Abelian Varieties

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Abelian Varieties

Serge Lang
Abelian Varieties Book PDF

Author : Serge Lang
Publisher : Courier Dover Publications
Page : 273 pages
File Size : 46,8 Mb
Release : 2019-02-13
Category : Mathematics
ISBN : 9780486839769

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Abelian Varieties by Serge Lang Pdf

Based on the work in algebraic geometry by Norwegian mathematician Niels Henrik Abel (1802–29), this monograph was originally published in 1959 and reprinted later in author Serge Lang's career without revision. The treatment remains a basic advanced text in its field, suitable for advanced undergraduates and graduate students in mathematics. Prerequisites include some background in elementary qualitative algebraic geometry and the elementary theory of algebraic groups. The book focuses exclusively on Abelian varieties rather than the broader field of algebraic groups; therefore, the first chapter presents all the general results on algebraic groups relevant to this treatment. Each chapter begins with a brief introduction and concludes with a historical and bibliographical note. Topics include general theorems on Abelian varieties, the theorem of the square, divisor classes on an Abelian variety, functorial formulas, the Picard variety of an arbitrary variety, the I-adic representations, and algebraic systems of Abelian varieties. The text concludes with a helpful Appendix covering the composition of correspondences.

Complex Tori and Abelian Varieties

Olivier Debarre
Complex Tori and Abelian Varieties Book PDF

Author : Olivier Debarre
Publisher : American Mathematical Soc.
Page : 124 pages
File Size : 50,6 Mb
Release : 2005
Category : Mathematics
ISBN : 0821831658

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Complex Tori and Abelian Varieties by Olivier Debarre Pdf

This graduate-level textbook introduces the classical theory of complex tori and abelian varieties, while presenting in parallel more modern aspects of complex algebraic and analytic geometry. Beginning with complex elliptic curves, the book moves on to the higher-dimensional case, giving characterizations from different points of view of those complex tori which are abelian varieties, i.e., those that can be holomorphically embedded in a projective space. This allows, on the one hand, for illuminating the computations of nineteenth-century mathematicians, and on the other, familiarizing readers with more recent theories. Complex tori are ideal in this respect: One can perform "hands-on" computations without the theory being totally trivial. Standard theorems about abelian varieties are proved, and moduli spaces are discussed. Recent results on the geometry and topology of some subvarieties of a complex torus are also included. The book contains numerous examples and exercises. It is a very good starting point for studying algebraic geometry, suitable for graduate students and researchers interested in algebra and algebraic geometry. Information for our distributors: SMF members are entitled to AMS member discounts.

Modular Curves and Abelian Varieties

John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet
Modular Curves and Abelian Varieties Book PDF

Author : John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet
Publisher : Birkhäuser
Page : 291 pages
File Size : 51,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879194

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Modular Curves and Abelian Varieties by John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet Pdf

This book presents lectures from a conference on "Modular Curves and Abelian Varieties'' at the Centre de Recerca Matemtica (Bellaterra, Barcelona). The articles in this volume present the latest achievements in this extremely active field and will be of interest both to specialists and to students and researchers. Many contributions focus on generalizations of the Shimura-Taniyama conjecture to varieties such as elliptic Q-curves and Abelian varieties of GL_2-type. The book also includes several key articles in the subject that do not correspond to conference lectures.

Complex Abelian Varieties

Herbert Lange,Christina Birkenhake
Complex Abelian Varieties Book PDF

Author : Herbert Lange,Christina Birkenhake
Publisher : Springer Science & Business Media
Page : 443 pages
File Size : 40,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662027882

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Complex Abelian Varieties by Herbert Lange,Christina Birkenhake Pdf

Abelian varieties are special examples of projective varieties. As such theycan be described by a set of homogeneous polynomial equations. The theory ofabelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jacobi. The subject of this book is the theory of abelian varieties over the field of complex numbers, and it covers the main results of the theory, both classic and recent, in modern language. It is intended to give a comprehensive introduction to the field, but also to serve as a reference. The focal topics are the projective embeddings of an abelian variety, their equations and geometric properties. Moreover several moduli spaces of abelian varieties with additional structure are constructed. Some special results onJacobians and Prym varieties allow applications to the theory of algebraic curves. The main tools for the proofs are the theta group of a line bundle, introduced by Mumford, and the characteristics, to be associated to any nondegenerate line bundle. They are a direct generalization of the classical notion of characteristics of theta functions.

Complex Abelian Varieties

Christina Birkenhake,Herbert Lange
Complex Abelian Varieties Book PDF

Author : Christina Birkenhake,Herbert Lange
Publisher : Springer Science & Business Media
Page : 638 pages
File Size : 54,7 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662063071

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Complex Abelian Varieties by Christina Birkenhake,Herbert Lange Pdf

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.

Degeneration of Abelian Varieties

Gerd Faltings,Ching-Li Chai
Degeneration of Abelian Varieties Book PDF

Author : Gerd Faltings,Ching-Li Chai
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 49,5 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783662026328

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Degeneration of Abelian Varieties by Gerd Faltings,Ching-Li Chai Pdf

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.

Abelian Varieties with Complex Multiplication and Modular Functions

Goro Shimura
Abelian Varieties with Complex Multiplication and Modular Functions Book PDF

Author : Goro Shimura
Publisher : Princeton University Press
Page : 232 pages
File Size : 40,6 Mb
Release : 2016-06-02
Category : Mathematics
ISBN : 9781400883943

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Abelian Varieties with Complex Multiplication and Modular Functions by Goro Shimura Pdf

Reciprocity laws of various kinds play a central role in number theory. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. The investigation of such algebraicity is relatively new, but has attracted the interest of increasingly many researchers. Many of the topics discussed in this book have not been covered before. In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.

Abelian Varieties

David Mumford
Abelian Varieties Book PDF

Author : David Mumford
Publisher : Debolsillo
Page : 0 pages
File Size : 42,5 Mb
Release : 2008
Category : Abelian varieties
ISBN : 8185931860

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Abelian Varieties by David Mumford Pdf

This is a reprinting of the revised second edition (1974) of David Mumford's classic 1970 book. It gives a systematic account of the basic results about abelian varieties. It includes expositions of analytic methods applicable over the ground field of complex numbers, as well as of scheme-theoretic methods used to deal with inseparable isogenies when the ground field has positive characteristic. A self-contained proof of the existence of the dual abelian variety is given. The structure of the ring of endomorphisms of an abelian variety is discussed. These are appendices on Tate's theorem on endomorphisms of abelian varieties over finite fields (by C. P. Ramanujam) and on the Mordell-Weil theorem (by Yuri Manin). David Mumford was awarded the 2007 AMS Steele Prize for Mathematical Exposition. According to the citation: ``Abelian Varieties ... remains the definitive account of the subject ... the classical theory is beautifully intertwined with the modern theory, in a way which sharply illuminates both ... [It] will remain for the foreseeable future a classic to which the reader returns over and over.''

Moduli of Supersingular Abelian Varieties

Ke-Zheng Li,Frans Oort
Moduli of Supersingular Abelian Varieties Book PDF

Author : Ke-Zheng Li,Frans Oort
Publisher : Springer
Page : 123 pages
File Size : 47,7 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540696667

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Moduli of Supersingular Abelian Varieties by Ke-Zheng Li,Frans Oort Pdf

Abelian varieties can be classified via their moduli. In positive characteristic the structure of the p-torsion-structure is an additional, useful tool. For that structure supersingular abelian varieties can be considered the most special ones. They provide a starting point for the fine description of various structures. For low dimensions the moduli of supersingular abelian varieties is by now well understood. In this book we provide a description of the supersingular locus in all dimensions, in particular we compute the dimension of it: it turns out to be equal to Äg.g/4Ü, and we express the number of components as a class number, thus completing a long historical line where special cases were studied and general results were conjectured (Deuring, Hasse, Igusa, Oda-Oort, Katsura-Oort).

Modular Curves and Abelian Varieties

John Cremona
Modular Curves and Abelian Varieties Book PDF

Author : John Cremona
Publisher : Springer Science & Business Media
Page : 308 pages
File Size : 52,6 Mb
Release : 2004-02-23
Category : Mathematics
ISBN : 3764365862

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Modular Curves and Abelian Varieties by John Cremona Pdf

This book presents lectures from a conference on "Modular Curves and Abelian Varieties'' at the Centre de Recerca Matemàtica (Bellaterra, Barcelona). The articles in this volume present the latest achievements in this extremely active field and will be of interest both to specialists and to students and researchers. Many contributions focus on generalizations of the Shimura-Taniyama conjecture to varieties such as elliptic Q-curves and Abelian varieties of GL_2-type. The book also includes several key articles in the subject that do not correspond to conference lectures.

Abelian Varieties, Theta Functions and the Fourier Transform

Alexander Polishchuk
Abelian Varieties Theta Functions and the Fourier Transform Book PDF

Author : Alexander Polishchuk
Publisher : Cambridge University Press
Page : 308 pages
File Size : 42,5 Mb
Release : 2003-04-21
Category : Mathematics
ISBN : 9780521808040

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Abelian Varieties, Theta Functions and the Fourier Transform by Alexander Polishchuk Pdf

Presents a modern treatment of the theory of theta functions in the context of algebraic geometry.

Compactifying Moduli Spaces for Abelian Varieties

Martin C. Olsson
Compactifying Moduli Spaces for Abelian Varieties Book PDF

Author : Martin C. Olsson
Publisher : Springer Science & Business Media
Page : 286 pages
File Size : 47,6 Mb
Release : 2008-08-25
Category : Mathematics
ISBN : 9783540705185

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Compactifying Moduli Spaces for Abelian Varieties by Martin C. Olsson Pdf

This volume presents the construction of canonical modular compactifications of moduli spaces for polarized Abelian varieties (possibly with level structure), building on the earlier work of Alexeev, Nakamura, and Namikawa. This provides a different approach to compactifying these spaces than the more classical approach using toroical embeddings, which are not canonical. There are two main new contributions in this monograph: (1) The introduction of logarithmic geometry as understood by Fontaine, Illusie, and Kato to the study of degenerating Abelian varieties; and (2) the construction of canonical compactifications for moduli spaces with higher degree polarizations based on stack-theoretic techniques and a study of the theta group.

Primality Testing and Abelian Varieties Over Finite Fields

Leonard M. Adleman,Ming-Deh A. Huang
Primality Testing and Abelian Varieties Over Finite Fields Book PDF

Author : Leonard M. Adleman,Ming-Deh A. Huang
Publisher : Springer
Page : 149 pages
File Size : 48,5 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540470212

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Primality Testing and Abelian Varieties Over Finite Fields by Leonard M. Adleman,Ming-Deh A. Huang Pdf

From Gauss to G|del, mathematicians have sought an efficient algorithm to distinguish prime numbers from composite numbers. This book presents a random polynomial time algorithm for the problem. The methods used are from arithmetic algebraic geometry, algebraic number theory and analyticnumber theory. In particular, the theory of two dimensional Abelian varieties over finite fields is developed. The book will be of interest to both researchers and graduate students in number theory and theoretical computer science.

Moduli of Abelian Varieties

Gerard van der Geer,C. Faber,Frans Oort
Moduli of Abelian Varieties Book PDF

Author : Gerard van der Geer,C. Faber,Frans Oort
Publisher : Birkhäuser
Page : 526 pages
File Size : 50,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883030

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Moduli of Abelian Varieties by Gerard van der Geer,C. Faber,Frans Oort Pdf

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.

Introduction to Abelian Varieties

V. Kumar Murty
Introduction to Abelian Varieties Book PDF

Author : V. Kumar Murty
Publisher : American Mathematical Soc.
Page : 136 pages
File Size : 49,8 Mb
Release : 1986
Category : Abelian groups
ISBN : 082187005X

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Introduction to Abelian Varieties by V. Kumar Murty Pdf

This book presents an elementary and self-contained approach to Abelian varieties, a subject that plays a central role in algebraic and analytic geometry, number theory, and complex analysis. The book is based on notes from a course given at Concordia University and would be useful for independent study or as a textbook for graduate courses in complex analysis, Riemann surfaces, number theory, or analytic geometry. Murty works mostly over the complex numbers, discussing the theorem of Abel-Jacobi and Lefschetz's theorem on projective embeddings. After presenting some examples, Murty touches on Abelian varieties over number fields, as well as the conjecture of Tate (Faltings's theorem) and its relation to Mordell's conjecture. References are provided to guide the reader in further study.