## Improved upper bounds for the number of points on curves over finite fields.(English)Zbl 1065.11043

Ann. Inst. Fourier 53, No. 6, 1677-1737 (2003); corrigendum ibid. 57, No. 3, 1019-1021 (2007).
Summary: 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 $${\mathbb F}_q$$, 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 $${\mathbb F}_q$$ 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 $${\mathbb F}_q$$ with $$N$$ points must have a low-degree map to another curve over $${\mathbb F}_q$$, and often this is enough to give us a contradiction. In particular, we are 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 $${\mathbb F}_8$$ 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 $$g/2$$.

### MSC:

 11G20 Curves over finite and local fields 14G05 Rational points 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G15 Finite ground fields in algebraic geometry

Magma
Full Text:

### References:

 [1] The magma algebra system I: the user language, J. Symbolic Comput., 24, 235-265, (1997) · Zbl 0898.68039 [2] The p-rank of curves and covers of curves, Courbes semi-stables et groupe fondamental en géométrie algébrique, 187, 267-277, (2000), Birkhäuser, Basel · Zbl 0979.14015 [3] Variétés abéliennes ordinaires sur un corps fini, Invent. Math., 8, 238-243, (1969) · Zbl 0179.26201 [4] Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory, 73, 426-450, (1998) · Zbl 0931.11023 [5] Corrigendum: real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory, 83, 1, 182 pp., (2000) · Zbl 0931.11023 [6] The genus of curves over finite fields with many rational points, Manuscripta Math, 89, 103-106, (1996) · Zbl 0857.11032 [7] Tables of curves with many points, Math. Comp., 69, 797-810, (2000) · Zbl 0965.11028 [8] Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc., 347, 2361-2401, (1995) · Zbl 0859.14016 [9] On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field, J. Number Theory, 92, 139-163, (2002) · Zbl 0998.11031 [10] On the genus of a maximal curve, Math. Ann., 323, 589-608, (2002) · Zbl 1018.11029 [11] Euclid’s algorithm in complex quartic fields, Acta Arith., 20, 393-400, (1972) · Zbl 0224.12001 [12] Improved upper bounds for the number of rational points on algebraic curves over finite fields, C. R. Acad. Sci. Paris, Sér. I Math., 328, 1181-1185, (1999) · Zbl 0948.11024 [13] Non-existence of a curve over $$\smallF_3$$ of genus 5 with 14 rational points, Proc. Amer. Math. Soc, 128, 369-374, (2000) · Zbl 0983.11036 [14] Zeta functions of curves over finite fields with many rational points, Coding Theory, Cryptography and Related Areas, 167-174, (2000), Springer-Verlag, Berlin · Zbl 1009.11049 [15] Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, J. Algebraic Geom., 10, 19-36, (2001) · Zbl 0982.14015 [16] The maximum or minimum number of rational points on genus three curves over finite fields, Compositio Math., 134, 87-111, (2002) · Zbl 1031.11038 [17] Abelian Varieties, 5, (1985), Oxford University Press, Oxford · Zbl 0583.14015 [18] Commutative group schemes, 15, (1966), Springer-Verlag, Berlin · Zbl 0216.05603 [19] The maximum number of rational points on a curve of genus 4 over $$\smallF_8$$ is 25, Canad. J. Math., 55, 331-352, (2003) · Zbl 1101.14026 [20] Sur le nombre des points rationnels d’une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris, Sér. I Math., 296, 397-402, (1983) · Zbl 0538.14015 [21] Nombres de points des courbes algébriques sur $$\smallF_q,$$ Sém. Théor. Nombres Bordeaux 1982/83, Exp. No. 22 · Zbl 0538.14016 [22] Résumé des cours de 1983–1984, Ann. Collège France, 79-83, (1984) [23] Rational points on curves over finite fields, (1985) [24] The trace of totally positive and real algebraic integers, Ann. of Math (2), 46, 302-312, (1945) · Zbl 0063.07009 [25] Totally positive algebraic integers of small trace, Ann. Inst. Fourier (Grenoble), 33, 3, 1-28, (1984) · Zbl 0534.12002 [26] On the Riemann hypothesis in hyperelliptic function fields, Analytic number theory, 24, 285-302, (1973), American Mathematical Society, Providence, R.I. · Zbl 0271.14012 [27] Algebraic Function Fields and Codes, (1993), Springer-Verlag, Berlin · Zbl 0816.14011 [28] Weierstrass points and curves over finite fields, Proc. London Math. Soc (3), 52, 1-19, (1986) · Zbl 0593.14020 [29] The p-rank of Artin-Schreier curves, Manuscripta Math., 16, 169-193, (1975) · Zbl 0321.14017 [30] Classes d’isogénie des variétés abéliennes sur un corps fini, Séminaire Bourbaki 1968/69, 179, 95-110, (1971), Springer-Verlag, Berlin · Zbl 0212.25702 [31] Improving the Oesterlé bound
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.