##
**Abelian \(\ell\)-adic representations and elliptic curves.**
*(English)*
Zbl 0902.14016

Research Notes in Mathematics. 7. Wellesley, MA: A K Peters. xviii, 180 p. (1998).

This classic monograph originally grew out of a series of lectures delivered by J.-P. Serre at McGill University (Montreal) in 1967 (see the first edition 1968; Zbl 0186.25701). Ever since, over the past thirty years, it provided a standard text on \(\ell\)-adic representation theory and elliptic curves in algebraic number theory and arithmetic geometry. Due to its fundamental depth and literary brilliance, which are of timelessly exemplary splendour, and in regard of the spectacular recent developments concerning the Taniyama-Weil conjecture and Fermat’s Last Theorem, which are closely related to the topics treated in this classic standard text, Serre’s book has maintained both mathematical actuality and outstanding significance in the existing literature.

In the first reprint of his treatise, which appeared in 1989 (Zbl 0709.14002), the author had enhanced the (otherwise unchanged) text by a few short remarks referring to new results obtained between 1967 and 1988, as well as by a supplementary bibliography listing up a selection of the most important works that appeared during this period. The present book under review is an accurate reproduction of that reprint from 1989.

In the first reprint of his treatise, which appeared in 1989 (Zbl 0709.14002), the author had enhanced the (otherwise unchanged) text by a few short remarks referring to new results obtained between 1967 and 1988, as well as by a supplementary bibliography listing up a selection of the most important works that appeared during this period. The present book under review is an accurate reproduction of that reprint from 1989.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14G25 | Global ground fields in algebraic geometry |

14H52 | Elliptic curves |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

22E05 | Local Lie groups |

11G05 | Elliptic curves over global fields |