, 11G20, 12G05, 12G10, 14G05, 14G27, 14J45, 14J70, 19C99, 19F05 |
| url | https://arxiv.org/abs/2206.04214 |