The fluctuations in the number of points on a family of curves over a finite field

Xiong, Maosheng

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 755-769.

Soit $l\ge 2$ un entier, ${𝔽}_q$ un corps fini de cardinal $q$ avec $q\equiv 1(modl)$. 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 ${ℂ}_F$ donnée par le modèle affine ${ℂ}_F:Y^l=F\left(X\right)$, où $F$ parcourt aléatoirement et uniformément l’ensemble des polynômes $F\in {𝔽}_q\left[X\right]$ unitaires, sans puissance $l$-ième, de degré $d$ quand $d\to \infty$. 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 $l\ge 2$ be a positive integer, ${𝔽}_q$ a finite field of cardinality $q$ with $q\equiv 1(modl)$. 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 ${ℂ}_F$ given by the affine model ${ℂ}_F:Y^l=F\left(X\right)$, where $F$ is drawn at random uniformly from the set of all monic $l$-th power-free polynomials $F\in {𝔽}_q\left[X\right]$ of degree $d$ as $d\to \infty$. 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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