Recovering p-adic valuations from pro-p Galois groups
| Autor: | Koenigsmann, Jochen, Strommen, Kristian |
|---|---|
| Rok vydání: | 2015 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1112/jlms.12901 |
| Popis: | 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 $\Gamma_v$ and $[\Gamma_v:2\Gamma_v]=2$. 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. Comment: Final version, published in the Journal of the London Mathematical Society (DOI: 10.1112/jlms.12901) |
| Databáze: | arXiv |
| Externí odkaz: |
|