We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant cq with the following property: for every integer g≥0, there is a genus-g curve over Fq with at least cqg rational points over Fq. Moreover, we show that there exists a positive constant d such that for every q we can choose cq=d log q. We show also that there is a constant c>0 such that for every q and every n>0, and for every sufficiently large g there is a genus-g curve over Fq that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)r for some r>cg/n.

Elkies, N; Howe, E; Kresch, A; Poonen, B; Wetherell, J; Zieve, M (2004). *Curves of every genus with many points. II. Asymptotically good families.* Duke Mathematical Journal, 122(2):399-422.

## Abstract

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant cq with the following property: for every integer g≥0, there is a genus-g curve over Fq with at least cqg rational points over Fq. Moreover, we show that there exists a positive constant d such that for every q we can choose cq=d log q. We show also that there is a constant c>0 such that for every q and every n>0, and for every sufficiently large g there is a genus-g curve over Fq that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)r for some r>cg/n.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | curves over finite fields with many rational points; asymptotic lower bounds; class field towers; degree-2 covering of curves |

Language: | English |

Date: | 2004 |

Deposited On: | 29 Nov 2010 16:26 |

Last Modified: | 05 Apr 2016 13:24 |

Publisher: | Duke University Press |

ISSN: | 0012-7094 |

Additional Information: | 2004 © Duke University Press |

Publisher DOI: | https://doi.org/10.1215/S0012-7094-04-12224-9 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=2053756 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1072.11041 |

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