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The fluctuations in the number of points on a family of curves over a finite field. (English. French summary) Zbl 1228.11089
Author’s summary: Let $$l\geq 2$$ be a positive integer,$$\mathbb F_q$$ a finite field of cardinality $$q$$ with $$q \equiv 1\pmod l$$. In this paper, the author studies the fluctuations in the number of $$\mathbb F_q$$-points on the curve $$C_F$$ given by the affine model $$C_F : Y^l = F(X)$$, where $$F$$ is drawn at random uniformly from the set of all monic $$l$$th power-free polynomials $$F\in \mathbb F_q[X]$$ of degree $$d$$ as $$d \rightarrow\infty$$. The method also enables us to study the fluctuations in the number of $$\mathbb F_q$$-points on the same family of curves arising from the set of monic irreducible polynomials.

##### MSC:
 11G20 Curves over finite and local fields 11T55 Arithmetic theory of polynomial rings over finite fields
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##### References:
 [1] J. Bergström, Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves. Preprint, http://arxiv.org/abs/math/0611813v1, 2006. [2] P. Billingsley, Probability and Measure. Third ed., Wiley Ser. Probab. Math. Stat., John Wiley & Sons Inc., Ney Youk, 1995, A Wiley-Interscience Publication. [3] A. Bucur, C. David, B. Feigon, M. Lalín, Statistics for traces of cyclic trigonal curves over finite fields. International Mathematics Research Notices (2010), 932-967. · Zbl 1201.11063 [4] A. Bucur, C. David, B. Feigon, M. Lalín, Biased statistics for traces of cyclic $$p$$-fold covers over finite fields. To appear in Proceedings of Women in Numbers, Fields Institute Communications. · Zbl 1258.11071 [5] P. Diaconis, M. Shahshahani, On the eigenvalues of random matrices. Studies in Applied Probability, J. Appl. Probab. 31A (1994), 49-62. · Zbl 0807.15015 [6] P. Kurlberg, Z. Rudnick, The fluctuations in the number of points on a hyperelliptic curve over a finite field. J. Number Theory Vol. 129 3 (2009), 580-587. · Zbl 1221.11141 [7] N. M. Katz, P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy. Amer. Math. Soc. Colloq. Publ., vol. 45, American Mathematical Socitey, Providence, RI, 1999. · Zbl 0958.11004 [8] N. M. Katz, P. Sarnak, Zeroes of zeta functions and symmetry. Bull. Am. Math. Soc. 36 (1999), 1-26. · Zbl 0921.11047 [9] L. A. Knizhnerman, V. Z. Sokolinskii, Some estimates for rational trigonometric sums and sums of Legendre symbols. Uspekhi Mat. Nauk 34 (3 (207))(1979), 199-200. · Zbl 0444.10030 [10] L. A. Knizhnerman, V. Z. Sokolinskii, Trigonometric sums and sums of Legendre symbols with large and small absolute values. Investigations in Number Theory, Saratov. Gos. Univ., Saratov, 1987, 76-89. · Zbl 0654.10036 [11] M. Larsen, The normal distribution as a limit of generalized sato-tate measures. Preprint. [12] M. Rosen, Number theory in function fields. Graduate Texts in Mathematics, 210, Springer-Verlag, New York, 2002. · Zbl 1043.11079 [13] A. Weil, Sur les Courbes Algébriques et les Variétés qui s’en Déduisent. Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, 1948. iv+85 pp. · Zbl 0036.16001
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