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Cover: 9783540520153 | Degeneration of Abelian Varieties | Ching-Li Chai (u. a.) | Buch | xii
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Degeneration of Abelian Varieties
Buch von Ching-Li Chai (u. a.)
Sprache: Englisch

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Beschreibung
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.
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.
Zusammenfassung
This book is concerned with the theory of degenerations of abelian varieties and its applications, and provides a new approach to Siegel modular forms. Faltings is a Fields Medal winner.
Inhaltsverzeichnis
I. Preliminaries.- II. Degeneration of Polarized Abelian Varieties.- III. Mumford's Construction.- IV. Toroidal Compactification of Ag.- V. Modular Forms and the Minimal Compactification.- VI. Eichler Integrals in Several Variables.- VII. Hecke Operators and Frobenii.- Glossary of Notations.- An Analytic Construction of Degenerating Abelian Varieties over Complete Rings.- David Mumford.
Details
Erscheinungsjahr: 1991
Fachbereich: Grundlagen
Genre: Mathematik, Medizin, Naturwissenschaften, Technik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Inhalt: xii
318 S.
ISBN-13: 9783540520153
ISBN-10: 3540520155
Sprache: Englisch
Einband: Gebunden
Autor: Chai, Ching-Li
Faltings, Gerd
Hersteller: Springer Berlin
Springer Berlin Heidelberg
Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße: 241 x 160 x 24 mm
Von/Mit: Ching-Li Chai (u. a.)
Erscheinungsdatum: 09.01.1991
Gewicht: 0,67 kg
Artikel-ID: 102141269
Zusammenfassung
This book is concerned with the theory of degenerations of abelian varieties and its applications, and provides a new approach to Siegel modular forms. Faltings is a Fields Medal winner.
Inhaltsverzeichnis
I. Preliminaries.- II. Degeneration of Polarized Abelian Varieties.- III. Mumford's Construction.- IV. Toroidal Compactification of Ag.- V. Modular Forms and the Minimal Compactification.- VI. Eichler Integrals in Several Variables.- VII. Hecke Operators and Frobenii.- Glossary of Notations.- An Analytic Construction of Degenerating Abelian Varieties over Complete Rings.- David Mumford.
Details
Erscheinungsjahr: 1991
Fachbereich: Grundlagen
Genre: Mathematik, Medizin, Naturwissenschaften, Technik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Inhalt: xii
318 S.
ISBN-13: 9783540520153
ISBN-10: 3540520155
Sprache: Englisch
Einband: Gebunden
Autor: Chai, Ching-Li
Faltings, Gerd
Hersteller: Springer Berlin
Springer Berlin Heidelberg
Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße: 241 x 160 x 24 mm
Von/Mit: Ching-Li Chai (u. a.)
Erscheinungsdatum: 09.01.1991
Gewicht: 0,67 kg
Artikel-ID: 102141269
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