The maximum number of rational points for a genus 4 curve over F_7 is 24

TL;DR

This paper establishes that the maximum number of rational points on a genus 4 curve over the finite field F_7 is 24, improving the known upper bound of 25, and provides an explicit example achieving this maximum.

Contribution

The paper proves the upper bound of 24 rational points for genus 4 curves over F_7 and constructs an explicit example attaining this maximum.

Findings

Maximum of 24 rational points on genus 4 curves over F_7

Proof that the upper bound is 24, not 25

Explicit example of a genus 4 curve with 24 points

Abstract

In this paper we show that the maximum number of rational points possible for a smooth, projective, absolutely irreducible genus 4 curve over a finite field F_7 is 24. It is known that a genus 4 curve over F_7 can have at most 25 points. In this paper we prove that such a curve can have at most 24. On the other hand we provide an explicit example of a genus 4 curve over F_7 having 24 points.