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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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