Rational points over homogeneous spaces and their compactifications
type: Item
instance of: scholarly article
title: Rational points over homogeneous spaces and their compactifications
zbMATH Open document ID: 1108.12003
author: N. E. Zubov
published in: Transformation Groups
publication date: 2007-01-24T00:00:00Z
review text: The main object of the paper under review is a homogeneous space \(X\) of an algebraic group \(G\) defined over a perfect field \(k\). The author's goal is to prove that if some \(k\)-variety birationally equivalent to \(X\) has a smooth rational point, then so does \(X\). In particular, one can deduce the existence of a rational point on \(X\) from the existence of a rational point on its smooth compactification. This result is a far-reaching generalization of a theorem by \textit{J.-L.~Colliot-Thélène} and \textit{J.-J.~Sansuc} [Duke Math. J. 54, 375--492 (1987; Zbl 0659.14028)] which concerned the case where \(G\) is an algebraic \(k\)-torus. The strategy of the proof is as follows. The author proves a stronger statement saying that the existence of a \(k\)-point follows from the existence of a \(k((t))\)-point. If \(X\) is a \(G\)-torsor, this statement is equivalent to the injectivity of the restriction map \(H^1(k,G)\to H^1(k((t)),G)\). To go over from torsors to arbitrary homogeneous spaces, one has, roughly speaking, to pass from nonabelian \(H^1\) to nonabelian \(H^2\). This is mainly what the author is doing. The main technical tool he uses is the notion of acyclic extension \(l/k\) (by definition, this means that the restriction map \(H^1(k,G)\to H^1(l,G)\) is bijective for all \(k\)-groups \(G\)). He notices that if \(l/k\) is acyclic, then the existence of a \(k\)-point on \(X\) follows from the existence of an \(l\)-point. It remains to construct an acyclic extension \(K_{\infty}\) of \(k\) containing \(k((t))\) (if the characteristic of \(k\) equals zero, it is enough to adjoin to \(k\) all roots of \(t\)). The proof can then be finished by a simple geometric argument allowing one, roughly, to move a smooth rational point along an integral curve. Apart from the main theorem, some of auxiliary results are of their own interest (as, for example, considerations around the restriction map for nonabelian \(H^2\)).
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 14E08
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 5118033
zbMATH Keywords: homogeneous space
zbMATH Keywords: rational point
zbMATH Keywords: nonabelian cohomology
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
full work available at URL: s00031-004-1107-9
OpenAlex ID: W1980366726
DOI: 10.1007/S00031-004-1107-9
description: scientific article; zbMATH DE number 5118033