Degeneration of Abelian Varieties

Gerd Faltings, Ching-Li Chai (Autoren)

Buch | Hardcover

XII, 318 Seiten

1991 | 1990
Springer Berlin (Verlag)
978-3-540-52015-3 (ISBN)

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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.

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.

Erscheint lt. Verlag	9.1.1991
Reihe/Serie	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Zusatzinfo	XII, 318 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	712 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Abelsche Mannigfaltigkeit • diophantine geometry • diophantische Geometrie • Hecke Operator • Moduli Raum • moduli space • Schema • schemes • Siegel modular form • Siegelsche Modulfunktion
ISBN-10	3-540-52015-5 / 3540520155
ISBN-13	978-3-540-52015-3 / 9783540520153
Zustand	Neuware