Silverman The Arithmetic of Elliptic Curves

2. Auflage 2009
ISBN: 978-0-387-09494-6
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 106, 514 Seiten

Reihe: Graduate Texts in Mathematics

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.

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 frequently invited 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 published 9 highly successful books with Springer, including the recently released, An Introduction to Mathematical Cryptography, for Undergraduate Texts in Mathematics.

Autoren/Hrsg.

Silverman, Joseph H.

Inhaltsverzeichnis

1;Preface to the Second Edition;5
1.1;Acknowledgments for the Second Edition;6
2;Preface to the First Edition;7
2.1;Acknowledgements;8
2.2;Acknowledgments for the Second Printing;8
3;Contents;10
4;Introduction;15
4.1;References;17
4.2;Standard Notation;18
5;I Algebraic Varieties;19
5.1;I.1 Affine Varieties;19
5.2;I.2 Projective Varieties;24
5.3;I.3 Maps Between Varieties;29
5.4;Exercises;32
6;II Algebraic Curves;35
6.1;II.1 Curves;35
6.2;II.2 Maps Between Curves;37
6.3;II.3 Divisors;45
6.4;II.4 Differentials;48
6.5;II.5 The Riemann–Roch Theorem;51
6.6;Exercises;55
7;III The Geometry of Elliptic Curves;59
7.1;III.1 Weierstrass Equations;60
7.2;III.2 The Group Law;69
7.3;III.3 Elliptic Curves;76
7.4;III.4 Isogenies;84
7.5;III.5 The Invariant Differential;93
7.6;III.6 The Dual Isogeny;98
7.7;III.7 The Tate Module;105
7.8;III.8 TheWeil Pairing;110
7.9;III.9 The Endomorphism Ring;117
7.10;III.10 The Automorphism Group;121
7.11;Exercises;122
8;IV The Formal Group of an Elliptic Curve;133
8.1;IV.1 Expansion Around O;133
8.2;IV.2 Formal Groups;138
8.3;IV.3 Groups Associated to Formal Groups;141
8.4;IV.4 The Invariant Differential;143
8.5;IV.5 The Formal Logarithm;145
8.6;IV.6 Formal Groups over Discrete Valuation Rings;147
8.7;IV.7 Formal Groups in Characteristic p;150
8.8;Exercises;153
9;V Elliptic Curves over Finite Fields;154
9.1;V.1 Number of Rational Points;154
9.2;V.2 The Weil Conjectures;157
9.3;V.3 The Endomorphism Ring;161
9.4;V.4 Calculating the Hasse Invariant;165
9.5;Exercises;170
10;VI Elliptic Curves over C;174
10.1;VI.1 Elliptic Integrals;175
10.2;VI.2 Elliptic Functions;178
10.3;VI.3 Construction of Elliptic Functions;182
10.4;VI.4 Maps Analytic and Maps Algebraic;188
10.5;VI.5 Uniformization;190
10.6;VI.6 The Lefschetz Principle;194
10.7;Exercises;195
11;VII Elliptic Curves over Local Fields;201
11.1;VII.1 MinimalWeierstrass Equations;201
11.2;VII.2 Reduction Modulo p;203
11.3;VII.3 Points of Finite Order;208
11.4;VII.4 The Action of Inertia;210
11.5;VII.5 Good and Bad Reduction;212
11.6;VII.6 The Group E/E0;215
11.7;VII.7 The Criterion of N´eron–Ogg–Shafarevich;217
11.8;Exercises;219
12;VIII Elliptic Curves over Global Fields;222
12.1;VIII.1 TheWeak Mordell–Weil Theorem;223
12.2;VIII.2 The Kummer Pairing via Cohomology;230
12.3;VIII.3 The Descent Procedure;233
12.4;VIII.4 The Mordell–Weil Theorem over Q;235
12.5;VIII.5 Heights on Projective Space;239
12.6;VIII.6 Heights on Elliptic Curves;249
12.7;VIII.7 Torsion Points;255
12.8;VIII.8 The Minimal Discriminant;258
12.9;VIII.9 The Canonical Height;262
12.10;VIII.10 The Rank of an Elliptic Curve;269
12.11;VIII.11 Szpiro’s Conjecture and ABC;270
12.12;Exercises;276
13;IX Integral Points on Elliptic Curves;283
13.1;IX.1 Diophantine Approximation;284
13.2;IX.2 Distance Functions;287
13.3;IX.3 Siegel’s Theorem;290
13.4;IX.4 The S-Unit Equation;295
13.5;IX.5 Effective Methods;300
13.6;IX.6 Shafarevich’s Theorem;307
13.7;IX.7 The Curve Y;310
13.8;IX.8 Roth’s Theorem—An Overview;313
13.9;Step I: An Auxiliary Polynomial;314
13.10;Step II: An Upper Bound for P;314
13.11;Step III: A Nonvanishing Result (Roth’s Lemma);314
13.12;Step IV: The Final Estimate;315
13.13;Exercises;316
14;X Computing the Mordell–Weil Group;322
14.1;X.1 An Example;323
14.2;X.2 Twisting—General Theory;331
14.3;X.3 Homogeneous Spaces;334
14.4;X.4 The Selmer and Shafarevich–Tate Groups;344
14.5;X.5 Twisting—Elliptic Curves;354
14.6;X.6 The Curve Y;357
14.7;Exercises;368
15;XI Algorithmic Aspects of Elliptic Curves;375
15.1;XI.1 Double-and-Add Algorithms;376
15.2;XI.2 Lenstra’s Elliptic Curve Factorization Algorithm;378
15.3;XI.3 Counting the Number of Points in E(Fq);384
15.4;XI.4 Elliptic Curve Cryptography;388
15.5;XI.5 Solving the Elliptic Curve Discrete Logarithm Problem: The General Case;393
15.6;XI.6 Solving the Elliptic Curve Discrete Logarithm Problem: Special Cases;398
15.7;XI.7 Pairing-Based Cryptography;402
15.8;XI.8 Computing the Weil Pairing;405
15.9;XI.9 The Tate–Lichtenbaum Pairing;409
15.10;Exercises;415
16;A Elliptic Curves in Characteristics 2 and 3;421
16.1;Exercises;426
17;B Group Cohomology (H0 and H1);427
17.1;B.1 Cohomology of Finite Groups;427
17.2;B.2 Galois Cohomology;430
17.3;B.3 Nonabelian Cohomology;433
17.4;Exercises;434
18;C Further Topics: An Overview;436
18.1;C.11 Complex Multiplication;436
18.2;C.12 Modular Functions;440
18.3;C.13 Modular Curves;450
18.4;C.14 Tate Curves;454
18.5;C.15 N´eron Models and Tate’s Algorithm;457
18.6;C.16 L-Series;460
18.7;C.17 Duality Theory;464
18.8;C.18 Local Height Functions;465
18.9;C.19 The Image of Galois;466
18.10;C.20 Function Fields and Specialization Theorems;467
18.11;C.21 Variation of ap and the Sato–Tate Conjecture;469
19;Notes on Exercises;472
20;List of Notation;478
21;References;483
22;Index;498