The Arithmetic of Elliptic Curves (Google eBook)

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Springer Science & Business Media, Apr 20, 2009 - Mathematics - 534 pages
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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 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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Contents

Algebraic Varieties
1
2 Projective Varieties
6
3 Maps Between Varieties
11
Exercises
14
Algebraic Curves
17
2 Maps Between Curves
19
3 Divisors
27
4 Differentials
30
7 Torsion Points
240
8 The Minimal Discriminant
243
9 The Canonical Height
247
10 The Rank of an Elliptic Curve
254
11 Szpiros Conjecture and ABC
255
Exercises
261
Integral Points on Elliptic Curves
269
1 Diophantine Approximation
270

5 The RiemannRoch Theorem
33
Exercises
37
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
The Formal Group of an Elliptic Curve
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
Elliptic Curves over Finite Fields
136
2 The Weil Conjectures
140
3 The Endomorphism Ring
144
4 Calculating the Hasse Invariant
148
Exercises
153
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
Elliptic Curves over Local Fields
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 EE
199
7 The Criterion of NeronOggShafarevich
201
Exercises
203
Elliptic Curves over Global Fields
206
1 The Weak MordellWeil Theorem
208
2 The Kummer Pairing via Cohomology
215
3 The Descent Procedure
218
4 The MordellWeil Theorem over Q
220
5 Heights on Projective Space
224
6 Heights on Elliptic Curves
234
2 Distance Functions
273
3 Siegels Theorem
276
4 The SUnit Equation
281
5 Effective Methods
286
6 Shafarevichs Theorem
293
7 The Curve Y² X³ + D
296
8 Roths TheoremAn Overview
299
Exercises
302
Computing the MordellWeil Group
308
1 An Example
310
2 TwistingGeneral Theory
318
3 Homogeneous Spaces
321
4 The Selmer and ShafarevichTate Groups
331
5 TwistingElliptic Curves
341
6 The Curve Y² X³ + DX
344
Exercises
355
Algorithmic Aspects of Elliptic Curves
363
1 DoubleandAdd Algorithms
364
2 Lenstras Elliptic Curve Factorization Algorithm
366
3 Counting the Number of Points in EFq
372
4 Elliptic Curve Cryptography
376
The General Case
381
Special Cases
386
7 PairingBased Cryptography
390
8 Computing the Weil Pairing
393
9 The TateLichtenbaum Pairing
397
Exercises
403
Elliptic Curves in Characteristics 2 and 3
409
Exercises
414
Group Cohomology H⁰ and H¹
415
2 Galois Cohomology
418
3 Nonabelian Cohomology
421
Exercises
422
Further Topics An Overview
424
12 Modular Functions
429
13 Modular Curves
439
14 Tate Curves
443
15 Neron Models and Tates Algorithm
446
16 LSeries
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 SatoTate Conjecture
458
Notes on Exercises
461
List of Notation
467
References
473
Index
489
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About the author (2009)

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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