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  • © 1990

Degeneration of Abelian Varieties

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  • ISBN: 978-3-662-02632-8
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Softcover Book USD 159.99
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Table of contents (7 chapters)

  1. Front Matter

    Pages I-XII
  2. Preliminaries

    • Gerd Faltings, Ching-Li Chai
    Pages 1-30
  3. Degeneration of Polarized Abelian Varieties

    • Gerd Faltings, Ching-Li Chai
    Pages 31-52
  4. Mumford’s Construction

    • Gerd Faltings, Ching-Li Chai
    Pages 53-92
  5. Toroidal Compactification of A g

    • Gerd Faltings, Ching-Li Chai
    Pages 93-135
  6. Modular Forms and the Minimal Compactification

    • Gerd Faltings, Ching-Li Chai
    Pages 136-193
  7. Eichler Integrals in Several Variables

    • Gerd Faltings, Ching-Li Chai
    Pages 194-242
  8. Hecke Operators and Frobenii

    • Gerd Faltings, Ching-Li Chai
    Pages 243-268
  9. Back Matter

    Pages 269-318

About this book

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

  • Department of Mathematics, Princeton University, Princeton, USA

    Gerd Faltings

  • Department of Mathematics, University of Pennsylvania, Philadelphia, USA

    Ching-Li Chai

Bibliographic Information

Buying options

eBook USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-3-662-02632-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 159.99
Price excludes VAT (USA)
Hardcover Book USD 159.99
Price excludes VAT (USA)