Rational points on Cubic, Quartic and Sextic Curves over Finite Fields

TL;DR

This paper develops character-based methods to count rational points on low-degree curves over finite fields, relating them to elliptic curves, and characterizes extremal curves of specific cubic and quartic forms.

Contribution

It introduces a character-based approach to count points on low-degree curves over finite fields and characterizes maximal and minimal curves of certain cubic and quartic forms.

Findings

Derived formulas for rational points on low-degree curves over finite fields.

Established methods to compute these numbers explicitly when the field size is prime.

Characterized maximal and minimal curves of specific cubic and quartic equations.

Abstract

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic curves. In the case where $q$ is a prime number, we give a way to calculate these numbers. As a consequence of these results, we characterize maximal and minimal curves given by equations of the forms $ax^3+by^3+cz^3=0$ and $ax^4+by^4+cz^4=0$.