The Arithmetic of Elliptic Curves / Edition 1

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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 algorthims 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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Editorial Reviews

From the Publisher
From the reviews of the second edition:

"This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."— MATHEMATICAL REVIEWS

“The book under review is the second, revised, enlarged, and updated edition of J. Silverman’s meanwhile classical primer of the arithmetic of elliptic curves. … All together, this enlarged and updated version of J. Silverman’s classic ‘The Arithmetic of Elliptic Curves’ significantly increases the unchallenged value of this modern primer as a standard textbook in the field. … This makes the entire text a perfect source for teachers and students, for courses and self-study, and for further studies in the arithmetic of elliptic curves likewise.” (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010)

“For the second edition of his masterly book, the author considerably updated and improved several results and proofs. … book contains a great many exercises, many of which develop or complement the results from the main body of the book. … The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. … Summarizing, this is an excellent book … . useful both for experienced mathematicians and for graduate students.”­­­ (Vasil' I. Andriĭchuk, Mathematical Reviews, Issue 2010 i)

“This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves … . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory text and a standard reference. … The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)

“The book is written for graduate students … and for researchers interested in standard facts about elliptic curves. … A wonderful textbook on the arithmetic theory of elliptic curves and it is a very popular introduction to the subject. … I recommend this book for anyone interested in the mathematical study of elliptic curves. It is an excellent introduction, elegant and very well written. It is one of the best textbooks to graduate level studies I have ever had contact yet.” (Book Inspections Blog, 2012)

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Product Details

  • ISBN-13: 9780387962030
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 12/28/1985
  • Series: Graduate Texts in Mathematics Series, #106
  • Edition description: 1st ed. 1986. Corr. 3rd printing
  • Edition number: 1
  • Pages: 400
  • Product dimensions: 6.32 (w) x 9.66 (h) x 0.94 (d)

Meet the Author

Dr. Joseph Silverman is a professor at Brown University and has been an instructor or professors since 1982. He was the Chair of the Brown Mathematics department from 2001-2004. He has received numerous fellowships, grants and awards, as well as being a frequentlyinvited lecturer. He is currently a member of the Council of the American Mathematical Society. His research areas of interest are number theory, arithmetic geometry, elliptic curves, dynamical systems and cryptography. He has co-authored over 120 publications and has had over 20 doctoral students under his tutelage. He has published9 highly successful books with Springer,including the recently released, An Introduction to Mathematical Cryptography, for Undergraduate Texts in Mathematics.

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Table of Contents

Preface to the Second Edition v

Preface to the First Edition vii

Introduction xvii

Chapter I Algebraic Varieties 1

1 Affine Varieties 1

2 Projective Varieties 6

3 Maps Between Varieties 11

Exercises 14

Chapter II Algebraic Curves 17

1 Curves 17

2 Maps Between Curves 19

3 Divisors 27

4 Differentials 30

5 The Riemann-Roch Theorem 33

Exercises 37

Chapter III The Geometry of Elliptic Curves 41

1 Weierstrass Equations 42

2 The Group Law 51

3 Elliptic Curves 58

4 Isogenies 66

5 The Invariant Differential 75

6 The Dual Isogeny 80

7 The Tate Module 87

8 The Weil Pairing 92

9 The Endomorphism Ring 99

10 The Automorphism Group 103

Exercises 104

Chapter IV The Formal Group of an Elliptic Curve 115

1 Expansion Around O 115

2 Formal Groups 120

3 Groups Associated to Formal Groups 123

4 The Invariant Differential 125

5 The Formal Logarithm 127

6 Formal Groups over Discrete Valuation Rings 129

7 Formal Groups in Characteristic p 132

Exercises 135

Chapter V Elliptic Curves over Finite Fields 137

1 Number of Rational Points 137

2 The Weil Conjectures 140

3 The Endomorphism Ring 144

4 Calculating the Hasse Invariant 148

Exercises 153

Chapter VI Elliptic Curves Over C 157

1 Elliptic Integrals 158

2 Elliptic Functions 161

3 Construction of Elliptic Functions 165

4 Maps Analytic and Maps Algebraic 171

5 Uniformization 173

6 The Lefschetz Principle 177

Exercises 178

Chapter VII Elliptic Curves over Local Fields 185

1 Minimal Weierstrass Equations 185

2 Reduction Modulo π 187

3 Points of Finite Order 192

4 The Action of Inertia 194

5 Good and Bad Reduction 196

6 The Group E/E0 199

7 The Criterionof Néron-Ogg-Shafarevich 201

Exercises 203

Chapter VIII Elliptic Curves over Global Fields 207

1 The Weak Mordell-Weil Theorem 208

2 The Kummer Pairing via Cohomology 215

3 The Descent Procedure 218

4 The Mordell-Weil Theorem over Q 220

5 Heights on Projective Space 224

6 Heights on Elliptic Curves 234

7 Torsion Points 240

8 The Minimal Discriminant 243

9 The Canonical Height 247

10 The Rank of an Elliptic Curve 254

11 Szpiro's Conjecture and ABC 255

Exercises 261

Chapter IX Integral Points on Elliptic Curves 269

1 Diophantine Approximation 270

2 Distance Functions 273

3 Siegel's Theorem 276

4 The S-Unit Equation 281

5 Effective Methods 286

6 Shafarevich's Theorem 293

7 The Curve Y2 = X3+ D 296

8 Roth's Theorem-An Overview 299

Exercises 302

Chapter X Computing the Mordell-Weil Group 309

1 An Example 310

2 Twisting-General Theory 318

3 Homogeneous Spaces 321

4 The Selmer and Shafarevich-Tate Groups 331

5 Twisting-Elliptic Curves 341

6 The Curve Y2 = X3 + DX 344

Exercises 355

Chapter XI Algorithmic Aspects of Elliptic Curves 363

1 Double-and-Add Algorithms 364

2 Lenstra's Elliptic Curve Factorization Algorithm 366

3 Counting the Number of Points in E(Fq) 372

4 Elliptic Curve Cryptography 376

5 Solving the ECDLP: The General Case 381

6 Solving the ECDLP: Special Cases 386

7 Pairing-Based Cryptography 390

8 Computing the Weil Pairing 393

9 The Tate-Lichtenbaum Pairing 397

Exercises 403

Appendix A Elliptic Curves in Characteristics 2 and 3 409

Exercises 414

Appendix B Group Cohomology (H0 and H1) 415

1 Cohomology of Finite Groups 415

2 Galois Cohomology 418

3 Nonabelian Cohomology 421

Exercises 422

Appendix C Further Topics: An Overview 425

11 Complex Multiplication 425

12 Modular Functions 429

13 Modular Curves 439

14 Tate Curves 443

15 Néron Models and Tate's Algorithm 446

16 L-Series 449

17 Duality Theory 453

18 Local Height Functions 454

19 The Image of Galois 455

20 Function Fields and Specialization Theorems 456

21 Variation of ap and the Sato-Tate Conjecture 458

Notes on Exercises 461

List of Notation 467

References 473

Index 489

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