Rational points and arithmetic of fundamental groups. Evidence for the section conjecture
type: Item
instance of: scholarly article
title: Rational points and arithmetic of fundamental groups. Evidence for the section conjecture
zbMATH Open document ID: 1272.14003
DOI: 10.1007/978-3-642-30674-7
author: Jakob Stix
published in: Lecture Notes in Mathematics
publication date: 2012-06-15T00:00:00Z
review text: For an algebraic variety \(X\) over a field \(k\) the étale fundamental group \(\pi_1(X)\) is an extension of the Galois-group \(\text{Gal}(k)\) by the geometric fundamental group \(\pi_1(\overline X)\). Rational points in \(X(k)\) induce \(\pi_1(\overline X)\)-conjugacy classes of sections of this extension. The section conjecture (due to Grothendieck) states that for suitable \(X\) and \(k\) this induces a bijection. ``Suitable'' of course implies that \(\pi_1(\overline X)\) and \(\text{Gal}(k)\) are sufficiently big, for example \(X\) could be a hyperbolic curve and \(k\) a number field. Except in cases where there are no sections the truth of conjecture is unknown. In the present book the author gives an overview of mathematics around the conjecture. For example it is shown that many aspects of rational points generalise to sections.
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 6046477
zbMATH Keywords: étale fundamental groups
zbMATH Keywords: rational points
describes a project that uses: SageMath
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-30674-7
OpenAlex ID: W2504834220
reviewed by: Gerd Faltings
description: scientific article; zbMATH DE number 6046477