Complex Abelian Varieties

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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 : 41,7 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 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 : 47,7 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.

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 : 42,9 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.

Complex Abelian Varieties and Theta Functions

George R. Kempf
Complex Abelian Varieties and Theta Functions Book PDF

Author : George R. Kempf
Publisher : Springer Science & Business Media
Page : 108 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642760792

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Complex Abelian Varieties and Theta Functions by George R. Kempf Pdf

Abelian varieties are a natural generalization of elliptic curves to higher dimensions, whose geometry and classification are as rich in elegant results as in the one-dimensional ease. The use of theta functions, particularly since Mumford's work, has been an important tool in the study of abelian varieties and invertible sheaves on them. Also, abelian varieties play a significant role in the geometric approach to modern algebraic number theory. In this book, Kempf has focused on the analytic aspects of the geometry of abelian varieties, rather than taking the alternative algebraic or arithmetic points of view. His purpose is to provide an introduction to complex analytic geometry. Thus, he uses Hermitian geometry as much as possible. One distinguishing feature of Kempf's presentation is the systematic use of Mumford's theta group. This allows him to give precise results about the projective ideal of an abelian variety. In its detailed discussion of the cohomology of invertible sheaves, the book incorporates material previously found only in research articles. Also, several examples where abelian varieties arise in various branches of geometry are given as a conclusion of the book.

Abelian Varieties

David Mumford
Abelian Varieties Book PDF

Author : David Mumford
Publisher : Debolsillo
Page : 0 pages
File Size : 44,7 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.''

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 : 51,8 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.

Introduction to Abelian Varieties

Vijaya Kumar Murty
Introduction to Abelian Varieties Book PDF

Author : Vijaya Kumar Murty
Publisher : American Mathematical Soc.
Page : 128 pages
File Size : 41,8 Mb
Release : 1993
Category : Abelian varieties
ISBN : 9780821811795

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Introduction to Abelian Varieties by Vijaya 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.

Analytic Theory of Abelian Varieties

H. P. F. Swinnerton-Dyer
Analytic Theory of Abelian Varieties Book PDF

Author : H. P. F. Swinnerton-Dyer
Publisher : Cambridge University Press
Page : 105 pages
File Size : 47,5 Mb
Release : 1974-12-12
Category : Mathematics
ISBN : 9780521205269

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Analytic Theory of Abelian Varieties by H. P. F. Swinnerton-Dyer Pdf

The study of abelian manifolds forms a natural generalization of the theory of elliptic functions, that is, of doubly periodic functions of one complex variable. When an abelian manifold is embedded in a projective space it is termed an abelian variety in an algebraic geometrical sense. This introduction presupposes little more than a basic course in complex variables. The notes contain all the material on abelian manifolds needed for application to geometry and number theory, although they do not contain an exposition of either application. Some geometrical results are included however.

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 : 50,8 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.

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 : 44,9 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 l-Adic Representations and Elliptic Curves

Jean-Pierre Serre
Abelian l Adic Representations and Elliptic Curves Book PDF

Author : Jean-Pierre Serre
Publisher : CRC Press
Page : 203 pages
File Size : 48,7 Mb
Release : 1997-11-15
Category : Mathematics
ISBN : 9781439863862

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Abelian l-Adic Representations and Elliptic Curves by Jean-Pierre Serre Pdf

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one

Abelian Varieties

David Mumford
Abelian Varieties Book PDF

Author : David Mumford
Publisher : Unknown
Page : 304 pages
File Size : 53,6 Mb
Release : 1970
Category : Language Arts & Disciplines
ISBN : UCSD:31822002537041

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Abelian Varieties by David Mumford Pdf

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.

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 : 44,8 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.

Complex Multiplication

S. Lang
Complex Multiplication Book PDF

Author : S. Lang
Publisher : Springer Science & Business Media
Page : 191 pages
File Size : 54,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461254850

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Complex Multiplication by S. Lang Pdf

The small book by Shimura-Taniyama on the subject of complex multi is a classic. It gives the results obtained by them (and some by Weil) plication in the higher dimensional case, generalizing in a non-trivial way the method of Deuring for elliptic curves, by reduction mod p. Partly through the work of Shimura himself (cf. [Sh 1] [Sh 2], and [Sh 5]), and some others (Serre, Tate, Kubota, Ribet, Deligne etc.) it is possible today to make a more snappy and extensive presentation of the fundamental results than was possible in 1961. Several persons have found my lecture notes on this subject useful to them, and so I have decided to publish this short book to make them more widely available. Readers acquainted with the standard theory of abelian varieties, and who wish to get rapidly an idea of the fundamental facts of complex multi plication, are advised to look first at the two main theorems, Chapter 3, §6 and Chapter 4, §1, as well as the rest of Chapter 4. The applications of Chapter 6 could also be profitably read early. I am much indebted to N. Schappacher for a careful reading of the manu script resulting in a number of useful suggestions. S. LANG Contents CHAPTER 1 Analytic Complex Multiplication 4 I. Positive Definite Involutions . . . 6 2. CM Types and Subfields. . . . . 8 3. Application to Abelian Manifolds. 4. Construction of Abelian Manifolds with CM 14 21 5. Reflex of a CM Type . . . . .

The Arithmetic of Elliptic Curves

Joseph H. Silverman
The Arithmetic of Elliptic Curves Book PDF

Author : Joseph H. Silverman
Publisher : Springer Science & Business Media
Page : 414 pages
File Size : 43,8 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475719208

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The Arithmetic of Elliptic Curves by Joseph H. Silverman Pdf

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.