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

Andrew Kresch·Joseph L. Wetherell·Michael E. Zieve
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Abstract

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.

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Andrew Kresch, Joseph L. Wetherell, Michael E. Zieve. 2001-08-29. Curves of every genus with many points, I: Abelian and toric families. https://arxiv.org/abs/math/9912069

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