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Distributions of traces of Frobenius for smooth plane curves over finite fields. (English) Zbl 1431.14047
Let $$k$$ be a finite field of order $$q$$. For $$C$$ a projective, non-singular, geometrically irreducible algebraic curve over $$k$$ of genus $$g$$, a basic problem is to compute $$N:=\# C(k)$$, the number of $$k$$-rational points of $$C$$. We have that $$N=q+1-t$$, where $$t$$ is the trace of the Frobenius map acting on the first étale cohomology group $$H^1(J_{\text{ét}},{\mathbb Z}_\ell)$$ of the Jacobian $$J$$ of $$C$$ being $$\ell$$ a prime different of the characteristic of $$k$$; moreover, $$|t|\leq g\lfloor 2\sqrt{q}\rfloor$$ (the Hasse-Weil-Serre bound). One is often interesting in computing $$N_q(g)$$, the maximal possible number of $$k$$-rational points on curves of genus $$g$$.
J.-P. Serre [Sémin. Théor. Nombres, Univ. Bordeaux I 1982–1983, Exp. No. 22, 8 p. (1983; Zbl 0538.14016)] computed $$N_q(1)$$ and $$N_q(2)$$, and consider the possibility to obtain a similar result for $$N_q(3)$$. He noticed that this task is more involved as here we have the so-called Serre obstruction, namely there are Jacobians of genus 3 curves over $$\bar k$$ which are not the Jacobian of curves over $$k$$; a reason for this obstruction is the existence of non-hyperelliptic curves of genus bigger than $$2$$. Although there have been many computational and conceptual approaches to Serre’s obstruction (see e.g. T. Ibukiyama [Tohoku Math. J. (2) 45, No. 3, 311–329 (1993; Zbl 0819.14007)], G. Lachaud and C. Ritzenthaler [Ser. Number Theory Appl. 5, 88–115 (2008; Zbl 1151.14321)]), there is no so far a closed formula for $$N_q(g)$$, $$g\geq 3$$.
In the paper under review, among other things, $$N_q(3)$$ is investigated via the possible values of the trace of Frobenius: for large negative value of $$t$$ it is shown that there are more curves with trace $$t$$ than with trace $$-t$$. Indeed, let $${\mathcal N}_{q,3}(t)$$ be the number of non-hyperelliptic genus $$3$$ curves over $$k$$ of trace $$t$$ up to $$k$$-isomorphism. Then the function $${\mathcal M}_{q,3}(t):={\mathcal N}_{q,3}(t)-{\mathcal N}_{q,3}(-t)$$ is considered for $$0\leq t\leq 6\sqrt{q}$$. It is shown that $${\mathcal M}_{q,3}(t)\leq 0$$ for $$t$$ large enough. For instance this is enough to show that there is a genus $$3$$-curve $$C$$ over $$k$$ such that $$\#C(k)\geq q+1+3\lfloor\sqrt{q}\rfloor-3$$ (cf. K. Lauter and J.-P. Serre [Compos. Math. 134, No. 1, 87–111 (2002; Zbl 1031.11038)]). See R. Lercier et al. [LMS J. Comput. Math. 17A, 128–147 (2014; Zbl 1333.14060)] for computations on prime fields of small order.
##### MSC:
 14Q05 Computational aspects of algebraic curves 14H10 Families, moduli of curves (algebraic) 14H25 Arithmetic ground fields for curves 14H37 Automorphisms of curves 14H45 Special algebraic curves and curves of low genus 14H50 Plane and space curves
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##### References:
 [1] Achter, [Achter et al. 15] J. D.; Erman, D.; Kedlaya, K. S.; Wood, M. Matchett; Zureick-Brown, D., A Heuristic for the Distribution of Point Counts for Random Curves Over a Finite Field., Phil. Trans. R. Soc. A, 373, 2040, (2015) · Zbl 1397.11108 [2] Beauville, [Beauville and Ritzenthaler 11] A.; Ritzenthaler, C., Jacobians Among Abelian Threefolds: A Geometric Approach., Math. Ann., 350, 4, 793-799, (2011) · Zbl 1228.14027 [3] Behrend, [Behrend 93] K. A., The Lefschetz Trace Formula for Algebraic Stacks., Invent. Math., 112, 1, 127-149, (1993) · Zbl 0792.14005 [4] Bucur, [Bucur et al. 10] A.; David, C.; Feigon, B.; Lalín, M., Fluctuations in the Number of Points on Smooth Plane Curves Over Finite Fields., J. Number Theory, 130, 11, 2528-2541, (2010) · Zbl 1210.11070 [5] Bucur, [Bucur et al. 10] A.; David, C.; Feigon, B.; Lalín, M., Statistics for Traces of Cyclic Trigonal Curves Over Finite Fields., Int. Math. Res. Not. IMRN, 2010, 5, 932-967, (2010) · Zbl 1201.11063 [6] Bucur, [Bucur et al. 11] A.; David, C.; Feigon, B.; Lalín, M., WIN—women in numbers, Biased statistics for traces of cyclic p-fold covers over finite fields., 121-143, (2011), Amer. Math. Soc.: Amer. Math. Soc., Providence, RI [7] Bucur, [Bucur et al. 12] A.; David, C.; Feigon, B.; Lalín, M.; Sinha, K., Distribution of Zeta Zeroes of Artin-Schreier Covers., Math. Res. Lett., 19, 6, 1329-1356, (2012) · Zbl 1348.11048 [8] Cheong, [Cheong et al. 15] G.; Wood, M. Matchett; Zaman, A., The Distribution of Points on Superelliptic Curves Over Finite Fields., Proc. Amer. Math. Soc., 143, 4, 1365-1375, (2015) · Zbl 1408.11063 [9] Entin, [Entin 12] A., On the Distribution of Zeroes of Artin-Schreier L-functions., Geom. Funct. Anal., 22, 5, 1322-1360, (2012) · Zbl 1321.11094 [10] Homma, [Homma and Kim 13] M.; Kim, S. J., Nonsingular Plane Filling Curves of Minimum Degree Over a Finite Field and their Automorphism Groups: Supplements to a Work of Tallini., Linear Algebra Appl., 438, 3, 969-985, (2013) · Zbl 1259.14023 [11] Homma, [Homma and Kim 15] M.; Kim, S. J., The second largest number of points of plane curves over finite fields, (2015) [12] Ibukiyama, [Ibukiyama 93] T., On Rational Points of Curves of Genus 3 Over Finite Fields., Tohoku Math. J. (2), 45, 3, 311-329, (1993) · Zbl 0819.14007 [13] Katz, [Katz and Sarnak 99] N. M.; Sarnak, P., Random matrices, Frobenius Eigenvalues, and Monodromy, (1999), Providence, RI: American Mathematical Society, Providence, RI [14] Kolassa, [Kolassa 97] J. E., Series Approximation Methods in Statistics, (1997), New York: Springer-Verlag, New York · Zbl 0877.62013 [15] Kurlberg, [Kurlberg and Rudnick 09] P.; Rudnick, Z., The Fluctuations in the Number of Points on a Hyperelliptic Curve Over a Finite Field., J. Number Theory, 129, 3, 580-587, (2009) · Zbl 1221.11141 [16] Kurlberg, [Kurlberg and Wigman 11] P.; Wigman, I., Gaussian Point Count Statistics for Families of Curves Over a Fixed Finite Field., Int. Math. Res. Not., 2011, 10, 2217, (2011) · Zbl 1297.11063 [17] Lachaud, [Lachaud and Ritzenthaler 08] G.; Ritzenthaler, C., Algebraic geometry and its applications, On Some Questions of Serre on Abelian Threefolds., 88-115, (2008), Hackensack, NJ: World Sci. Publ., Hackensack, NJ · Zbl 1151.14321 [18] Lachaud, [Lachaud et al. 10] G.; Ritzenthaler, C.; Zykin, A., Jacobians Among Abelian Threefolds: A Formula of Klein and A Question of Serre., Math. Res. Lett., 17, 2, 323-333, (2010) · Zbl 1228.14028 [19] Lauter, [Lauter 02] K., The Maximum or Minimum Number of Rational Points on Genus Three Curves Over Finite Fields., Compos. Math., 134, 1, 87-111, (2002) · Zbl 1031.11038 [20] Lercier, [Lercier 14] R.; Ritzenthaler, C.; Rovetta, F.; Sijsling, J., Parametrizing the Moduli Space of Curves and Applications to Smooth Plane Quartics Over Finite Fields., LMS J. Comput. Math., 17, Suppl. A, 128-147, (2014) · Zbl 1333.14060 [21] Wood, [Matchett Wood 12] M. Matchett, The Distribution of the Number of Points on Trigonal Curves Over ., Int. Math. Res. Not. IMRN, 2012, 23, 5444-5456, (2012) · Zbl 1276.11106 [22] Mestre, [Mestre 10] J.-F., Courbes de genre 3 avec comme groupe d’automorphismes, (2010) [23] Nart, [Nart and Ritzenthaler 08] E.; Ritzenthaler, C., Jacobians in Isogeny Classes of Supersingular Abelian Threefolds in Characteristic 2., Finite Fields Appl., 14, 676-702, (2008) · Zbl 1159.14023 [24] Nart, [Nart and Ritzenthaler 10] E.; Ritzenthaler, C., Proceedings of AGCT-12, 521, Genus Three Curves with Many Involutions and Application to Maximal Curves in Characteristic 2., 71-85, (2010), Contemporary Mathematics · Zbl 1219.14036 [25] Poonen, [Poonen 04] B., Bertini Theorems Over Finite Fields., Ann. Math. (2), 160, 3, 1099-1127, (2004) · Zbl 1084.14026 [26] Ritzenthaler, [Ritzenthaler 09] C., Aspects arithmétiques et algorithmiques des courbes de genre 1, 2 et 3, (2009), Habilitation à Diriger des Recherches, Université de la Méditerranée [27] Ritzenthaler, [Ritzenthaler 10] C., Explicit Computations of Serre’s Obstruction for Genus-3 Curves and Application to Optimal Curves., LMS J. Comput. Math., 13, 192-207, (2010) · Zbl 1278.11068 [28] Serre, [Serre 83] J.-P., Seminar on number theory, 1982–1983 (Talence, 1982/1983), Nombres de points des courbes algébriques sur ., (1983), Univ. Bordeaux I: Univ. Bordeaux I, Talence [29] Sørensen, [Sørensen 94] A. B., On the Number of Rational Points on Codimension-1 Algebraic Sets in ., Discrete Math., 135, 1-3, 321-334, (1994) · Zbl 0816.14009 [30] Tallini, [Tallini 61] G., Le ipersuperficie irriducibili d’ordine minimo che invadono uno spazio di Galois., Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 30, 706-712, (1961) · Zbl 0107.38104 [31] Tallini, [Tallini 61] G., Sulle ipersuperficie irriducibili d’ordine minimo che contengono tutti i punti di uno spazio di Galois ., Rend. Mat. e Appl. (5), 20, 431-479, (1961) · Zbl 0106.35604 [32] van der Geer, [van der Geer and van der Vlugt 92] G.; van der Vlugt, M., Supersingular Curves of Genus 2 Over Finite Fields of Characteristic 2., Mathematische Nachrichten, 159, 73-81, (1992) · Zbl 0774.14045 [33] Xiong, [Xiong 10] M., The Fluctuations in the Number of Points on a Family of Curves Over a Finite Field., J. Théor. Nombres Bordeaux, 22, 3, 755-769, (2010) · Zbl 1228.11089 [34] Xiong, [Xiong 15] M., Distribution of Zeta Zeroes for Abelian Covers of Algebraic Curves Over a Finite Field., J. Number Theory, 147, 789-823, (2015) · Zbl 1395.11101
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