Model Theory and Algebraic Geometry An introduction to E. Hrushovski
- PDF / 12,660,512 Bytes
- 223 Pages / 439.32 x 666.12 pts Page_size
- 50 Downloads / 266 Views
1696
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Singapore Tokyo
Elisabeth Bouscaren (Ed.)
Model Theory and Algebraic Geometry An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture
Springer
Editor Elisabeth Bouscaren Universite Paris 7 - C.N.R.S. UFR de Mathematiques 2 Place Jussieu, case 7012 F-75251 Paris Cedex 05, France e-mail: [email protected]
Cataloging-in Publication Data available.
Corrected 2nd Printing 1999 Mathematics Subject Classification (1991): lIU09,03C60, IIGI0, 14G99, 03C45 ISSN 0075-8434 ISBN 3-540-64863-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1998 Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10735063 41/3143-543210 - Printed on acid-free paper
Preface Introduction Model theorists have often joked in recent years that the part of mathematical 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 of mathematics and to yield genuine 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 proof yields the first effective bound for the cardinality of the finite sets involved (The Manin-Mumford conjecture, preprint). There have been previous instances of applications of model theory to algebra 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... Hrushovski's results a
Data Loading...
