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

Stephen D. Cohen; Giorgos Kapetanakis; Lucas Reis

doi:10.5802/crmath.328

Number theory

The existence of

𝔽_{q}

-primitive points on curves using freeness

Stephen D. Cohen ¹ ; Giorgos Kapetanakis ² ; Lucas Reis ³

¹ 6 Bracken Road, Portlethen, Aberdeen AB12 4TA, Scotland, UK
² Department of Mathematics, University of Thessaly, 3rd km Old National Road Lamia-Athens, 35100 Lamia, Greece
³ Departamento de Matemática, Universidade Federal de Minas Gerais, UFMG, Belo Horizonte MG, 31270901, Brazil

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 641-652

Abstract

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 $θ \in 𝔽_{q}$ for which $f (θ)$ is $(r, n)$ -free and $F (θ)$ is $(R, N)$ -free, where $f, F \in 𝔽_{q} [x]$ . As an application, we consider the existence of $𝔽_{q}$ -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 $𝔽_{q}$ -primitive point.

Received: 2021-12-06
Revised: 2022-01-13
Accepted: 2022-01-13
Published online: 2022-06-22

Zbl

DOI: 10.5802/crmath.328

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

Author's affiliations:

Stephen D. Cohen ¹; Giorgos Kapetanakis ²; Lucas Reis ³

¹ 6 Bracken Road, Portlethen, Aberdeen AB12 4TA, Scotland, UK
² Department of Mathematics, University of Thessaly, 3rd km Old National Road Lamia-Athens, 35100 Lamia, Greece
³ Departamento de Matemática, Universidade Federal de Minas Gerais, UFMG, Belo Horizonte MG, 31270901, Brazil

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Stephen D. Cohen and Giorgos Kapetanakis and Lucas Reis},
     title = {The existence of $\protect \mathbb{F}_q$-primitive points on curves using freeness},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {641--652},
     year = {2022},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     doi = {10.5802/crmath.328},
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     language = {en},
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Stephen D. Cohen; Giorgos Kapetanakis; Lucas Reis. The existence of $\protect \mathbb{F}_q$-primitive points on curves using freeness. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 641-652. doi: 10.5802/crmath.328

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