## Elliptic Curves: Diophantine AnalysisIt is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points. |

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### Contents

II | 3 |

III | 6 |

IV | 10 |

V | 13 |

VI | 17 |

VII | 19 |

VIII | 23 |

IX | 26 |

XLII | 148 |

XLIII | 151 |

XLIV | 155 |

XLV | 159 |

XLVII | 162 |

XLVIII | 164 |

XLIX | 166 |

L | 169 |

X | 33 |

XII | 37 |

XIII | 43 |

XIV | 47 |

XV | 48 |

XVI | 54 |

XVII | 55 |

XVIII | 62 |

XIX | 68 |

XX | 73 |

XXI | 77 |

XXIII | 84 |

XXIV | 85 |

XXV | 88 |

XXVI | 93 |

XXVII | 98 |

XXVIII | 101 |

XXX | 105 |

XXXI | 107 |

XXXII | 109 |

XXXIII | 115 |

XXXIV | 128 |

XXXV | 129 |

XXXVI | 137 |

XXXVII | 140 |

XXXVIII | 142 |

XXXIX | 144 |

XL | 146 |

XLI | 147 |

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abelian varieties absolute value algebraic integers algebraic number analytic apply assume Baker Chapter VII coefficients complex multiplication complex numbers concludes the proof conjecture constant depending coordinates degree denominator denote derivatives diophantine approximation divisible elements elliptic curve elliptic curve defined elliptic function equation v2 estimate exists factor finite extension finite number follows formula Galois group give given height Hence homomorphism induction inequality infinity integral points isomorphism Kummer theory lattice linear combinations linear equations linearly independent log max lower bound main lemma Math Neron Neron function Neron-Tate notation number field number of equations obtained p-adic parallelogram parameter positive integer power series prime number quadratic form quotient radius rational functions rational numbers reader ring roots of unity satisfying shows Siegel solutions subgroup Suppose system of linear Tate curve Theorem 1.2 torsion points units Weierstrass equation Weierstrass form Weierstrass function whence write yields zeros