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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.

11G20 Curves over finite and local fields
11T55 Arithmetic theory of polynomial rings over finite fields
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