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

Cohen, Stephen D.; Kapetanakis, Giorgos; Reis, Lucas

Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 641-652.

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

Classification : 11T30, 11A07, 11T23
Mots clés : finite fields, character sums, elliptic curves

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