Duke Mathematical Journal

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

Noam D. Elkies, Everett W. Howe, Andrew Kresch, Bjorn Poonen, Joseph L. Wetherell, and Michael E. Zieve

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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 (/n)r for some r>cg/n.

Article information

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

First available in Project Euclid: 14 April 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G20: Curves over finite and local fields [See also 14H25]
Secondary: 14G05, 14G15: Finite ground fields


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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See also

  • First in series: A. Kresch, J. Wetherell, M.E. Zieve. Curves of every genus with many points, 1: Abelian and toric families. J Algebra 250 (2002), 353-370.