Introduction To Abelian Varieties

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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 : 52,6 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.

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 : 43,6 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.

O-Minimality and Diophantine Geometry

G. O. Jones,A. J. Wilkie
O Minimality and Diophantine Geometry Book PDF

Author : G. O. Jones,A. J. Wilkie
Publisher : Cambridge University Press
Page : 235 pages
File Size : 55,6 Mb
Release : 2015-08-13
Category : Mathematics
ISBN : 9781107462496

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O-Minimality and Diophantine Geometry by G. O. Jones,A. J. Wilkie Pdf

This book brings the researcher up to date with recent applications of mathematical logic to number theory.

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 : 50,8 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.

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 : 54,8 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).

Arithmetic Geometry

G. Cornell,J. H. Silverman
Arithmetic Geometry Book PDF

Author : G. Cornell,J. H. Silverman
Publisher : Springer Science & Business Media
Page : 359 pages
File Size : 54,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461386551

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Arithmetic Geometry by G. Cornell,J. H. Silverman Pdf

This volume is the result of a (mainly) instructional conference on arithmetic geometry, held from July 30 through August 10, 1984 at the University of Connecticut in Storrs. This volume contains expanded versions of almost all the instructional lectures given during the conference. In addition to these expository lectures, this volume contains a translation into English of Falt ings' seminal paper which provided the inspiration for the conference. We thank Professor Faltings for his permission to publish the translation and Edward Shipz who did the translation. We thank all the people who spoke at the Storrs conference, both for helping to make it a successful meeting and enabling us to publish this volume. We would especially like to thank David Rohrlich, who delivered the lectures on height functions (Chapter VI) when the second editor was unavoidably detained. In addition to the editors, Michael Artin and John Tate served on the organizing committee for the conference and much of the success of the conference was due to them-our thanks go to them for their assistance. Finally, the conference was only made possible through generous grants from the Vaughn Foundation and the National Science Foundation.

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 : 54,5 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.

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 : 48,9 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.

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 : 48,7 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 : 51,5 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.

Abelian Varieties

Serge Lang
Abelian Varieties Book PDF

Author : Serge Lang
Publisher : Courier Dover Publications
Page : 273 pages
File Size : 52,8 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.

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 : 47,9 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.

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 : 44,8 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

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 : 53,6 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.

Diophantine Geometry

Marc Hindry,Joseph H. Silverman
Diophantine Geometry Book PDF

Author : Marc Hindry,Joseph H. Silverman
Publisher : Springer Science & Business Media
Page : 574 pages
File Size : 42,6 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461212102

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Diophantine Geometry by Marc Hindry,Joseph H. Silverman Pdf

This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.