## The Arithmetic of Elliptic Curves (Google eBook)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

1 | |

6 | |

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 |

473 | |

489 | |

### Common terms and phrases

ABC conjecture abelian afﬁne algebraic algorithm Alice assume Aut(E Chapter char(K choose coefﬁcients cohomology compute conjecture constant Corollary deﬁned Deﬁnition degree denote discriminant divisor E(Fq ECDLP element elliptic curve deﬁned End(E exact sequence example Exercise ﬁeld ﬁnd ﬁnite ﬁnite extension ﬁnite ﬁeld ﬁnite set ﬁrst ﬁxed formal group formula Frobenius Galois genus gives GK/K group law height function Hence homogeneous space homomorphism ideal implies inﬁnitely integer isogeny isomorphism j-invariant lattice Lemma Let E/K logarithm minimal Weierstrass equation modular modulo Mordell–Weil theorem morphism nonconstant nonsingular nonzero notation number ﬁeld pairing polynomial power series prime Proposition Prove quadratic rational map Remark result Riemann–Roch theorem ring satisﬁes satisfying says solutions Springer Science+Business Media subgroup sufﬁces supersingular Suppose Tate module Tl(E torsion unramiﬁed Weierstrass equation Weil pairing well-deﬁned φˆ