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

Abstract

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.

12 pages

References in corpus (1)

• Statistics for traces of cyclic trigonal curves over finite fields

Cited by in corpus (6)

• Semiample Bertini theorems over finite fields
• A heuristic for the distribution of point counts for random curves over a finite field
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• Gaussian point count statistics for families of curves over a fixed finite field
• Refinements of Katz-Sarnak theory for the number of points on curves over finite fields