Curves of Every Genus with Many Points, II: Asymptotically Good Families

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Curves of Every Genus with Many Points, II: Asymptotically Good Families

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Title: Curves of Every Genus with Many Points, II: Asymptotically Good Families
Author: Zieve, Michael E.; Wetherell, Joseph L.; Poonen, Bjorn; Kresch, Andrew; Howe, Everett W.; Elkies, Noam David

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Citation: Zieve, Michael E., Joseph L. Wetherell, Bjorn Poonen, Andrew Kresch, Everett W. Howe, and Noam D. Elkies. 2004. “Curves of Every Genus with Many Points, II: Asymptotically Good Families.” Duke Mathematical Journal 122 (2) (April): 399–422.
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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 c_q with the following property: for every non-negative integer g, there is a genus-g curve over F_q with at least c_q * g rational points over F_q. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q = 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 F_q that has at least c*g/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)^r for some r > c*g/n.
Published Version: http://dx.doi.org/10.1215/S0012-7094-04-12224-9
Other Sources: http://arxiv.org/abs/math/0208060
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:12211469
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