The fluctuations in the number of points of smooth plane curves over finite fields

Details The fluctuations in the number of points of smooth plane curves over finite fields The fluctuations in the number of points of smooth plane curves over finite fields

2009-12-24

arxiv.org/abs/0912.4761v1

J. Number Theory 130 (2010), pp. 2528-2541

Mathematics

Number Theory

12 pages

Scientific paper

10.1016/j.jnt.2010.05.009

In this note, we study the fluctuations in the number of points of smooth projective plane curves over finite fields $\mathbb{F}_q$ as $q$ is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen.

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