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

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


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 limsup g→ ∞ Nq(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 limg→ ∞ Nq(g) = ∞. More precisely, we use abelian covers of P 1 to prove that liminfg→ ∞ Nq(g)/(g / logg)> 0, and we use curves on toric surfaces to prove that liminfg→ ∞ Nq(g)/g 1/3> 0; we also show that these results are the best possible that can be proved with these families of curves

Year: 1999
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