04013nam 22006855 450 991014630620332120250731082137.03-540-68521-910.1007/978-3-540-68521-0(CKB)1000000000437316(SSID)ssj0000324886(PQKBManifestationID)12124433(PQKBTitleCode)TC0000324886(PQKBWorkID)10322481(PQKB)11216380(DE-He213)978-3-540-68521-0(MiAaPQ)EBC3088526(MiAaPQ)EBC6485915(PPN)155184598(BIP)47731173(EXLCZ)99100000000043731620121227d1998 u| 0engurnn#008mamaatxtccrModel Theory and Algebraic Geometry An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture /edited by Elisabeth Bouscaren1st ed. 1998.Berlin, Heidelberg :Springer Berlin Heidelberg :Imprint: Springer,1998.1 online resource (XVI, 216 p.)Lecture Notes in Mathematics,1617-9692 ;1696Bibliographic Level Mode of Issuance: Monograph3-540-64863-1 Includes bibliographical references at the end of each chapters and index.to model theory -- to stability theory and Morley rank -- Omega-stable groups -- Model theory of algebraically closed fields -- to abelian varieties and the Mordell-Lang conjecture -- The model-theoretic content of Lang’s conjecture -- Zariski geometries -- Differentially closed fields -- Separably closed fields -- Proof of the Mordell-Lang conjecture for function fields -- Proof of Manin’s theorem by reduction to positive characteristic.Introduction Model theorists have often joked in recent years that the part of mathemat­ ical logic known as "pure model theory" (or stability theory), as opposed to the older and more traditional "model theory applied to algebra" , turns out to have more and more to do with other subjects ofmathematics and to yield gen­ uine applications to combinatorial geometry, differential algebra and algebraic geometry. We illustrate this by presenting the very striking application to diophantine geometry due to Ehud Hrushovski: using model theory, he has given the first proof valid in all characteristics of the "Mordell-Lang conjecture for function fields" (The Mordell-Lang conjecture for function fields, Journal AMS 9 (1996), 667-690). More recently he has also given a new (model theoretic) proof of the Manin-Mumford conjecture for semi-abelian varieties over a number field. His proofyields the first effective bound for the cardinality ofthe finite sets involved (The Manin-Mumford conjecture, preprint). There have been previous instances of applications of model theory to alge­ bra or number theory, but these appl~cations had in common the feature that their proofs used a lot of algebra (or number theory) but only very basic tools and results from the model theory side: compactness, first-order definability, elementary equivalence...Lecture Notes in Mathematics,1617-9692 ;1696Geometry, AlgebraicLogic, Symbolic and mathematicalNumber theoryAlgebraic GeometryMathematical Logic and FoundationsNumber TheoryGeometry, Algebraic.Logic, Symbolic and mathematical.Number theory.Algebraic Geometry.Mathematical Logic and Foundations.Number Theory.516.3503C60mscBouscaren Elisabeth1956-MiAaPQMiAaPQMiAaPQBOOK9910146306203321Model theory and algebraic geometry78161UNINA