## Rational Points on Elliptic CurvesIn 1961 the second author deliv1lred a series of lectures at Haverford Col lege on the subject of "Rational Points on Cubic Curves. " These lectures, intended for junior and senior mathematics majors, were recorded, tran scribed, and printed in mimeograph form. Since that time they have been widely distributed as photocopies of ever decreasing legibility, and por tions have appeared in various textbooks (Husemoller [1], Chahal [1]), but they have never appeared in their entirety. In view of the recent inter est in the theory of elliptic curves for subjects ranging from cryptogra phy (Lenstra [1], Koblitz [2]) to physics (Luck-Moussa-Waldschmidt [1]), as well as the tremendous purely mathematical activity in this area, it seems a propitious time to publish an expanded version of those original notes suitable for presentation to an advanced undergraduate audience. We have attempted to maintain much of the informality of the orig inal Haverford lectures. Our main goal in doing this has been to write a textbook in a technically difficult field which is "readable" by the average undergraduate mathematics major. We hope we have succeeded in this goal. The most obvious drawback to such an approach is that we have not been entirely rigorous in all of our proofs. In particular, much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than to rigorously prove. |

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### Contents

V | 9 |

VI | 15 |

VII | 22 |

VIII | 28 |

IX | 32 |

X | 38 |

XI | 41 |

XII | 47 |

XXX | 147 |

XXXI | 152 |

XXXII | 157 |

XXXIII | 165 |

XXXIV | 168 |

XXXV | 171 |

XXXVI | 174 |

XXXVII | 177 |

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### Common terms and phrases

abelian group affine curve algebraic algorithm auxiliary polynomial Bezout's theorem C(FP calculate Chapter common completes the proof complex multiplication complex numbers compute conic cubic curve cubic equation curve given curves of degree cyclic group define denominator Diophantine Approximation elements elliptic curve endomorphism equation y2 example factor Fermat's finite number finite order Galois extension Galois group geometry gives group law group of order group of rational Hence homomorphism inequality integer coefficients integer coordinates integer points integer solutions intersection point isomorphism Lemma look matrix modulo Mordell's theorem Nagell-Lutz theorem non-singular cubic curve number theory one-to-one order three point at infinity points of finite points of order polynomial F(X projective curve projective plane Prove quadratic rational numbers rational points rational solutions real numbers satisfy Section solutions in integers subgroup Suppose tangent line third intersection values Weierstrass equation write zero

### Popular passages

Page vi - Mordell's theorem, which says that the group of rational points on an elliptic curve is finitely generated...