The Rational Points On Elliptic Curves
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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.
Rational Points On Elliptic Curves
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Author : Joseph H. Silverman
language : en
Publisher: Springer Science & Business Media
Release Date : 1994-11-18
Rational Points On 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 1994-11-18 with Mathematics categories.
The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.
Rational Points On Modular Elliptic Curves
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Author : Henri Darmon
language : en
Publisher: American Mathematical Soc.
Release Date : 2004
Rational Points On Modular Elliptic Curves written by Henri Darmon 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 2004 with Mathematics categories.
The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture.
The Rational Points On Elliptic Curves
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Author : Julian Rojter
language : en
Publisher:
Release Date : 1974
The Rational Points On Elliptic Curves written by Julian Rojter and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1974 with categories.
Rational Points On Elliptic Curves
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Author : Kendra Hershey
language : en
Publisher:
Release Date : 1993
Rational Points On Elliptic Curves written by Kendra Hershey and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with categories.
Rational Points On Elliptic Curves
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Author : David Grant
language : en
Publisher:
Release Date : 1981
Rational Points On Elliptic Curves written by David Grant and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1981 with categories.
Elliptic Curves
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Author : Dale Husemöller
language : en
Publisher: Springer Science & Business Media
Release Date : 2004
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 2004 with Mathematics categories.
First Edition sold over 2500 copies in the Americas; New Edition contains three new chapters and two new appendices
Elliptic Curves And Their Arithmetic Geometry
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Author : Mr. Rohit Manglik
language : en
Publisher: EduGorilla Publication
Release Date : 2024-07-28
Elliptic Curves And Their Arithmetic Geometry written by Mr. Rohit Manglik and has been published by EduGorilla Publication this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-07-28 with Mathematics categories.
EduGorilla Publication is a trusted name in the education sector, committed to empowering learners with high-quality study materials and resources. Specializing in competitive exams and academic support, EduGorilla provides comprehensive and well-structured content tailored to meet the needs of students across various streams and levels.
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 Mathematics Of Ciphers
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Author : S.C. Coutinho
language : en
Publisher: CRC Press
Release Date : 1999-01-15
The Mathematics Of Ciphers written by S.C. Coutinho and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-01-15 with Mathematics categories.
This book is an introduction to the algorithmic aspects of number theory and its applications to cryptography, with special emphasis on the RSA cryptosys-tem. It covers many of the familiar topics of elementary number theory, all with an algorithmic twist. The text also includes many interesting historical notes.