Modular Curves and Abelian Varieties

Modular Curves and Abelian Varieties
Author :
Publisher : Birkhäuser
Total Pages : 291
Release :
ISBN-10 : 9783034879194
ISBN-13 : 3034879199
Rating : 4/5 (94 Downloads)

Synopsis Modular Curves and Abelian Varieties by : John Cremona

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

Complex Abelian Varieties
Author :
Publisher : Springer Science & Business Media
Total Pages : 443
Release :
ISBN-10 : 9783662027882
ISBN-13 : 3662027887
Rating : 4/5 (82 Downloads)

Synopsis Complex Abelian Varieties by : Herbert Lange

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.

Curves and Abelian Varieties

Curves and Abelian Varieties
Author :
Publisher : American Mathematical Soc.
Total Pages : 290
Release :
ISBN-10 : 9780821843345
ISBN-13 : 0821843346
Rating : 4/5 (45 Downloads)

Synopsis Curves and Abelian Varieties by : Valery Alexeev

"This book is devoted to recent progress in the study of curves and abelian varieties. It discusses both classical aspects of this deep and beautiful subject as well as two important new developments, tropical geometry and the theory of log schemes." "In addition to original research articles, this book contains three surveys devoted to singularities of theta divisors. of compactified Jucobiuns of singular curves, and of "strange duality" among moduli spaces of vector bundles on algebraic varieties."--BOOK JACKET.

Complex Tori and Abelian Varieties

Complex Tori and Abelian Varieties
Author :
Publisher : American Mathematical Soc.
Total Pages : 124
Release :
ISBN-10 : 0821831658
ISBN-13 : 9780821831656
Rating : 4/5 (58 Downloads)

Synopsis Complex Tori and Abelian Varieties by : Olivier Debarre

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.

The Arithmetic of Elliptic Curves

The Arithmetic of Elliptic Curves
Author :
Publisher : Springer Science & Business Media
Total Pages : 414
Release :
ISBN-10 : 9781475719208
ISBN-13 : 1475719205
Rating : 4/5 (08 Downloads)

Synopsis The Arithmetic of Elliptic Curves by : Joseph H. Silverman

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.

Heights in Diophantine Geometry

Heights in Diophantine Geometry
Author :
Publisher : Cambridge University Press
Total Pages : 676
Release :
ISBN-10 : 0521712297
ISBN-13 : 9780521712293
Rating : 4/5 (97 Downloads)

Synopsis Heights in Diophantine Geometry by : Enrico Bombieri

This monograph is a bridge between the classical theory and modern approach via arithmetic geometry.

Abelian l-Adic Representations and Elliptic Curves

Abelian l-Adic Representations and Elliptic Curves
Author :
Publisher : CRC Press
Total Pages : 203
Release :
ISBN-10 : 9781439863862
ISBN-13 : 1439863865
Rating : 4/5 (62 Downloads)

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

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

Abelian Varieties
Author :
Publisher : Dover Publications
Total Pages : 273
Release :
ISBN-10 : 9780486828053
ISBN-13 : 0486828050
Rating : 4/5 (53 Downloads)

Synopsis Abelian Varieties by : Serge Lang

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.

Abelian Varieties

Abelian Varieties
Author :
Publisher : Debolsillo
Total Pages : 0
Release :
ISBN-10 : 8185931860
ISBN-13 : 9788185931869
Rating : 4/5 (60 Downloads)

Synopsis Abelian Varieties by : David Mumford

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

Abelian Varieties, Theta Functions and the Fourier Transform
Author :
Publisher : Cambridge University Press
Total Pages : 308
Release :
ISBN-10 : 9780521808040
ISBN-13 : 0521808049
Rating : 4/5 (40 Downloads)

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

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