The fluctuations in the number of points on a family of curves over a finite field
Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 755-769.

Soit l2 un entier, 𝔽 q un corps fini de cardinal q avec q1(modl). Dans cet article, inspiré par [6, 3, 4] et en utilisant une méthode légèrement différente, nous étudions les fluctuations du nombre de 𝔽 q -points de la courbe F donnée par le modèle affine F :Y l =F(X), où F parcourt aléatoirement et uniformément l’ensemble des polynômes F𝔽 q [X] unitaires, sans puissance l-ième, de degré d quand d. La méthode nous permet aussi d’étudier les fluctuations du nombre de 𝔽 q -points de la même famille de courbes provenant de l’ensemble des polynômes unitaires irréductibles.

Let l2 be a positive integer, 𝔽 q a finite field of cardinality q with q1(modl). In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of 𝔽 q -points on the curve F given by the affine model F :Y l =F(X), where F is drawn at random uniformly from the set of all monic l-th power-free polynomials F𝔽 q [X] of degree d as d. The method also enables us to study the fluctuations in the number of 𝔽 q -points on the same family of curves arising from the set of monic irreducible polynomials.

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DOI : https://doi.org/10.5802/jtnb.745
Classification : 11G20,  11T55
@article{JTNB_2010__22_3_755_0,
     author = {Maosheng Xiong},
     title = {The fluctuations in the number of points on a family of curves over a finite field},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {755--769},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {3},
     year = {2010},
     doi = {10.5802/jtnb.745},
     zbl = {1228.11089},
     mrnumber = {2769344},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.745/}
}
Maosheng Xiong. The fluctuations in the number of points on a family of curves over a finite field. Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 755-769. doi : 10.5802/jtnb.745. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.745/

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