On the number of points of a plane curve over a finite field
Masaaki Homma
Department of Mathematics, Kanagawa University Yokohama 221-8686, Japan
homma@n.kanagawa-u.ac.jp Seon Jeong Kim
Department of Mathematics and RINS Gyeongsang National University
Jinju 660-701, Korea skim@gnu.kr
In the paper [7], Sziklai posed a conjecture on the number of points of a plane curve over a finite field. LetC be a plane curve of degreedoverFq
without an Fq-linear component. Then he conjectured that the number of Fq-pointsNq(C) ofC would be at most (d−1)q+ 1. But he had overlooked the known example of a curve of degree 4 over F4 with 14 points ([6], [1]).
So we must modify this conjecture.
Modified Sziklai’s Conjecture UnlessCis a curve overF4which is pro- jectively equivalent to
X4+Y4+Z4+X2Y2+Y2Z2+Z2X2+X2Y Z+XY2Z+XY Z2= 0 (1) overF4, we might have
Nq(C)≤(d−1)q+ 1. (2)
This conjecture makes sense only if 2≤d≤q+1 because the conjectural bound exceeds the obvious boundNq(C)≤#P2(Fq) =q2+q+ 1 ifd≥q+ 2.
In [3], we proved the inequality
Nq(C)≤(d−1)q+ (q+ 2−d),
which guarantees the truth of the conjecture for d= q+ 1, and presented an example of a curve of degree q+ 1 having q2 + 1 Fq-points. Moreover, we observed that if a curve of degree 4 overF4 has more than 13F4-points, then this curve is projectively equivalent to the curve (1) overF4.
Recently we have found two facts concerning this conjecture. The first one is that the inequality (2) holds ifd=q >4, and for eachq, there exists
1
a nonsingulsr curve of degree q over Fq with (q−1)q+ 1 rational points.
Note that the truth of the inequality (2) for d=q = 3 is classical [4], and it is well known ford=q= 2. The second one is that (2) holds if the curve C is nonsingular of degreed≤q−1, which is an easy cosequence of results of St¨ohr - Voloch [5] and Hefez - Voloch [2]. Therefore, together with our previous results, the following theorem has been established.
Theorem The modified Sziklai’s conjecture is true for nonsingular curves.
Moreover there is an example of a nonsingular curve for which equality holds in (2) for d=q+ 2, q+ 1, q, q−1,√
q+ 1(when q is square), and2.
References
[1] G. van der Geer and M. van der Vlugt, Tables of curves with many points,http://www.science.uva.nl/~geer/
[2] A. Hefez and J. F. Voloch,Frobenius nonclassical curves, Arch. Math.
(Basel) 54 (1990), 263–273; Correction, Arch. Math. (Basel) 57 (1991), 416.
[3] M. Homma and S. J. Kim, Around Sziklai’s conjecture on the number of points of a plane curve over a finite field, to appear in Finite Fields and Their Applications.
[4] B. Segre, Le geometrie di Galois, Ann. Mat. Pura Appl. (4) 48 (1959) 1–96.
[5] K.-O. St¨ohr and J. F. Voloch,Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3) 52 (1986), 1–19.
[6] J. P. Serre, Nombres de points des courbes alg´ebriques sur Fq , Sem.
de Th´eorie des Nombres de Bordeaux 1982–1983, exp. 22; Oeuvres III, No. 129, 664–668.
[7] P. Sziklai, A bound on the number of points of a plane curve, Finite Fields Appl. 14 (2008) 41–43.
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