## The Arithmetic of Elliptic CurvesThe 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. |

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### Contents

1 | |

17 | |

III The Geometry of Elliptic Curves | 41 |

IV The Formal Group of an Elliptic Curve | 115 |

V Elliptic Curves over Finite Fields | 137 |

VI Elliptic Curves over C | 157 |

VII Elliptic Curves over Local Fields | 184 |

VIII Elliptic Curves over Global Fields | 207 |

XI Algorithmic Aspects of Elliptic Curves | 362 |

A Elliptic Curves in Characteristics 2 and 3 | 409 |

B Group Cohomology H0 and H1 | 415 |

An Overview | 425 |

Notes on Exercises | 458 |

List of Notation | 467 |

472 | |

489 | |

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### Common terms and phrases

a e GR/k ABC conjecture abelian absolute value algebraic algorithm Alice assume Aut(E bad reduction Chapter char(K choose coefficients cohomology compute conjecture constant Corollary Definition degree denote discriminant div(f divisor E(Fq ECDLP element elliptic curve defined elliptic curve E/K End(E Eſm exact sequence example Exercise finite extension finite field formal group formula Frobenius Galois genus gives group law height function Hence homogeneous space homomorphism ideal implies integer invariant differential 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 satisfying says solutions Springer Science+Business Media subgroup supersingular Suppose Tate module torsion unramified Weierstrass equation Weil pairing