A quantitative primitive divisor result for points on elliptic curves

Ingram, Patrick

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 609-634

Let $E/K$ be an elliptic curve defined over a number field, and let $P\in E\left(K\right)$ be a point of infinite order. It is natural to ask how many integers $n\ge 1$ fail to occur as the order of $P$ modulo a prime of $K$. For $K=\mathbb{Q}$, $E$ a quadratic twist of $y^2=x^3-x$, and $P\in E\left(\mathbb{Q}\right)$ as above, we show that there is at most one such $n\ge 3$.

Soient $E/K$ une courbe elliptique définie sur un corps de nombres et $P\in E\left(K\right)$ un point d’ordre infini. Il est naturel de se demander combien de nombres entiers $n\ge 1$ n’apparaissent pas comme ordre du point $P$ modulo un idéal premier de $K$. Dans le cas où $K=\mathbb{Q}$, $E$ une tordue quadratique de $y^2=x^3-x$ et $P\in E\left(\mathbb{Q}\right)$ comme ci-dessus, nous démontrons qu’il existe au plus un tel $n\ge 3$.

Classification : 11G05, 11B39

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