Many rational points. Coding theory and algebraic geometry.

*(English)*Zbl 1072.11042
Mathematics and its Applications (Dordrecht) 564. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1766-9/hbk). xxii, 346 p. (2003).

This book gives a nice overview of background and recent results on curves over finite fields. The theorem of Hasse-Weil gives bounds on the possible number of points on a curve over a finite field depending on the genus. For applications in coding theory one is interested in curves with many points (hence the name of the book) as that rules the size of the code.

Hurt collects results on the mathematical background as well as on the applications in coding theory and cryptography. The book also comprises results on Deligne-Lusztig spaces, Drinfeld modules, and Shimura curves.

The main advantage of this book is that it provides a huge bibliography and takes into account even very recent results which are so far only presented at conferences or in preprints. So it serves well to get an update on recent results for the experienced reader and links to the original results for more details.

Obviously, there is not enough space for providing detailed background or even proofs – however, sometimes it would be advantageous for the reader if the relevance of the distinct results would have been stressed. This might be asking too much in the case of new results for which the relevance is not yet established or is hard to estimate – but the reader should be warned that this task remains on him.

Hurt collects results on the mathematical background as well as on the applications in coding theory and cryptography. The book also comprises results on Deligne-Lusztig spaces, Drinfeld modules, and Shimura curves.

The main advantage of this book is that it provides a huge bibliography and takes into account even very recent results which are so far only presented at conferences or in preprints. So it serves well to get an update on recent results for the experienced reader and links to the original results for more details.

Obviously, there is not enough space for providing detailed background or even proofs – however, sometimes it would be advantageous for the reader if the relevance of the distinct results would have been stressed. This might be asking too much in the case of new results for which the relevance is not yet established or is hard to estimate – but the reader should be warned that this task remains on him.

Reviewer: Tanja Lange (Lyngby)

##### MSC:

11G20 | Curves over finite and local fields |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

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

14G05 | Rational points |

14G15 | Finite ground fields in algebraic geometry |

11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |

14G50 | Applications to coding theory and cryptography of arithmetic geometry |

94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |

94A60 | Cryptography |