Mordell-Weil groups and the Galois module structure of rings of integers.

*(English)*Zbl 0631.14033This is a foundational article for the study of the rings of integers of fields obtained by the division of points on abelian varieties with complex multiplication. It is shown that the arising Galois module structures possess a natural group law, and that they are self-dual, with bounded order in the relevant Grothendieck group. Moreover there is an exciting, new conjectural link made between the L-function of the abelian variety and the triviality of the Galois module structure of such rings of integers.

Particular consideration is then given to the case of elliptic curves: firstly it is shown that, under certain conditions, the above mentioned conjecture is satisfied; secondly it is shown that, in a qualitative sense, trivial Galois module structure is relatively rare.

Particular consideration is then given to the case of elliptic curves: firstly it is shown that, under certain conditions, the above mentioned conjecture is satisfied; secondly it is shown that, in a qualitative sense, trivial Galois module structure is relatively rare.

##### MSC:

14K22 | Complex multiplication and abelian varieties |

14G05 | Rational points |

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |

14H45 | Special algebraic curves and curves of low genus |

14H52 | Elliptic curves |

11R32 | Galois theory |