Complex Multiplication
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Complex Multiplication
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Author : Reinhard Schertz
language : en
Publisher: Cambridge University Press
Release Date : 2010-04-29
Complex Multiplication written by Reinhard Schertz and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-04-29 with Mathematics categories.
This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between elliptic units and class numbers.
Complex Multiplication
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Author : S. Lang
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Complex Multiplication written by S. Lang and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
The small book by Shimura-Taniyama on the subject of complex multi is a classic. It gives the results obtained by them (and some by Weil) plication in the higher dimensional case, generalizing in a non-trivial way the method of Deuring for elliptic curves, by reduction mod p. Partly through the work of Shimura himself (cf. [Sh 1] [Sh 2], and [Sh 5]), and some others (Serre, Tate, Kubota, Ribet, Deligne etc.) it is possible today to make a more snappy and extensive presentation of the fundamental results than was possible in 1961. Several persons have found my lecture notes on this subject useful to them, and so I have decided to publish this short book to make them more widely available. Readers acquainted with the standard theory of abelian varieties, and who wish to get rapidly an idea of the fundamental facts of complex multi plication, are advised to look first at the two main theorems, Chapter 3, §6 and Chapter 4, §1, as well as the rest of Chapter 4. The applications of Chapter 6 could also be profitably read early. I am much indebted to N. Schappacher for a careful reading of the manu script resulting in a number of useful suggestions. S. LANG Contents CHAPTER 1 Analytic Complex Multiplication 4 I. Positive Definite Involutions . . . 6 2. CM Types and Subfields. . . . . 8 3. Application to Abelian Manifolds. 4. Construction of Abelian Manifolds with CM 14 21 5. Reflex of a CM Type . . . . .
Complex Multiplication
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Author : Serge Lang
language : en
Publisher:
Release Date : 1983-01-01
Complex Multiplication written by Serge Lang and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1983-01-01 with Multiplication complexe categories.
Seminar On Complex Multiplication
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Author : A. Borel
language : en
Publisher: Springer
Release Date : 2007-08-16
Seminar On Complex Multiplication written by A. Borel and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-08-16 with Mathematics categories.
Arithmetic On Elliptic Curves With Complex Multiplication
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Author : B.H. Gross
language : en
Publisher: Springer
Release Date : 2006-11-14
Arithmetic On Elliptic Curves With Complex Multiplication written by B.H. Gross and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.
Primes Of The Form X2 Ny2 Fermat Class Field Theory And Complex Multiplication Third Edition With Solutions
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Author : David A. Cox
language : en
Publisher: American Mathematical Soc.
Release Date : 2022-11-16
Primes Of The Form X2 Ny2 Fermat Class Field Theory And Complex Multiplication Third Edition With Solutions written by David A. Cox and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-11-16 with Education categories.
This book studies when a prime p can be written in the form x2+ny2. It begins at an elementary level with results of Fermat and Euler and then discusses the work of Lagrange, Legendre and Gauss on quadratic reciprocity and the genus theory of quadratic forms. After exploring cubic and biquadratic reciprocity, the pace quickens with the introduction of algebraic number fields and class field theory. This leads to the concept of ring class field and a complete but abstract solution of p=x2+ny2. To make things more concrete, the book introduces complex multiplication and modular functions to give a constructive solution. The book ends with a discussion of elliptic curves and Shimura reciprocity. Along the way the reader will encounter some compelling history and marvelous formulas, together with a complete solution of the class number one problem for imaginary quadratic fields. The book is accessible to readers with modest backgrounds in number theory. In the third edition, the numerous exercises have been thoroughly checked and revised, and as a special feature, complete solutions are included. This makes the book especially attractive to readers who want to get an active knowledge of this wonderful part of mathematics.
Seminar On Complex Multiplication
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Author : A. Borel
language : en
Publisher:
Release Date : 2014-09-01
Seminar On Complex Multiplication written by A. Borel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-01 with categories.
Complex Multiplication And Lifting Problems
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Author : Ching-Li Chai
language : en
Publisher: American Mathematical Soc.
Release Date : 2013-12-19
Complex Multiplication And Lifting Problems written by Ching-Li Chai and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-19 with Mathematics categories.
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.
Arithmetic On Elliptic Curves With Complex Multiplication
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Author : B. H. Gross
language : en
Publisher:
Release Date : 2014-09-01
Arithmetic On Elliptic Curves With Complex Multiplication written by B. H. Gross and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-01 with categories.
Seminar On Complex Multiplication
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Author : Armand Borel
language : en
Publisher:
Release Date : 1966
Seminar On Complex Multiplication written by Armand Borel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1966 with Class field theory categories.