Improved upper bounds for the number of points on curves over finite fields
Everett W. Howe, Kristin E. Lauter
备注：
LaTex, 40 pages. There was a mistake in Section 7 that invalidated the proofs of two of our results. We correct the error in Section 7, and add an appendix with new proofs of the two results
发表信息：
Ann. Inst. Fourier (Grenoble) 53, fasc. 6 (2003), 1677-1737; Corrigendum, Ann. Inst. Fourier (Grenoble) 57 (2007) 1019-1021
We give new arguments that improve the known upper bounds on the maximal number N_q(g) of rational points of a curve of genus g over a finite field F_q for a number of pairs (q,g). Given a pair (q,g) and an integer N, we determine the possible zeta functions of genus-g curves over F_q with N points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-g curve over F_q with N points must have a low-degree map to another curve over F_q, and often this is enough to give us a contradiction. In particular, we able to provide eight previously unknown values of N_q(g), namely: N_4(5) = 17, N_4(10) = 27, N_8(9) = 45, N_{16}(4) = 45, N_{128}(4) = 215, N_3(6) = 14, N_9(10) = 54, and N_{27}(4) = 64. Our arguments also allow us to give a non-computer-intensive proof of the recent result of Savitt that there are no genus-4 curves over F_8 having exactly 27 rational points. Furthermore, we show that there is an infinite sequence of q's such that for every g with 0 < g < log_2 q, the difference between the Weil-Serre bound on N_q(g) and the actual value of N_q(g) is at least g/2.
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Yangzhou University（扬州大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；University of Bristol（布里斯托大学）；University of Science and Technology of China（中国科学技术大学）
