Model Theory and Algebraic Geometry: An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture: 1696 - Brossura

Brossura

ISBN 10: 3540648631 ISBN 13: 9783540648635

Casa editrice: Springer, 2009

Riassunto

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...

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Contenuti

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.

Product Description

Book by Bouscaren Elisabeth Ed

Dettagli dell'edizione

• Editore: Springer
• Data di pubblicazione: 2009
• Lingua: Inglese
• ISBN 10: 3540648631
• ISBN 13: 9783540648635
• Rilegatura: Copertina flessibile
• Numero edizione: 1
• Numero di pagine: 228
• Redattore: Bouscaren Elisabeth
• Contatto del produttore: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3; Heidelberg; 69115; Germania