## Duke Mathematical Journal

### Curves of every genus with many points, II: Asymptotically good families

#### 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 $\mathbb{F}_q$ with at least cqg rational points over $\mathbb{F}_q$. 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 $\mathbb{F}_q$ that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to $(\mathbb{Z}/n\mathbb{Z})^r$ for some r>cg/n.

#### Article information

Source
Duke Math. J., Volume 122, Number 2 (2004), 399-422.

Dates
First available in Project Euclid: 14 April 2004

https://projecteuclid.org/euclid.dmj/1081971771

Digital Object Identifier
doi:10.1215/S0012-7094-04-12224-9

Mathematical Reviews number (MathSciNet)
MR2053756

Zentralblatt MATH identifier
1072.11041

Subjects
Secondary: 14G05, 14G15: Finite ground fields

#### Citation

Elkies, Noam D.; Howe, Everett W.; Kresch, Andrew; Poonen, Bjorn; Wetherell, Joseph L.; Zieve, Michael E. Curves of every genus with many points, II: Asymptotically good families. Duke Math. J. 122 (2004), no. 2, 399--422. doi:10.1215/S0012-7094-04-12224-9. https://projecteuclid.org/euclid.dmj/1081971771

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