Abelian Varieties With Complex Multiplication And Modular Functions

Author : Goro Shimura
Publisher : Princeton University Press
Page : 232 pages
File Size : 40,7 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.

Author : John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet
Publisher : Birkhäuser
Page : 291 pages
File Size : 50,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879194

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Modular Curves and Abelian Varieties by John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet Pdf

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,Brian Conrad, Frans Oort
Publisher : American Mathematical Soc.
Page : 402 pages
File Size : 51,5 Mb
Release : 2013-12-19
Category : Mathematics
ISBN : 9781470410148

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Complex Multiplication and Lifting Problems by Ching-Li Chai,Brian Conrad, Frans Oort Pdf

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,Zvi Goren
Publisher : American Mathematical Soc.
Page : 282 pages
File Size : 45,8 Mb
Release : 2002
Category : Abelian varieties
ISBN : 9780821819951

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Lectures on Hilbert Modular Varieties and Modular Forms by Eyal Zvi Goren,Zvi Goren Pdf

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 : B.J. Birch,W. Kuyk
Publisher : Springer
Page : 158 pages
File Size : 55,9 Mb
Release : 2006-12-08
Category : Mathematics
ISBN : 9783540375883

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Modular Functions of One Variable IV by B.J. Birch,W. Kuyk Pdf

Author : Herbert Lange,Christina Birkenhake
Publisher : Springer Science & Business Media
Page : 443 pages
File Size : 46,7 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.

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.

Author : Gorō Shimura
Publisher : Princeton University Press
Page : 292 pages
File Size : 46,8 Mb
Release : 1971-08-21
Category : Mathematics
ISBN : 0691080925

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Introduction to the Arithmetic Theory of Automorphic Functions by Gorō Shimura Pdf

The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles.

Author : Tiago J. Fonseca
Publisher : American Mathematical Society
Page : 158 pages
File Size : 43,9 Mb
Release : 2023-01-18
Category : Mathematics
ISBN : 9781470460198

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Higher Ramanujan Equations and Periods of Abelian Varieties by Tiago J. Fonseca Pdf

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Author : James Arthur
Publisher : American Mathematical Soc.
Page : 658 pages
File Size : 53,9 Mb
Release : 2011
Category : Mathematics
ISBN : 9780821852040

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On Certain $L$-Functions by James Arthur Pdf

Illuminate various areas of the study of geometric, analytic, and number theoretic aspects of automorphic forms and their $L$-functions, and both local and global theory are addressed. Topics discussed in the articles include Langlands functoriality, the Rankin-Selberg method, the Langlands-Shahidi method, motivic Galois groups, Shimura varieties, orbital integrals, representations of $p$-adic groups, Plancherel formula and its consequences, and the Gross-Prasad conjecture.

Author : Gerd Faltings,Ching-Li Chai
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 52,7 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783662026328

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Degeneration of Abelian Varieties by Gerd Faltings,Ching-Li Chai Pdf

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 : Moshe Jarden,Tony Shaska
Publisher : American Mathematical Soc.
Page : 200 pages
File Size : 40,6 Mb
Release : 2021-05-03
Category : Education
ISBN : 9781470452070

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Abelian Varieties and Number Theory by Moshe Jarden,Tony Shaska Pdf

This book is a collection of articles on Abelian varieties and number theory dedicated to Gerhard Frey's 75th birthday. It contains original articles by experts in the area of arithmetic and algebraic geometry. The articles cover topics on Abelian varieties and finitely generated Galois groups, ranks of Abelian varieties and Mordell-Lang conjecture, Tate-Shafarevich group and isogeny volcanoes, endomorphisms of superelliptic Jacobians, obstructions to local-global principles over semi-global fields, Drinfeld modular varieties, representations of etale fundamental groups and specialization of algebraic cycles, Deuring's theory of constant reductions, etc. The book will be a valuable resource to graduate students and experts working on Abelian varieties and related areas.

Author : Jean-Pierre Serre
Publisher : CRC Press
Page : 203 pages
File Size : 45,8 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

Author : Haruzo Hida
Publisher : World Scientific
Page : 376 pages
File Size : 43,5 Mb
Release : 2000-09-27
Category : Mathematics
ISBN : 9789814492898

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Geometric Modular Forms and Elliptic Curves by Haruzo Hida Pdf

This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the Shimura-Taniyama conjecture, is given. In addition, the book presents an outline of the proof of diverse modularity results of two-dimensional Galois representations (including that of Wiles), as well as some of the author's new results in that direction. Contents:An Algebro-Geometric Tool BoxElliptic CurvesGeometric Modular FormsJacobians and Galois RepresentationsModularity Problems Readership: Graduates and researchers in number theory. Keywords:Control Theorems;Shimura-Taniyama-Weil Conjecture;Elliptic Curve;Modular Curve;Deformation Rings;Hecke Algebras;Modular Galois Representations;Moduli Spaces of Elliptic Curves;Modular FormsReviews:“… this is a welcome addition to the literature in a field difficult to penetrate. This book should obviously be carefully studied by advanced students and by professional mathematicians in arithmetic algebraic geometry or (modern) number theory.”Mathematical Reviews “Geometric Modular Forms and Elliptic Curves is suited for both the (advanced and specialized) classroom and (well-prepared and highly motivated) reader bent of serious self-study. Beyond this, the book's prose is clear, there are examples and exercises available, and, as always, the serious student should have a go at them: he will reap wonderful benefits.”MAA Reviews

Author : B.H. Gross
Publisher : Springer
Page : 100 pages
File Size : 41,7 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540385752

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Arithmetic on Elliptic Curves with Complex Multiplication by B.H. Gross Pdf