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| Main Authors: | , |
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| Format: | Preprint |
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2015
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1506.05956 |
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| _version_ | 1866909226062839808 |
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| author | Koenigsmann, Jochen Strommen, Kristian |
| author_facet | Koenigsmann, Jochen Strommen, Kristian |
| contents | Let $K$ be a field with $G_K(2) \simeq G_{\mathbb{Q}}(2)$, where $G_F(2)$ denotes the maximal pro-2 quotient of the absolute Galois group of a field $F$. We prove that then $K$ admits a (non-trivial) valuation $v$ which is 2-henselian and has residue field $\mathbb{F}_2$. Furthermore, $v(2)$ is a minimal positive element in the value group $Γ_v$ and $[Γ_v:2Γ_v]=2$. This forms the first positive result on a more general conjecture about recovering $p$-adic valuations from pro-$p$ Galois groups which we formulate precisely. As an application, we show how this result can be used to easily obtain number-theoretic information, by giving an independent proof of a strong version of the birational section conjecture for smooth, complete curves $X$ over $\mathbb{Q}_2$, as well as an analogue for varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1506_05956 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Recovering p-adic valuations from pro-p Galois groups Koenigsmann, Jochen Strommen, Kristian Number Theory Let $K$ be a field with $G_K(2) \simeq G_{\mathbb{Q}}(2)$, where $G_F(2)$ denotes the maximal pro-2 quotient of the absolute Galois group of a field $F$. We prove that then $K$ admits a (non-trivial) valuation $v$ which is 2-henselian and has residue field $\mathbb{F}_2$. Furthermore, $v(2)$ is a minimal positive element in the value group $Γ_v$ and $[Γ_v:2Γ_v]=2$. This forms the first positive result on a more general conjecture about recovering $p$-adic valuations from pro-$p$ Galois groups which we formulate precisely. As an application, we show how this result can be used to easily obtain number-theoretic information, by giving an independent proof of a strong version of the birational section conjecture for smooth, complete curves $X$ over $\mathbb{Q}_2$, as well as an analogue for varieties. |
| title | Recovering p-adic valuations from pro-p Galois groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/1506.05956 |