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arXiv math/9912069math.AGmath.NT
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每种多点曲线I：阿贝尔和复曲面族

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

Andrew Kresch, Joseph L. Wetherell, Michael E. Zieve

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LaTeX, 20 pages
发表信息：
J. Algebra 250, no. 1 (2002), 353--370
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摘要：

设N_q（g）表示有限域F_q上亏格为g的任意曲线上F_q-有理点的最大个数。Ihara（对平方q）和Serre（对一般q）证明了对任意固定q，limsup_{g-->infinity} N_q（g）/g > 0.在他们的证明中，他们用无穷多个属中的许多点构造曲线;然而，他们的属序列有些稀疏。本文证明了lim_{g-->infinity} N_q（g）=无穷大.更准确地说，我们使用P^1的阿贝尔覆盖证明了liminf_{g-->infinity} N_q（g）/（g/log g）> 0，我们使用复曲面上的曲线证明了liminf_{g-->infty} N_q（g）/g^{1/3} > 0;我们还证明了这些结果是可以用这些曲线族证明的最佳可能。

英文摘要：

Let N_q(g) denote the maximal number of F_q-rational points on any curve of genus g over the finite field F_q. Ihara (for square q) and Serre (for general q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their proofs they constructed curves with many points in infinitely many genera; however, their sequences of genera are somewhat sparse. In this paper, we prove that lim_{g-->infinity} N_q(g) = infinity. More precisely, we use abelian covers of P^1 to prove that liminf_{g-->infinity} N_q(g)/(g/log g) > 0, and we use curves on toric surfaces to prove that liminf_{g-->infty} N_q(g)/g^{1/3} > 0; we also show that these results are the best possible that can be proved with these families of curves.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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