The existence of $\protect \mathbb{F}_q$-primitive points on curves using freeness

Abstract

Let $\mathcal{C}_Q$ be the cyclic group of order $Q$, $n$ a divisor of $Q$ and $r$ a divisor of $Q/n$. We introduce the set of $(r,n)$-free elements of $\mathcal{C}_Q$ and derive a lower bound for the number of elements $\theta \in \mathbb{F}_q$ for which $f(\theta )$ is $(r,n)$-free and $F(\theta )$ is $(R,N)$-free, where $ f, F \in \mathbb{F}_q[x]$. As an application, we consider the existence of $\mathbb{F}_q$-primitive points on curves like $y^n=f(x)$ and find, in particular, all the odd prime powers $q$ for which the elliptic curves $y^2=x^3 \pm x$ contain an $\mathbb{F}_q$-primitive point.

Keywords

• finite fields
• character sums
• elliptic curves