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

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

doi:10.5802/crmath.328

Geodesic

Parcourir par

Théorie des nombres

The existence of

𝔽_{q}

-primitive points on curves using freeness

Cohen, Stephen D.¹ ; Kapetanakis, Giorgos ² ; Reis, Lucas 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, Tome 360 (2022) no. G6, pp. 641-652.

Voir la notice de l'article provenant de la source Numdam

Résumé

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.

Reçu le : 2021-12-06
Révisé le : 2022-01-13
Accepté le : 2022-01-13
Publié le : 2022-06-22

Zbl

DOI : 10.5802/crmath.328

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

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The existence of $\protect \mathbb{F}_q$-primitive points on curves using freeness},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {641--652},
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Cohen, Stephen D.; Kapetanakis, Giorgos; Reis, Lucas. The existence of $\protect \mathbb{F}_q$-primitive points on curves using freeness. Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 641-652. doi : 10.5802/crmath.328. https://geodesic-test.mathdoc.fr/articles/10.5802/crmath.328/

Bibliographie
Cité par

[1] Booker, Andrew R.; Cohen, Stephen D.; Sutherland, Nicole; Trudgian, Tim Primitive values of quadratic polynomials in a finite field, Math. Comput., Volume 88 (2019) no. 318, pp. 1903-1912 | MR | DOI | Zbl

[2] Brochero Martínez, F. E.; Reis, Lucas Elements of high order in Artin–Schreier extensions of finite fields $𝔽_{q}$ , Finite Fields Appl., Volume 41 (2016), pp. 24-33 | Zbl | MR | DOI

[3] Carlitz, Leonard Primitive roots in a finite field, Trans. Am. Math. Soc., Volume 73 (1952), pp. 373-382 | Zbl | MR | DOI

[4] Carvalho, Cícero; Guardieiro, João Paulo; Neumann, Victor G. L.; Tizziotti, Guilherme On special pairs of primitive elements over a finite field, Finite Fields Appl., Volume 73 (2021), 101839, 10 pages | Zbl | MR

[5] Cohen, Stephen D. The orders of related elements of a finite field, Ramanujan J., Volume 7 (2003) no. 1-3, pp. 169-183 | Zbl | MR | DOI

[6] Cohen, Stephen D.; Huczynska, Sophie The primitive normal basis theorem — without a computer, J. Lond. Math. Soc., Volume 67 (2003) no. 1, pp. 41-56 | Zbl | MR | DOI

[7] Cohen, Stephen D.; Kapetanakis, Giorgos The trace of 2-primitive elements of finite fields, Acta Arith., Volume 192 (2020) no. 4, pp. 397-419 | Zbl | MR | DOI

[8] Cohen, Stephen D.; Kapetanakis, Giorgos The translate and line properties for 2-primitive elements in quadratic extensions, Int. J. Number Theory, Volume 16 (2020) no. 9, pp. 2027-2040 | Zbl | MR | DOI

[9] Cohen, Stephen D.; Kapetanakis, Giorgos Finite field extensions with the line or translate property for $r$ -primitive elements, J. Aust. Math. Soc., Volume 111 (2021) no. 3, pp. 311-319 | Zbl | MR

[10] Cohen, Stephen D.; Oliveira e Silva, Tomás; Sutherland, Nicole; Trudgian, Tim Linear combinations of primitive elements of a finite field, Finite Fields Appl., Volume 51 (2018), pp. 388-406 | Zbl | MR | DOI

[11] Cohen, Stephen D.; Oliveira e Silva, Tomás; Trudgian, Tim A proof of the conjecture of Cohen and Mullen on sums of primitive roots, Math. Comput., Volume 84 (2015) no. 296, pp. 2979-2986 | Zbl | MR | DOI

[12] Gao, Shuhong Elements of provable high orders in finite fields, Proc. Am. Math. Soc., Volume 127 (1999) no. 6, pp. 1615-1623 | Zbl | MR

[13] Huczynska, Sophie; Mullen, Gary L.; Panario, Daniel; Thomson, David Existence and properties of $k$ -normal elements over finite fields, Finite Fields Appl., Volume 24 (2013), pp. 170-183 | Zbl | MR | DOI

[14] Kapetanakis, Giorgos; Reis, Lucas Variations of the primitive normal basis theorem, Des. Codes Cryptography, Volume 87 (2018) no. 7, pp. 1459-1480 | Zbl | MR | DOI

[15] Lang, Serge; Trotter, Hale Primitive points on elliptic curves, Bull. Am. Math. Soc., Volume 83 (1977), pp. 289-292 | Zbl | MR | DOI

[16] Lidl, Rudolf; Niederreiter, Harald Finite Fields, Encyclopedia of Mathematics and Its Applications, 20, Cambridge University Press, 1996 | DOI

[17] Handbook of Finite Fields (Mullen, Gary L.; Panario, Daniel, eds.), Discrete Mathematics and its Applications, CRC Press, 2013

[18] Popovych, Roman Elements of high order in finite fields of the form $F_{q} [x] / (x^{m} - a)$ , Finite Fields Appl., Volume 19 (2013), p. 96-92 | Zbl | MR

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