Recovering p-adic valuations from pro-p Galois groups
Let (Formula presented.) be a field with (Formula presented.), where (Formula presented.) denotes the maximal pro-2 quotient of the absolute Galois group of a field (Formula presented.). We prove that then (Formula presented.) admits a (non-trivial) valuation (Formula presented.) which is 2-henselia...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Wiley
2024
|
| _version_ | 1811140533451489280 |
|---|---|
| author | Koenigsmann, J Strommen, K |
| author_facet | Koenigsmann, J Strommen, K |
| author_sort | Koenigsmann, J |
| collection | OXFORD |
| description | Let (Formula presented.) be a field with (Formula presented.), where (Formula presented.) denotes the maximal pro-2 quotient of the absolute Galois group of a field (Formula presented.). We prove that then (Formula presented.) admits a (non-trivial) valuation (Formula presented.) which is 2-henselian and has residue field (Formula presented.). Furthermore, (Formula presented.) is a minimal positive element in the value group (Formula presented.) and (Formula presented.). This forms the first positive result on a more general conjecture about recovering (Formula presented.) -adic valuations from pro- (Formula presented.) 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 (Formula presented.) over (Formula presented.), as well as an analogue for varieties. |
| first_indexed | 2024-09-25T04:23:30Z |
| format | Journal article |
| id | oxford-uuid:5abb8998-484e-4ef8-a9b1-a8d3794d4f39 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:23:30Z |
| publishDate | 2024 |
| publisher | Wiley |
| record_format | dspace |
| spelling | oxford-uuid:5abb8998-484e-4ef8-a9b1-a8d3794d4f392024-08-21T16:19:26ZRecovering p-adic valuations from pro-p Galois groupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5abb8998-484e-4ef8-a9b1-a8d3794d4f39EnglishSymplectic ElementsWiley2024Koenigsmann, JStrommen, KLet (Formula presented.) be a field with (Formula presented.), where (Formula presented.) denotes the maximal pro-2 quotient of the absolute Galois group of a field (Formula presented.). We prove that then (Formula presented.) admits a (non-trivial) valuation (Formula presented.) which is 2-henselian and has residue field (Formula presented.). Furthermore, (Formula presented.) is a minimal positive element in the value group (Formula presented.) and (Formula presented.). This forms the first positive result on a more general conjecture about recovering (Formula presented.) -adic valuations from pro- (Formula presented.) 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 (Formula presented.) over (Formula presented.), as well as an analogue for varieties. |
| spellingShingle | Koenigsmann, J Strommen, K Recovering p-adic valuations from pro-p Galois groups |
| title | Recovering p-adic valuations from pro-p Galois groups |
| title_full | Recovering p-adic valuations from pro-p Galois groups |
| title_fullStr | Recovering p-adic valuations from pro-p Galois groups |
| title_full_unstemmed | Recovering p-adic valuations from pro-p Galois groups |
| title_short | Recovering p-adic valuations from pro-p Galois groups |
| title_sort | recovering p adic valuations from pro p galois groups |
| work_keys_str_mv | AT koenigsmannj recoveringpadicvaluationsfrompropgaloisgroups AT strommenk recoveringpadicvaluationsfrompropgaloisgroups |