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

Maosheng Xiong — 2010

Journal de Théorie des Nombres de Bordeaux

Let l 2 be a positive integer, 𝔽 q a finite field of cardinality q with q 1 ( mod l ) . In this paper, inspired by [, , ] 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 F 𝔽 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...

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