The number of rational points of a class of superelliptic curves

Abstract

In this paper, we study the number of Fqn-rational points on the affine curve Xd,a,b given by the equationyd=axTr(x)+b, where Tr denote the trace function from Fqn to Fq and d is a positive integer. In particular, we present bounds for the number of Fq-rational points on Xd,a,b and, for the cases where d satisfies a natural condition, explicit formulas for the number of rational points are obtained. Particularly, a complete characterization is given for the case d=2. As a consequence of our results, we compute the number of elements α in Fqn such that α and Tr(α) are quadratic residues in Fqn.