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

TL;DR

This paper investigates how the number of points on smooth projective plane curves over finite fields varies with genus, demonstrating that fluctuations align with a probabilistic model based on independent point conditions.

Contribution

It introduces a probabilistic model predicting point fluctuations on curves over finite fields and employs Poonen's geometric sieving method for analysis.

Findings

Fluctuations match the probabilistic model predictions.

Points impose independent conditions on the curves.

Uses geometric sieving to analyze point distributions.

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.