The number of rational points of a class of superelliptic curves

Abstract

In this paper, we study the number of ${\mathbb{F}}_{q^n}$-rational points on the affine curve ${\mathcal{X}}_{d,a,b}$ given by the equation $$y^d=ax\text{Tr}(x)+b,$$ where $\text{Tr}$ denote the trace function from ${\mathbb{F}}_{q^n}$ to ${\mathbb{F}}_q$ and $d$ is a positive integer. In particular, we present bounds for the number of ${\mathbb{F}}_q$-rational points on ${\mathcal{X}}_{d,a,b}$ 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 $\alpha$ in ${\mathbb{F}}_{q^n}$ such that $\alpha$ and $\text{Tr}(\alpha )$ are quadratic residues in ${\mathbb{F}}_{q^n}$.