Small exponent point groups on elliptic curves

Luca, Florian; McKee, James; Shparlinski, Igor E.

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

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

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

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