Abelian Functions
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| Author | : Serge Lang |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 178 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461257409 |
Introduction to Algebraic and Abelian Functions is a self-contained presentation of a fundamental subject in algebraic geometry and number theory. For this revised edition, the material on theta functions has been expanded, and the example of the Fermat curves is carried throughout the text. This volume is geared toward a second-year graduate course, but it leads naturally to the study of more advanced books listed in the bibliography.
| 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.
| Author | : Alekse_ Ivanovich Markushevich |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 188 |
| Release | : 2006-07-26 |
| Genre | : Mathematics |
| ISBN | : 9780821898369 |
Historical introduction. The Jacobian inversion problem Periodic functions of several complex variables Riemann matrices. Jacobian (intermediate) functions Construction of Jacobian functions of a given type. Theta functions and Abelian functions. Abelian and Picard manifolds Appendix A. Skew-symmetric determinants Appendix B. Divisors of analytic functions Appendix C. A summary of the most important formulas
| 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 | : Henry Frederick Baker |
| Publisher | : |
| Total Pages | : 712 |
| Release | : 1897 |
| Genre | : Functions, Abelian |
| ISBN | : |
| Author | : Henry Frederick Baker |
| Publisher | : Cambridge University Press |
| Total Pages | : 724 |
| Release | : 1995-12-14 |
| Genre | : Mathematics |
| ISBN | : 9780521498777 |
Classical algebraic geometry, inseparably connected with the names of Abel, Riemann, Weierstrass, Poincaré, Clebsch, Jacobi and other outstanding mathematicians of the last century, was mainly an analytical theory. In our century it has been enriched by the methods and ideas of topology, commutative algebra and Grothendieck's schemes seemed to have replaced once and forever the somewhat naive language of classical algebraic geometry. This book contains more than its modest title suggests. Written in 1897, its scope was as broad as it could possibly be, namely to cover the whole of algebraic geometry, and associated theories. The subject is discussed by Baker in terms of transcendental functions, and in particular theta functions. Many of the ideas put forward are of continuing relevance today, and some of the most exciting ideas from theoretical physics draw on work presented here.
| Author | : Carl Ludwig Siegel |
| Publisher | : John Wiley & Sons |
| Total Pages | : 260 |
| Release | : 1989-01-18 |
| Genre | : Mathematics |
| ISBN | : 9780471504016 |
Develops the higher parts of function theory in a unified presentation. Starts with elliptic integrals and functions and uniformization theory, continues with automorphic functions and the theory of abelian integrals and ends with the theory of abelian functions and modular functions in several variables. The last topic originates with the author and appears here for the first time in book form.
| Author | : Henry Frederick Baker |
| Publisher | : Palala Press |
| Total Pages | : 714 |
| Release | : 2018-02-22 |
| Genre | : History |
| ISBN | : 9781378525517 |
This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
| 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 | : George R. Kempf |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 108 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642760791 |
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.
