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1 April 2004 Curves of every genus with many points, II: Asymptotically good families
Noam D. Elkies, Everett W. Howe, Andrew Kresch, Bjorn Poonen, Joseph L. Wetherell, Michael E. Zieve
Duke Math. J. 122(2): 399-422 (1 April 2004). DOI: 10.1215/S0012-7094-04-12224-9


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.


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Noam D. Elkies. Everett W. Howe. Andrew Kresch. Bjorn Poonen. Joseph L. Wetherell. Michael E. Zieve. "Curves of every genus with many points, II: Asymptotically good families." Duke Math. J. 122 (2) 399 - 422, 1 April 2004.


Published: 1 April 2004
First available in Project Euclid: 14 April 2004

zbMATH: 1072.11041
MathSciNet: MR2053756
Digital Object Identifier: 10.1215/S0012-7094-04-12224-9

Primary: 11G20
Secondary: 14G05, , 14G15

Rights: Copyright © 2004 Duke University Press


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Vol.122 • No. 2 • 1 April 2004
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