Comptes Rendus
Number theory
The existence of 𝔽 q -primitive points on curves using freeness
Comptes Rendus. Mathématique, Volume 360 (2022), 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 θ𝔽 q for which f(θ) is (r,n)-free and F(θ) is (R,N)-free, where f,F𝔽 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 ±x contain an 𝔽 q -primitive point.

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DOI: 10.5802/crmath.328
Classification: 11T30, 11A07, 11T23
Keywords: finite fields, character sums, elliptic curves

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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     title = {The existence of $\protect \mathbb{F}_q$-primitive points on curves using freeness},
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.328/

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