Small exponent point groups on elliptic curves

Florian Luca; James McKee; Igor E. Shparlinski

doi:10.5802/jtnb.554

Florian Luca ¹ ; James McKee ² ; Igor E. Shparlinski ³

¹ Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
² Department of Mathematics Royal Holloway, University of London Egham, Surrey, TW20 0EX, UK
³ Department of Computing Macquarie University Sydney, NSW 2109, Australia

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 471-476.

Résumé
Abstract

Soit $E$ une courbe elliptique définie sur $F_{q}$ , le corps fini à $q$ éléments. Nous montrons que pour une constante $η > 0$ dépendant seulement de $q$ , il existe une infinité d’entiers positifs $n$ tels que l’exposant de $E (F_{q^{n}})$ , le groupe des points $F_{q^{n}}$ -rationnels sur $E$ , est au plus $q^{n} exp (- n^{η / log log n})$ . Il s’agit d’un analogue d’un résultat de R. Schoof sur l’exposant du groupe $E (F_{p})$ des points $F_{p}$ -rationnels, lorsqu’une courbe elliptique fixée $E$ est définie sur $ℚ$ et le nombre premier $p$ tend vers l’infini.

Let $E$ be an elliptic curve defined over $F_{q}$ , the finite field of $q$ elements. We show that for some constant $η > 0$ depending only on $q$ , there are infinitely many positive integers $n$ such that the exponent of $E (F_{q^{n}})$ , the group of $F_{q^{n}}$ -rational points on $E$ , is at most $q^{n} exp (- n^{η / log log n})$ . This is an analogue of a result of R. Schoof on the exponent of the group $E (F_{p})$ of $F_{p}$ -rational points, when a fixed elliptic curve $E$ is defined over $ℚ$ and the prime $p$ tends to infinity.

MR Zbl

DOI : 10.5802/jtnb.554

Affiliations des auteurs :

Florian Luca ¹ ; James McKee ² ; Igor E. Shparlinski ³

¹ Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
² Department of Mathematics Royal Holloway, University of London Egham, Surrey, TW20 0EX, UK
³ Department of Computing Macquarie University Sydney, NSW 2109, Australia

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     author = {Florian Luca and James McKee and Igor E. Shparlinski},
     title = {Small exponent point groups on elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {471--476},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {2},
     year = {2006},
     doi = {10.5802/jtnb.554},
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     zbl = {05135399},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.554/}
}

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Florian Luca; James McKee; Igor E. Shparlinski. Small exponent point groups on elliptic curves. Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 471-476. doi : 10.5802/jtnb.554. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.554/

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