Elliptic Curves Second Edition
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Elliptic Curves
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Author : James S Milne
language : en
Publisher: World Scientific Publishing Company
Release Date : 2023-02
Elliptic Curves written by James S Milne and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-02 with categories.
This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. It assumes only a knowledge of the basic algebra, complex analysis, and topology usually taught in first-year graduate courses. An elliptic curve is a plane curve defined by a cubic polynomial. Although the problem of finding the rational points on an elliptic curve has fascinated mathematicians since ancient times, it was not until 1922 that Mordell proved that the points form a finitely generated group. There is still no proven algorithm for finding the rank of the group, but in one of the earliest important applications of computers to mathematics, Birch and Swinnerton-Dyer discovered a relation between the rank and the numbers of points on the curve computed modulo a prime. Chapter IV of the book proves Mordell's theorem and explains the conjecture of Birch and Swinnerton-Dyer. Every elliptic curve over the rational numbers has an L-series attached to it.Hasse conjectured that this L-series satisfies a functional equation, and in 1955 Taniyama suggested that Hasse's conjecture could be proved by showing that the L-series arises from a modular form. This was shown to be correct by Wiles (and others) in the 1990s, and, as a consequence, one obtains a proof of Fermat's Last Theorem. Chapter V of the book is devoted to explaining this work. The first three chapters develop the basic theory of elliptic curves. For this edition, the text has been completely revised and updated.
The Arithmetic Of Elliptic Curves
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Author : Joseph H. Silverman
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-04-20
The Arithmetic Of Elliptic Curves written by Joseph H. Silverman 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 2009-04-20 with Mathematics categories.
The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.
Geometric Modular Forms And Elliptic Curves 2nd Edition
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Author : Haruzo Hida
language : en
Publisher: World Scientific
Release Date : 2011-12-28
Geometric Modular Forms And Elliptic Curves 2nd Edition written by Haruzo Hida and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-28 with Mathematics categories.
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.In this new second edition, a detailed description of Barsotti-Tate groups (including formal Lie groups) is added to Chapter 1. As an application, a down-to-earth description of formal deformation theory of elliptic curves is incorporated at the end of Chapter 2 (in order to make the proof of regularity of the moduli of elliptic curve more conceptual), and in Chapter 4, though limited to ordinary cases, newly incorporated are Ribet's theorem of full image of modular p-adic Galois representation and its generalization to ‘big’ Λ-adic Galois representations under mild assumptions (a new result of the author). Though some of the striking developments described above is out of the scope of this introductory book, the author gives a taste of present day research in the area of Number Theory at the very end of the book (giving a good account of modularity theory of abelian ℚ-varieties and ℚ-curves).
Elliptic Curves
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Author : Dale Husemöller
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-06-06
Elliptic Curves written by Dale Husemöller 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 2006-06-06 with Mathematics categories.
There are three new appendices, one by Stefan Theisen on the role of Calabi– Yau manifolds in string theory and one by Otto Forster on the use of elliptic curves in computing theory and coding theory. In the third appendix we discuss the role of elliptic curves in homotopy theory. In these three introductions the reader can get a clue to the far-reaching implications of the theory of elliptic curves in mathematical sciences. During the ?nal production of this edition, the ICM 2002 manuscript of Mike Hopkins became available. This report outlines the role of elliptic curves in ho- topy theory. Elliptic curves appear in the form of the Weierstasse equation and its related changes of variable. The equations and the changes of variable are coded in an algebraic structure called a Hopf algebroid, and this Hopf algebroid is related to a cohomology theory called topological modular forms. Hopkins and his coworkers have used this theory in several directions, one being the explanation of elements in stable homotopy up to degree 60. In the third appendix we explain how what we described in Chapter 3 leads to the Weierstrass Hopf algebroid making a link with Hopkins’ paper.
Rational Points On Elliptic Curves
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Author : Joseph H. Silverman
language : en
Publisher: Springer
Release Date : 2015-06-02
Rational Points On Elliptic Curves written by Joseph H. Silverman and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-06-02 with Mathematics categories.
The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This volume stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry. Most concretely, an elliptic curve is the set of zeroes of a cubic polynomial in two variables. If the polynomial has rational coefficients, then one can ask for a description of those zeroes whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of Rational Points on Elliptic Curves. Topics covered include the geometry and group structure of elliptic curves, the Nagell–Lutz theorem describing points of finite order, the Mordell–Weil theorem on the finite generation of the group of rational points, the Thue–Siegel theorem on the finiteness of the set of integer points, theorems on counting points with coordinates in finite fields, Lenstra's elliptic curve factorization algorithm, and a discussion of complex multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat's Last Theorem by Wiles et al. via the use of elliptic curves.
Elliptic Curves
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Author : J. S. Milne
language : en
Publisher:
Release Date : 1996
Elliptic Curves written by J. S. Milne and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with categories.
Elliptic Curves Second Edition
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Author : James S Milne
language : en
Publisher: World Scientific
Release Date : 2020-08-20
Elliptic Curves Second Edition written by James S Milne and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-08-20 with Mathematics categories.
This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. It assumes only a knowledge of the basic algebra, complex analysis, and topology usually taught in first-year graduate courses.An elliptic curve is a plane curve defined by a cubic polynomial. Although the problem of finding the rational points on an elliptic curve has fascinated mathematicians since ancient times, it was not until 1922 that Mordell proved that the points form a finitely generated group. There is still no proven algorithm for finding the rank of the group, but in one of the earliest important applications of computers to mathematics, Birch and Swinnerton-Dyer discovered a relation between the rank and the numbers of points on the curve computed modulo a prime. Chapter IV of the book proves Mordell's theorem and explains the conjecture of Birch and Swinnerton-Dyer.Every elliptic curve over the rational numbers has an L-series attached to it.Hasse conjectured that this L-series satisfies a functional equation, and in 1955 Taniyama suggested that Hasse's conjecture could be proved by showing that the L-series arises from a modular form. This was shown to be correct by Wiles (and others) in the 1990s, and, as a consequence, one obtains a proof of Fermat's Last Theorem. Chapter V of the book is devoted to explaining this work.The first three chapters develop the basic theory of elliptic curves.For this edition, the text has been completely revised and updated.
Elliptic Curves
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Author : Lawrence C. Washington
language : en
Publisher: CRC Press
Release Date : 2008-04-03
Elliptic Curves written by Lawrence C. Washington and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-04-03 with Computers categories.
Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and application
Elliptic Curves
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Author : James S Milne
language : en
Publisher:
Release Date : 2020
Elliptic Curves written by James S Milne and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with Mathematics categories.
This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. It assumes only a knowledge of the basic algebra, complex analysis, and topology usually taught in first-year graduate courses. An elliptic curve is a plane curve defined by a cubic polynomial. Although the problem of finding the rational points on an elliptic curve has fascinated mathematicians since ancient times, it was not until 1922 that Mordell proved that the points form a finitely generated group. There is still no proven algorithm for finding the rank of the group, but in one of the earliest important applications of computers to mathematics, Birch and Swinnerton-Dyer discovered a relation between the rank and the numbers of points on the curve computed modulo a prime. Chapter IV of the book proves Mordell's theorem and explains the conjecture of Birch and Swinnerton-Dyer. Every elliptic curve over the rational numbers has an L-series attached to it.Hasse conjectured that this L-series satisfies a functional equation, and in 1955 Taniyama suggested that Hasse's conjecture could be proved by showing that the L-series arises from a modular form. This was shown to be correct by Wiles (and others) in the 1990s, and, as a consequence, one obtains a proof of Fermat's Last Theorem. Chapter V of the book is devoted to explaining this work. The first three chapters develop the basic theory of elliptic curves. For this edition, the text has been completely revised and updated.
Contemporary Cryptography Second Edition
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Author : Rolf Oppliger
language : en
Publisher: Artech House
Release Date : 2011
Contemporary Cryptography Second Edition written by Rolf Oppliger and has been published by Artech House this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Computers categories.
Whether you're new to the field or looking to broaden your knowledge of contemporary cryptography, this newly revised edition of an Artech House classic puts all aspects of this important topic into perspective. Delivering an accurate introduction to the current state-of-the-art in modern cryptography, the book offers you an in-depth understanding of essential tools and applications to help you with your daily work. The second edition has been reorganized and expanded, providing mathematical fundamentals and important cryptography principles in the appropriate appendixes, rather than summarized at the beginning of the book. Now you find all the details you need to fully master the material in the relevant sections. This allows you to quickly delve into the practical information you need for your projects. Covering unkeyed, secret key, and public key cryptosystems, this authoritative reference gives you solid working knowledge of the latest and most critical concepts, techniques, and systems in contemporary cryptography. Additionally, the book is supported with over 720 equations, more than 60 illustrations, and numerous time-saving URLs that connect you to websites with related information.