Analytic Theory Of Abelian Varieties
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| Author | : H. P. F. Swinnerton-Dyer |
| Publisher | : Cambridge University Press |
| Total Pages | : 105 |
| Release | : 1974-12-12 |
| Genre | : Mathematics |
| ISBN | : 0521205263 |
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.
| Author | : David Mumford |
| Publisher | : Debolsillo |
| Total Pages | : 0 |
| Release | : 2008 |
| Genre | : Abelian varieties |
| ISBN | : 9788185931869 |
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.''
| Author | : Alexander Polishchuk |
| Publisher | : Cambridge University Press |
| Total Pages | : 308 |
| Release | : 2003-04-21 |
| Genre | : Mathematics |
| ISBN | : 0521808049 |
Presents a modern treatment of the theory of theta functions in the context of algebraic geometry.
| Author | : Herbert Lange |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 443 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 3662027887 |
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.
| Author | : Olivier Debarre |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 124 |
| Release | : 2005 |
| Genre | : Mathematics |
| ISBN | : 9780821831656 |
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.
| Author | : Gerd Faltings |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 328 |
| Release | : 2013-04-17 |
| Genre | : Mathematics |
| ISBN | : 3662026325 |
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.
| Author | : Leonard M. Adleman |
| Publisher | : Springer |
| Total Pages | : 149 |
| Release | : 2006-11-15 |
| Genre | : Mathematics |
| ISBN | : 3540470212 |
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.
| Author | : Serge Lang |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 267 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642514472 |
From the reviews: "This book gives a thorough introduction to several theories that are fundamental to research on modular forms. Most of the material, despite its importance, had previously been unavailable in textbook form. Complete and readable proofs are given... In conclusion, this book is a welcome addition to the literature for the growing number of students and mathematicians in other fields who want to understand the recent developments in the theory of modular forms." #Mathematical Reviews# "This book will certainly be indispensable to all those wishing to get an up-to-date initiation to the theory of modular forms." #Publicationes Mathematicae#
| Author | : Vladimir G. Berkovich |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 181 |
| Release | : 2012-08-02 |
| Genre | : Mathematics |
| ISBN | : 0821890204 |
The purpose of this book is to introduce a new notion of analytic space over a non-Archimedean field. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. This makes it possible to apply the usual notions of homotopy and singular homology. The book includes a homotopic characterization of the analytic spaces associated with certain classes of algebraic varieties and an interpretation of Bruhat-Tits buildings in terms of these analytic spaces. The author also studies the connection with the earlier notion of a rigid analytic space. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a non-Archimedean spectral theory of bounded linear operators. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. It would be of interest to research mathematicians and graduate students working in algebraic geometry, number theory, and -adic analysis.
| Author | : Goro Shimura |
| Publisher | : Princeton University Press |
| Total Pages | : 232 |
| Release | : 2016-06-02 |
| Genre | : Mathematics |
| ISBN | : 1400883946 |
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.
