## Elliptic Curves: A Computational Approach (Google eBook)The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory. The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available. |

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### Contents

xi | |

Chapter 2 Elliptic curves over the complex numbers | 33 |

Chapter 3 Elliptic curves over finite fields | 63 |

Chapter 4 Elliptic curves over local fields | 87 |

Chapter 5 The MordellWeil theorem and heights | 103 |

Chapter 6 Torsion group | 147 |

Chapter 7 The rank | 198 |

Chapter 8 Basis | 242 |

Chapter 9 Sintegral points | 263 |

Appendix A Algorithmic theory of diophantine equations | 294 |

Appendix B Multiquadratic number fields | 316 |

Backmatter | 351 |

### Common terms and phrases

abelian group algebraic number algorithm biquadratic Birch and Swinnerton-Dyer canonical height Chapter char(K coefficients complex multiplication complex numbers compute conjecture consider cubic curve in long curves over number curves over Q define deﬁned Deﬁnition denote diophantine equation discriminant divisor E(K)tors elliptic curve elliptic logarithms End(E endomorphism endomorphism ring estimate exists ﬁeld ﬁnite finite fields ﬁrst follows formula Galois group Heegner point height functions Hence integral basis isogeny isomorphism j-invariant L-series lattice Lemma Let E|K long Weierstraß normal Math method minimal modulo Mordell curve Mordell–Weil Mordell–Weil theorem multiplicative reduction norm equations number field Number Theory obtain ord(x ordp p-adic Peth˝o point of order points on elliptic points P1 polynomial prime ideal prime number Proof Proposition proved quadratic field quartic rank residue degree root S-integral points Section Siksek Silverman 204 solutions Theorem torsion group torsion point upper bound Weierstraß normal form Zimmer