The number of rational points of a class of superelliptic curves

TL;DR

This paper investigates the number of rational points on a class of superelliptic curves over finite fields, providing bounds, explicit formulas, and a complete characterization for the case d=2, with applications to quadratic residues.

Contribution

It introduces new bounds and explicit formulas for counting rational points on superelliptic curves, especially for the case d=2, advancing understanding of their arithmetic properties.

Findings

Bounds for the number of rational points over finite fields.

Explicit formulas for cases where d satisfies certain conditions.

Complete characterization of rational points when d=2.

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 $\mathbb F_{q^n}$.