The maximum or minimum number of rational points on curves of genus three over finite fields

TL;DR

This paper investigates the bounds on the number of rational points on genus 3 curves over finite fields, showing existence results near theoretical bounds and improving upper bounds for certain fields.

Contribution

It demonstrates the existence of genus 3 curves with rational points close to Serre-Weil bounds over all finite fields and improves upper bounds for some specific fields.

Findings

Existence of genus 3 curves within 3 points of Serre-Weil bounds over all finite fields

Improved upper bounds for the number of rational points on genus 3 curves for certain finite fields

Provides new insights into the distribution of rational points on algebraic curves over finite fields

Abstract

We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre-Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.