## Elliptic curves"This book is an introduction to the theory of elliptic curves, ranging from its most elementary aspects to current research. The first part, which grew out of Tate's Haverford lectures, covers the elementary arithmetic theory of elliptic curves over the r" |

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### Contents

Introduction to Rational Points on Plane Curves | 1 |

3 Pythagoras Diophantus and Fermat | 7 |

5 The Group Law on Cubic Curves and Elliptic Curves | 14 |

Copyright | |

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### Common terms and phrases

abelian group algebraic curve algebraically closed assertion automorphism bad reduction calculate change of variable Chapter characteristic coefficients cohomology complex multiplication complex numbers conic conjecture consider cubic curve cubic equation curve of degree Definition denote discriminant divides divisors elements elliptic curve elliptic curve defined elliptic function equation in normal equation y2 equivalent Euler product example extension factor Fermat field of fractions form y2 formula Galois Galois extension given group homomorphism group law Hence homogeneous integer intersection points irreducible isogeny isomorphic L-function minimal modular modulo morphism nonzero normal form notations number field ordp ordp(x p-adic plane curve point of order polynomial prime projective space Proof proves the proposition proves the theorem Pythagorean triple Q)tors quadratic quotient rational numbers rational points relation Remark roots satisfies singular point solution square subgroup supersingular tangent line torsion points valuation Weierstrass equation zero zeta function