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Degeneration of Abelian Varieties

  • Gerd Faltings
  • Ching-Li Chai

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 22)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Gerd Faltings, Ching-Li Chai
    Pages 1-30
  3. Gerd Faltings, Ching-Li Chai
    Pages 31-52
  4. Gerd Faltings, Ching-Li Chai
    Pages 53-92
  5. Gerd Faltings, Ching-Li Chai
    Pages 93-135
  6. Gerd Faltings, Ching-Li Chai
    Pages 136-193
  7. Gerd Faltings, Ching-Li Chai
    Pages 194-242
  8. Gerd Faltings, Ching-Li Chai
    Pages 243-268
  9. Back Matter
    Pages 269-318

About this book

Introduction

The topic of this book is the theory of degenerations of abelian varieties and its application to the construction of compactifications of moduli spaces of abelian varieties. These compactifications have applications to diophantine problems and, of course, are also interesting in their own right. Degenerations of abelian varieties are given by maps G - S with S an irre­ ducible scheme and G a group variety whose generic fibre is an abelian variety. One would like to classify such objects, which, however, is a hopeless task in this generality. But for more specialized families we can obtain more: The most important theorem about degenerations is the stable reduction theorem, which gives some evidence that for questions of compactification it suffices to study semi-abelian families; that is, we may assume that G is smooth and flat over S, with fibres which are connected extensions of abelian varieties by tori. A further assumption will be that the base S is normal, which makes such semi-abelian families extremely well behaved. In these circumstances, we give a rather com­ plete classification in case S is the spectrum of a complete local ring, and for general S we can still say a good deal. For a complete base S = Spec(R) (R a complete and normal local domain) the main result about degenerations says roughly that G is (in some sense) a quotient of a covering G by a group of periods.

Keywords

Hecke operator Moduli Raum Schema Siegel modular form Siegelsche Modulfunktion diophantine geometry diophantische Geometrie moduli space schemes

Authors and affiliations

  • Gerd Faltings
    • 1
  • Ching-Li Chai
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-02632-8
  • Copyright Information Springer-Verlag Berlin Heidelberg 1990
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08088-3
  • Online ISBN 978-3-662-02632-8
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site