Model theory and algebraic geometry. An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture
type: Item
instance of: scholarly article
title: Model theory and algebraic geometry. An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture
zbMATH Open document ID: 0920.03046
DOI: 10.1007/978-3-540-68521-0
published in: Lecture Notes in Mathematics
publication date: 1998-10-06T00:00:00Z
review text: The aim of the volume is to give a more or less self-contained exposition of E. Hrushovski's remarkable model-theoretic proof of the Mordell-Lang conjecture for function fields [\textit{E. Hrushovski}, J. Am. Math. Soc. 9, 667-690 (1996; Zbl 0864.03026)]. The conjecture can be roughly formulated as follows. Let \(k\) and \(K\) be algebraically closed fields, and \(k\) be a proper subfield of \(K\). Let \(A\) be an abelian variety and \(X\) its subvariety both defined over \(K\), and \(\Gamma\) be a subgroup of finite rank of \(A(K)\). Then either \(X\cap\Gamma\) is a finite union of cosets of subgroups of \(\Gamma\), or one can `descend' the whole situation to \(k\), by finding a homomorphism from \(A\) to an abelian variety \(S\) defined over \(k\) such that \(X\) is a translate of the preimage of a subvariety of \(S\) also defined over \(k\). For the case of characteristic zero the result had been proven by Yu. Manin in 1963; for the case of positive characteristic only partial results were known before Hrushovski's proof. Note that it is possible to deduce the result in characteristic zero from the corresponding result in positive characteristic. Hrushovski showed that this and some other questions of diophantine geometry can be naturally integrated in the abstract framework which has been developed in model theory. It turned out that the Mordell-Lang conjecture can be considered as an instance of a general model-theoretic trichotomy the idea of which had been discovered in the early eighties by B. Zilber. Note that more recently, by using a similar model-theoretic approach, Hrushovski found a new proof of the Manin-Mumford conjecture over a number field. The volume presents an exposition of Hrushovski's proof of the geometric Mordell-Lang conjecture as a series of coordinated papers written by experts in model theory (except one, written by M. Hindry, an algebraic geometer). To avoid technical difficulties, the presentation is restricted to the case of abelian v…
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 00B15
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11U09
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 1206424
zbMATH Keywords: Model theory
zbMATH Keywords: Algebraic geometry
zbMATH Keywords: Mordell-Lang conjecture
zbMATH Keywords: abelian variety
zbMATH Keywords: model-theoretic stability
zbMATH Keywords: omega-stable group
zbMATH Keywords: Zariski geometry
zbMATH Keywords: differentially closed field
zbMATH Keywords: separably closed field
reviewed by: Oleg Belegradek
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-540-68521-0
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description: scientific article; zbMATH DE number 1206424