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Curves of every genus with many points. I: Abelian and toric families


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

Abstract

Let Nq(g) denote the maximal number of Fq-rational points on any curve of genus g over Fq. Ihara (for square q) and Serre (for general q) proved that lim supg→∞ Nq(g)/g > 0 for any fixed q. Here we prove limg→∞ Nq(g) = ∞. More precisely, we use abelian covers of ℙ1 to prove lim infg→∞ Nq(g)/(g/log g) > 0, and we use curves on toric surfaces to prove lim infg→∞ Nq(g)/g1/3 > 0; we also show that these results are the best possible that can be proved using these families of curves. © 2002 Elsevier Science (USA).

Let Nq(g) denote the maximal number of Fq-rational points on any curve of genus g over Fq. Ihara (for square q) and Serre (for general q) proved that lim supg→∞ Nq(g)/g > 0 for any fixed q. Here we prove limg→∞ Nq(g) = ∞. More precisely, we use abelian covers of ℙ1 to prove lim infg→∞ Nq(g)/(g/log g) > 0, and we use curves on toric surfaces to prove lim infg→∞ Nq(g)/g1/3 > 0; we also show that these results are the best possible that can be proved using these families of curves. © 2002 Elsevier Science (USA).

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:abelian covers; toric surfaces
Language:English
Date:2002
Deposited On:29 Nov 2010 16:27
Last Modified:05 Apr 2016 13:25
Publisher:Elsevier
ISSN:0021-8693
Publisher DOI:10.1006/jabr.2001.9081
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1898389
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1062.14027
Permanent URL: http://doi.org/10.5167/uzh-21959

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