Fluctuations in the number of points on smooth plane curves over finite fields

2009/12/24 by Alina Bucur, Chantal David, Brooke Feigon +2
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Limits and Structures in Graph Theory #math.NT #msc:11G20 #msc:11G25 #msc:11T55

paper · pdf · doi:10.1016/j.jnt.2010.05.009

published as J. Number Theory 130 (2010), pp. 2528-2541 · 12 pages

arxiv created 2009/12/24 · openalex publication_date 2010/07/21 · arxiv updated 2010/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this note, we study the fluctuations in the number of points of smooth projective plane curves over finite fields \mathbbF_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.

Citations

• Statistics for Traces of Cyclic Trigonal Curves over Finite Fields