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

TL;DR

This paper introduces a new method using freeness to analyze the existence of primitive points on algebraic curves over finite fields, providing lower bounds and specific existence results for elliptic curves.

Contribution

It develops the concept of (r,n)-free elements in cyclic groups and applies this to establish conditions for primitive points on curves like y^n=f(x).

Findings

Lower bounds for the number of elements with certain freeness properties.

Identification of all odd prime powers q where specific elliptic curves have primitive points.

Application of freeness to determine primitive points on algebraic curves.

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 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.