The $m$-step solvable anabelian geometry of mixed-characteristic local fields
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914154626940928 |
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| author | Hyeon, Seung-Hyeon |
| author_facet | Hyeon, Seung-Hyeon |
| contents | Let $K$ be a mixed-characteristic local field. For an integer $m \geq 0$, we denote by $K^m / K$ the maximal $m$-step solvable extension of $K$, and by $G_K^m$ the maximal $m$-step solvable quotient of the absolute Galois group $G_K$ of $K$. We regard $G_K$ and its quotients as filtered profinite groups via the respective upper-numbering ramification filtrations. It is known from the previous result due to Mochizuki that the isomorphism class of $K$ is determined by the isomorphism class of the filtered profinite group $G_K$. In this paper, we prove that the isomorphism class of $K$ is determined by the isomorphism class of the maximal $2$-step solvable quotient $G_K^2$ as a filtered profinite group, and furthermore, that $K^m / K$ is determined functorially by the filtered profinite group $G_K^{m + 2}$ (resp. $G_K^{m + 3}$) for $m \geq 2$ (resp. $m = 0, 1$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16950 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $m$-step solvable anabelian geometry of mixed-characteristic local fields Hyeon, Seung-Hyeon Number Theory Algebraic Geometry 11S20, 11S31, 11S15, 11F80 Let $K$ be a mixed-characteristic local field. For an integer $m \geq 0$, we denote by $K^m / K$ the maximal $m$-step solvable extension of $K$, and by $G_K^m$ the maximal $m$-step solvable quotient of the absolute Galois group $G_K$ of $K$. We regard $G_K$ and its quotients as filtered profinite groups via the respective upper-numbering ramification filtrations. It is known from the previous result due to Mochizuki that the isomorphism class of $K$ is determined by the isomorphism class of the filtered profinite group $G_K$. In this paper, we prove that the isomorphism class of $K$ is determined by the isomorphism class of the maximal $2$-step solvable quotient $G_K^2$ as a filtered profinite group, and furthermore, that $K^m / K$ is determined functorially by the filtered profinite group $G_K^{m + 2}$ (resp. $G_K^{m + 3}$) for $m \geq 2$ (resp. $m = 0, 1$). |
| title | The $m$-step solvable anabelian geometry of mixed-characteristic local fields |
| topic | Number Theory Algebraic Geometry 11S20, 11S31, 11S15, 11F80 |
| url | https://arxiv.org/abs/2405.16950 |