Small exponent point groups on elliptic curves
Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 471-476.

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/loglogn . 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 and the prime p tends to infinity.

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/loglogn . 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 et le nombre premier p tend vers l’infini.

DOI: 10.5802/jtnb.554
Luca, Florian 1; McKee, James 2; Shparlinski, Igor E. 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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Luca, Florian; McKee, James; Shparlinski, Igor E. Small exponent point groups on elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 471-476. doi : 10.5802/jtnb.554. http://www.numdam.org/articles/10.5802/jtnb.554/

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