Reconstruction of function fields
type: Item
instance of: scholarly article
title: Reconstruction of function fields
zbMATH Open document ID: 1155.14016
author: Yuri Tschinkel
author: Fedor Bogomolov
published in: Geometric and Functional Analysis. GAFA
publication date: 2008-08-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0303075
review text: Let \(K=k(X)\) be the rational function field of a surface \(X\) over \(k\) an algebraic closure of the field with \(p\) elements. The authors show that \(K\) can be recovered from the maximal pro-\(\ell\)-quotient \(\mathcal G_K\) of its absolute Galois group; namely, if there exists an isomorphism \(\Psi\) from the abelianization \(\mathcal G_K ^a\) to \(\mathcal G_L ^a\) of abelian pro-\(\ell\)-groups including a bijection of sets \(\Sigma _K =\Sigma _L\) for another function field \(L/k\), then \(c\Psi\) is induced by an isomorphism between the perfect closures of \(K\) and \(L\), where \(\Sigma _K\) is determined by the second lower central series quotient of \(\mathcal G_K\), and \(c\in \mathbb{Z}^* _\ell\). This long paper has contents including projective structures, flag maps, valuations, Galois groups of curves, valuations on surfaces, \(\ell\)-adic analysis (curves, surfaces, etc), and ends with a 3-page proof.
Mathematics Subject Classification ID: 14G32
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 5313042
zbMATH Keywords: function field
zbMATH Keywords: surface
zbMATH Keywords: Galois group
zbMATH Keywords: pro-L-quotient
zbMATH Keywords: projective structure
zbMATH Keywords: flag map
zbMATH Keywords: L-adic analysis
reviewed by: Xianke Zhang
MaRDI profile type: Publication
OpenAlex ID: W2119948034
arXiv ID: math/0303075
DOI: 10.1007/S00039-008-0665-8
description: scientific article