The Arithmetic of Elliptic Curves [electronic resource] / by Joseph H. Silverman.

By: Silverman, Joseph H [author.]
Contributor(s): SpringerLink (Online service)
Material type: TextTextSeries: Graduate Texts in Mathematics: 106Publisher: New York, NY : Springer New York : Imprint: Springer, 2009Edition: Second EditionDescription: XX, 514 p. 14 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9780387094946Subject(s): Mathematics | Algebra | Algebraic geometry | Number theory | Mathematics | Algebraic Geometry | Algebra | Number TheoryAdditional physical formats: Printed edition:: No titleDDC classification: 516.35 LOC classification: QA564-609Online resources: Click here to access online
Contents:
Algebraic Varieties -- Algebraic Curves -- The Geometry of Elliptic Curves -- The Formal Group of an Elliptic Curve -- Elliptic Curves over Finite Fields -- Elliptic Curves over C -- Elliptic Curves over Local Fields -- Elliptic Curves over Global Fields -- Integral Points on Elliptic Curves -- Computing the Mordell#x2013;Weil Group -- Algorithmic Aspects of Elliptic Curves.
In: Springer eBooksSummary: 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 theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel's theorem on finiteness of integral points. For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra's factorization algorithm, Schoof's point counting algorithm, Miller's algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro's conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises. The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.
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Algebraic Varieties -- Algebraic Curves -- The Geometry of Elliptic Curves -- The Formal Group of an Elliptic Curve -- Elliptic Curves over Finite Fields -- Elliptic Curves over C -- Elliptic Curves over Local Fields -- Elliptic Curves over Global Fields -- Integral Points on Elliptic Curves -- Computing the Mordell#x2013;Weil Group -- Algorithmic Aspects of Elliptic Curves.

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 theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel's theorem on finiteness of integral points. For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra's factorization algorithm, Schoof's point counting algorithm, Miller's algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro's conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises. The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.

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