## LMSST: 24 Lectures on Elliptic CurvesThe study of special cases of elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centers of research in number theory. This book, addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Wei finite basis theorem, points of finite order (Nagell-Lutz), etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the "Riemann hypothesis for function fields") and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch. Many examples and exercises are included for the reader, and those new to elliptic curves, whether they are graduate students or specialists from other fields, will find this a valuable introduction. |

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### Contents

Introduction | 1 |

Curves of genus 0 Introduction | 3 |

padic numbers | 6 |

The localglobal principle for conies | 13 |

Geometry of numbers | 17 |

Localglobal principle Conclusion of proof | 20 |

Cubic curves | 23 |

Nonsingular cubics The group law | 27 |

Remedial mathematics Resultants | 75 |

Heights Finite Basis Theorem | 78 |

Localglobal for genus 1 | 85 |

Elements of Galois cohomology | 89 |

Construction of the jacobian | 92 |

Some abstract nonsense | 98 |

Principal homogeneous spaces and Galois cohomology | 104 |

The TateShafarevich group | 108 |

Elliptic curves Canonical Form | 32 |

Degenerate laws | 39 |

Reduction | 42 |

The padic case | 46 |

Global torsion | 50 |

Finite Basis Theorem Strategy and comments | 54 |

A 2isogeny | 58 |

The weak finite basis theory | 66 |

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### Common terms and phrases

algebraic closure algebraic number theory algorithm automorphism AXZ2 birational equivalence birationally equivalent canonical form clearly coboundary cocycle coefficients commutative conic consider construct Corollary cubic curve curve Y2 curves of genus deduce defined over Q denote Diophantine divisor elliptic curve elliptic curve defined equation equivalent over Q everwhere locally example Exercises fields Qp Finite Basis Theorem finite fields follows formula function field fundamental sequence further Gal(Q/Q Galois cohomology geometry give given ground field group law Hence Hensel's Lemma homogeneous co-ordinates homomorphism integer isogeny isomorphic jacobian kernel Let F(X local-global principle Mordell-Weil multiple neutral element nonsingular cubic Note number of points point defined point of order precisely prime factor principal homogeneous space prove quadratic form rational point rational root reduced Show simple poles simple zeros singular point Sketch proof solution square free subgroup third intersection transformation trivial