About the conjecture of Zilber-Pink
type: Item
instance of: scholarly article
title: About the conjecture of Zilber-Pink
zbMATH Open document ID: 1196.11083
author: Gaël Rémond
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-10-26T00:00:00Z
full work available at URL: https://eudml.org/doc/10887
review text: This is a survey paper from the author's talk at the 2007 Journées Arithmétiques. Let \(A\) denote a semi-abelian variety over a field \(K\) of characteristic zero, \(X\) a closed integral subscheme of \(A\) that is not contained in any translate of a genuine subgroup scheme of \(A\). The author outlines the main steps of a strategy to prove the following conjecture: For any subgroup \(\Gamma\) of finite rank in \(A(K)\), the intersection \(X(K) \cap \left( \Gamma + A^{[\dim X+1]}\right)\) is not dense in \(X\). This is equivalent to the following conjecture, attributed to Pink and Zilber: The intersection of \(X(K) \cap \bigcup G(K)\) is not dense in \(X\), where the union runs over all subgroup schemes \(G\) of \(A\) such that \(\dim X + \dim G < \dim A\). As is explained, the most difficult step appears to lie in diophantine approximation.
reviewed by: Gunther Cornelissen
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11J95
zbMATH DE Number: 5620657
zbMATH Keywords: Zilber-Pink conjecture
zbMATH Keywords: semi-abelian variety
zbMATH Keywords: diophantine approximation
Wikidata QID: Q123197564
MaRDI profile type: Publication
OpenAlex ID: W2000129294
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DOI: 10.5802/JTNB.677
description: scientific article; zbMATH DE number 5620657