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

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 $a{x}^3+b{y}^3+c{z}^3=0$ and $a{x}^4+b{y}^4+c{z}^4=0$.