Abelian Varieties With Complex Multiplication And Modular Functions
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Author | : Goro Shimura |
Publisher | : Princeton University Press |
Total Pages | : 232 |
Release | : 2016-06-02 |
Genre | : Mathematics |
ISBN | : 1400883946 |
Download Abelian Varieties with Complex Multiplication and Modular Functions Book in PDF, Epub and Kindle
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.
Author | : Gorō Shimura |
Publisher | : |
Total Pages | : 217 |
Release | : 1998 |
Genre | : Abelian varieties |
ISBN | : 9780691016566 |
Download Abelian Varieties with Complex Multiplication and Modular Functions Book in PDF, Epub and Kindle
Author | : John Cremona |
Publisher | : Birkhäuser |
Total Pages | : 291 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3034879199 |
Download Modular Curves and Abelian Varieties Book in PDF, Epub and Kindle
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.
Author | : Ching-Li Chai |
Publisher | : American Mathematical Soc. |
Total Pages | : 402 |
Release | : 2013-12-19 |
Genre | : Mathematics |
ISBN | : 1470410141 |
Download Complex Multiplication and Lifting Problems Book in PDF, Epub and Kindle
Abelian varieties with complex multiplication lie at the origins of class field theory, and they play a central role in the contemporary theory of Shimura varieties. They are special in characteristic 0 and ubiquitous over finite fields. This book explores the relationship between such abelian varieties over finite fields and over arithmetically interesting fields of characteristic 0 via the study of several natural CM lifting problems which had previously been solved only in special cases. In addition to giving complete solutions to such questions, the authors provide numerous examples to illustrate the general theory and present a detailed treatment of many fundamental results and concepts in the arithmetic of abelian varieties, such as the Main Theorem of Complex Multiplication and its generalizations, the finer aspects of Tate's work on abelian varieties over finite fields, and deformation theory. This book provides an ideal illustration of how modern techniques in arithmetic geometry (such as descent theory, crystalline methods, and group schemes) can be fruitfully combined with class field theory to answer concrete questions about abelian varieties. It will be a useful reference for researchers and advanced graduate students at the interface of number theory and algebraic geometry.
Author | : Eyal Zvi Goren |
Publisher | : American Mathematical Soc. |
Total Pages | : 282 |
Release | : 2002 |
Genre | : Mathematics |
ISBN | : 082181995X |
Download Lectures on Hilbert Modular Varieties and Modular Forms Book in PDF, Epub and Kindle
This book is devoted to certain aspects of the theory of $p$-adic Hilbert modular forms and moduli spaces of abelian varieties with real multiplication. The theory of $p$-adic modular forms is presented first in the elliptic case, introducing the reader to key ideas of N. M. Katz and J.-P. Serre. It is re-interpreted from a geometric point of view, which is developed to present the rudiments of a similar theory for Hilbert modular forms. The theory of moduli spaces of abelianvarieties with real multiplication is presented first very explicitly over the complex numbers. Aspects of the general theory are then exposed, in particular, local deformation theory of abelian varieties in positive characteristic. The arithmetic of $p$-adic Hilbert modular forms and the geometry ofmoduli spaces of abelian varieties are related. This relation is used to study $q$-expansions of Hilbert modular forms, on the one hand, and stratifications of moduli spaces on the other hand. The book is addressed to graduate students and non-experts. It attempts to provide the necessary background to all concepts exposed in it. It may serve as a textbook for an advanced graduate course.
Author | : Gorō Shimura |
Publisher | : |
Total Pages | : 180 |
Release | : 1961 |
Genre | : Abelian groups |
ISBN | : |
Download Complex Multiplication of Abelian Varieties and Its Applications to Number Theory Book in PDF, Epub and Kindle
Author | : George R. Kempf |
Publisher | : Springer Science & Business Media |
Total Pages | : 108 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3642760791 |
Download Complex Abelian Varieties and Theta Functions Book in PDF, Epub and Kindle
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.
Author | : B.J. Birch |
Publisher | : Springer |
Total Pages | : 158 |
Release | : 2006-12-08 |
Genre | : Mathematics |
ISBN | : 3540375880 |
Download Modular Functions of One Variable IV Book in PDF, Epub and Kindle
Author | : Vijaya Kumar Murty |
Publisher | : American Mathematical Soc. |
Total Pages | : 128 |
Release | : 1993 |
Genre | : Mathematics |
ISBN | : 0821811797 |
Download Introduction to Abelian Varieties Book in PDF, Epub and Kindle
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.
Author | : Herbert Lange |
Publisher | : Springer Science & Business Media |
Total Pages | : 443 |
Release | : 2013-03-09 |
Genre | : Mathematics |
ISBN | : 3662027887 |
Download Complex Abelian Varieties Book in PDF, Epub and Kindle
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.