The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics Book 106)

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. 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. Ch. I. Algebraic Varieties -- Ch. Ii. Algebraic Curves -- Ch. Iii. Geometry Of Elliptic Curves -- Ch. Iv. Formal Group Of An Elliptic Curve -- Ch. V. Elliptic Curves Over Finite Fields -- Ch. Vi. Elliptic Curves Over C -- Ch. Vii. Elliptic Curves Over Local Fields -- Ch. Viii. Elliptic Curves Over Global Fields -- Ch. Ix. Integral Points On Elliptic Curves -- Ch. X. Computing The Mordell-weil Group -- Ch. Xi. Algorithmic Aspects Of Elliptic Curves -- App. A. Elliptic Curves In Characteristics 2 And 3 -- App. B. Group Cohomology (h[superscript 0] And H[superscript 1]) -- App. C. Further Topics: An Overview. Joseph H. Silverman. Includes Bibliographical References (p. 473-487) And Index. 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, elliptic curves over finite fields, the complex numbers, local fields, and global fields. The last two chapters deal with integral and rational points, including Siegel's theorem and explicit computations for the curve Y 2 = X 3 + DX. 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. 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. Front Matter....Pages 1-17 Algebraic Varieties....Pages 1-16 Algebraic Curves....Pages 17-40 The Geometry of Elliptic Curves....Pages 41-114 The Formal Group of an Elliptic Curve....Pages 115-135 Elliptic Curves over Finite Fields....Pages 137-156 Elliptic Curves over C....Pages 157-183 Elliptic Curves over Local Fields....Pages 185-205 Elliptic Curves over Global Fields....Pages 207-267 Integral Points on Elliptic Curves....Pages 269-307 Computing the Mordell–Weil Group....Pages 309-361 Algorithmic Aspects of Elliptic Curves....Pages 363-408 Back Matter....Pages 1-102 Treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. This book discusses the necessary algebro-geometric results, and offers an exposition of the geometry of elliptic curves, and the formal group of an elliptic curve