Small exponent point groups on elliptic curves
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 471-476.

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.

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.

Reçu le : 2004-12-31
Publié le : 2008-12-02
DOI : https://doi.org/10.5802/jtnb.554
@article{JTNB_2006__18_2_471_0,
     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},
     mrnumber = {2289434},
     zbl = {05135399},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2006__18_2_471_0/}
}
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/item/JTNB_2006__18_2_471_0/

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