*(English)*Zbl 0752.14034

The book gives a good introduction for students which are interested in Diophantine equations and arithmetic geometry. It is based on lectures of J. Tate from 1961. It contains a lot of exercises. Often further developments and applications are explained, for instance Lenstra’s algorithm for factorisation of integers using elliptic curves.

The book starts with the geometry and group structure of elliptic curves. It contains the Nagell-Lutz theorem describing points if finite order, the Mordell-Weil theorem on the finite generation of the group of rational points, the Thue-Siegel theorem on the finiteness of the set of integer points. Also points over finite fields are considered. — At the end one finds complex multiplication and Galois representations associated to torsion points.

The algebraic geometry needed for the purpose of the book (for instance Bézout’s theorem) is presented in an appendix.

##### MSC:

11G05 | Elliptic curves over global fields |

11-01 | Textbooks (number theory) |

14-01 | Textbooks (algebraic geometry) |

11G15 | Complex multiplication and moduli of abelian varieties |

11D25 | Cubic and quartic diophantine equations |

14G05 | Rational points |

14G15 | Finite ground fields |

14H52 | Elliptic curves |