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Author: Gerd Faltings Publisher: Springer Science & Business Media ISBN: 9783540520153 Category : Mathematics Languages : en Pages : 348
Book Description
A new and complete treatment of semi-abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of Siegel moduli space, with most of the results being published for the first time. Highlights of the book include a classification of semi-abelian schemes, construction of the toroidal and the minimal compactification over the integers, heights for abelian varieties over number fields, and Eichler integrals in several variables, together with a new approach to Siegel modular forms. A valuable source of reference for researchers and graduate students interested in algebraic geometry, Shimura varieties or diophantine geometry.
Author: Gerd Faltings Publisher: Springer Science & Business Media ISBN: 3662026325 Category : Mathematics Languages : en Pages : 328
Book Description
A new and complete treatment of semi-abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of Siegel moduli space, with most of the results being published for the first time. Highlights of the book include a classification of semi-abelian schemes, construction of the toroidal and the minimal compactification over the integers, heights for abelian varieties over number fields, and Eichler integrals in several variables, together with a new approach to Siegel modular forms. A valuable source of reference for researchers and graduate students interested in algebraic geometry, Shimura varieties or diophantine geometry.
Author: Kai-Wen Lan Publisher: Princeton University Press ISBN: 1400846013 Category : Mathematics Languages : en Pages : 584
Book Description
By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary. This book explains in detail the following topics about PEL-type Shimura varieties and their compactifications: A construction of smooth integral models of PEL-type Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structures An analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base rings A construction of toroidal and minimal compactifications of smooth integral models of PEL-type Shimura varieties, with detailed descriptions of their structure near the boundary Through these topics, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai).
Author: David Mumford Publisher: Oxford University Press, USA ISBN: Category : Language Arts & Disciplines Languages : en Pages : 304
Book Description
Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. Mumford describes the analytic methods and results applicable when the ground field k is the complex field C and discusses the scheme-theoretic methods and results used to deal with inseparable isogenies when the ground field k has characteristic p. The author also provides a self-contained proof of the existence of a dual abeilan variety, reviews the structure of the ring of endormorphisms, and includes in appendices "The Theorem of Tate" and the "Mordell-Weil Thorem." This is an established work by an eminent mathematician and the only book on this subject.
Author: Gerard van der Geer Publisher: Birkhäuser ISBN: 303488303X Category : Mathematics Languages : en Pages : 526
Book Description
Abelian varieties and their moduli are a topic of increasing importance in today`s mathematics, applications ranging from algebraic geometry and number theory to mathematical physics. This collection of 17 refereed articles originates from the third "Texel Conference" held in 1999. Leading experts discuss and study the structure of the moduli spaces of abelian varieties and related spaces, giving an excellent view of the state of the art in this field.
Author: Werner Lütkebohmert Publisher: Springer ISBN: 331927371X Category : Mathematics Languages : en Pages : 398
Book Description
This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.
Author: Siegfried Bosch Publisher: Springer Science & Business Media ISBN: 3642514383 Category : Mathematics Languages : en Pages : 336
Book Description
Néron models were invented by A. Néron in the early 1960s in order to study the integral structure of abelian varieties over number fields. Since then, arithmeticians and algebraic geometers have applied the theory of Néron models with great success. Quite recently, new developments in arithmetic algebraic geometry have prompted a desire to understand more about Néron models, and even to go back to the basics of their construction. The authors have taken this as their incentive to present a comprehensive treatment of Néron models. This volume of the renowned "Ergebnisse" series provides a detailed demonstration of the construction of Néron models from the point of view of Grothendieck's algebraic geometry. In the second part of the book the relationship between Néron models and the relative Picard functor in the case of Jacobian varieties is explained. The authors helpfully remind the reader of some important standard techniques of algebraic geometry. A special chapter surveys the theory of the Picard functor.
Author: Carel Faber Publisher: Springer Science & Business Media ISBN: 3322901726 Category : Technology & Engineering Languages : en Pages : 205
Book Description
The Dutch Intercity Seminar on Moduli, which dates back to the early eighties, was an initiative of G. van der Geer, F. Oort and C. Peters. Through the years it became a focal point of Dutch mathematics and it gained some fame, also outside Holland, as an active biweekly research seminar. The tradition continues up to today. The present volume, with contributions of R. Dijkgraaf, C. Faber, G. van der Geer, R. Hain, E. Looijenga, and F. Oort, originates from the seminar held in 1995--96. Some of the articles here were discussed, in preliminary form, in the seminar; others are completely new. Two introductory papers, on moduli of abelian varieties and on moduli of curves, accompany the articles.
Author: Christina Birkenhake Publisher: Springer Science & Business Media ISBN: 9783540204886 Category : Mathematics Languages : en Pages : 658
Book Description
This book explores the theory of abelian varieties over the field of complex numbers, explaining both classic and recent results in modern language. The second edition adds five chapters on recent results including automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. ". . . far more readable than most . . . it is also much more complete." Olivier Debarre in Mathematical Reviews, 1994.
Author: Gerd Faltings
Author: Gerd Faltings
Author: Gerd Faltings
Author: Kai-Wen Lan
Author: S. Ogata
Author: David Mumford
Author: Gerard van der Geer
Author: Werner Lütkebohmert
Author: Siegfried Bosch
Author: Carel Faber
Author: Christina Birkenhake