Rational points on cubic, quartic and sextic curves over finite fields

Abstract

Let Fq 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 Fq 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 ax3+by3+cz3=0 and ax4+by4+cz4=0.